Further properties of Tsallis extropy and some of its related measures

This article introduces the concept of residual and past Tsallis extropy as a continuous information measure within the context of continuous distribution. Moreover, the characteristics and their relationships with other models are evaluated. Several stochastic comparisons are provided, along with outcomes concerning order statistics. Additionally, the models acquired include instances such as uniform and power function distributions. The measure incorporates its monotonic traits, and the outcomes defining its characteristics are presented. On the other hand, a different portrayal of the Tsallis extropy is introduced, expressed in relation to the hazard rate function. The Tsallis extropy of the lifetime for both mixed and coherent systems is explored. In the case of mixed systems, components' lifetimes are considered independent and identically distributed. Additionally, constraints on the Tsallis extropy of these systems are established, along with a clarification of their practical applicability. Non-parametric estimation using an alternative form of Tsallis function extropy for simulated and real data is performed

A new least squares method for estimation and prediction based on the cumulative Hazard function

In this paper, the cumulative hazard function is used to solve estimation and prediction problems for generalized ordered statistics (defined in a general setup) based on any continuous distribution. The suggested method makes use of Rényi representation. The method can be used with type Ⅱ right-censored data as well as complete data. Extensive simulation experiments are implemented to assess the efficiency of the proposed procedures. Some comparisons with the maximum likelihood (ML) and ordinary weighted least squares (WLS) methods are performed. The comparisons are based on both the root mean squared error (RMSE) and Pitman's measure of closeness (PMC). Finally, two real data sets are considered to investigate the applicability of the presented methods

On Convergence of Intermediate Order Statistics under Power Normalization

We discuss the convergence of the moments of intermediate order statistics under power normalization. The moments convergence is established for four p-max-stable laws according to conditions imposed on the considered distribution and on the rank sequence

Limit theory for bivariate central and bivariate intermediate dual generalized order statistics

Burkschat et al. 2003 have introduced the concept of dual generalized order statistics dgos to unify several models that produce descendingly ordered random variables rv’s like reversed order statistics, lower k-records and lower Pfeifer records. In this paper we derive the limit distribution functions df’s of bivariate central and bivariate intermediate m-dgos. It is revealed that the convergence of the marginals of the m-dgos implies the convergence of the joint df. Moreover, we derive the conditions under which the asymptotic independence between the two marginals occurs.Burkschat et al. 2003 have introduced the concept of dual generalized order statistics dgos to unify several models that produce descendingly ordered random variables rv’s like reversed order statistics, lower k-records and lower Pfeifer records. In this paper we derive the limit distribution functions df’s of bivariate central and bivariate intermediate m-dgos. It is revealed that the convergence of the marginals of the m-dgos implies the convergence of the joint df. Moreover, we derive the conditions under which the asymptotic independence between the two marginals occurs

An Extension of the Skew-Normal Distribution

The stable symmetric family of distribution functions (DF’s) suggested by [1] is a family that contains the reverse of every DF belonging to it. It is revealed that the stable families are capable of describing many types of statistical data. We introduce a new stable family via a mixture of the skew-normal distribution and its reverse, after inserting a scale parameter and its reciprocal to the skew-normal distribution and its reverse, respectively. We show that this family contains all the possible types of DFs. Besides, it has a very remarkable wide range of the indices of skewness and kurtosis. Computational technique using EM algorithm is implemented for estimating the model parameters. Moreover, an application with a real data set is presented

Generalized Order Statistics with Random Indices in a Stationary Gaussian Sequence

In this paper we study the limit distributions of extreme, intermediate and central m-generalized order statistics (gos), as well as m-dual generalized order statistics (dgos), of a stationary Gaussian sequence (sGs) under equi-correlated set up, when the random sample size is assumed to converge weakly. Moreover, the result of extremes is extended to a wide subclass of gos (as well as dgos) which contains the most important models of ordered random variables (rv’s)

Cumulative Residual Tsallis Entropy-Based Test of Uniformity and Some New Findings

The Tsallis entropy is an extension of the Shannon entropy and is used extensively in physics. The cumulative residual Tsallis entropy, which is a generalization of the Tsallis entropy, plays an important role in the measurement uncertainty of random variables and has simple relationships with other important information and reliability measures. In this paper, some novel properties of the cumulative residual Tsallis entropy are disclosed. Moreover, this entropy measure is applied to testing the uniformity, where the limit distribution and an approximation of the distribution of the test statistic are derived. In addition, the property of stability is discussed. Furthermore, the percentage points and power against seven alternative distributions of this test statistic are presented. Finally, to compare the power of the suggested test with that of other tests of uniformity, a simulation study is conducted

Cumulative Residual Tsallis Entropy-Based Test of Uniformity and Some New Findings

The Tsallis entropy is an extension of the Shannon entropy and is used extensively in physics. The cumulative residual Tsallis entropy, which is a generalization of the Tsallis entropy, plays an important role in the measurement uncertainty of random variables and has simple relationships with other important information and reliability measures. In this paper, some novel properties of the cumulative residual Tsallis entropy are disclosed. Moreover, this entropy measure is applied to testing the uniformity, where the limit distribution and an approximation of the distribution of the test statistic are derived. In addition, the property of stability is discussed. Furthermore, the percentage points and power against seven alternative distributions of this test statistic are presented. Finally, to compare the power of the suggested test with that of other tests of uniformity, a simulation study is conducted