These are some notes on a very simple comparative introduction to four basic approaches of statistical inference—Fisher, Neyman–Pearson, Fisher/Neyman–Pearson hybrid, and Bayes—from a course on Quantitative & Statistical Reasoning at OU in Fall 2016. In particular, I hope to give a rough understanding of the differences between the frequentist and Bayesian paradigms, though they are not entirel...

Tolerance intervals are widely used in industrial applications. So far attention has been mainly focused on the construction of tolerance intervals for continuous distributions. In this paper we introduce a unified analytical approach to the construction of tolerance intervals for discrete distributions in exponential families with quadratic variance functions. These tolerance intervals are sho...

A useful discrete distribution (the Conway–Maxwell–Poisson distribution) is revived and its statistical and probabilistic properties are introduced and explored. This distribution is a two-parameter extension of the Poisson distribution that generalizes some well-known discrete distributions (Poisson, Bernoulli and geometric). It also leads to the generalization of distributions derived from th...

Measuring the dependence between random variables is one of the most fundamental problems in statistics, and therefore, determining the joint distribution of the relevant variables is crucial. Copulas have recently become an important tool for properly inferring the joint distribution of the variables of interest. Although many studies have addressed the case of continuous variables, few studie...

Testing of hypotheses for discrete distributions is considered in this paper. The goal is to develop conditional frequentist tests that allow the reporting of datadependent error probabilities such that the error probabilities have a strict frequentist interpretation and also reflect the actual amount of evidence in the observed data. The resulting randomized tests are also seen to be Bayesian ...

Complete and joint mixability has raised considerable interest in recent few years, in both the theory of distributions with given margins, and applications in discrete optimization and quantitative risk management. We list various open questions in the theory of complete and joint mixability, which are mathematically concrete, and yet accessible to a broad range of researchers without specific...

How can we generate realistic graphs? In addition, how can we do so with a mathematically tractable model that makes it feasible to analyze their properties rigorously? Real graphs obey a long list of surprising properties: Heavy tails for the inand out-degree distribution; heavy tails for the eigenvalues and eigenvectors; small diameters; and the recently discovered “Densification Power Law” (...