Characterization of continuous symmetric distributions using information measures of records
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Characterization of continuous symmetric distributions using information measures of records Jafar Ahmadi1 Received: 5 May 2020 / Revised: 2 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper, several characterizations of continuous symmetric distributions are provided. The results are based on the properties of some information measures of k-records. These include cumulative residual (past) entropy, Shannon entropy, Rényi entropy, Tsallis entropy, also some common Kerridge inaccuracy measures. It is proved that the equality of information in upper and lower k-records is a characteristic property of continuous symmetric distributions. Completeness properties of certain function sequences are also used to obtain some characterization results. Keywords Completeness properties · Cross entropy · Cumulative entropy · Kerridge inaccurac · k-Records · Rényi entropy · Symmetric distribution · Tsallis entropy Mathematics Subject Classification 62E10 · 60E05 · 62G30
1 Introduction We recall that the class of symmetric distributions is so broad and includes several well-known distributions such as arc-sine, Bates, beta (with equal shape parameters), Cauchy, double Weibull distribution, Laplace, logistic, normal, Student’s t, Tukey lambda and uniform distributions. We refer the reader to the book by Johnson et al. (1995) which includes many examples of characterization and applications of common symmetric distributions. Also, in the recent years, several authors introduced skew-symmetric models which are generated starting from a symmetric distribution. Suppose X is a continuous random variable with cumulative distribution function (cdf) FX and support S X . It is known that if there exists a finite c such that FX (c − x) + FX (c + x) = 1, for all x ∈ S X , then it is said that X has a symmetric distribution around c. If the probability density function (pdf) f X exists then the
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Jafar Ahmadi [email protected] Department of Statistics, Ferdowsi University of Mashhad, P. O. Box 1159, Mashhad 91775, Iran
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J. Ahmadi
equivalent condition for symmetry is: f X (c − x) = f X (c + x), for all x ∈ S X . Several authors studied the properties of symmetric distributions. Mahdizadeh and Zamanzade (2020) proposed a nonparametric estimator for symmetric distribution function under multi-stage ranked set sampling. It should be mentioned that, in the literature, there are various characterization results for symmetric distributions based on the properties of order statistics, see for examples, Miloševi´c and Obradovi´c (2016) and Balakrishnan and Selvitella (2017), Ahmadi and Fashandi (2019a, b) and Ahmadi et al. (2020). The main aim of this work is to provide some new characterizations for continuous symmetric distributions based on the properties of information measures of record data extracted from a sequence of independent and identically distributed (iid) continuous random variables. Usually, criteria for checking whether a distribution is symmetric or not can be formulated ba
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