Statistics

In this paper, we prove that the Renyi entropy of linearly normalized partial maxima of independent and identically distributed random variables is convergent to the corresponding limit Renyi entropy when the linearly normalized partial maxima converges to some nondegenerate random variable.

Harmonic, Geometric, Arithmetic, Heronian and Contraharmonic means have been studied by many mathematicians. In 2003, H. Evens studied these means from geometrical point of view and established some of the inequalities between them in using a circle and its radius. In 1961, E. Beckenback and R. Bellman introduced several inequalities corresponding to means. In this paper, we will introduce the concept of mean functions and integral means and give bounds on some of these mean functions and integral means.

The objective of this study is to find the underlying pattern of how perception of emotions has evolved in India. Here Hindustani Music has been used as a reference frame for tracking the changing perception of emotions. It has been found that different emotions perceived from Hindustani Music form a particular sequential pattern when their corresponding pitch periods are analyzed using the standard deviations and mean successive squared differences.This sequential pattern of emotions coincides with their corresponding sequential pattern of tempos or average number of steady states. On the basis of this result we further found that the range of perception of emotions has diminished significantly these days compared to what it was before. The proportion of responses for the perceived emotions like Anger, Serenity, Romantic and Sorrow has also decreased to a great extent than what it was previously. The proportion of responses for the perceived emotion Anxiety has increased phenomenally. Both standard deviation and mean successive squared difference are two very good measures in tracking the changing perception of emotions. The overall pattern of the change of perceived emotions has corresponded to the psychological and sociological change of human life.

Significance tests are probably the most extended form of inference in empirical research, and significance is often interpreted as providing greater informational content than non-significance. In this article we show, however, that rejection of a point null often carries very little information, while failure to reject may be highly informative. This is particularly true in empirical contexts where data sets are large and where there are rarely reasons to put substantial prior probability on a point null. Our results challenge the usual practice of conferring point null rejections a higher level of scientific significance than non-rejections. In consequence, we advocate a visible reporting and discussion of non-significant results in empirical practice.

This paper is motivated by the questions of how to give the concept of probability an adequate real-world meaning, and how to explain a certain type of phenomenon that can be found, for instance, in Ellsberg's paradox. It attempts to answer these questions by constructing an alternative theory to one that was proposed in earlier papers on the basis of various important criticisms that were raised against this earlier theory. The conceptual principles of the corresponding definition of probability are laid out and explained in detail. In particular, what is required to fully specify a probability distribution under this definition is not just the distribution function of the variable concerned, but also an assessment of the internal and/or the external strength of this function relative to other distribution functions of interest. This way of defining probability is applied to various examples and problems including, perhaps most notably, to a long-running controversy concerning the distinction between Bayesian and fiducial inference. The characteristics of this definition of probability are carefully evaluated in terms of the issues that it sets out to address.

An often-cited fact regarding mixing or mixture distributions is that their density functions are able to approximate the density function of any unknown distribution to arbitrary degrees of accuracy, provided that the mixing or mixture distribution is sufficiently complex. This fact is often not made concrete. We investigate and review theorems that provide approximation bounds for mixing distributions. Connections between the approximation bounds of mixing distributions and estimation bounds for the maximum likelihood estimator of finite mixtures of location- scale distributions are reviewed.

The analysis of adverse events (AEs) is a key component in the assessment of a drug's safety profile. Inappropriate analysis methods may result in misleading conclusions about a therapy's safety and consequently its benefit-risk ratio. The statistical analysis of AEs is complicated by the fact that the follow-up times can vary between the patients included in a clinical trial. This paper takes as its focus the analysis of AE data in the presence of varying follow-up times within the benefit assessment of therapeutic interventions. Instead of approaching this issue directly and solely from an analysis point of view, we first discuss what should be estimated in the context of safety data, leading to the concept of estimands. Although the current discussion on estimands is mainly related to efficacy evaluation, the concept is applicable to safety endpoints as well. Within the framework of estimands, we present statistical methods for analysing AEs with the focus being on the time to the occurrence of the first AE of a specific type. We give recommendations which estimators should be used for the estimands described. Furthermore, we state practical implications of the analysis of AEs in clinical trials and give an overview of examples across different indications. We also provide a review of current practices of health technology assessment (HTA) agencies with respect to the evaluation of safety data. Finally, we describe problems with meta-analyses of AE data and sketch possible solutions.

We derive a simple mathematical "theory" to show that two decision-making entities can work better together only if at least one of them is occasionally willing to stay neutral. This provides a mathematical "justification" for an age-old cliche among marriage counselors.

This paper attempts to situate statistics in relation to qualitative research methods and other means of "finding out". It compares and contrasts aspects of qualitative research methods and statistical inquiry and attempts to answer the question of whether and how elements of qualitative research methods should be included in statistics teaching.

The Benini distribution is a lognormal-like distribution generalizing the Pareto distribution. Like the Pareto and the lognormal distributions it was originally proposed for modeling economic size distributions, notably the size distribution of personal income. This paper explores a probabilistic property of the Benini distribution, showing that it is not determined by the sequence of its moments although all the moments are finite. It also provides explicit examples of distributions possessing the same set of moments. Related distributions are briefly explored.