The purpose of this paper is to introduce a discrete distribution named Poisson-area-biased Lindley distribution and its applications on biological data. Poisson area-biased Lindley distribution is introduced with some of its basic properties including moments, coefficient of skewness and kurtosis are discussed. The method of moments and maximum likelihood estimation of the parameters of Poisson area-biased Lindley distribution are investigated. It is found that the parameter estimated by method of moments is positively biased, consistent and asymptotically normal. Application of the model to some biological data sets is compared with Poisson distribution.

Keywords: PABLD, PD, PLD, area-biased, MOM, MLE; factorial moments

Lindley¹ introduced a single parameter distribution named as Lindley distribution with probability distribution function (pdf)

$f\left(x;\theta \right)=\frac{{\theta }^2}{\theta +1}\left(1+x\right)e^{-\theta x},\begin{array}{ccc}& & x>0,\theta >0.\end{array}$ (1.1)

The pdf (1.1) is the mixture of exponential $\left(\theta \right)$ and gamma $\left(2,\theta \right)$ distributions. The cumulative distribution function (cdf) of the Lindley distribution is

$F\left(x\right)=1-\frac{\theta +1+\theta x}{\theta +1}{e}^{-\theta x},\begin{array}{ccc}& & x>0,\theta >0.\end{array}$  (1.2)

The first two moments of the Lindley distribution are

${{\mu }^{\prime }}_1=\frac{\theta +2}{\theta \left(\theta +1\right)},$   ${{\mu }^{\prime }}_2=\frac{2\left(\theta +3\right)}{{\theta }^2\left(\theta +1\right)}.$

Sankaran² introduced the Lindley mixture of Poisson distribution named Poisson-Lindley distribution with the following pdf

$f\left(x;\theta \right)=\frac{{\theta }^2\left(x+\theta +2\right)}{{\left(\theta +1\right)}^{.}},\begin{array}{cccc}& & x=0,1,2,\mathrm{.......},& \theta >0.\end{array}$ (1.3)

The pdf (1.3) is applied to count data and arises from Poisson distribution when its parameter $\lambda$ follows a Lindley distribution. Ghitany & Al-Mutairi³ discussed various properties of the Lindley distribution. Ghitany & Al-Mutairi³ introduced size-biased Poisson Lindley distribution with applications. They considered the size biased form of the Poisson-Lindley distribution. Ghitany & Al-Mutairi⁴ discussed estimation methods for the discrete Poisson-Lindley distribution. Srivastava & Adhikari⁵ introduced a size-biased Poisson-Lindley distribution which is obtained by considering the size-biased form of the Poisson distribution with Lindley distribution without its size-biased form. Adhikari & Srivastava⁶ proposed a Poisson size-biased Lindley distribution which is obtained by computing Poisson distribution without its size-biased form with size-biased Lindley distribution. Shanker & Fesshaye⁷ discussed Poisson-Lindley distribution with several of its properties including factorial moments and parameter estimation. They applied the Poisson-Lindley distribution on ecology and genetics data sets and showed that it can be an important tool for modeling biological science data.

Rao⁸ introduced the distributions that are used in situations when the recorded observations do not have an equal probability of selection and do not have the original distribution. The distributions used to handle such situations are called weighted distributions. Suppose that the original distribution comes from a distribution with pdf $f_0\left(x\right)$  and the observations is recorded to a probability re-weighted by a weight function $w\left(x\right)>0,$ then the weighted distribution is defined as

$f\left(x\right)=\frac{w\left(x\right)}{E\left[w\left(X\right)\right]}{f}_0\left(x\right)$ (1.4)

The weighted distribution with $w\left(x\right)=x$  is called size-biased/length-biased distributions and $w\left(x\right)=x^2$ is called area-biased distribution. Patil & Ord⁹ discussed size-biased sampling and related form-invariant weighted distributions. Patil & Rao¹⁰ discussed some models leading to weighted distributions and showed applications of weighted distributions in many real sampling problems. Mir & Ahmad¹¹ introduced size-biased form of some discrete distributions with their applications.

The Poisson area-biased Lindley distribution (PABLD) arises from the Poisson distribution with pdf

$f\left(x;\lambda \right)=\frac{e^{-\lambda }{\lambda }^x}{x!},\begin{array}{cccc}& & x=0,1,2,\mathrm{......}& \lambda >0,\end{array}$  (2.1)

when its parameter $\lambda$ follows the area-biased Lindley distribution (ABLD) in (2.1) with pdf

$f\left(x;\theta \right)=\frac{{\theta }^4}{2\left(\theta +3\right)}{x}^2\left(1+x\right)e^{-\theta x},\begin{array}{ccc}& & x>0,\theta >0.\end{array}$  (2.3)

So

${\displaystyle \underset{0}{\overset{\infty }{\int }}f\left(x;\lambda \right)f\left(\lambda ;\theta \right)}d\lambda =\frac{{\theta }^4}{2\left(\theta +3\right)x!}{\displaystyle \underset{0}{\overset{\infty }{\int }}{e}^{-\lambda \left(\theta +1\right)}\left({\lambda }^{x+2}+{\lambda }^{x+3}\right)}d\lambda .$

After simplifying it the pdf of PABLD is obtained

$f\left(x;\theta \right)={\left(\frac{\theta }{\theta +1}\right)}^4\frac{\left(x+1\right)\left(x+2\right)\left(\theta +x+4\right)}{2\left(\theta +3\right){\left(\theta +1\right)}^x},\begin{array}{cccc}& & x=0,1,2,\mathrm{.....}& \theta >0.\end{array}$  (2.4)

Properties of the poisson-area-biased-lindley distribution

The factorial moments of the PABLD in (2.1)

${{\mu }^{\prime }}_{\left(r\right)}=E\left[E\left(x^{\left(r\right)}/\lambda \right)\right],$

${{\mu }^{\prime }}_{\left(r\right)}=\frac{\left(\theta +r+3\right)\left(r+2\right)!}{2\left(\theta +3\right){\theta }^r}.$ (2.5)

For $r=1,2,3\&4$ in (2.5), the first four factorial moments of the PABLD are

${{\mu }^{\prime }}_{\left(1\right)}=\frac{3\left(\theta +4\right)}{\theta \left(\theta +3\right)}$ , ${{\mu }^{\prime }}_{\left(2\right)}=\frac{12\left(\theta +5\right)}{{\theta }^2\left(\theta +3\right)}$ , ${{\mu }^{\prime }}_{\left(3\right)}=\frac{60\left(\theta +6\right)}{{\theta }^3\left(\theta +3\right)}$ , ${{\mu }^{\prime }}_{\left(4\right)}=\frac{360\left(\theta +7\right)}{{\theta }^4\left(\theta +3\right)}$ (2.6)

Since the first four raw moments of the PABLD are

${{\mu }^{\prime }}_1=\frac{3\left(\theta +4\right)}{\theta \left(\theta +3\right)}$ , $${{\mu }^{\prime }}_2=\frac{3\left({\theta }^2+8\theta +30\right)}{{\theta }^2\left(\theta +3\right)}$$ (2.7)

${{\mu }^{\prime }}_3=\frac{3\left({\theta }^3+16{\theta }^2+80\theta +120\right)}{{\theta }^3\left(\theta +3\right)}$ , ${{\mu }^{\prime }}_4=\frac{3\left({\theta }^4+32{\theta }^3+260{\theta }^2+840\theta +840\right)}{{\theta }^4\left(\theta +3\right)}$ (2.8)

The mean moments of PABLD are

${\mu }_2={\sigma }^2=\frac{3\left({\theta }^3+8{\theta }^2+30\theta +42\right)}{{\theta }^2{\left(\theta +3\right)}^2}.$  (2.9)

${\mu }_3=\frac{3\left({\theta }^5+10{\theta }^4+14{\theta }^3+36{\theta }^2-2160\theta +2664\right)}{{\theta }^3{\left(\theta +3\right)}^3}.$  (2.10)

${\mu }_4=\frac{3\left({\theta }^7+20{\theta }^6+2{\theta }^5+61122{\theta }^4-366276{\theta }^3-548280{\theta }^2+19224\theta +41688\right)}{{\theta }^4{\left(\theta +3\right)}^4}.$  (2.11)

The coefficient of skewness and kurtosis of the PABLD are

${\gamma }_1=\sqrt{{\beta }_1}=\frac{\left({\theta }^5+10{\theta }^4+14{\theta }^3+36{\theta }^2-2160\theta -2664\right)}{\sqrt{3{\left({\theta }^3+8{\theta }^2+30\theta +42\right)}^3}}.$  (2.12)

${\beta }_2=\frac{\left({\theta }^7+20{\theta }^6+2{\theta }^5+61122{\theta }^4-366276{\theta }^3-548280{\theta }^2+19224\theta +41688\right)}{3{\left({\theta }^3+8{\theta }^2+30\theta +42\right)}^2}.$  (2.13)

For the PABLD, from (2.12) and (2.13) it can be seen that $\left({\gamma }_1,{\beta }_2\right)\to \left(-5.65,7.88\right)$  as $\theta \to 0$ , the model is negatively skewed and leptokurtic.

Some more properties of the PABLD are

$\frac{f\left(x+1;\theta \right)}{f\left(x;\theta \right)}=\frac{\left(x+3\right)\left(\theta +x+5\right)}{\left(\theta +1\right)\left(x+1\right)\left(\theta +x+4\right)}.$

$\frac{f\left(x+1;\theta \right)}{f\left(x;\theta \right)}=\frac{\left(1+\frac{3}{x}\right)\left(\theta +\frac{1}{x}+5\right)}{\left(\theta +1\right)\left(1+\frac{1}{x}\right)\left(\theta +\frac{1}{x}+4\right)}.$  (2.15)

The dispersion of the PABLD is defined to be

From equation (2.14) and Table 1, it can be observed that the PABLD is over-dispersed but as $\theta \to \infty$ then $\mu ={\sigma }^2$ and the PABLD is equi-dispersed. Therefore for large $\theta$ the PABLD is equi-dispersed.

If $x_1,x_2,\mathrm{....},x_n$  be the random sample from PABLD with pdf (2.4), the method of moments (MOM) estimate $\tilde{\theta }$ of the parameter $\theta$  is given by

$\tilde{\theta }=\frac{-3\left(\overline{x}-1\right)+\sqrt{9{\left(\overline{x}-1\right)}^2+48\overline{x}}}{2\overline{x}}$  (3.1)

Theorem 1: The MOM estimator $\tilde{\theta }$ of $\theta$ is positively biased.

Proof: Let $\tilde{\theta }=\psi \left(\overline{x}\right)$ , where $\Psi \left(z\right)=\frac{-3\left(z-1\right)+\sqrt{9{\left(z-1\right)}^2+48z}}{2z}.$

So,

${\psi }^{″}\left(z\right)=\frac{78z+69z^2+297z^3+108z^4+\left(108z+405z^2+135z^3\right)\sqrt{9{\left(z-1\right)}^2+48z}}{4z^4{\left[9{\left(z-1\right)}^2+48z\right]}^{3/2}}>0,$  (3.2)

Then $\Psi \left(z\right)$  is strictly convex. By using the Jensen’s inequality we have

$E\left\{\psi \left(\overline{X}\right)\right\}>\psi \left\{E\left(\overline{X}\right)\right\}.$

Since $\psi \left\{E\left(\overline{X}\right)\right\}=\psi \left(\mu \right)=\psi \left(\frac{3\left(\theta +4\right)}{\theta \left(\theta +3\right)}\right)=\theta$ , therefore $E\left(\tilde{\theta }\right)>\theta .$

Theorem 2: The MOM estimator $\tilde{\theta }$ of $\theta$  is consistent and asymptotically normal:

$\sqrt{n}\left(\tilde{\theta }-\theta \right)\stackrel{d}{\to }N\left(0,{\nu }^2\left(\theta \right)\right).$

Where

${\nu }^2\left(\theta \right)=\frac{{\theta }^2{\left(\theta +3\right)}^2\left({\theta }^3+8{\theta }^2+30\theta +42\right)}{3\left({\theta }^2+8\theta +12\right)}.$  (3.3)

Proof: -

Consistency: Since $\mu <\infty ,$  then $\overline{X}\stackrel{P}{\to }\mu .$  And $\psi \left(z\right)$  is a continuous function at $z=\mu$ , then $\psi \left(\overline{X}\right)\stackrel{P}{\to }\psi \left(\mu \right),$  i-e. $\tilde{\theta }\stackrel{P}{\to }\theta .$

Asymptotic normality: as ${\sigma }^2<\infty$  then by using the central limit theorem we have

$\sqrt{n}\left(\overline{X}-\mu \right)\stackrel{d}{\to }N\left(0,{\sigma }^2\right).$

$\psi \left(\mu \right)$  is a differentiable function and ${\psi }^{\prime }\left(\mu \right)\ne 0,$  then by using the delta-method we have

$\sqrt{n}\left(\psi \left(\overline{X}\right)-\psi \left(\mu \right)\right)\stackrel{d}{\to }N\left(0,{\left[{\psi }^{\prime }\left(\mu \right)\right]}^2{\sigma }^2\right).$

Finally we have $\psi \left(\overline{X}\right)=\tilde{\theta }\begin{array}{cc},& \psi \left(\mu \right)=\theta \end{array}$  and

${\psi }^{\prime }\left(\mu \right)=\frac{-1-6\mu -6\sqrt{9{\left(\mu -1\right)}^2+48\mu }}{4{\mu }^2\sqrt{9{\left(\mu -1\right)}^2+48\mu }}=-\frac{{\theta }^2{\left(\theta +3\right)}^2}{3\left({\theta }^2+8\theta +12\right)}.$  (3.4)

The theorem 2 follow the asymptotic $100\left(1-\alpha \right)\%$  confidence interval for $\theta$  is

$\tilde{\theta }\pm z_{\frac{\alpha }{2}}\frac{\nu \left(\tilde{\theta }\right)}{\sqrt{n}}.$  (3.5)

Let $x_1,x_2,\mathrm{....},x_n$  be the random sample on size n from PABLD with pdf (2.4), the maximum likelihood estimate (MLE) $\widehat{\theta }$ of the parameter $\theta$  is the solution of the non-linear equation:

$\frac{4n}{\theta }-\frac{n\left(4-\overline{x}\right)}{\left(\theta +1\right)}-\frac{n}{\left(\theta +3\right)}+{\displaystyle \sum _{i=1}^n\frac{1}{\theta +x_i+4}}=0$  (4.1)

In this section the PABLD is applied to some biological data sets and compared with PD.

1. Guire, et al.¹² gave data on European corn borers per plant with 0, 1, 2, 3 and 4 and counts 83, 36, 14, 2, and 1.

Form Table 2, it can be seen that the PABLD gives much closer fit than the PD and PLD to the data set of number of bores per plant . Thus PABLD provides a better alternative to PD and PLD for modeling count data sets.

2. Beall¹³ gave the distribution of Pyrausta nublilalis in 1937, no of insects 0, 1, 2, 3, 4 and 5 with counts 33, 12, 6, 3, 1 and 1.

Form Table 3, it can be seen that the PABLD gives better fit than the PD to the data set of number of insects. Thus PABLD provides a better alternative to PD for modeling count data sets.

3. Juday¹⁴ and Thomas ¹⁵ gave data on macroscopic fresh-water fauna in dredge samples from the bottom of Weber Lake.

Form Table 4 it can be seen that the PABLD gives better fit than PD and PLD to the animal distribution of microcalanus nauplii. Thus PABLD provides a better alternative to PD and PLD for modeling count data sets.

4. Archibald¹⁶ gave data on plant populations. The distribution of representing salicornia stricta.

Form Table 5, it can be seen that the PABLD gives better fit than the PD and PLD. Thus PABLD provides a better alternative to PD and PLD for modeling count data sets.

5. Archibald^16-18 gave data on plant populations. The distribution of representing Plantago maritime.

From Table 6 it is concluded that the PABLD gives better fit than the PD and almost equally good fit as PLD distribution to the distribution of Plantago maritime. Therefore the PABLD is better alternative to PD and PLD to model discrete data sets.

Note: The highlighted expected frequencies from Table 2-6 are the pooled frequencies that are less than 5, so the degrees of freedom are calculated according to them.

From Table 2-7, it is observed that the PABLD gives better fit than PD and PLD to the some biological count data sets. PD is a discrete distribution with parameter $\lambda$ . Lindley distribution is a continuous life time distribution and PLD is the mixture of Poisson and Lindley distributions with parameter $\theta$ . The proposed model named PABLD is obtained by the mixture of the Poisson distribution and the area biased form of the Lindley distribution. The area biased distribution is a type of the weighted distribution with weight $w\left(x\right)=x^2$ , due to mixture of PD and LD with this weight, the proposed model is showing applications better than PD and PLD to biological data sets. Mostly the applications of the weighted distributions to the data relating biology can be found in Patil & Rao [10].

f. Interval Estimation: By using equation (3.5) the parameter $\theta$ of PABLD is estimated by the interval estimation for the Biological data sets. The estimated interval for $\theta$ of PABLD by the interval estimation is closer to the estimated value by MOM.

The Poisson area-biased Lindley distribution (PABLD) is discrete distribution that is obtained by mixture of the Poisson distribution and area-biased Lindley distribution. Some important properties of the PABLD are derived. From Figure 1 it can be seen that the PABLD is positively skewed moreover it can be seen that as $\theta \to 0$ , $\left({\gamma }_1,{\beta }_2\right)\to \left(-5.65,7.88\right)$  and the PABLD is negatively skewed and leptokurtic. Furthermore it is found that the PABLD is over-dispersed but as $\theta \to \infty$ the PABLD is equi-dispersed. The parameter of the PABLD is estimated by the method of moments (MOM) and it is proved that the $\tilde{\theta }$  of $\theta$  is positively biased, consistent and asymptotically normal. In section 4, the proposed model PABLD is applied to some biological data sets and compared with PD and PLD. It is observed that the PABLD gives better approach to the given data sets. Therefore it is concluded that PABLD is a better alternative to PD and PLD and it has useful applications in real life biological data sets. The asymptotic $95\%$  confidence interval (C.I) for $\theta$ of PABLD is also found on these data sets and it is observed that the estimated interval for $\theta$ of PABLD by the interval estimation is closer to the estimated value obtained by MOM.

Figure 1 Plots of the pdf of PABLD for θ = 0.5, θ = 1, θ = 2, θ = 8.