Estimation of the Poisson Parameter with Moment Generating Method

A new estimator of the Poisson parameter is proposed using the moment generating function. Some statistical properties of the proposed estimator are studied. The performance of the new estimator is compared with the maximum likelihood estimator (MLE) via examples and simulation in terms of goodness of fit and relative efficiency. Simulation and examples to real-life data suggest that the new estimator has higher relative efficiency compared to the MLE, while both are comparable in goodness of fit. The R program utilized in all computation and simulation is incorporated to facilitate the implementation of the new estimator in computation.

A discrete random variable is said to follow a Poisson distribution with parameter if the probability mass function is given by In general, is unknown and estimated using a sample. Let 1 , 2 , ⋯ , be a random sample of size . Then, the maximum likelihood function of ( ) is given by Taking logarithm on both sides Taking derivative of with respect to , and setting equal to zero, a maximum likelihood estimator (MLE) of , ̂ is given by It is easy to see that ̂ is an unbiased estimator of , i.e.,

() =
The variance of ̂ given by While the estimator ̂ of by the MLE method is unbiased, there is a possibility that other form of estimator of , even biased, might have smaller variance. Indeed, in classical statistics, it is well-known that estimators might be biased but have increased accuracy (i.e., smaller variance), which is termed as bias-variance trade-off. For example, one might refer to James-Stein estimator (Stein, 1956;James and Stein, 1961) or LASSO (Tibshirani, 1996(Tibshirani, , 1997, where the estimator achieves lower mean square error (MSE) than the ML estimator.
In this paper, we proposed a new estimator using the moment generating function, which is biased but provides an increased efficiency compared to the MLE estimator. Method of moments is widely used in different areas of statistics, such as causal inference (e.g., Lu, 2016).

Proposed Estimator
In this section, we propose a new estimator of the Poisson parameter using the moment generating function. The moment generating function of ~( ) is ( ) = ( ) = ( −1) Given a random sample 1 , 2 , ⋯ , of size , the moment generating function of ∑ =1 is given by By the method of moments, the proposed estimator of , ̃ follows from the solving the equation (1) After an algebraic manipulation of (1), we have the following new estimator ̃ of A similar method of estimation exists in literature; for example, see Sidhu, Tailor and Singh (2009).

Properties of New Estimator
In this section, we study some properties of the proposed estimator, which we state in terms of the following theorems:  with respect to ̂ is It is easy to see that as → 0, ̃ and ̂ are the same. If ≠ 0, then there may exist a non-zero such that (̃) < () or, 2 + ( − + 1) 2 < ( − 1) 2 (2) We can easily search for values of , for selected values of and , satisfying the relation (2) and estimate the percent relative efficiency of the proposed estimator ̃ with respect to .
In section 4, we provide an example of a Poisson distributional fit using the two estimators ̂ and ̃. We have utilized an R program to search for values of while using ̃ for estimating and assessing goodness of fit.

Applications: Fitting Poisson Distributions to Real-Life Data
In this section, we fit a Poisson distribution model to the number of land-falling hurricane in the USA in 98-year period from 1900 to 1997, appeared in Glover and Mitchell (2002). The data in Table 1, refers to number of hurricanes per year ( ) and their frequencies ( ).  , we execute a search of satisfying the equation (2) using an R program. By searching, we consider a value of = 0.0125. Then, we have The R program that we have used for the search is provided in the Appendix.
Using the estimator ̃, the Poisson distribution to hurricane data and the expected frequency (̃), take the form: The estimated expected frequency corresponding to the observed frequency using two estimators ̂ and ̃ are provided in Table 2.
We computed the chi-square value by amalgamating expected frequencies for four or more hurricanes per year so that the expected frequency for each cell is at least 5. With this modification, we have a chi-square = 5 − 1 − 1 = 3, and ( = 3, = 0.05) = 7.81. Then, by comparing the observed value of chi-square (1.262, -value = 0.7382) with 7.81, we may accept the null hypothesis that the US land-falling hurricane follows a Poisson (1.622) distribution.
The value of the Chi-squared test statistic under the proposed estimator is given by International Journal of Statistics and Probability Vol. 7, No. 6;2018 As before, by comparing the observed value of chi-square (1.227, -value = 0.7465) with 7.81, we may accept the null hypothesis that the US land-falling hurricane follows a Poisson (1.612) distribution.
Note that using both the estimators we accept the null hypothesis that the US land-falling hurricane follows a Poisson with mean 1.622 using the MLE estimator and mean 1.612 using the proposed estimator, with comparable p-value. Therefore, the proposed estimator is better (higher p-value) or at least as good as the MLE estimator in goodness of fit of Poisson distribution to the given example case.

Relative Efficiency of the Proposed Estimator
In this section, we consider an empirical study to evaluation relative efficiency ( ) of the proposed estimator ̃ compared to the MLE ̂ for varying values of sample size and . We consider four fixed values of estimator ̂ at 0.25, 0.50, 0.75 and 1.25, chosen arbitrarily, and sample size ranging between 5 and 50 at =5, 6, 7, 8, 9, 10, 15, 20, 25, 30, 35, 40, 45 and 50. For each combination of ̂ and , we consider values of between and with an increment of 0.01, denoted by ∈ , : @0.01-, where = 0.01 and values of are evaluated using the search so as to satisfy the equation (2). The estimated of ̃ compared to the MLE ̂ are reported in Table 3 for varying sample size and . As appears in Table 3, the estimated of the proposed estimator ̃ as compared to the MLE ̂ is sensitive to sample size. For example, when = 5, the ranges from 100.07 to 180, when ranges from 0.01 to 3.47, with an optimum relative efficiency of 180 observed at optimum = 1.08. However, when = 10, the ranges from 100.07 to 140, when ranges from 0.01 to 1.50, with an optimum relative efficiency of 140 observed at optimum = 0.64. Overall, as ̂ increases the decreases.
The estimated of the proposed estimator ̃ as compared to the MLE ̂ for selected sample sizes between 5 and 50, and satisfying equation (2)   From Figures 1-4, it follows that we need to search in an interval of the positive neighborhood of 0 for higher value of the relative efficiency of the proposed estimator compared to the MLE. Also, the estimated relative efficiency decreases as the sample size increases. It is also evident that there is an optimum value of , , at which the attains its maximum value, for a given sample size and the MLE estimator .

Results and Discussion
In order to search for satisfying equation (2), we have utilized a program written in R, which is incorporated in the Appendix. Once values of are obtained, we have evaluated the relative efficiency of the proposed estimator as compared to the MLE. It appears that the values of for the example data model remain positive for relative efficiency to be more than 100% for the proposed estimator compared to the MLE estimator. In empirical study, we have restricted our search for nearing 0 at positive values using trial and error method, by choosing values of in some interval. Theoretically, since the proposed estimate is unbiased as → 0, we wish to achieve efficiency as well as nearing unbiased estimate by choosing values of nearing 0. For example, when ̂ = 0.50 and the sample size = 5, the relative efficiency of the proposed estimator ranges from 100.45 to 140 as ranges from 0.01 to 1.50, with a maximum relative efficiency of 140 observed at = 0.64. However, when the sample size increases to = 10 (̂= 0.50), the relative efficiency ranges from 100.20 to 120, when ranges from 0.01 to 0.76, with a maximum of 120 occurring at 0.35. From the reported results, it appears that for a fixed estimator of the Poisson parameter, the lower sample size provides better efficiency for the proposed estimator. It makes sense because as sample size gets larger, the values of (̃) and () both get smaller to lead to equally efficient estimator ̃ and . It also follows that relative efficiency of the proposed estimate is better when the value of MLE estimator ̂ is fixed at a lower value, a rare event rate. For example, the maximum value of the estimated of the proposed estimator ̃ is 180 when ̂ is 0.25 and = 5, whereas the maximum estimated decreases to 116 when the value of ̂ decreases to 1.25 and = 5. Therefore, the proposed estimate is efficiently applicable to the rare events data.

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