Inferences on the difference between future observations for comparing two treatments

The comparison of two treatments with normally distributed data is considered. Inferences are considered based upon the difference between single potential future observations from each of the two treatments, which provides a useful and easily interpretable assessment of the difference between the two treatments. These methodologies combine information from a standard confidence interval analysis of the difference between the two treatment means, with information available from standard prediction intervals of future observations. Win-probabilities, which are the probabilities that a future observation from one treatment will be superior to a future observation from the other treatment, are a special case of these methodologies. The theoretical derivation of these methodologies is based upon inferences about the non-centrality parameter of a non-central t-distribution. Equal and unequal variance situations are addressed, and extensions to groups of future observations from the two treatments are also considered. Some examples and discussions of the methodologies are presented.     

Prediction bands for the EDF of a future sample

We develop both nonparametric and parametric methods for obtaining prediction bands for the empirical distribution function (EDF) of a future sample. These methods yield simultaneous prediction intervals for all order statistics of the future sample, and they also correspond to tests for the two-sample problem. The nonparametric prediction bands correspond to the two-sample Kolmogorov-Smirnov test and related nonparametric tests, but the parametric prediction bands correspond to entirely new parametric two-sample tests. The parametric prediction bands tend to outperform the nonparametric bands when the parametric assumptions hold, but they may have true coverage probabilities well below their nominal levels when the parametric assumptions fail. A new computational algorithm is used to obtain critical values in the nonparametric case.     

On families of beta- and generalized gamma-generated distributions and associated inference

A general family of univariate distributions generated by beta random variables, proposed by Jones, has been discussed recently in the literature. This family of distributions possesses great flexibility while fitting symmetric as well as skewed models with varying tail weights. In a similar vein, we define here a family of univariate distributions generated by Stacy’s generalized gamma variables. For these two families of univariate distributions, we discuss maximum entropy characterizations under suitable constraints. Based on these characterizations, an expected ratio of quantile densities is proposed for the discrimination of members of these two broad families of distributions. Several special cases of these results are then highlighted. An alternative to the usual method of moments is also proposed for the estimation of the parameters, and the form of these estimators is particularly amenable to these two families of distributions.     

Visualizing Mean,Median, Mean Deviation,and Standard Deviation of a Set of Numbers

We review the existing visualizations of the mean and the median of a given set of numbers. Then we give an alternative visualization of the mean using the empirical cumulative distribution function of the given numbers. Next, we visualize the mean deviation (MD) and the mean square deviation (MSD) of the given numbers from any arbitrary value, including the variance. In light of these new visualizations, we revisit the well-known optimal properties of the MD from the median and the MSD from the mean. We also give a more elementary explanation of why the denominator of the sample variance of a set of numbers is one less than the sample size.     

Interval Estimations for the Mean of the Generalized Poisson and Generalized Negative Binomial Distributions

In this article, we extend the Wald, score, skewness-corrected score, likelihood ratio, and mid-P intervals for the means of the generalized Poisson and generalized negative binomial distributions. These distributions are the members of the discrete version of the natural exponential family (NEF) with cubic variance function (CVF). Also, the coverage probabilities, the distal and mesial noncoverage probabilities, and the lengths of the proposed confidence intervals are estimated by means of a Monte Carlo simulation study. Finally, some practical examples are provided to show the applicability of the proposed intervals in applied studies.     

A new flexible hazard rate distribution: Application and estimation of its parameters

We introduce a new class of flexible hazard rate distributions which have constant, increasing, decreasing, and bathtub-shaped hazard function. This class of distributions obtained by compounding the power and exponential hazard rate functions, which is called the power-exponential hazard rate distribution and contains several important lifetime distributions. We obtain some distributional properties of the new family of distributions. The estimation of parameters is obtained by using the maximum likelihood and the Bayesian methods under squared error, linear-exponential, and Stein’s loss functions. Also, approximate confidence intervals and HPD credible intervals of parameters are presented. An application to real dataset is provided to show that the new hazard rate distribution has a better fit than the other existing hazard rate distributions and some four-parameter distributions. Finally , to compare the performance of proposed estimators and confidence intervals, an extensive Monte Carlo simulation study is conducted.