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5. Asymptotic Relative Efficiency

Since the MLEs of the parameters θ and in the ZIP model have no closed form expressions, their exact standard errors are unlikely. Hence we are left with the asymptotic relative efficiencies of the estimators for the analytical comparison. Since the ZIP model in (1.1) belongs to two parameter exponential family, the MLEs of θ and are also asymptotically normal and

Hence the asymptotic relative efficiency of with respect to is

Therefore, the MMEs and the MLEs of θ are asymptotically equally efficient. The same is true in the case of too.

6. Simulation Study

Using R software, 1000 samples of various sizes were simulated fixing θ = 2.5 and = 0.3. For each of the samples the MLEs and the MMEs of θ and were computed. The histograms of the MLEs and the MMEs were separately drawn for θ and. The histograms are given in the Figures 1-4.

Nanjundan et al. [8] have carried out an elaborate simulation study by considering various values of θ and and varying the number of samples. From the above histograms, it can be observed that the MMEs are also normally distributed even for moderate sample sizes. This gives the graphical evidence for the asymptotic normality of the MLEs and the MMEs of both the parameters in the model.

The mean squared errors (MSEs) computed from the simulation study are shown in the Appendix. The following observations about the performance of the estimates are made from the mean squared errors.

1) Though the MSEs corresponding to the MLEs are less than the MSEs of MMEs, the difference is very insignificant. That is the MMEs also perform equally good when compared the MLEs.

2) For the majority of the combinations of θ and,

Figure 1. Histograms of the MMEs and the MLEs of θ and φ based on 1000 samples of size 25 each drawn from the distribution 0.3p0(x) + 0.7p1(x, 2.5).

Figure 2. Histograms of the moment estimators and the MLEs of θ and φ based on 1000 samples of size 50 each drawn from the distribution 0.3p0(x) + 0.7p1(x, 2.5).

Figure 3. Histograms of the moment estimators and the MLEs of θ and φ based on 1000 samples of size 100 each drawn from the distribution 0.3p0(x) + 0.7p1(x, 2.5).

the MSEs corresponding to θ are less than the MSEs corresponding to in the case of both the estimates.

7. Discussion and Conclusions

Zero-inflated Poisson models are readily applicable in many biological and social contexts. Two such situations are briefly discussed in this section.

Insects live on the leaves of a tree when they are found

Figure 4. Histograms of the moment estimators and the MLEs of θ and φ based on 1000 samples of size 250 each drawn from the distribution 0.3p0(x) + 0.7p1(x, 2.5).

to be suitable for feeding and they do not live on those which are unsuitable for feeding. Suppose that the proportion of unsuitable leaves in a tree is and the number of insects on a suitable leaf has a Poisson distribution with mean θ. If an observed leaf has any insects on it, then it is definitely a suitable one. On the other hand, if a leaf has no insects, then it may or may not be suitable for feeding. Let X denote the number of insects on any leaf. Then X has the p.m.f. given in (1.1).

A social group under study may have fertile and sterile couples. If the proportion of sterile couples is and the number of children per fertile couple has a Poisson distribution with mean θ. Then X, the number of children of a randomly chosen couple, has the ZIP distribution specified in (1.1).

For more applications, one can refer to Lambert [1] and Kale [4].

The MLEs of the parameters in the ZIP model have no closed form expressions and computing them even by the EM algorithm needs computer facility. Whereas the MMEs have simple closed form expressions and they can be computed even with pocket calculators. The MMEs and the MLEs are asymptotically equally efficient. Hence MMEs can easily be used instead of the MLEs when the sample size is sufficiently large.

8. Acknowledgements

Part of this work was done during a short visit of the first author to the Dept. of Statistics, University of Poona, Pune during Jan, 2008. He is grateful to Prof. B. K. Kale for his guidance in this direction. He is also thankful to Prof. Naik Nimbalkar for providing research facility. The authors appreciate Prof. K. Suresh Chandra for useful suggestions in the presentation of the results. The authors record their deep sense of gratitude to the referee for valuable suggestions which improved the content of the paper.

REFERENCES

  1. D. Lambert, “Zero-Inflated Poisson Regression with an Application to Detects in Manufacturing,” Technometrics, Vol. 34, No. 1, 1992, pp. 1-14. doi:10.2307/1269547
  2. D.-L. Couturier and M.-P. Victoria-Feser, “Zero-Inflated Truncated Generalized Pareto Distribution for the Analysis of Radio Audience Data,” The Annals of Applied Statistics, Vol. 4, No. 4, 2010, pp. 1824-1846. doi:10.1214/10-AOAS358
  3. P. Yip, “Inference about the Mean of Poisson Distribution in the Presence of a Nuisance Parameter,” Australian Journal of Statistics, Vol. 30, No. 3, 1998, pp. 299-306. doi:10.1111/j.1467-842X.1988.tb00624.x
  4. B. K. Kale, “Optimal Estimating Equations for Discrete Data with Higher Frequencies at a Point,” Journal of the Indian Statistical Association, Vol. 36, 1998, pp. 125- 136.
  5. A. P. Dempster, N. M. Laird and D. B. Rubin, “Maximum Likelihood Estimation from Incomplete Data via the EM Algorithm (with Discussion),” Journal of the Royal Statistical Society: Series B, Vol. 39, No. 1, 1977, pp. 1- 38.
  6. G. Nanjundan, “An EM Algorithmic Approach to Maximum Likelihood Estimation in a Mixture Model,” Vignana Bharathi, Vol. 18, 2006, pp. 7-13.
  7. G. Nanjundan, “On the Computation of the Maximum Likelihood Estimates of the Parameters in a Mixture Model,” Mapana Journal of Sciences, Vol. 6, No. 2, 2007, pp. 57-66.
  8. G. Nanjundan, A. Loganathan and T. R. Naika, “An Empirical Comparison of Maximum Likelihood and Moment Estimators of Parameters in a Zero-Inflated Poisson Model,” Mapana Journal of Sciences, Vol. 8, No. 2, 2009, pp. 59-72.

Appendix

In the following table, the upper and the lower cells give the mean squared errors of the MLEs of θ and φ respectively. The values within the brackets are the mean squared errors corresponding to MMEs. Sample size = 100 and number of samples = 1000.

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