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In this work, we estimated the different entropies like Shannon entropy, Rényi divergences, Csiszár divergence by using Jensen’s type functionals. The Zipf’s–Mandelbrot law and hybrid Zipf’s–Mandelbrot law are used to estimate the Shannon entropy. The Abel–Gontscharoff Green functions and Fink’s Identity are used to construct new inequalities and generalized them for m-convex function.

Read (1984) presented an asymptotic expansion for the distribution function of the power divergence statistics whose speed of convergence is dependent on the parameter of the family. Generalizes that result by considering the family of (h, φ)‐divergence measures. Considers two other closer approximations to the exact distribution. Compares these three approximations for the Renyi’s statistic in small samples.

To provide a new family of test statistics to solve the Behrens‐Fisher problem and to compare it with the classic test statistics through a different simulation studies.

A general procedure for testing composite hypothesis to k samples of different size problems on the basis of the Renyi's divergence is used to develop a new parametric family of test statistics that contains as a particular case the classical likelihood ratio test. The scope of the paper is to find out if some member of the new family of test statistics it is preferable to the classical ones.

Some members of the new parametric family of test statistics behave remarkably well in comparison to the classic ones, as the different computational studies reveal.

This paper offers a new way to solve the Behrens‐Fisher problem that it is preferable in some cases to the known procedures.

By combining the subjective probabilistic viewpoint of fuzziness with the entropy of deterministic functions, it is possible to expand an information theory of fuzzy sets which is fully compatible and consistent with the classical Shannonian information theoretic framework. A model of transinformation between fuzzy sets, which could be of help in approximate reasoning can be obtained, an interesting feature of which is that it can be duplicated in the framework of fuzzy set theory.

Methods of testing simple hypotheses about lifetime parameters from doubly censored data are given on the basis of the maximum likelihood principle. It is shown that, under the assumptions of standard type, the asymptotic distribution of proposed statistics is chi‐square or linear combination of chi‐square distributions. The choice of statistics optimal from the point of view of power is discussed and illustrated by several examples.

Several coefficients, called divergences, have been suggested in the statistical literature to reflect the fact that some probability distributions are “closer together” than others and consequently that it may be easier to distinguish between the distributions of one pair than between those of another. When comparing three biological populations, it is often interesting to measure how two of them “move apart” from the third. Deals with the statistical analysis of this problem by means of bivariate divergence statistics. Provides a unified study, depicting the behaviour and relative merits of traditional divergences, by using the (h,ø), divergence family of statistics introduced by Menéndez et al.

Goodness‐of‐fit test based on K_ϕ‐divergence between observed and theoretical frequencies are considered. The asymptotic chi‐square null distribution and three alternative approximations to the exact distribution function of this family are compared in small samples. Numerical results are presented for the symmetric null hypothesis for different multinomial sample sizes with various cell numbers. Exact power under specific alternatives to the symmetric null hypothesis are calculated and a comparison with the family of power divergence statistics is made.

Surveys some of the important contributions of information theory (IT) to the understanding of systems science and cybernetics. Presents a short background on the main definitions of IT, and examines in which way IT could be thought of as a unified approach to general systems. Analyses the topics: syntax and semantics in information, information and self‐organization, entropy of forms (entropy of non‐random functions), and information in dynamical systems. Enumerates some suggestions for further research and takes this opportunity to describe new points of view, mainly by using entropy of non‐random functions.

Weighted (h,φ) – divergence statistics are obtained by either replacing both distributions involved in the argument by their nonparametric estimators or replacing one distribution and considering the other as given. Asymptotic properties of weighted (h,φ) – divergence statistics are obtained and some tests constructed on the basis of these results are presented.

Proposes a test of goodness‐of‐fit with composite null hypotheses and weights in the classes based on weighted (h,φ)‐divergences.

The weighted (h,φ)‐divergence between an empirical distribution and the probability of the estimated model is here investigated for large simple random samples.

The unknown parameters of the model are estimated using minimum (h,φ)‐divergences estimators with weights as studied in previous works by the authors.

Research makes an important contribution to (h,φ)‐divergences and their applications in statistical and other areas.