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In this paper on the basis of the f*‐Divergence, a comparison criterion between fuzzy Information Systems is presented. This criterion is called “f*‐Divergence Criterion”.

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.

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.

The decision rule which minimizes the probability of error, in the discrimination problem, is the Bayes decision rule which assigns x to the class with the highest a posteriori probability. This rule leads to a partial probability of error which is given by Pe(x) = 1−max p(C2lx) for each x e X. Prior to observing X, the probability of error associated with X is defined as Pe = EX [Pe(x)]. Tanaka, Okuda and Asai formulated the discrimination problem with fuzzy classes and fuzzy information using the probability of fuzzy events and derived a bound for the average error probability, when the decision in the classifier is made according to the fuzzified Bayes method. The aim is to obtain bounds for the average error probability in terms of (αβ)‐information energy, when the decision in the classifier is made according to the fuzzified Bayes method.

Presents (h, ø)‐entropies as a generalization of ø‐entropies. Studies some applications of this function in Bayesian inference, especially in the comparison of experiments. Also studies the relationship of the (h,ø)‐entropy criterion to the classical approaches of Blackwell (1951) and Lehmann (1959).

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.

Concerning trends in graduate education, such as high attrition and underdeveloped skills, drive toward a new doctoral education approach. This paper aims to describe and propose a transformative doctoral education model (TDEM), incorporating elements that potentially address these challenges and expand the current practice. The model envisions discipline-specific knowledge coupled with a broader interdisciplinary perspective and addresses the transferable skills necessary to successfully navigate an ever-changing workforce and global landscape. The overarching goal of TDEM is to transform the doctoral student into a multi-dimensional and adaptive scholar, so the students of today can effectively and meaningfully solve the problems of tomorrow.

The foundation of TDEM is transformative learning theory, supporting the notion learner transformation occurs throughout the doctoral educational experience.

Current global doctoral education models and literature were reviewed. These findings informed the new TDEM.

Designed as a customizable framework for learner-centered doctoral education, TDEM promotes a mentor network on and off-campus, interdisciplinarity and agile career scope preparedness.

Within the TDEM framework, doctoral students develop valuable knowledge and transferable skills. These developments increase doctoral student career adaptability and preparedness, as well as enables graduates to appropriately respond to global and societal complex problems.

This proposed doctoral education framework was formulated through a review of the literature and experiences with curricular design and pedagogical practices at a research-intensive university’s teaching and learning center. TDEM answers the call to develop frameworks that address issues in doctoral education and present a flexible and more personalized training. TDEM encourages doctoral student transformation into adaptive, forward-thinking scholars and thriving in an ever-changing workforce.

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.

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.

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.