Goodness-of-fit testing for the inverse Gaussian distribution based on new entropy estimation using ranked set sampling and double ranked set sampling

Amer Al-Omari,Abdul Haq

doi:10.1186/2193-2697-1-8

Amer Al-Omari, Abdul Haq

Open Access

https://doi.org/10.1186/2193-2697-1-8

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Abstract

Entropy is a measure of uncertainty and dispersion associated with a random variable. Several goodness-of-fit tests based on entropy are available in literature and the entropy been widely used in many applications. Goodness-of-fit test for the inverse Gaussian distribution is studied based on new entropy estimation using simple random sampling (SRS), ranked set sampling (RSS) and double ranked set sampling (DRSS) methods. The critical values of the new tests are obtained using Monte Carlo simulations. The power values of the suggested tests based on several alternative hypotheses using SRS, RSS, and DRSS are also presented. It is observed that the proposed tests are more powerful as compared to the test under SRS. Also, it turns out that the test based on DRSS is superior to the RSS test for all of the cases considered in this study. Since the suggested goodness-of-fit tests for the inverse Gaussian distribution using DRSS are more efficient than that based on RSS, one may consider them using multistage RSS.

Highlights

Entropy is a measure of uncertainty and dispersion associated with a random variable
The root mean square errors (RMSEs) and the bias values are obtained for the estimators based on 10,000 samples of sizes n = 10, 20, 30 with window sizes 1 ≤ m ≤5, 1 ≤ m ≤10 and 1 ≤ m ≤ 15, respectively
Comparison between VE(m,n) and AE(m,n) The samples are selected from the uniform, exponential and the standard normal distributions using simple random sampling (SRS), ranked set sampling (RSS) and double ranked set sampling (DRSS) methods

Summary

Introduction

Entropy is a measure of uncertainty and dispersion associated with a random variable. It is not uniquely defined, there exist axiom systems that justify the particular entropies. Shannon (1948) defined the entropy H(f ) of the random variable X as Z1. À1 where X is a continuous random variable with probability density function (pdf ) f(x) and cumulative. D log FÀ1ðpÞ dp: dp Vasicek (1976) ð2Þ. ; Xn be a simple random sample of size n from F(x) and let Xð1Þ ≤ Xð2Þ ≤ ⋯ ≤ XðnÞ be the order statistics of the sample. Vasicek (1976) estimator of H(f ). Let X1; X2; . . . ; Xn be a simple random sample of size n from F(x) and let Xð1Þ ≤ Xð2Þ ≤ ⋯ ≤ XðnÞ be the order statistics of the sample. Vasicek (1976) estimator of H(f )

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