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eISSN: 2378-315X

Biometrics & Biostatistics International Journal

Research Article Volume 6 Issue 3

The long term fréchet distribution: estimation, properties and its application

Pedro Luiz Ramos, Diego Nascimento, Francisco Louzada

Institute of Mathematical Science and Computing, University of São Paulo, Brazil

Correspondence: Pedro Luiz Ramos,Institute of Mathematical Science and Computing, University of São Paulo, Brazil

Received: June 28, 2017 | Published: September 14, 2017

Citation: Ramos PL, Nascimento D, Louzada F. The long term fréchet distribution: estimation, properties and its application. Biom Biostat Int J. 2017;6(3):357-362. DOI: 10.15406/bbij.2017.06.00170

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Abstract

In this paper a new long-term survival distribution is proposed. The so called long term Fréchet distribution allows us to fit data where a part of the population is not susceptible to the event of interest. This model may be used, for example, in clinical studies where a portion of the population can be cured during a treatment. It is shown an account of mathematical properties of the new distribution such as its moments and survival properties. As well is presented the maximum likelihood estimators (MLEs) for the parameters. A numerical simulation is carried out in order to verify the performance of the MLEs. Finally, an important application related to the leukemia free-survival times for transplant patients are discussed to illustrates our proposed distribution

Keywords: Fréchet distribution; Long-term survival distribution; Survival model

Introduction

Extreme value models play an important role in statistic. The generalized extreme value (GEV) distribution [1] and its sub-models are widely used in application involving extreme events. These sub-models are the well known Weibull, Fréchet and Gumbel distributions. The Fréchet distribution can be seen as the inverse Weibull distribution which gives a probability density function (PDF) such as

f( t,λ,α )= α λ ( t λ ) ( α+1 ) e ( t λ ) α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOzamaabmaapaqaa8qacaWG0bGaaiilaiabeU7aSjaacYca cqaHXoqyaiaawIcacaGLPaaacqGH9aqpdaWcaaWdaeaapeGaeqySde gapaqaa8qacqaH7oaBaaWaaeWaa8aabaWdbmaalaaapaqaa8qacaWG 0baapaqaa8qacqaH7oaBaaaacaGLOaGaayzkaaqcga4damaaCaaaju aGbeqaaKqzadWdbiabgkHiTKGbaoaabmaajuaGpaqaaKqzadWdbiab eg7aHjabgUcaRiaaigdaaKqbakaawIcacaGLPaaaaaGaamyza8aada ahaaqabeaajugWa8qacqGHsisljyaGdaqadaqcfa4daeaajyaGpeWa aSaaaKqba+aabaqcLbmapeGaamiDaaqcfa4daeaajugWa8qacqaH7o aBaaaajuaGcaGLOaGaayzkaaqcga4damaaCaaajuaGbeqaaKqzadWd biabgkHiTiabeg7aHbaaaaaaaa@65AC@ (1)

The survival function is given by

S( t,λ,α )=1 e ( t λ ) α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaam4uamaabmaapaqaa8qacaWG0bGaaiilaiabeU7aSjaacYca cqaHXoqyaiaawIcacaGLPaaacqGH9aqpcaaIXaGaeyOeI0Iaamyza8 aadaahaaqabeaajugWa8qacqGHsisljyaGdaqadaqcfa4daeaajyaG peWaaSaaaKqba+aabaqcLbmapeGaamiDaaqcfa4daeaajugWa8qacq aH7oaBaaaajuaGcaGLOaGaayzkaaqcga4damaaCaaajuaGbeqaaKqz adWdbiabgkHiTiabeg7aHbaaaaaaaa@53C7@ (2)

Although the GEV distribution is the most used generalization of the Fréchet model, other distributions has been proposed in the literature. De Gusmão [2] proposed a three parameter generalized inverse Weibull distribution in which includes the Fréchet distribution. Krishna et al. [3] proposed the Marshall-Olkin Fréchet distribution. Barreto-Souza et al. [4] discussed some results for beta Fréchet distribution. However, in survival studies habitually the researches may consider a portion of the population as cured during a given treatment, this type of distribution is called long-term (LT) survival models.

In this study, a long-term survival novel proposing a mixture model introduced by Berkson and Gage [5], hereafter we shall call it the long-term Fréchet distribution or simplistically the LF distribution. Some mathematical properties about the LF distribution were provided such as moments, survival properties and hazard function. The maximum likelihood estimators of the parameters and its asymptotic properties are discussed likewise. Similar studies were presented by Roman et al. [6] for the geometric exponential distribution and by Louzada and Ramos [7] for the weighted Lindley distribution. It was performed a numerical simulation towards to examine the performance of the MLEs. Finally, our proposed methodology is illustrated in a real data set related to the leukemia free-survival times (in years) for the 50 autologous transplant patients.

The paper is organized as follows. Section 2 presents the long term Fréchet distribution and its mathematical properties. Section 3 discusses the parameter estimation under the maximum likelihood approach. Section 4 presents a simulation study under different values of the parameters and different levels of censorship. The proposed methodology is also fully illustrated in a real data set. Lastly, Section 6 summarizes the founds in this study and its potential contribution.

Long Term Fréchet distribution

Long-term survivors are an important feature to incorporate in the modeling process, since a portion of the population may no longer be eligible to the event of interest (according to Maller and Zhou, [8]; or Perdona and Louzada, [9]). Hence the population can be segregate as a not eligible to the event of interest with probability p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiCaaaa@372F@ and as eligible (in risk) to the event of interest with probability ( 1p ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaeWaa8aabaWdbiaaigdacqGHsislcaWGWbaacaGLOaGaayzk aaaaaa@3A7F@ . The long-term survivor is expressed as

S( t;p,θ )=p+( 1p ) S 0 ( t;θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYlf9irVeeu0dXdh9LjY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaake aajuaGqaaaaaaaaaWdbiaadofadaqadaWdaeaapeGaamiDaiabgUda 7iaadchacqGHSaaliiqacqWF4oqCaiaawIcacaGLPaaacqGH9aqpca WGWbGaey4kaSYaaeWaa8aabaWdbiabggdaXiabgkHiTiaadchaaiaa wIcacaGLPaaacaWGtbqcga4damaaBaaajuaGbaqcLbmapeGaeyimaa dajuaGpaqabaWdbmaabmaapaqaa8qacaWG0bGaey4oaSJae8hUdeha caGLOaGaayzkaaaaaa@5419@ ,  (3)

where p( 0,1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiCaiabgIGiopaabmaapaqaa8qacaaIWaGaaiilaiaaigda aiaawIcacaGLPaaaaaa@3C80@ and S 0 ( t;θ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaam4ua8aadaWgaaqaaKqzadWdbiaaicdaaKqba+aabeaapeWa aeWaa8aabaWdbiaadshacaGG7aqeduuDJXwAKbYu51MyVXgaiuqacq WF4oqCaiaawIcacaGLPaaaaaa@4406@ is the survival function related to the eligible group. The obtained survival function (not conditional) is improper and its limit corresponds to the individual proportion cure. From the survival function one can easily derive the PDF (improper) given by

f( t;p,θ )= t S( t;p,θ )=( 1π ) f 0 ( t;θ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOzamaabmaapaqaa8qacaWG0bGaai4oaiaadchacaGGSaqe duuDJXwAKbYu51MyVXgaiuqacqWF4oqCaiaawIcacaGLPaaacqGH9a qpcqGHsisldaWcaaWdaeaapeGaeyOaIylapaqaa8qacqGHciITcaWG 0baaaiaadofadaqadaWdaeaapeGaamiDaiaacUdacaWGWbGaaiilai ab=H7aXbGaayjkaiaawMcaaiabg2da9maabmaapaqaa8qacaaIXaGa eyOeI0IaeqiWdahacaGLOaGaayzkaaGaamOzaKGba+aadaWgaaqcfa yaaKqzadWdbiaaicdaaKqba+aabeaapeWaaeWaa8aabaWdbiaadsha caGG7aGae8hUdehacaGLOaGaayzkaaaaaa@607E@ ,  (4)

where f 0 ( t;θ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOza8aadaWgaaqaaKqzadWdbiaaicdaaKqba+aabeaapeWa aeWaa8aabaWdbiaadshacaGG7aqeduuDJXwAKbYu51MyVXgaiuqacq WF4oqCaiaawIcacaGLPaaaaaa@4419@ is the PDF related to the susceptible group.

Figure 1: It shows some cases about the PDF and the survival function shapes applied to LF distribution.
In Left panel: Probability density function of the LF distribution. Right panel: Survival function of the LF distribution.

Considering that f 0 ( t;θ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOza8aadaWgaaqaaKqzadWdbiaaicdaaKqba+aabeaapeWa aeWaa8aabaWdbiaadshacaGG7aqeduuDJXwAKbYu51MyVXgaiuqacq WF4oqCaiaawIcacaGLPaaaaaa@4419@ follows a Fréchet distribution, then the PDF of the Long Term Fréchet (LF) distribution is given by

f( t;λ,α,p )= α( 1p ) λ ( t λ ) ( α+1 ) e ( t λ ) α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOzamaabmaapaqaa8qacaWG0bGaai4oaiabeU7aSjaacYca cqaHXoqycaGGSaGaamiCaaGaayjkaiaawMcaaiabg2da9maalaaapa qaa8qacqaHXoqydaqadaWdaeaapeGaaGymaiabgkHiTiaadchaaiaa wIcacaGLPaaaa8aabaWdbiabeU7aSbaadaqadaWdaeaapeWaaSaaa8 aabaWdbiaadshaa8aabaWdbiabeU7aSbaaaiaawIcacaGLPaaapaWa aWbaaeqajyaGbaWdbiabgkHiTmaabmaapaqaa8qacqaHXoqycqGHRa WkcaaIXaaacaGLOaGaayzkaaaaaKqbakaadwgajyaGpaWaaWbaaKqb agqabaqcLbmapeGaeyOeI0scga4aaeWaaKqba+aabaqcga4dbmaala aajuaGpaqaaKqzadWdbiaadshaaKqba+aabaqcLbmapeGaeq4UdWga aaqcfaOaayjkaiaawMcaaKGba+aadaahaaqcfayabeaajugWa8qacq GHsislcqaHXoqyaaaaaaaa@68BB@ ,  (5)

where λ>0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeq4UdWMaeyOpa4JaaGimaaaa@39B0@ , α>0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqySdeMaeyOpa4JaaGimaaaa@399B@ and p( 0,1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiCaiabgIGiopaabmaapaqaa8qacaaIWaGaaiilaiaaigda aiaawIcacaGLPaaaaaa@3C80@ . The cumulative distribution function is given by

F( t;λ,α,p )=( 1p ) e ( t λ ) α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOramaabmaapaqaa8qacaWG0bGaai4oaiabeU7aSjaacYca cqaHXoqycaGGSaGaamiCaaGaayjkaiaawMcaaiabg2da9maabmaapa qaa8qacaaIXaGaeyOeI0IaamiCaaGaayjkaiaawMcaaiaadwgajyaG paWaaWbaaeqabaWdbiabgkHiTmaabmaapaqaa8qadaWcaaWdaeaape GaamiDaaWdaeaapeGaeq4UdWgaaaGaayjkaiaawMcaa8aadaahaaqa beaapeGaeyOeI0IaeqySdegaaaaaaaa@4F6F@ .  (6)

In this case, the LF has the quantile function in closed-form and is given by

t u =λlog ( ( 1p ) u ) 1 α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiDaKGba+aadaWgaaqaa8qacaWG1baapaqabaqcfa4dbiab g2da9iabeU7aSjGacYgacaGGVbGaai4zamaabmaapaqaa8qadaWcaa WdaeaapeWaaeWaa8aabaWdbiaaigdacqGHsislcaWGWbaacaGLOaGa ayzkaaaapaqaa8qacaWG1baaaaGaayjkaiaawMcaaKGba+aadaahaa qcfayabeaajugWa8qacqGHsisljyaGdaWcaaqcfa4daeaajugWa8qa caaIXaaajuaGpaqaaKqzadWdbiabeg7aHbaaaaaaaa@5090@    (7)

where 0u<1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaaGimaiabgsMiJkaadwhacqGH8aapcaaIXaaaaa@3B62@ . The r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOCaaaa@3731@  -th moments of T about the origin is

E( T r ;λ,α,p )=( 1p ) λ r Γ( 1 r α ) , α>r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamyramaabmaapaqaa8qacaWGubqcga4damaaCaaabeqaa8qa caWGYbaaa8aacaGG7aqcfa4dbiabeU7aSjaacYcacqaHXoqycaGGSa GaamiCaaGaayjkaiaawMcaaiabg2da9maabmaapaqaa8qacaaIXaGa eyOeI0IaamiCaaGaayjkaiaawMcaaiabeU7aSLGba+aadaahaaqabe aapeGaaeOCaaaapaGaeu4KdCucfa4dbmaabmaapaqaa8qacaaIXaGa eyOeI0YaaSaaa8aabaWdbiaadkhaa8aabaWdbiabeg7aHbaaaiaawI cacaGLPaaacaGGGcGaaiilaiaacckacqaHXoqycqGH+aGpcaWGYbaa aa@5A02@ , (8)

For r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOCaiabgIGioprr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhA Gq1DVbacfaGae8xfH4eaaa@4361@ and Γ( x )= 0 e y y x1 dy MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeu4KdC0aaeWaa8aabaWdbiaadIhaaiaawIcacaGLPaaacqGH 9aqpdaGfWbqabKGba+aabaWdbiaaicdaa8aabaWdbiabg6HiLcqcfa 4daeaapeGaey4kIipaaiaadwgajyaGpaWaaWbaaeqabaWdbiabgkHi TiaadMhaaaqcfaOaamyEaKGba+aadaahaaqabeaapeGaamiEaiabgk HiTiaaigdaaaqcfaOaamizaiaadMhaaaa@4C65@ is called gamma function. Along with some algebraic manipulation the mean and variance of the LF distribution are given, respectively, by

E( T;λ,α,p )=( 1p )λΓ( 1 1 α ) ,  α>1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamyramaabmaapaqaa8qacaWGubGaai4oaiabeU7aSjaacYca cqaHXoqycaGGSaGaamiCaaGaayjkaiaawMcaaiabg2da9maabmaapa qaa8qacaaIXaGaeyOeI0IaamiCaaGaayjkaiaawMcaaiabeU7aSjab fo5ahnaabmaapaqaa8qacaaIXaGaeyOeI0YaaSaaa8aabaWdbiaaig daa8aabaWdbiabeg7aHbaaaiaawIcacaGLPaaacaGGGcGaaiilaiaa cckacaGGGcGaeqySdeMaeyOpa4JaaGymaaaa@55C8@

and

V( T;λ,α,p )=( 1p ) λ 2 ( Γ( 1 2 α )( 1p )Γ ( 1 1 α ) 2 ),   α>2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOvamaabmaapaqaa8qacaWGubGaai4oaiabeU7aSjaacYca cqaHXoqycaGGSaGaamiCaaGaayjkaiaawMcaaiabg2da9maabmaapa qaa8qacaaIXaGaeyOeI0IaamiCaaGaayjkaiaawMcaaiabeU7aS9aa daahaaqabKGbagaapeGaaGOmaaaajuaGdaqadaWdaeaapeGaeu4KdC 0aaeWaa8aabaWdbiaaigdacqGHsisldaWcaaWdaeaapeGaaGOmaaWd aeaapeGaeqySdegaaaGaayjkaiaawMcaaiabgkHiTmaabmaapaqaa8 qacaaIXaGaeyOeI0IaamiCaaGaayjkaiaawMcaaiabfo5ahnaabmaa paqaa8qacaaIXaGaeyOeI0YaaSaaa8aabaWdbiaaigdaa8aabaWdbi abeg7aHbaaaiaawIcacaGLPaaapaWaaWbaaeqajyaGbaWdbiaaikda aaaajuaGcaGLOaGaayzkaaGaaiilaiaacckacaGGGcGaaiiOaiabeg 7aHjabg6da+iaaikdaaaa@6849@ .

The survival and hazard functions of LF( λ,α,p ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamitaiaadAeadaqadaWdaeaapeGaeq4UdWMaaiilaiabeg7a HjaacYcacaWGWbaacaGLOaGaayzkaaaaaa@3F25@ distribution is given by

S( t;λ,α,p )=p+( 1p )( 1 e ( t λ ) α ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaam4uamaabmaapaqaa8qacaWG0bGaai4oaiabeU7aSjaacYca cqaHXoqycaGGSaGaamiCaaGaayjkaiaawMcaaiabg2da9iaadchacq GHRaWkdaqadaWdaeaapeGaaGymaiabgkHiTiaadchaaiaawIcacaGL PaaadaqadaWdaeaapeGaaGymaiabgkHiTiaadwgajyaGpaWaaWbaae qabaWdbiabgkHiTmaabmaapaqaa8qadaWcaaWdaeaapeGaamiDaaWd aeaapeGaeq4UdWgaaaGaayjkaiaawMcaa8aadaahaaqabeaapeGaey OeI0IaeqySdegaaaaaaKqbakaawIcacaGLPaaaaaa@5530@    (9)

and

h( t;λ,α,p )= α λ ( t λ ) ( α+1 ) e ( t λ ) α p+( 1p )( 1exp( ( t λ ) α ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiAamaabmaapaqaa8qacaWG0bGaai4oaiabeU7aSjaacYca cqaHXoqycaGGSaGaamiCaaGaayjkaiaawMcaaiabg2da9maalaaapa qaa8qadaWcaaWdaeaapeGaeqySdegapaqaa8qacqaH7oaBaaWaaeWa a8aabaWdbmaalaaapaqaa8qacaWG0baapaqaa8qacqaH7oaBaaaaca GLOaGaayzkaaqcga4damaaCaaabeqaa8qacqGHsisldaqadaWdaeaa peGaeqySdeMaey4kaSIaaGymaaGaayjkaiaawMcaaaaajuaGcaWGLb qcga4damaaCaaabeqaa8qacqGHsisldaqadaWdaeaapeWaaSaaa8aa baWdbiaadshaa8aabaWdbiabeU7aSbaaaiaawIcacaGLPaaapaWaaW baaeqabaWdbiabgkHiTiabeg7aHbaaaaaajuaGpaqaa8qacaWGWbGa ey4kaSYaaeWaa8aabaWdbiaaigdacqGHsislcaWGWbaacaGLOaGaay zkaaWaaeWaa8aabaWdbiaaigdacqGHsislciGGLbGaaiiEaiaaccha daqadaWdaeaapeGaeyOeI0YaaeWaa8aabaWdbmaalaaapaqaa8qaca WG0baapaqaa8qacqaH7oaBaaaacaGLOaGaayzkaaWdamaaCaaabeqc gayaa8qacqGHsislcqaHXoqyaaaajuaGcaGLOaGaayzkaaaacaGLOa Gaayzkaaaaaaaa@7303@ . (10)

Parameter Estimation

For each failure time related to the i-th individual, it may not be perceived or subject by the right censoring. Furthermore, the random censoring times C i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaam4qa8aadaWgaaqcgayaa8qacaWGPbaajuaGpaqabaaaaa@395B@ s are independent of T i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamivaKGba+aadaWgaaqaa8qacaWGPbaapaqabaaaaa@38DE@ s (non-censored time) and their distribution does not depend on the parameters. In a scenario of a n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOBaaaa@372C@  sample of size, the data set will be describe by D=( t i , δ i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamrr1ngBPrwtHr hAXaqeguuDJXwAKbstHrhAG8KBLbacfaqcfaieaaaaaaaaa8qacqWF deprcqGH9aqpdaqadaWdaeaapeGaamiDaKGba+aadaWgaaqaa8qaca WGPbaapaqabaqcfa4dbiaacYcacqaH0oazjyaGpaWaaSbaaeaapeGa amyAaaWdaeqaaaqcfa4dbiaawIcacaGLPaaaaaa@4C50@ , where t i =min( T i , C i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiDaKGba+aadaWgaaqaa8qacaWGPbaapaqabaqcfa4dbiab g2da9iGac2gacaGGPbGaaiOBamaabmaapaqaa8qacaWGubqcga4dam aaBaaabaWdbiaadMgaa8aabeaajuaGpeGaaiilaiaadoeajyaGpaWa aSbaaeaapeGaamyAaaWdaeqaaaqcfa4dbiaawIcacaGLPaaaaaa@4641@ and δ i =I( T i C i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqiTdq2damaaBaaajyaGbaWdbiaadMgaaKqba+aabeaapeGa eyypa0Jaamysamaabmaapaqaa8qacaWGubqcga4damaaBaaabaWdbi aadMgaa8aabeaajuaGpeGaeyizImQaam4qaKGba+aadaWgaaqaa8qa caWGPbaapaqabaaajuaGpeGaayjkaiaawMcaaaaa@45EE@ . This general random censoring scheme has as special case type I and II censoring mechanism. The likelihood function is given by

L( θ;D )= i=1 n f ( t i ;θ ) δ i S ( t i ;θ ) 1 δ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamitamaabmaapaqaaeXafv3ySLgzGmvETj2BSbacfeWdbiab =H7aXjaacUdatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaG Gbbiab+nq8ebGaayjkaiaawMcaaiabg2da9maawahabeqcga4daeaa peGaamyAaiabg2da9iaaigdaa8aabaWdbiaad6gaaKqba+aabaWdbi abg+GivdaacaWGMbWaaeWaa8aabaWdbiaadshajyaGpaWaaSbaaeaa peGaamyAaaWdaeqaaKqba+qacaGG7aGae8hUdehacaGLOaGaayzkaa WdamaaCaaabeqcgayaa8qacqaH0oazpaWaaSbaaeaapeGaamyAaaWd aeqaaaaajuaGpeGaam4uamaabmaapaqaa8qacaWG0bqcga4damaaBa aabaWdbiaadMgaa8aabeaajuaGpeGaai4oaiab=H7aXbGaayjkaiaa wMcaa8aadaahaaqabeaapeGaaGymaiabgkHiTiabes7aK9aadaWgaa qcgayaa8qacaWGPbaajuaGpaqabaaaaaaa@6EAB@

Let T 1 , , T n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamivaKGba+aadaWgaaqaa8qacaaIXaaapaqabaqcfa4dbiaa cYcacaGGGcGaeyOjGWRaaiilaiaadsfajyaGpaWaaSbaaeaapeGaam OBaaWdaeqaaaaa@4005@ be a random sample of LF distribution, the likelihood function considering data with random censoring is given by

L( λ,α,p;D )= α d ( 1p ) d λ dα i=1 n t i δ i ( α+1 ) exp( i=1 n δ i ( t i λ ) α )× ( p+( 1p )( 1 e ( t λ ) α ) ) 1 δ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamitamaabmaapaqaa8qacqaH7oaBcaGGSaGaeqySdeMaaiil aiaadchacaGG7aWefv3ySLgznfgDOfdaryqr1ngBPrginfgDObYtUv gaiuqacqWFdepraiaawIcacaGLPaaacqGH9aqpdaWcaaWdaeaapeGa eqySde2damaaCaaabeqcgayaa8qacaWGKbaaaKqbaoaabmaapaqaa8 qacaaIXaGaeyOeI0IaamiCaaGaayjkaiaawMcaa8aadaahaaqabKGb agaapeGaamizaaaaaKqba+aabaWdbiabeU7aSLGba+aadaahaaqabe aapeGaeyOeI0Iaamizaiabeg7aHbaaaaqcfa4aaybCaeqajyaGpaqa a8qacaWGPbGaeyypa0JaaGymaaWdaeaapeGaamOBaaqcfa4daeaape Gaey4dIunaaiaadshapaWaa0baaeaapeGaamyAaaWdaeaapeGaeyOe I0IaeqiTdqwcga4damaaBaaabaWdbiaadMgaa8aabeaajuaGpeWaae Waa8aabaWdbiabeg7aHjabgUcaRiaaigdaaiaawIcacaGLPaaaaaGa ciyzaiaacIhacaGGWbWaaeWaa8aabaWdbiabgkHiTmaawahabeqcga 4daeaapeGaamyAaiabg2da9iaaigdaa8aabaWdbiaad6gaaKqba+aa baWdbiabggHiLdaacqaH0oazjyaGpaWaaSbaaeaapeGaamyAaaWdae qaaKqba+qadaqadaWdaeaapeWaaSaaa8aabaWdbiaadshajyaGpaWa aSbaaeaapeGaamyAaaWdaeqaaaqcfayaa8qacqaH7oaBaaaacaGLOa Gaayzkaaqcga4damaaCaaabeqaa8qacqGHsislcqaHXoqyaaaajuaG caGLOaGaayzkaaGaey41aq7aaeWaa8aabaWdbiaadchacqGHRaWkda qadaWdaeaapeGaaGymaiabgkHiTiaadchaaiaawIcacaGLPaaadaqa daWdaeaapeGaaGymaiabgkHiTiaadwgapaWaaWbaaeqabaqcga4dbi abgkHiTmaabmaapaqaa8qadaWcaaWdaeaapeGaamiDaaWdaeaapeGa eq4UdWgaaaGaayjkaiaawMcaa8aadaahaaqabeaapeGaeyOeI0Iaeq ySdegaaaaaaKqbakaawIcacaGLPaaaaiaawIcacaGLPaaajyaGpaWa aWbaaeqabaWdbiaaigdacqGHsislcqaH0oazpaWaaSbaaeaapeGaam yAaaWdaeqaaaaaaaa@A8E1@

where d= i=1 n δ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamizaiabg2da9maawahabeqcga4daeaapeGaamyAaiabg2da 9iaaigdaa8aabaWdbiaad6gaaKqba+aabaWdbiabggHiLdaacqaH0o azjyaGpaWaaSbaaeaapeGaamyAaaWdaeqaaaaa@42E8@ . The log-likelihood function is given as

l( λ,α,p;D )=dlog( α )+dlog( 1p )+dαlogλ( α+1 ) i=1 n δ i log t i i=1 n δ i ( λ t i ) α + i=1 n ( 1 δ i )log( p+( 1p )( 1 e ( t λ ) α ) ). MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiBamaabmaapaqaa8qacqaH7oaBcaGGSaGaeqySdeMaaiil aiaadchacaGG7aWefv3ySLgznfgDOfdaryqr1ngBPrginfgDObYtUv gaiuqacqWFdepraiaawIcacaGLPaaacqGH9aqpcaWGKbGaciiBaiaa c+gacaGGNbWaaeWaa8aabaWdbiabeg7aHbGaayjkaiaawMcaaiabgU caRiaadsgaciGGSbGaai4BaiaacEgadaqadaWdaeaapeGaaGymaiab gkHiTiaadchaaiaawIcacaGLPaaacqGHRaWkcaWGKbGaeqySdeMaci iBaiaac+gacaGGNbGaeq4UdWMaeyOeI0YaaeWaa8aabaWdbiabeg7a HjabgUcaRiaaigdaaiaawIcacaGLPaaadaGfWbqabKGba+aabaWdbi aadMgacqGH9aqpcaaIXaaapaqaa8qacaWGUbaajuaGpaqaa8qacqGH ris5aaGaeqiTdqwcga4damaaBaaabaWdbiaadMgaa8aabeaajuaGpe GaciiBaiaac+gacaGGNbGaamiDaKGba+aadaWgaaqaa8qacaWGPbaa paqabaqcfa4dbiabgkHiTmaawahabeqcga4daeaapeGaamyAaiabg2 da9iaaigdaa8aabaWdbiaad6gaaKqba+aabaWdbiabggHiLdaacqaH 0oazjyaGpaWaaSbaaeaapeGaamyAaaWdaeqaaKqba+qadaqadaWdae aapeWaaSaaa8aabaWdbiabeU7aSbWdaeaapeGaamiDa8aadaWgaaqc gayaa8qacaWGPbaajuaGpaqabaaaaaWdbiaawIcacaGLPaaapaWaaW baaeqabaWdbiabeg7aHbaacqGHRaWkdaGfWbqabKGba+aabaWdbiaa dMgacqGH9aqpcaaIXaaapaqaa8qacaWGUbaajuaGpaqaa8qacqGHri s5aaWaaeWaa8aabaWdbiaaigdacqGHsislcqaH0oazjyaGpaWaaSba aeaapeGaamyAaaWdaeqaaaqcfa4dbiaawIcacaGLPaaaciGGSbGaai 4BaiaacEgadaqadaWdaeaapeGaamiCaiabgUcaRmaabmaapaqaa8qa caaIXaGaeyOeI0IaamiCaaGaayjkaiaawMcaamaabmaapaqaa8qaca aIXaGaeyOeI0IaamyzaKGba+aadaahaaqabeaapeGaeyOeI0YaaeWa a8aabaWdbmaalaaapaqaa8qacaWG0baapaqaa8qacqaH7oaBaaaaca GLOaGaayzkaaWdamaaCaaabeqaa8qacqGHsislcqaHXoqyaaaaaaqc faOaayjkaiaawMcaaaGaayjkaiaawMcaaiaac6caaaa@B8B8@ (11)

The maximum likelihood estimators (MLEs) are widely explored as statistical inferential methodology due its many desirable properties, in which includes consistency, asymptotic efficiency and invariance. The MLEs are obtained from the maximization of the log-likelihood function (11). Before we derive the MLEs of the LF, let us define the following function

η j ( λ,α,p;D )= i=1 n ( 1 δ i ) logS( t i ;θ ) θ j ,  j=1,2,3. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeq4TdGwcga4damaaBaaajuaGbaqcLbmapeGaamOAaaqcfa4d aeqaa8qadaqadaWdaeaapeGaeq4UdWMaaiilaiabeg7aHjaacYcaca WGWbGaai4oamrr1ngBPrwtHrhAXaqeguuDJXwAKbstHrhAG8KBLbac feGae83aXteacaGLOaGaayzkaaGaeyypa0ZaaybCaeqajyaGpaqaa8 qacaWGPbGaeyypa0JaaGymaaWdaeaapeGaamOBaaqcfa4daeaapeGa eyyeIuoaamaabmaapaqaa8qacaaIXaGaeyOeI0IaeqiTdq2damaaBa aajyaGbaWdbiaadMgaaKqba+aabeaaa8qacaGLOaGaayzkaaWaaSaa a8aabaWdbiGacYgacaGGVbGaai4zaiaadofadaqadaWdaeaapeGaam iDaKGba+aadaWgaaqaa8qacaWGPbaapaqabaqcfa4dbiaacUdarmqr 1ngBPrgitLxBI9gBaGGbbiab+H7aXbGaayjkaiaawMcaaaWdaeaape GaeyOaIyRaeqiUde3damaaBaaajyaGbaWdbiaadQgaaKqba+aabeaa aaWdbiaacYcacaGGGcGaaiiOaiaadQgacqGH9aqpcaaIXaGaaiilai aaikdacaGGSaGaaG4maiaac6caaaa@7E50@

Then, the likelihood equations are given by

dα λ α λ α1 i=1 n δ i t i α + η 1 ( λ,α,p;D )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaSaaa8aabaWdbiaadsgacqaHXoqya8aabaWdbiabeU7aSbaa cqGHsislcqaHXoqycqaH7oaBjyaGpaWaaWbaaeqabaWdbiabeg7aHj abgkHiTiaaigdaaaqcfa4aaybCaeqajyaGpaqaa8qacaWGPbGaeyyp a0JaaGymaaWdaeaapeGaamOBaaqcfa4daeaapeGaeyyeIuoaaiabes 7aKLGba+aadaWgaaqaa8qacaWGPbaapaqabaqcfa4dbiaadshajyaG paWaa0baaeaapeGaamyAaaWdaeaapeGaeyOeI0IaeqySdegaaKqbak abgUcaRiabeE7aOLGba+aadaWgaaqaa8qacaaIXaaapaqabaqcfa4d bmaabmaapaqaa8qacqaH7oaBcaGGSaGaeqySdeMaaiilaiaadchaca GG7aWefv3ySLgznfgDOfdaryqr1ngBPrginfgDObYtUvgaiuqacqWF depraiaawIcacaGLPaaacqGH9aqpcaaIWaaaaa@6ED1@ ,

d α +dlogλ i=1 n δ i ( λ t i ) α log( λ t i ) i=1 n δ i log t i + η 2 ( λ,α,p;D )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaSaaa8aabaWdbiaadsgaa8aabaWdbiabeg7aHbaacqGHRaWk caWGKbGaciiBaiaac+gacaGGNbGaeq4UdWMaeyOeI0YaaybCaeqajy aGpaqaa8qacaWGPbGaeyypa0JaaGymaaWdaeaapeGaamOBaaqcfa4d aeaapeGaeyyeIuoaaiabes7aKLGba+aadaWgaaqaa8qacaWGPbaapa qabaqcfa4dbmaabmaapaqaa8qadaWcaaWdaeaapeGaeq4UdWgapaqa a8qacaWG0bqcga4damaaBaaabaWdbiaadMgaa8aabeaaaaaajuaGpe GaayjkaiaawMcaa8aadaahaaqabKGbagaapeGaeqySdegaaKqbakGa cYgacaGGVbGaai4zamaabmaapaqaa8qadaWcaaWdaeaapeGaeq4UdW gapaqaa8qacaWG0bqcga4damaaBaaabaWdbiaadMgaa8aabeaaaaaa juaGpeGaayjkaiaawMcaaiabgkHiTmaawahabeqcga4daeaapeGaam yAaiabg2da9iaaigdaa8aabaWdbiaad6gaaKqba+aabaWdbiabggHi LdaacqaH0oazjyaGpaWaaSbaaeaapeGaamyAaaWdaeqaaKqba+qaci GGSbGaai4BaiaacEgacaWG0bqcga4damaaBaaabaWdbiaadMgaa8aa beaajuaGpeGaey4kaSIaeq4TdGwcga4damaaBaaabaWdbiaaikdaa8 aabeaajuaGpeWaaeWaa8aabaWdbiabeU7aSjaacYcacqaHXoqycaGG SaGaamiCaiaacUdatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5 wzaGqbbiab=nq8ebGaayjkaiaawMcaaiabg2da9iaaicdaaaa@8BE9@ and

d p1 + η 3 ( λ,α,p;D )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaSaaa8aabaWdbiaadsgaa8aabaWdbiaadchacqGHsislcaaI XaaaaiabgUcaRiabeE7aOLGba+aadaWgaaqaa8qacaaIZaaapaqaba qcfa4dbmaabmaapaqaa8qacqaH7oaBcaGGSaGaeqySdeMaaiilaiaa dchacaGG7aWefv3ySLgznfgDOfdaryqr1ngBPrginfgDObYtUvgaiu qacqWFdepraiaawIcacaGLPaaacqGH9aqpcaaIWaaaaa@53EB@ .

The maximization of the log-likelihood function can be performed directly by using existing statistical packages. Further information about the numerical procedures will be discussed in the next section.

According to Migon et al. [10], under mild conditions the obtained estimators are consistent and efficient with an asymptotically normal joint distribution given by

( λ ^ , α ^ , p ^ )~ N 3 ( ( λ,α,p ), H 1 ( λ,α,p ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaeWaa8aabaWaaCbiaeaapeGafq4UdWMbaKaaa8aabeqaaaaa peGaaiila8aadaWfGaqaa8qacuaHXoqygaqcaaWdaeqabaaaa8qaca GGSaWdamaaxacabaWdbiqadchagaqcaaWdaeqabaaaaaWdbiaawIca caGLPaaacaGG+bGaamOtaKGba+aadaWgaaqaa8qacaaIZaaapaqaba qcfa4dbmaabmaapaqaa8qadaqadaWdaeaapeGaeq4UdWMaaiilaiab eg7aHjaacYcacaWGWbaacaGLOaGaayzkaaGaaiilaiaadIeapaWaaW baaeqajyaGbaWdbiabgkHiTiaaigdaaaqcfa4aaeWaa8aabaWdbiab eU7aSjaacYcacqaHXoqycaGGSaGaamiCaaGaayjkaiaawMcaaaGaay jkaiaawMcaaaaa@57E7@ ,

where Η( λ,α,P ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGGabKqbacbaaa aaaaaapeGae83LdG0aaeWaa8aabaaccmWdbiab+T7aSjaacYcacqGF XoqycaGGSaGaaCiuaaGaayjkaiaawMcaaaaa@3EE0@ , is the 3×3 observed Fisher information matrix and H ij ( λ,α,p ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqadKqbacbaaa aaaaaapeGaa8hsaKGba+aadaWgaaqaa8qacaWFPbGaa8NAaaWdaeqa aKqba+qadaqadaWdaeaaiiWapeGae43UdWMaaiilaiab+f7aHjaacY cacaWFWbaacaGLOaGaayzkaaaaaa@41AD@ is the Fisher information given by

H ij ( θ )= θ i θ j l( θ;D ),   i,j=1,2,3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamisa8aadaWgaaqaa8qacaWGPbqcgaOaamOAaaqcfa4daeqa a8qadaqadaWdaeaacaWH4oaapeGaayjkaiaawMcaaiabg2da9iabgk HiTmaalaaapaqaa8qacqGHciITa8aabaWdbiabgkGi2kabeI7aXLGb a+aadaWgaaqaa8qacaWGPbaapaqabaqcfa4dbiabgkGi2kabeI7aX9 aadaWgaaqcgayaa8qacaWGQbaajuaGpaqabaaaa8qacaWGSbWaaeWa a8aabaWdbiaahI7acaGG7aWefv3ySLgznfgDOfdaryqr1ngBPrginf gDObYtUvgaiuqacqWFdepraiaawIcacaGLPaaacaGGSaGaaiiOaiaa cckacaGGGcGaamyAaiaacYcacaWGQbGaeyypa0JaaGymaiaacYcaca aIYaGaaiilaiaaiodaaaa@6727@ .

Note that, the observed Fisher information matrix was used since it is not possible to compute the expected Fisher information matrix due its lack of closed form expression. For large samples, confidence intervals approximation can be constructed for the individual parameters θ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaeXafv3ySLgzGm vETj2BSbacfeqcfaieaaaaaaaaa8qacqWF4oqCjyaGpaWaaSbaaKqb agaajugWa8qacaWHPbaajuaGpaqabaaaaa@4113@ i=1,2,3, assuming a confidence coefficient 100( 1γ )% MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaaGymaiaaicdacaaIWaWaaeWaa8aabaWdbiaaigdacqGHsisl iiqacqWFZoWzaiaawIcacaGLPaaaieWacaGFLaaaaa@3E17@ the marginal distributions are given by

θ ^ i ~N( θ i , H ii 1 ( θ ) ),  i=1,2,3. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaxacaba aeaaaaaaaaa8qacuaH4oqCgaqcaaWdaeqabaaaamaaBaaajyaGbaWd biaadMgaaKqba+aabeaapeGaaiOFaiaad6eadaqadaWdaeaapeGaeq iUde3damaaBaaajyaGbaWdbiaadMgaaKqba+aabeaapeGaaiilaiaa dIeajyaGpaWaa0baaeaapeGaamyAaiaadMgaa8aabaWdbiabgkHiTi aaigdaaaqcfa4aaeWaa8aabaqeduuDJXwAKbYu51MyVXgaiuqapeGa e8hUdehacaGLOaGaayzkaaaacaGLOaGaayzkaaGaaiilaiaacckaca GGGcGaamyAaiabg2da9iaaigdacaGGSaGaaGOmaiaacYcacaaIZaGa aiOlaaaa@5A5C@

Simulation Study

The maximum likelihood method efficiency was analyzed through a simulation study on the LF distribution. This procedure was conducted by computing the mean relative errors (MRE) and the mean square errors (MSE) given by

MR E i = 1 N j=1 N θ ^ i,j θ i ,    MS E i = 1 N j=1 N ( θ ^ i,j θ i ) 2 ,  for i=1,2,3.  MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamytaiaadkfacaWGfbqcga4damaaBaaabaWdbiaadMgaa8aa beaajuaGpeGaeyypa0ZaaSaaa8aabaWdbiaaigdaa8aabaWdbiaad6 eaaaWaaybCaeqapaqaa8qacaWGQbGaeyypa0JaaGymaaqcga4daeaa peGaamOtaaqcfa4daeaapeGaeyyeIuoaamaalaaapaqaa8qacuaH4o qCgaqcaKGba+aadaWgaaqaa8qacaWGPbGaaiilaiaadQgaa8aabeaa aKqbagaapeGaeqiUdexcga4damaaBaaabaWdbiaadMgaa8aabeaaaa qcfa4dbiaacYcacaGGGcGaaiiOaiaacckacaGGGcGaamytaiaadofa caWGfbqcga4damaaBaaabaWdbiaadMgaa8aabeaajuaGpeGaeyypa0 ZaaSaaa8aabaWdbiaaigdaa8aabaWdbiaad6eaaaWaaybCaeqajyaG paqaa8qacaWGQbGaeyypa0JaaGymaaWdaeaapeGaamOtaaqcfa4dae aapeGaeyyeIuoaamaabmaapaqaa8qacuaH4oqCgaqca8aadaWgaaqa aKGba+qacaWGPbqcfaOaaiilaKGbakaadQgaaKqba+aabeaapeGaey OeI0IaeqiUdexcga4damaaBaaabaWdbiaadMgaa8aabeaaaKqba+qa caGLOaGaayzkaaqcga4damaaCaaabeqaa8qacaaIYaaaaKqbakaacY cacaGGGcGaaiiOaiaabAgacaqGVbGaaeOCaiaacckacaWGPbGaeyyp a0JaaGymaiaacYcacaaIYaGaaiilaiaaiodacaGGUaGaaiiOaaaa@817A@

as N is the number of estimates obtained through the MLE approach. The 95% coverage probabilities of the asymptotic confidence intervals were also evaluated. The adopted approach prioritize that the expected MLEs returns the MREs closer to one with smaller MSEs. Additionally, by considering a 95% confidence level, the interval covers the true values of θ closer to 95%.Considering scenarios with sample sizes n=(10,25,50,100, 200) and N=100,000 for the simulation study, two situations are presented by considering the proportion of cure in the population of 0.3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaaGimaiaac6cacaaIZaaaaa@3863@ and 0.5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaaGimaiaac6cacaaI1aaaaa@3865@ . In these cases, the censored proportions are observed in different levels.

In pursuance to find the maximization of the log-likelihood function, described in the equation (11), the package called maxLik available in R developed by Henningsen and Toomet [11] was used. The numerical results are well-behaved since was not found numerical problems using the SANN method (Simulated-annealing), such as failure evidence of convergence or end on multiple maxima. The programs can be obtained, upon request.

The estimates obtained from Tables 1-4 for α, λ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGGadKqbacbaaa aaaaaapeGae8xSdeMaaiilaGqadiaa+bkacqWF7oaBaaa@3B6C@  and p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqadKqbacbaaa aaaaaapeGaa8hCaaaa@3737@ are asymptotically unbiased, implying that MREs tend to one when n increases and the MSEs decrease to zero for n large. Analyzing the MLEs performance, with a coverage probabilities tending to 0.95, good coverage properties may be deliberated for the parameter estimators. In practical applications, those estimation procedures will be relevant as shown in the next section.

θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGGadiab=H7aXb aa@374A@

α=0.5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHXoqycqGH9aqpcaaIWaGaaiOlaiaaiwdaaaa@3B75@

λ=2.0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaH7oaBcqGH9aqpcaaIYaGaaiOlaiaaicdaaaa@3B87@

p=0.3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGWbGaeyypa0JaaGimaiaac6cacaaIZaaaaa@3AC9@

0.457

n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGUbaaaa@3798@

MRE

MSE

C 95% MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGdbWdamaaBaaabaqcLbmapeGaaGyoaiaaiwdacaGGLaaa juaGpaqabaaaaa@3BA1@

MRE

MSE

C 95% MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGdbWdamaaBaaabaqcLbmapeGaaGyoaiaaiwdacaGGLaaa juaGpaqabaaaaa@3BA1@

MRE

MSE

C 95% MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGdbWdamaaBaaabaqcLbmapeGaaGyoaiaaiwdacaGGLaaa juaGpaqabaaaaa@3BA1@

M p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGnbWcpaWaaSbaaKqbagaajugWa8qacaWGWbaajuaGpaqa baaaaa@3B10@

25

1.265

0.060

0.948

1.100

3.344

0.810

1.100

0.021

0.925

0.461

50

1.114

0.020

0.952

1.117

2.002

0.860

1.024

0.013

0.938

0.458

100

1.048

0.008

0.952

1.098

1.087

0.893

0.992

0.008

0.948

0.457

200

1.022

0.004

0.953

1.059

0.488

0.919

0.991

0.004

0.952

0.457

300

1.014

0.003

0.951

1.039

0.293

0.928

0.993

0.003

0.952

0.457

Table 1: MREs, MSEs, C 95% MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGdbWdamaaBaaabaqcLbmapeGaaGyoaiaaiwdacaGGLaaa juaGpaqabaaaaa@3BA1@ estimates for 100,000 considering n = (25, 50, 100, 200, 300) and 45.7% of censorship.

 

α=0.5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHXoqycqGH9aqpcaaIWaGaaiOlaiaaiwdaaaa@3B75@

λ=2.0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaH7oaBcqGH9aqpcaaIYaGaaiOlaiaaicdaaaa@3B87@

p=0.5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGWbGaeyypa0JaaGimaiaac6cacaaI1aaaaa@3ACB@

0.612

n

MRE

MSE

C 95% MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGdbWdamaaBaaabaqcLbmapeGaaGyoaiaaiwdacaGGLaaa juaGpaqabaaaaa@3BA1@

MRE

MSE

C 95% MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGdbWdamaaBaaabaqcLbmapeGaaGyoaiaaiwdacaGGLaaa juaGpaqabaaaaa@3BA1@

MRE

MSE

C 95% MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGdbWdamaaBaaabaqcLbmapeGaaGyoaiaaiwdacaGGLaaa juaGpaqabaaaaa@3BA1@

M p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGnbWcpaWaaSbaaKqbagaajugWa8qacaWGWbaajuaGpaqa baaaaa@3B0F@

25

1.384

0.144

0.939

1.263

8.714

0.788

1.027

0.022

0.914

0.612

50

1.158

0.033

0.940

1.238

5.967

0.838

1.001

0.014

0.928

0.612

100

1.070

0.013

0.946

1.156

2.683

0.877

0.994

0.007

0.940

0.612

200

1.031

0.006

0.951

1.085

0.841

0.906

0.994

0.004

0.948

0.612

300

1.020

0.004

0.951

1.056

0.459

0.921

0.996

0.002

0.951

0.612

Table 2: MREs, MSEs, C 95% MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGdbWdamaaBaaabaqcLbmapeGaaGyoaiaaiwdacaGGLaaa juaGpaqabaaaaa@3BA1@ estimates for 100,000 considering n = (25, 50, 100, 200, 300) and 61% of censorship.

 

α=2.0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHXoqycqGH9aqpcaaIYaGaaiOlaiaaicdaaaa@3B72@

λ=4.0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaH7oaBcqGH9aqpcaaI0aGaaiOlaiaaicdaaaa@3B89@

p=0.3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGWbGaeyypa0JaaGimaiaac6cacaaIZaaaaa@3AC9@

0.35

n

MRE

MSE

C 95% MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGdbWdamaaBaaabaqcLbmapeGaaGyoaiaaiwdacaGGLaaa juaGpaqabaaaaa@3BA1@

MRE

MSE

C 95% MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGdbWdamaaBaaabaqcLbmapeGaaGyoaiaaiwdacaGGLaaa juaGpaqabaaaaa@3BA1@

MRE

MSE

C 95% MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGdbWdamaaBaaabaqcLbmapeGaaGyoaiaaiwdacaGGLaaa juaGpaqabaaaaa@3BA1@

M p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGnbWcpaWaaSbaaKqbagaajugWa8qacaWGWbaajuaGpaqa baaaaa@3B0F@

25

1.103

0.316

0.951

1.022

0.347

0.921

0.997

0.010

0.927

0.349

50

1.047

0.116

0.951

1.011

0.155

0.938

0.998

0.005

0.937

0.348

100

1.023

0.050

0.950

1.005

0.074

0.945

1.000

0.002

0.945

0.349

200

1.011

0.023

0.950

1.003

0.036

0.947

1.000

0.001

0.947

0.349

300

1.008

0.015

0.951

1.002

0.024

0.947

0.999

0.001

0.947

0.348

Table 3: MREs, MSEs, C 95% MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGdbWdamaaBaaabaqcLbmapeGaaGyoaiaaiwdacaGGLaaa juaGpaqabaaaaa@3BA1@ estimates for 100,000 considering n = (25, 50, 100, 200, 300) and 35% of censorship.

 

α=2.0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHXoqycqGH9aqpcaaIYaGaaiOlaiaaicdaaaa@3B72@

λ=4.0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaH7oaBcqGH9aqpcaaI0aGaaiOlaiaaicdaaaa@3B89@

p=0.3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGWbGaeyypa0JaaGimaiaac6cacaaIZaaaaa@3AC9@

0.535

n

MRE

MSE

C 95% MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGdbWdamaaBaaabaqcLbmapeGaaGyoaiaaiwdacaGGLaaa juaGpaqabaaaaa@3BA1@

MRE

MSE

C 95% MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGdbWdamaaBaaabaqcLbmapeGaaGyoaiaaiwdacaGGLaaa juaGpaqabaaaaa@3BA1@

MRE

MSE

C 95% MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGdbWdamaaBaaabaqcLbmapeGaaGyoaiaaiwdacaGGLaaa juaGpaqabaaaaa@3BA1@

M p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGnbWcpaWaaSbaaKqbagaajugWa8qacaWGWbaajuaGpaqa baaaaa@3B0F@

25

1.158

0.619

0.953

1.034

0.546

0.910

0.998

0.011

0.933

0.535

50

1.068

0.182

0.953

1.016

0.230

0.933

0.999

0.006

0.942

0.535

100

1.033

0.075

0.950

1.007

0.105

0.942

1.000

0.003

0.947

0.535

200

1.016

0.034

0.950

1.004

0.051

0.946

0.999

0.001

0.947

0.534

300

1.011

0.022

0.949

1.002

0.034

0.948

1.000

0.001

0.948

0.535

Table 4: MREs, MSEs, C 95% MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGdbWdamaaBaaabaqcLbmapeGaaGyoaiaaiwdacaGGLaaa juaGpaqabaaaaa@3BA1@ estimates for 100,000 considering n = (25, 50, 100, 200, 300) and 53.5% of censorship.

Application

In this section, we considered the data set presented by Kersey et al. [12]. The results were collected in a group of 46 patients, per years, upon the recurrence of leukemia whom received autologous marrow. Table 5 shows the full data set (+ indicates censored observations).

0.0301

0.0384

0.0630

0.0849

0.0877

0.0959

0.1397

0.1616

0.1699

0.2137

0.2137

0.2164

0.2384

0.2712

0.2740

0.3863

0.4384

0.4548

0.5918

0.6000

0.6438

0.6849

0.7397

0.8575

0.9096

0.9644

1.0082

1.2822

1.3452

1.4000

1.5260

1.7205+

1.9890+

2.2438

2.5068+

2.6466+

3.0384

3.1726+

3.4411

4.4219+

4.4356+

4.5863+

4.6904+

4.7808+

4.9863+

5.0000+

 

 

Table 5: Leukemia free-survival times (in years) for the 46 autologous transplant patients (where + indicates censored observations).

The proposed model is compared with some usual long-term survival models, such as the LT Weibull and LT weighted Lindley (Louzada and Ramos, [13]). Different discrimination criterion methods are considered:  the negative of the maximum value of the likelihood function l( θ ^ ;t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiBamaabmaapaqaamaaxacabaqeduuDJXwAKbYu51MyVXga iuqapeGaf8hUdeNbaKaaa8aabeqaaaaapeGaai4oaGqadiaa+rhaai aawIcacaGLPaaaaaa@41BE@ , the Akaike information criterion ( AIC=2l( θ ^ ;t )+2k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaeWaaeaacaWGbbGaamysaiaadoeacqGH9aqpcqGHsislcaaI YaGaamiBamaabmaapaqaamaaxacabaqeduuDJXwAKbYu51MyVXgaiu qapeGaf8hUdeNbaKaaa8aabeqaaaaapeGaai4oaGqadiaa+rhaaiaa wIcacaGLPaaacqGHRaWkcaaIYaGaam4AaaGaayjkaiaawMcaaaaa@4AE0@ and the corrected AIC ( AIC+2k( k+1 )/ ( nk1 ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaeWaaeaacaWGbbGaamysaiaadoeacqGHRaWkcaaIYaGaam4A amaabmaapaqaa8qacaWGRbGaey4kaSIaaGymaaGaayjkaiaawMcaai aac+cacaGGGcWaaeWaa8aabaWdbiaad6gacqGHsislcaWGRbGaeyOe I0IaaGymaaGaayjkaiaawMcaaaGaayjkaiaawMcaaaaa@48D9@ , where k is the number of parameters to be fitted. The best model is the one which provides the minimum criterion method values.

Figure 2 presents the empirical survival function adjusted by the Kaplan-Meier estimator and different LT survival distributions.

Figure 2: Survival function adjusted by the empirical survival function (Kaplan-Meier estimator), LT Fréchet, LT Weibull and LT WL distribution.

Table 6 presents the results of the different discrimination criteria for different probability distributions. Comparing the results of the different discrimination methods, we observed that the LT Fréchet distribution has better fit then the LT models under the Weibull and weighted Lindley baseline distribution.

Method

LT Fréchet

LT Weibull

LT WL

logL MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqGHsislciGGSbGaai4BaiaacEgacaWGmbaaaa@3B33@

45.33

46.15

46.56

AIC

96.66

98.30

99.12

AICc

97.23

98.87

99.69

Table 6: Represents the results of the different discrimination criteria for different probability distributions.

The MLEs were obtained through the same procedure as described in Section 3. The standard error (SE) and the confidence intervals, considering a 95% confidence level for α, λ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqySdeMaaiilaiaacckacqaH7oaBaaa@3B61@ and p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiCaaaa@372F@  are displays in Table 7.

θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcfaieaa aaaaaaa8qacaWF4oaaaa@37ED@

MLE

SE

C I 95% ( θ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGdbGaamysa8aadaWgaaqaaKqzadWdbiaaiMdacaaI1aGa aiyjaaqcfa4daeqaa8qadaqadaWdaeaaieWapeGaa8hUdaGaayjkai aawMcaaaaa@3F71@

α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHXoqyaaa@3844@

0.65682

0.01975

( 0.38140;0.93225 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaqadaWdaeaapeGaaGimaiaac6cacaaIZaGaaGioaiaaigda caaI0aGaaGimaiaacUdacaaIWaGaaiOlaiaaiMdacaaIZaGaaGOmai aaikdacaaI1aaacaGLOaGaayzkaaaaaa@434D@

λ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaH7oaBaaa@3859@

0.31358

0.01531

(  0.07106;0.55609 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaqadaWdaeaapeGaaiiOaiaaicdacaGGUaGaaGimaiaaiEda caaIXaGaaGimaiaaiAdacaGG7aGaaGimaiaac6cacaaI1aGaaGynai aaiAdacaaIWaGaaGyoaaGaayjkaiaawMcaaaaa@4473@

p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGWbaaaa@379A@

0.12476

0.01597

( 0.00000;0.37245 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaqadaWdaeaapeGaaGimaiaac6cacaaIWaGaaGimaiaaicda caaIWaGaaGimaiaacUdacaaIWaGaaiOlaiaaiodacaaI3aGaaGOmai aaisdacaaI1aaacaGLOaGaayzkaaaaaa@433D@

Table 7: MLE, Standard Error (SE), and confidence interval under 95% confidence level for α, λ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHXoqycaGGSaGaaiiOaiabeU7aSbaa@3BCC@ and p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGWbaaaa@379A@ .

Note that, in Kersey et al. [12] they use the non-parametric KM estimate of the cure fraction in which was 0.20 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaaGimaiaac6cacaaIYaGaaGimaaaa@391C@ where ( 0.08;0.32 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba aeaaaaaaaaa8qacaaIWaGaaiOlaiaaicdacaaI4aGaai4oaiaaicda caGGUaGaaG4maiaaikdaa8aacaGLOaGaayzkaaaaaa@3E5E@ is the 95% confidence interval. Therefore, results showed to be consistence with Kersey et al. [12] results while our estimate was contained in the non-parametric interval. By using our parametric model the estimate obtained for p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiCaaaa@372F@ was 0.125 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaaGimaiaac6cacaaIXaGaaGOmaiaaiwdaaaa@39DC@ showing an overestimation of the long term survival patients. As it can be seen, through our proposed methodology the data related to the leukemia free-survival times (in months) for the 50 autologous transplant patients can be described by the LF distribution.

Discussion

In this paper, we have proposed a new long-term survival distribution called long term Fréchet distribution and its mathematical properties were studied. It was presented results towards the maximum likelihood parameters’ estimators and their asymptotic properties. The estimators’ efficient were present in the simulation study as the MLEs for the three unknown parameters obtained acceptable results even for small sample sizes. As such of the real dataset problem, related to the leukemia, free-survival times (in months) for the 50 autologous transplant patients. Many extensions from this present work can be considered, for instance, the parameters estimation may also be studied under an objective Bayesian analysis (Ramos et al., [14,15]) or using different classical methods (Louzada et al., Bakouch et al. [16]). Other approach should be to include covariates under the assumption of Cox model, i.e., proportional hazards. In conclusion, this regression model can be extended for the Bayesian approach as well.

Acknowledgements

The authors are thankful to the Editorial Board and to the reviewers for their valuable comments and suggestions which led to this improved version.

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