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Classical and Bayesian Estimation of Entropy for Pareto Distribution in Presence of Outliers with Application

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Abstract

The measure of entropy has a pivotal role in the information theory area. In this paper, estimation of differential entropy for Pareto distribution in presence of r outliers is considered. In this regard, the classical and Bayesian estimation techniques of differential entropy are employed. In classical setup, we obtain the maximum likelihood estimators of the differential entropy as well as assessing their performance via a simulation study. The entropy Bayesian estimator is derived using squared error, linear exponential, weighted squared error and K loss functions. The Metropolis-Hastings algorithm is used to generate posterior random variables. Monte Carlo simulations are designed to implement the precision of estimates for different sample sizes and number of outliers. Furthermore, performance of estimates is planned by experiments with real data. Generally, we conclude that the entropy Bayesian estimates of simulated data tend to the true value as the number of outliers increases. Further, the entropy Bayesian estimate under weighted squared error loss function is preferable to the other estimates in majority of situations.

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Authors and Affiliations

  1. Department of Mathematical Statistics, Faculty of Graduate Studies for Statistical Research, Giza, 12613, Egypt

    Amal S. Hassan, E. A. Elsherpieny & Rokaya E. Mohamed

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  1. Amal S. Hassan
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  2. E. A. Elsherpieny
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  3. Rokaya E. Mohamed
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Correspondence to Rokaya E. Mohamed.

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Appendices

Appendix A.

Table 1 Statistics measures of maximum likelihood entropy estimates in homogenous and outliers cases
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Table 2 Statistics measures of maximum likelihood entropy estimates in homogenous and outliers cases
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Table 3 Statistics measures of entropy BEs in homogenous and outliers cases
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Table 4 Statistics measures of entropy BEs in homogenous and outliers cases
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Table 5 Statistics measures of entropy BEs in homogenous and outliers cases
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Table 6 Statistics measures of entropy BEs in homogenous and outliers cases
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Table 7 Estimates of differential entropy and their MSEs (in brackets) for 20 claim amounts
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Appendix B

Figure 1
figure 1

MSEs of MLEs of entropy at r = 0,1,2 and 4 for Set 1

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Figure 2
figure 2

MSEs of MLEs of entropy at r = 0,1,2 and 4 for Set 3

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Figure 3
figure 3

MSEs of entropy BEs at r = 1 for Set 2

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Figure 4
figure 4

MSEs of entropy BEs at r = 2 for Set 4

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Figure 5
figure 5

MSEs of \(\overline {S}(X)\) for Set 3 at r = 0,1,2,4 and n = 10

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Figure 6
figure 6

MSEs of \(\overline {S}(X)\) for Set 1 at r = 0,1,2,4 and n = 30

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Figure 7
figure 7

The MCMC plots for data for the Pareto distribution in presence of outliers (r = 1)

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Figure 8
figure 8

The MCMC plots for data for the Pareto distribution in presence of outliers (r = 4)

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Hassan, A.S., Elsherpieny, E.A. & Mohamed, R.E. Classical and Bayesian Estimation of Entropy for Pareto Distribution in Presence of Outliers with Application. Sankhya A 85, 707–740 (2023). https://doi.org/10.1007/s13171-021-00274-z

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  • DOI: https://doi.org/10.1007/s13171-021-00274-z

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