In this paper, an area–biased form of the single parameter Poisson exponential distribution (PED) is obtained by area biasing the discrete Poisson exponential distribution (PED) introduced by Fazal & Bashir.¹ Poisson–exponential distribution is an important discrete distribution which has many applications in countable datasets. The first four moments (about origin) and the central moments (about mean) have been obtained and hence expression for coefficient of variation (CV), skewness, kurtosis and index of dispersion are derived. To estimate the parameters of Area–biased Poisson exponential distribution (ABPED), maximum likelihood method (MLE) and method of moments (MOM) are also developed. The goodness of fit for (ABPED) has been discussed using three real data–sets the fit shows a better fit over size–biased Poisson Lindley distribution (SBPLD).

Keywords: poisson exponential distribution, weighted distributions, moments, estimation of parameters, goodness of fit

Fazal & Bashir¹ have obtained discrete Poisson exponential distribution (PED) for modelling count data with probability mass function (p.m.f)

$P(X=x)=\frac{\theta }{{(1+\theta )}^{x+1}},\theta >0,x=0,1,2, \mathrm{...} \mathrm{...}$ (1.1)

The Poisson exponential distribution (PED) in (1.1) is a mixture of Poisson and exponential distribution when the parameter of Poisson distribution (λ) follows exponential distribution. The first four moments (about origin) and the variance of (PED) obtained by Fazal & Bashir¹ are given as

${{\mu }^{\prime }}_1=\frac{1}{\theta }$

${\mu }^{\text{'}}{}_2=\frac{2+\theta }{{\theta }^2}$ (1.2)

${{\mu }^{\prime }}_3=\frac{{\theta }^2+6\theta +6}{{\theta }^3}$

${{\mu }^{\prime }}_4=\frac{{\theta }^3+14{\theta }^2+36\theta +24}{{\theta }^4}$

${\mu }_2=\frac{1+\theta }{{\theta }^2}$

The mathematical properties and estimation of parameter have been discussed by Fazal & Bashir¹ and its application proves that it is a good replacement of poisson distribution and Lindley distribution. The size–biased form of (PED) has been discussed by Fazal & Bashir¹ and its goodness of fit gives quite satisfactory fit over size–biased poisson distribution, size–biased lindley distribution and size–biased geometric distribution. The mixture of Poisson and Size–biased exponential distribution has been discussed by Fazal & Bashir¹ with properties and applications.

The size–biased and Area–biased distributions were discussed earlier by Fisher² when sample observations have unequal probability of selection, therefore we apply weights to the distribution to model bias.

If the rv ‘x’ had pdf f(x,θ); x=0,1,2,…..,; θ>0 , then the weighted distribution is of the form

$P\left(x;\theta \right)=\frac{x^m{f}_o\left(x;\theta \right)}{{{\mu }^{\prime }}_m}$ (1.3)

For m=1 and m=2 we get the size–biased and area–biased distributions respectively. Area–biased distributions are applicable for sampling in forestry, medical sciences, psychology etc, different discrete mixed distributions have been size–biased and discussed with their applications in real data–sets. Shankar & Kumar³ obtained size–biased Poisson Garima distribution with mathematical properties to analyze genetics datasets. Shankar⁴ introduced size–biased Poisson Shankar distribution with applications. Shankar & Fasshaye⁵ considered the size–biased form of Poisson Sujata distribution which was first introduced by Shankar⁵ for modelling count data in various fields of knowledge. Shakila & Mujahid Rasul⁶ derived the Poisson Area–Biased Lindley distribution with its applications in biological data to prove that it gives a better fit than Poisson distribution. Shankar & Fassahe⁷ proposed the size–biased form of Poisson Amarenda distribution and its applications proved that it is a good replacement of size–biased Poisson distribution (SBPD) size–biased Poisson Lindley distribution (SBPLD) and Size–biased Poisson.

Sujhata Distribution (SBPSD).Shakila and Mujahid⁸ proposed the size–biased form of Poisson Janardhan distribution and derived its mathematical properties, whereas Janardhan distribution is a two parameter distribution obtained by Rama & Mishra⁹ as a mixture of exponential and gamma distribution. Rama & Mishra⁹ obtained the size–biased form of Qaussi Poisson–Lindley distribution of which size–biased poisson Lindley distribution is a particular case (SBPLD). Ahmed & Munir¹⁰ have discussed few size–biased discrete distributions and their generalizations with properties and application. The size–biased version of Poisson Lindley distribution has been discussed by Ghitanni & Mutairi¹¹ and the new distribution introduced in this paper i.e Area–biased poisson exponential distribution (ABPED) gives more satisfactory fit as compared to size–biased Poisson Lindley distribution. The mathematical properties and estimation of parameters has been discussed and goodness of fit is also presented.^12–16

Using (1.1), (1.2), (1.3) the pmf of the Area–biased Poisson exponential distribution can be obtained as

$P(x;\theta )=\frac{x^2{f}_o\left(x;\theta \right)}{{{\mu }^{\prime }}_2}=\frac{x^2\theta /{\left(1+\theta \right)}^{x+1}}{\left(2+\theta \right)/{\theta }^2}$

Where ${{\mu }^{\prime }}_2=\frac{\left(2+\theta \right)}{{\theta }^2}$  is the second raw moment of discrete poisson exponential distribution.

With simplifications we get the pmf of Area–biased Poisson exponential distribution with parameter θ as

$P\left(X=x\right)=\frac{x^2{\theta }^3}{{\left(1+\theta \right)}^{x+1}\left(2+\theta \right)} \theta >0 , x=1,2,3,4,\dots \dots \dots .$ (1.4)

Graphs of Area– biased Poisson exponential distribution for different values of θ are shown in Figure 1 below

Figure 1 Graphs of Area–biased Poisson exponential distribution for different values of θ.

We start the mathematical derivations with moments and moment measures.
The first four raw moments of Area–Biased Poisson exponential distribution (ABPED) are

${{\mu }^{\prime }}_1=\frac{{\theta }^2+6\theta +\theta 6}{\theta \left(\theta +2\right)}$

${{\mu }^{\prime }}_2=\frac{{\theta }^2+12\theta +12}{{\theta }^2}$

${{\mu }^{\prime }}_3=\frac{{\theta }^4+30{\theta }^3+150{\theta }^2+240\theta +120}{{\theta }^3\left(\theta +2\right)}$

${{\mu }^{\prime }}_4=\frac{{\theta }^4+60{\theta }^3+420{\theta }^2+720\theta +360}{{\theta }^4}$

The mean moments of ABPED are obtained by using the relationship between moments about mean and moments about origin

${\mu }_2=\frac{4{\theta }^3+16{\theta }^2-24\theta +12}{{\theta }^2{\left(\theta +2\right)}^2}$

${\mu }_3=\frac{4{\theta }^5+28{\theta }^4-336{\theta }^3+168{\theta }^2+144\theta +48}{{\theta }^3{\left(\theta +2\right)}^3}$

${\mu }_4=\frac{760{\theta }^7+938{\theta }^6+6200{\theta }^5+32992{\theta }^4+97632{\theta }^3+132672{\theta }^2+65088\theta +720}{{\theta }^4{\left(\theta +2\right)}^4}$

The Harmonic mean of Area–biased Poisson exponential distribution is

$H.M=\frac{\theta }{\theta +2}$

The coefficient of variation (C.V), coefficient of skewness $\left(\sqrt{{\beta }_1}\right)$ , coefficient of kurtosis $\left(\sqrt{{\beta }_1}\right)$  and index of dispersion (γ) of Area–biased Poisson exponential distribution ABPED are obtained as:

$C.V=\frac{\sigma }{{{\mu }^{\prime }}_1}=\frac{\sqrt{4{\theta }^3+16{\theta }^2-24\theta +12}}{{\theta }^2+6\theta +6}$

$\sqrt{{\beta }_1}=\frac{{\mu }_3}{{\mu }_2{}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}=\frac{\sqrt{4{\theta }^5+28{\theta }^4-336{\theta }^3+168{\theta }^2+144\theta +48}}{{\left(4{\theta }^3+16{\theta }^2-24\theta +12\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}$

${\beta }_2=\frac{{\mu }_4}{{\mu }_2{}^2}=\frac{76{\theta }^7+938{\theta }^6+6200{\theta }^5+32992{\theta }^4+97632{\theta }^3+132672{\theta }^2+65088\theta +720}{{\left(4{\theta }^3+16{\theta }^2-24\theta +12\right)}^2}$

$\gamma =\frac{{\sigma }^2}{{\mu }_1{}^{\text{'}}}=\frac{4{\theta }^3+16{\theta }^2-24\theta +12}{\theta \left(\theta +2\right)\left({\theta }^2+6\theta +6\right)}$

For the Area–biased Poisson exponential distribution, (ABPED), from equations (1.6) and (1.7) it can be seen that $({\gamma }_1,{\beta }_2)\to \left(1.15,5\right)$ as θ→0, the model is positively skewed and leptokurtic.

To study the characteristics and comparative behavior of ABPED and SBPLD, a table of$({\gamma }_1,{\beta }_2)\to \left(1.15,5\right)$ for varying values of the parameter θ, has been prepared and presented in the table below (Table 1 & 2):

The comparative graphs of coefficient of variation, coefficient of skewness, coefficient of kurtosis and index of dispersion of ABPED and SBPLD are shown in Figure 2.

Figure 2 Graphs of for ABPED and SBPLD.

Using pmf of ABPED from (1.4), we have

$\frac{P\left(x+1,\theta \right)}{P\left(x.\theta \right)}=\frac{{\left(1+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$x$}\right.\right)}^2}{\left(1+\theta \right)}$

Which is a decreasing function of x, therefore ABPED is unimodal and has an increasing failure rate.

Probability generating function: the probability generating function of ABPED can be obtained as

$P_x\left(t\right)=E\left(t^x\right)={\displaystyle \sum _{x=1}^{\infty }\frac{x^2{\theta }^3{t}^x}{{\left(1+\theta \right)}^{x+1}\left(2+\theta \right)}}$

$=\frac{{\theta }^3{t}^2+t\left({\theta }^4+{\theta }^3\right)}{{\theta }^4+5{\theta }^3+9{\theta }^2+7\theta -t^3\left(\theta +2\right)+3t^2\left({\theta }^2+3\theta +2\right)-3t\left({\theta }^3+4{\theta }^2+5\theta +2\right)+2}$

Moment generating function: the moment generating function of ABPED is obtained as

$M_x\left(t\right)=E\left(e^{tx}\right)=\frac{{\theta }^3{e}^{2t}+\left({\theta }^4+{\theta }^3\right)e^t}{{\theta }^4+5{\theta }^3+9{\theta }^2+7\theta -\left(\theta +2\right)e^{3t}+\left(3{\theta }^2+9\theta +6\right)e^{2t}-\left(3{\theta }^3+12{\theta }^2+15\theta +6\right)e^t}$

Fisher information matrix
If $x~f(x/\theta )$  where $f(x/\theta )=\frac{x^2{\theta }^3}{{\left(1+\theta \right)}^{x+1}\left(2+\theta \right)}$  is the pmf of Area–biased Poisson exponential distribution with θ>0, then

$I_x\left(\theta \right)=\frac{20{\theta }^4-8{\theta }^3-192{\theta }^2-309\theta -144}{{\theta }^4{\left(1+\theta \right)}^2{\left(2+\theta \right)}^3}$

is the Fisher Information Matrix of ABPED, with

Method of moment (MOM) estimate: let x1, x2,x3…….xn be a sample of size n from ABPED (1.6) then equating the population parameter to the sample mean we obtain the MOM estimate $\tilde{\theta }$  of <$\theta$ of ABPED as

$\tilde{\theta }=\frac{-\left(\overline{x}-3\right)+\sqrt{{\left(\overline{x}-3\right)}^2+6\left(\overline{x}-1\right)}}{\left(\overline{x}-1\right)}$

Where x is the sample mean.

Maximum likelihood estimate (MLE): let x1, x2, x3…….xn be a sample of size n from ABPED (1.6) , the MLE estimate $\tilde{\theta }$  of $\theta$  is obtained as

$\tilde{\theta }=\frac{-\left(\overline{x}-3\right)+\sqrt{{\left(\overline{x}-3\right)}^2+6\left(\overline{x}-1\right)}}{\left(\overline{x}-1\right)}$

The Area–biased Poisson exponential distribution has been fitted to a number of countable datasets, and compared with size–biased Poisson lindley distribution. The following examples are used to illustrate a few situations generating the Area–biased distribution and–their applications. Microsoft Excel has been used to facilitate the use of the size–biased models to real life data. The MOM and MLE estimates are used to fit the distributions and is presented in the tables below. The datasets include number of observations of size distributions i:e small groups in various public situations reported by James,¹⁷ Coleman and James,¹⁸ and Simnoff,¹⁹ thunderstorm datasets reported by Shankar et al.,¹⁶ for all these datasets the ABPED distribution gives much closer fit than SBPLD (Tables 3–6).

Area–biased Poisson exponential distribution (ABPED) has been derived by area biasing the Discrete Poisson exponential distribution (PED) introduced by Fazal & Bashir.¹ The discussion on estimation and applications of the area–biased distribution demonstrates that ABPED has a practical use to real life data. Form AIC and BIC measures the proposed Area–biased model appears to offer substantial improvements in fit over the size–biased poisson Lindley model. Also the fitting in these tables reveal that the Area–biased distribution provides us with better fits in the situations where zero–class is missing gives a better fit than size–biased Poisson Lindley distribution (SBPLD) and size–biased Poisson distribution(SBPD) therefore we conclude that area–biased poisson exponential distribution is a good replacement of (SBPLD).

The author expresses thankfulness to the learned referee for her valuable suggestions which improved the quality of the paper.

None.

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