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eISSN: 2378-315X

Biometrics & Biostatistics International Journal

Research Article Volume 9 Issue 3

A note on size– biased new quasi Poisson– Lindley distribution

Rama Shanker,1 Kamlesh Kumar Shukla,2 Ravi Shanker3

1Department of Statistics, Assam University, Silcher, India
2Department of Statistics, Mainefhi College of Science, Asmara, Eritrea
3 Department of Mathematics, G.L.A.College, N.P University, India

Correspondence: Kamlesh Kumar Shukla, Mainefhi College of Science, Asmara, Eritrea

Received: April 24, 2020 | Published: June 9, 2020

Citation: Shanker R, Shukla KK, Shanker R. A note on size– biased new quasi poisson– lindley distribution. Biom Biostat Int J. 2020;9(3):97-104. DOI: 10.15406/bbij.2020.09.00306

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Abstract

In this paper some important properties including coefficients of variation, skewness, kurtosis and index of dispersion of size–biased new quasi Poisson–Lindley distribution (SBNQPLD) have been discussed and their behaviors have been explained graphically for varying values of parameters. Some applications of SBNQPLD have also been discussed.

Keywords: Size–biased new Quasi Poisson–Lindley distribution, moments based measures, maximum likelihood estimation, goodness of fit

Introduction

The size– biased version of Poisson–Lindley distribution (SBPLD) proposed by Sankaran1 has been introduced by Ghitany and Mutairi2 and is defined by its probability mass function (pmf)

P1(x,θ)=θ3θ+2x(x+θ+2)(θ+1)x+2;θ>0,x=1,2,3,...

Shanker et al.3 have proposed a simple method of deriving moments of SBPLD and the applications of SBPLD to model thunderstorms events. The Poisson –Lindley distribution (PLD), a Poisson mixture of Lindley distribution of Lindley,4 is defined by its pmf

P2(x;θ)=θ2(x+θ+2)(θ+1)x+3;x= 0, 1, 2,,  > 0.                                     

The Lindley distribution is defined by its probability density function (pdf)                        f1(x,θ)=θ2θ+1(1+x)eθx ; x > 0, θ > 0                                                           

The size–biased quasi Poisson–Lindley distribution (SBQPLD), size–biased version of quasi Poisson–Lindley distribution (QPLD) of Shanker and Mishra,5 suggested by Shanker and Mishra6 with parameters θ and α is defined by its pmf

P3(x;θ,α)=θ2α+2x(θx+θα+θ+α)(θ+1)x+2;x=1,2,3,...,θ>0,α>2         

The QPLD, a Poisson mixture of quasi Lindley distribution proposed by Shanker and Mishra,7 is defined by its pmf

P4(x;θ,α)=θ(θx+θα+θ+α)(α+1)(θ+1)x+2;x=0,1,2,...;θ>0,α>1          

The QLD is defined by its pdf

f2(x;θ,α)=θα+1(α+xθ)eθx;x>0,θ > 0, α>1                        

Shanker and Amanuel8 proposed a new quasi Lindley distribution (NQLD) having pdf

f3(x;θ,α)=θ2θ2+α(θ+αx)eθx

whereθ+αx>0andθ2+α>0orθ+αx<0andθ2+α<0 for x>0 θ>0 . Lindley distribution is a particular case of NQLD at α=θ . A new quasi Poisson–Lindley distribution (NQPLD), a Poisson mixture of NQLD, has been suggested by Shanker and Tekie9 and defined by its pmf

P5(x;θ,α)=θ2(θ+1)x+2[1+θ+αxθ2+α]

where θ+αx>0andθ2+α>0orθ+αx<0andθ2+α<0 for x=0,1,2,...; θ>0 .

It can be seen that the PLD is a particular case of it at α=θ . Shanker et al.10 derived the pmf of size biased new quasi Poisson–Lindley distribution (SBNQPLD) having pmf

P6(x;θ,α)=θ3θ2+2αx(θ2+θ+α+αx)(θ+1)x+2;x=1,2,3,...,θ>0,θ2+2α>0

Shanker et al.10 discussed various statistical properties, parameters estimation and applications of SBNQPLD. Shanker et al.10 have shown that SBNQPLD can also be obtained from the size–biased Poisson distribution when its parameter λ follows a SBNQLD with pdf       

f4(λ;θ,α)=θ3θ2+2αλ(θ+αx)eθx

 where θ+αx>0andθ2+α>0orθ+αx<0andθ2+α<0 for x>0 θ>0 . That is

P(X=x)=0eλλx1(x1)!θ3θ2+2αλ(θ+αλ)eθλdλ

=θ3θ2+2αx(θ2+θ+α+αx)(θ+1)x+2 ;x=1,2,3,...

The r th factorial moment about origin μ(r) of SBNQPLD obtained by Shanker et al.10 as

μ(r)=r!{rθ3+(r+1)θ2+r(r+1)αθ+(r+1)(r+2)α}θr(θ2+2α);r=1,2,3,...

Thus, the first four moments about origin obtained by Shanker et al.10 are

μ1=1+2(θ2+3α)θ(θ2+2α)

μ2=1+6(θ2+3α)θ(θ2+2α)+6(θ2+4α)θ2(θ2+2α)

μ3=1+14(θ2+3α)θ(θ2+2α)+36(θ2+4α)θ2(θ2+2α)+24(θ2+5α)θ3(θ2+2α)

μ4=1+30(θ2+3α)θ(θ2+2α)+126(θ2+4α)θ2(θ2+2α)+240(θ2+5α)θ3(θ2+2α)+120(θ2+6α)θ4(θ2+2α) .

It has been observed that the central moments (moments about the mean) has not been given by et al.10 and hence many important characteristics including coefficient of variation, skewness, kurtosis and index of dispersion of SBNQPLD has not been studied by Shanker et al.10

The main purpose of this paper is to derive expressions for coefficients of variation, skewness, kurtosis and index of dispersion of SBNQPLD and study their behaviour graphically. The goodness of fit of the distribution has been presented with a number of count datasets using maximum likelihood estimates from various fields of knowledge.    

Moments based measures

Using the relationship between moments about the mean and the moments about the origin, the moments about mean of SBNQPLD can be obtained as

μ2=2(θ5+θ4+5αθ3+6αθ2+6α2θ+6α2)θ2(θ2+2α)2

μ3=2{θ8+3θ7+(7α+2)θ6+24αθ5+(16α2+18α)θ4+54α2θ3+(12α3+36α2)θ2+36α3θ+24α3}θ3(θ2+2α)3

μ4=2{θ11+13θ10+(9α+24)θ9+(130α+12)θ8+(30α2+264α)θ7+(460α2+144α)θ6+(44α3+936α2)θ5+(696α3+504α2)θ4+(24α4+1368α3)θ3+(384α4+720α3)θ2+720α4θ+360α4}θ4(θ2+2α)4 .

The coefficient of variation (C.V), coefficient of Skewness (β1) , coefficient of Kurtosis (β2) and Index of dispersion (γ) of SBNQPLD are obtained as

C.V.=σμ1=2(θ5+θ4+5αθ3+6αθ2+6α2θ+6α2)θ3+2θ2+2αθ+6α

β1=μ3(μ2)3/2={θ8+3θ7+(7α+2)θ6+24αθ5+(16α2+18α)θ4+54α2θ3+(12α3+36α2)θ2+36α3θ+24α3}2(θ5+θ4+5αθ3+6αθ2+6α2θ+6α2)3/2

β2=μ4μ22={θ11+13θ10+(9α+24)θ9+(130α+12)θ8+(30α2+264α)θ7+(460α2+144α)θ6+(44α3+936α2)θ5+(696α3+504α2)θ4+(24α4+1368α3)θ3+(384α4+720α3)θ2+720α4θ+360α4}2(θ5+θ4+5αθ3+6αθ2+6α2θ+6α2)2

γ=σ2μ1=2(θ5+θ4+5αθ3+6αθ2+6α2θ+6α2)θ(θ2+2α)(θ3+2θ2+2αθ+6α) .

Shapes of coefficient of variation, skewness, kurtosis and index of dispersion of SBNQPLD for varying values of parameters have been shown in figure 1.

Figure 1 Behaviors of C.V, Skewness, Kurtosis and Index of dispersion of SBNQPLD for values of θ and α

Maximum likelihood estimation of parameters

Suppose (x1,x2,,xn) as random samples of size n from the SBNQPLD and fx , the observed frequency in the sample corresponding to X=x (x=1,2,...,k) such that kx=1fx=n , where k being the largest observed value having non–zero frequency. The log likelihood function of SBNQPLD can be presented as

logL=nlog(θ3θ2+2α)kx=1fx(x+2)log(θ+1)+kx=1fxlog[αx2+x(θ2+θ+α)]

The two log likelihood equations are thus obtained as

logLθ=3nθ2nθθ2+2αkx=1(x+2)fxθ+1+kx=1(2θ+1)xfx[αx2+x(θ2+θ+α)]=0

logLα=2nθ2+2α+kx=1x(x+1)fx[αx2+x(θ2+θ+α)]=0 .

These two log likelihood equations seems difficult to solve directly as these cannot be expressed in closed forms. The (MLE’s) (ˆθ,ˆα) of parameters (θ,α) can be computed directly by solving the log likelihood equation using Newton–Raphson iteration available in R–software till sufficiently close values of ˆθandˆα are obtained. The initial values of parameters θ and α are the MOME (˜θ,˜α) of the parameters (θ,α) , given in Shanker et al.10

Goodness of fit

To test the goodness of fit of SBNQPLD along with SBPD, SBPLD and SBQPLD, several cont datasets have been considered from various fields of knowledge. The expected frequencies of SBPD, SBPLD and SBQPLD have also been given in the tables (Table 1–10). The estimates of the parameters have been obtained by the method of maximum likelihood. It is obvious from the goodness of fit of SBNQPLD that it provides better fit over SBPD and SBPLD and competing well with SBQPLD in majority of datasets. The following datasets have been considered for testing the goodness of fit of SBNQPLD.

Group Size

Observed frequency

Expected frequency

SBPD

SBPLD

SBQPLD

SBNQPLD

1

2

3

4

5

6

1486

694

195

37

10

1

1452.4

743.3

190.2

32.4

4.1

0.6

1532.5

630.6

191.9

51.3

12.8

3.9

1485.4

697.2

189.7

41.1

7.8

1.8

1505.5

656.8

202.5

49.2

9.0

0.0

Total

2423

2423.0

2423.0

2423

 

ML Estimate

 

ˆθ=0.5118 ˆθ=4.5082 ˆθ=7.14063
ˆα=0.79104
ˆθ=2.69606
ˆα=1.39128
χ2

 

7.370

13.760

0.776

6.1

d.f.

 

2

3

2

2

p-value

 

0.0251

0.003

0.6804

0.04735

2logL

 

10445.34

4622.36

4607.8

4610.0

AIC

 

10447.34

4624.36

4611.8

4614.0

Table 1 Pedestrians-Eugene, Spring, Morning, available in Coleman and James11

Group Size

Observed frequency

Expected frequency

SBPD

SBPLD

SBQPLD

SBNQPLD

1

2

3

4

5

316

141

44

5

4

306.3

156.1

39.8

6.7

1.1

322.9

132.5

40.2

10.7

3.7

315.7

142.7

40.1

9.1

2.4

313.5

141.4

44.1

10.4

0.6

Total

510

510.0

510.0

510.0

 

ML Estimate

 

ˆθ=0.5098 ˆθ=4.5211 ˆθ=6.5501
ˆα=0.5069
ˆθ=2.4693
ˆα=1.2977
χ2

 

2.39

3.07

0.94

0.38

d.f.

 

2

2

1

1

p-value

 

0.3027

0.2154

0.3322

0.5376

2logL

 

916.63

972.78

971.07

970.24

AIC

 

918.63

974.78

975.07

974.24

Table 2 Play Groups-Eugene, Spring, Public Playground A, available in Coleman and James11

Group Size

Observed frequency

Expected frequency

SBPD

SBPLD

SBQPLD

SBNQPLD

1

2

3

4

5

306

132

47

10

2

292.2

155.2

41.2

7.3

1.1

309.4

131.2

41.1

11.3

4.0

304.4

137.9

41.3

10.3

3.1

306.4

134.4

41.6

11.0

3.6

Total

497

497.0

497.0

 

 

ML Estimate

 

ˆθ=0.5312 ˆθ=4.3548 ˆθ=5.71547
ˆα=4.9998

ˆθ=4.9998
ˆα=25.6948

χ2

 

6.479

0.932

1.19

1.2

d.f.

 

2

2

1

1

p-value

 

0.039

0.6281

0.2753

0.2733

2logL

 

2142.03

971.86

970.96

971.25

AIC

 

2144.03

973.86

974.96

975.25

Table 3 Play Groups-Eugene, Spring, Public Playground A, available in Coleman and James11

Group Size

Observed frequency

Expected frequency

SBPD

SBPLD

SBQPLD

SBNQPLD

1

2

3

4

5

6

305

144

50

5

2

1

296.5

159.0

42.6

7.6

1.0

0.3

314.4

134.4

42.5

11.8

3.1

0.8

304.3

148.2

42.3

9.6

1.9

0.7

310.1

138.8

43.1

11.3

2.7

1.0

Total

507

507.0

507.0

507.0

507.0

ML Estimate

 

ˆθ=0.5365 ˆθ=4.3179 ˆθ=6.70804
ˆα=0.74907
ˆθ=5.1516
ˆα=48.6067
χ2

 

3.035

6.415

2.96

4.64

d.f.

 

2

2

1

1

p-value

 

0.219

0.040

0.0853

0.0312

2logL

 

2376.75

993.10

990.02

991.51

AIC

 

2378.75

995.1

994.02

995.51

Table 4 Play Groups-Eugene, Spring, Public Playground D, available in Coleman and James11

Group Size

Observed frequency

Expected frequency

SBPD

SBPLD

SBQPLD

SBNQPLD

1

2

3

4

5

 

276

229

61

12

3

 

292.3

200.7

68.9

15.8

3.3

 

319.6

166.5

63.8

21.4

9.7

 

276.0

228.3

61.9

12.2

2.6

313.7

173.1

65.2

20.7

8.3

Total

581

581.0

581.0

581.0

581.0

ML Estimate

 

ˆθ=0.6867 ˆθ=3.4359 ˆθ=8.6724
ˆα=1.4944
ˆθ=4.1645
ˆα=61.0287
χ2

 

6.68

37.86

0.017

29.6

d.f.

 

2

2

1

1

p-value

 

0.0354

0.00

0.8962

0.000

2logL

 

1146.7

1277.42

1238.11

1268.77

AIC

 

1148.7

1279.42

1242.11

1272.77

Table 5 Play Groups-Eugene, Spring, Public Playground D, available in Coleman and James11

No. of sites with particles

Observed Frequency

Expected Frequency

SBPD

SBPLD

SBQPLD

SBNQPLD

1

2

3

4

5

122

50

18

4

4

111.3

64.1

119.0

53.8

18.0

119.2

53.5

17.9

5.3

2.1

119.3

53.3

17.8

5.3

2.3

Total

198

198.0

198.0

198.0

198.0

ML estimate

 

ˆθ=0.575758 ˆθ=4.050987 ˆθ=3.7564
ˆα=10.1281
ˆθ=3.4795
ˆα=0.0216
χ2

 

4.64

0.43

0.34

0.28

d.f.

 

1

2

1

1

p-value

 

0.0312

0.8065

0.5598

0.5967

2logL

 

393.95

409.28

409.17

409.13

AIC

 

395.95

411.28

413.17

413.13

Table 6 Distribution of number of counts of sites with particles from Immunogold data, available in Mathews and Appleton12

No. times hares caught

Observed Frequency

Expected Frequency

SBPD

SBPLD

SBQPLD

SBNQPLD

1

2

3

4

5

184

55

14

4

4

170.6

72.5

177.3

62.5

177.4

62.3

16.3

3.8

1.2

177.5

62.2

16.3

3.8

1.2

Total

261

261.0

261.0

261

261.0

ML estimate

 

ˆθ=0.425287 ˆθ=5.351256 ˆθ=4.9800
ˆα=14.9193
ˆθ=4.6959
ˆα=0.0302
χ2

 

6.22

1.18

3.2

3.19

d.f.

 

1

1

1

1

p-value

 

0.0126

0.2773

0.0736

0.07409

2logL

 

452.40

457.10

456.86

456.80

AIC

 

454.40

459.10

460.86

460.80

Table 7 Distribution of snowshoe hares captured over 7 days, available in Keith and Meslow13  

Number of pairs of running shoes

Observed frequency

Expected Frequency

SBPD

SBPLD

SBQPLD

SBNQPLD

1

18

15.0

20.3

17.4

19.5

2

18

20.8

17.4

19.6

18.0

3

12

14.4

10.9

12.3

11.3

4

5

7

5

 

5.9

5.5

6.1

4.6

6.0

5.2

Total

60

60.0

60.0

60.0

60

ML Estimate

 

ˆθ=1.383333 ˆθ=1.818978 ˆθ=2.5858
ˆα=0.7318
ˆθ=2.08739
ˆα=17.3228
χ2

 

1.87

0.64

0.31

0.33

d.f.

 

2

3

1

2

P-value

 

0.3926

0.8872

0.5777

0.8478

2logL

 

147.1

187.08

185.55

186.33

AIC

 

149.1

189.08

189.55

190.33

Table 8 Number of counts of pairs of running shoes owned by 60 members of an athletics club, reported by Simonoff14

Number of fly eggs

Observed Frequency

Expected Frequency

SBPD

SBPLD

SBQPLD

SBNQPLD

1

2

3

4

5

6

7

8

9

22

18

18

11

9

6

3

0

1

11.3

23.2

23.8

16.2

8.3

3.4

1.1

0.3

0.4

20.3

22.0

17.2

11.6

7.2

4.2

2.4

1.3

1.8

19.8

22.1

17.5

11.8

7.3

4.2

2.3

1.3

1.7

19.8

22.1

17.5

11.8

7.3

4.2

2.3

1.3

1.7

Total

88

 

 

88.0

88.0

ML estimate

 

ˆθ=2.0454 ˆθ=1.2822 ˆθ=1.3483
ˆα=0.6925
ˆθ=1.3465
ˆα=2.5654
χ2

 

18.8

1.39

1.49

1.49

d.f.

 

4

4

3

3

p-value

 

0.0008

0.8459

0.6845

0.6845

2logL

 

206.59

329.92

329.86

329.86

AIC

 

208.59

331.92

333.86

333.86

Table 9 The numbers of counts of flower heads as per the number of fly eggs reported by Finney and Varley15

x

Observed frequency

Expected Frequency

SBPD

SBPLD

SBQPLD

SBNQPLD

1

375

341.2

262.8

363.3

363.6

2

143

186.8

157.4

156.5

156.3

3

49

51.1

50.4

50.4

50.4

4

5

6

7

8

17

2

2

1

1

9.3

1.2

0.1

0.2

0.1

14.2

3.7

0.9

0.2

0.3

14.4

3.9

1.0

0.2

0.4

14.4

3.8

1.0

0.2

0.3

Total

590

 

590.0

590.0

590.0

ML Estimate

 

ˆθ=0.5474 ˆθ=4.24 ˆθ=3.8386
ˆα=17.2968
ˆθ=3.6534
ˆα=0.00067
χ2

 

14.1

2.48

2.11

2.08

d.f.

 

2

3

2

2

P-value

 

0.0008

0.4789

0.3481

0.3534

2logL

 

1124.3

1190.4

1189.67

1189.57

AIC

 

1126.3

1192.4

1193.67

1193.57

Table 10 Number of households having at least one migrant according to the number of migrants, reported by Singh and Yadav16

Conclusion

In this paper expressions based ob central moments including coefficients of variation, skewness, kurtosis and index of dispersion of SBNQPLD have been derived and their behaviors have been explained graphically for varying values of the parameters. Some important applications of SBNQPLD have also been discussed and its goodness of fit has been compared with other discrete distributions. It has been observed that SBNQPLD provides much better fit over SBPD, SBPLD and competing well with SBQPLD in majority of datasets.

Acknowledgments

None.

Conflicts of interest

None.

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