Size-biased new Quasi-Poisson-Lindley distribution: Properties and applications

Introduction

Size-biased distributions are a special case of weighted distributions. Fisher [2] introduced weighted distributions to model ascertainment biased and was later formalized by Rao [5] . It has been observed that in many situations the experimenters do not work with the random samples from the population in which they are interested. When an observation is recorded by nature according to certain stochastic model, the recorded observation will not have the original distribution unless every observation is given an equal chance of being recorded. Since the observations are recorded with unequal probability, the resulting sampled distribution does not follow the original distribution. If a random variable 𝑋𝑋 has distribution 𝑃𝑃 0 (𝑥𝑥; 𝜃𝜃) with unknown parameter 𝜃𝜃, then the corresponding weighted distribution is of the form 𝑃𝑃(𝑥𝑥; 𝜃𝜃) = 𝑤𝑤(𝑥𝑥)𝑃𝑃 0 (𝑥𝑥; 𝜃𝜃)

𝐸𝐸[𝑤𝑤(𝑥𝑥)]

, where, 𝑤𝑤(𝑥𝑥) is a non-negative function such that 𝐸𝐸[𝑤𝑤(𝑥𝑥)] exists. The weighted distribution with weight 𝑤𝑤(𝑥𝑥) = 𝑥𝑥 is known as size biased distribution having the form, 𝑃𝑃 * (𝑥𝑥; 𝜃𝜃, 𝛼𝛼) = 𝑥𝑥𝑃𝑃 0 (𝑥𝑥; 𝜃𝜃,𝛼𝛼) 𝐸𝐸 (𝑥𝑥) .

In this chapter, a size-biased new quasi-Poisson-Lindley (SBNQPL) distribution has been obtained in section II and has been compared with size-biased quasi-Poisson-Lindley (SBQPL) distribution. In section III graphical representations of SBNQPL distribution have been shown. Statistical properties of the distribution have been studied in section IV. In Section V, the estimation of parameters of SBNQPL distribution has been discussed. Goodness of fit has been discussed in section VI. The two-parameter size-biased quasi-Poisson-Lindley distribution obtained by Shanker and Mishra (2014) [12] has the probability mass function (pmf) as,

The 𝑟𝑟 𝑡𝑡ℎ factorial moments have been obtained as,

The variance has been obtained as

The probability generating function has been obtained as,

The recurrence relation for probability may be written as

Derivation of the proposed distribution

Let us consider 𝑋𝑋 to be a random variable following NQPL distribution having the probability mass function (pmf),

Then, the probability mass function (pmf) of (SBNQPL) distribution may be obtained as

where, 𝑃𝑃 0 (𝑥𝑥; 𝜃𝜃, 𝛼𝛼) is the p.m.f of NQPL distribution and 𝜇𝜇 =

, is the mean of the NQPL distribution.

Thus, the pmf of SBNQPL distribution has been obtained as,

SBNQPL distribution may also be obtained from the size-biased Poisson distribution having the probability mass function (pmf) as,

When the parameter 𝜆𝜆 of size-biased Poisson distribution follows the size-biased new quasi-Lindley distribution having the density function,

Then, the pmf of size-biased new quasi-Poisson-Lindley distribution may also be obtained from equation ( 1) and ( 2) by integrating the mixture model as,

It has been observed that when 𝛼𝛼 = 𝜃𝜃, SBNQPL distribution reduces to size-biased Poisson-Lindley (SBPL) distribution having the pmf as,

Graphical representation of SBNQPL distribution

To study the behavior of SBNQPL distribution for different values of parameter 𝛼𝛼 and 𝜃𝜃, the probability for possible values of 𝑥𝑥 are computed using equation (3). Figure 1 represents the pmf plot of SBNQPL distribution for fixed 𝛼𝛼 i.e 𝛼𝛼 = 0.1 and 𝜃𝜃 = 1.5, 2.0 and 3.0 and Figure 2 represents the pmf plot of SBNQPL distribution for 𝛼𝛼 = 0.9 and 𝜃𝜃 = 1.5, 2.0 and 3.0 respectively.

Statistical properties of SBNQPL distribution A. Shape of the probability function

� which is a decreasing function in ′x′ and hence SBNQPL distribution is log-concave. Thus, we may conclude that SBNQPL distribution is unimodal and has an increasing failure rate. [Johnson et al.] [4] .

B. Moments and related measures

The r th factorial moment of SBNQPL distribution may be obtained as,

Taking ′x + r′ in place of x, we get,

The expression in the bracket is (λ + r) and hence we have,

Hence the solution of the gamma integral will be,

Thus, from equation (4) the r th factorial moments may be obtained by substituting r = 1,2,3 and 4 as, μ

. Now, using the relationship between raw and factorial moments we have obtained the raw moments as,

The variance of SBNQPL distribution may be obtained as,

C. Recurrence Relations for Probabilities of SBNQPL distribution

The probability generating function which is considered as a useful tool for dealing with discrete random variables has been used to generate all the probabilities of the distribution. The probability generating function (pgf) of size biased new quasi-Poisson-Lindley distribution may be obtained as,

, where, P * (x) is the pmf of SBNQPL distribution.

Equating the coefficient of ′t r ′ on both sides of equation ( 5), the expression for recurrence relation for probabilities of SBNQPL distribution has been obtained as,

The higher probabilities may be obtained from equation (6) substituting by r = 4, 5, … etc. It has been observed that when α = θ SBNQPL distribution reduces to SBPL distribution. and

and 𝑏𝑏 � may be obtained from equation (8).

D. Maximum Likelihood Estimates

Let 𝑥𝑥 1 , 𝑥𝑥 2 , … … , 𝑥𝑥 𝑛𝑛 be a random sample of size n from the SBNQPL distribution and let 𝑓𝑓 𝑥𝑥 be the observed frequency in the sample corresponding to 𝑋𝑋 = 𝑥𝑥 (𝑥𝑥 = 1,2, … … , 𝑘𝑘) such that ∑ 𝑓𝑓 𝑥𝑥 = 𝑛𝑛 𝑘𝑘 𝑥𝑥=1

. The Likelihood function 𝐿𝐿 of the new quasi-Poisson Lindley distribution may be given as

. The log likelihood function becomes

Then the log likelihood equations are obtained as

The above two equations could be solved using Fisher's scoring method. We have

The following equations can be solved for 𝜃𝜃 � and 𝛼𝛼 � iteratively till sufficiently close values of 𝜃𝜃 and 𝛼𝛼 are obtained. Thus, we have

Where 𝜃𝜃 0 and 𝛼𝛼 0 are the initials value of 𝜃𝜃 and 𝛼𝛼 respectively.

Goodness of fit

To illustrate the applications and to justify suitability of SBNQPL distribution, the SBNQPL distribution has been fitted to some published datasets for which various distributions were fitted earlier by different author. In fitting of probability distributions, the estimation of parameters plays a very important role. The parameters of SBNQPL distribution have been estimated by the method of moment. The parameters are then used to obtain the expected frequencies which are later used to calculate the 𝜒𝜒 2 value. The SBNQPL distribution is then compared with other distributions as measured by 𝜒𝜒 2 criterion. Here, the SBNQPL distribution has been fitted to three data sets and the distribution has been compared with size-biased Poisson-Lindley, size-biased quasi-Poisson-Lindley. The SBNQPL distribution has also been fitted to data reported by James [3] in Table 1 which is regarding the distribution for the Counts of people in public places on a spring afternoon in Portland. The parameters are obtained by the method of moment as 𝛼𝛼 � = -63.6352 and 𝜃𝜃 � = 7.2136. The derived distribution has been compared with SBPL and SBNQPL distributions. In Table 2 the SBNQPL distribution has been fitted to data reported by Coleman and James [1] for the Counts of Shopping Groups-Eugene, spring, Department Store and Public Market. From the data set we have obtained the expected frequency and 𝜒𝜒 2 value for goodness of fit. Table 3 is regarding the sets of data reported by Simonoff [13] for counts of Play Groups-Eugene, Spring Public Playground D where there are six groups and the observed frequency corresponding to the groups are given.

Conclusion

The observed frequencies and expected frequencies of SBPL, SBQPL and SBNQPL distribution for the three data sets have been shown in Table 1, Table 2 and Table 3 respectively for its comparison. Comparing the observed and expected frequencies the 𝜒𝜒 2 values and 𝑝𝑝-values have been calculated for testing the goodness of fit. It has been observed that based on chi-square value and 𝑝𝑝-value from Table 1, Table 2, and Table 3 the SBNQPL distribution provide a closer fit than SBPL and SBQPL distribution.

References

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