The Lindley distribution has been generalized by many authors in recent years. Here, we introduce a new generalization that provides better fits than the Lindley distribution and all of its known generalizations. The distribution contains Lindley and weighted Lindley (Ghitany et al. (Math. Comput. Simulation 81 (2011) 1190–1201)) distributions as special cases. Also, the distribution can be represented as a mixture of weighted exponential (Gupta and Kundu (Statistics 43 (2009) 621–634)) and weighted gamma distributions, and as a negative mixture of Lindley distributions with different parameters. Various properties of the distribution (including quantiles, moments, moment generating function, hazard rate function, mean residual lifetime, Lorenz curve, Gini index, Rényi entropy and Mathai–Haubold entropy) are derived. Maximum likelihood estimators of the distribution parameters are derived and their behavior is assessed via simulation. Fisher’s information matrix and asymptotic confidence intervals for the distribution parameters are given. Finally, a real data application is presented.

In the independence setup, when the responses exhibit high degree of asymmetry, the median regression model is preferred to the mean regression model to obtain consistent and efficient regression estimates. However, when this type of asymmetric data are collected repeatedly over time, developing median regression model for such correlated asymmetric data may not be easy. As a remedy, there exist some studies where the longitudinal correlations of this type of asymmetric data have been computed using the moment estimates for all pairwise correlations and these correlations of repeated (multi-dimensional) data used to develop a median based quasi-likelihood approach for estimation of the regression effects. By considering an autoregressive order 1 (AR(1)) model for longitudinal exponential responses, in this paper, it is however, demonstrated that the existing pairwise estimates of correlations under median regression model may yield inefficient estimates as compared to the simpler independence assumption based estimates. We illustrate the inference techniques discussed in the paper by reanalyzing the well-known labor pain data.

The estimation of the size of a population is, in general, performed using capture–recapture experiments. In this paper, we consider a closed population capture–recapture model in which individuals are captured independently and with the same probability in each sampling occasion, but the probabilities may vary from occasion to occasion. The unknown number of individuals is the parameter of interest, while the capture probabilities are the nuisance ones. Four likelihood functions free of nuisance parameters, namely the profile, conditional, uniform and Jeffrey’s integrated likelihood functions are derived and procedures for point and interval estimation are discussed. The estimation of population size is illustrated on a real dataset. The frequentist properties of the estimators are evaluated by means of a simulation study. The Jeffrey’s integrated likelihood achieved the best performance over all considered estimators for both point and interval estimation, particularly in situations with little information with small number of elements, small capture probabilities and small number of capture occasions.

Our purpose is to recommend a change in the paradigm of testing by generalizing a very natural idea, originated perhaps in Jeffreys [Proceedings of the Cambridge Philosophy Society 31 (1935) 203–222; The Theory of Probability (1961) Oxford Univ. Press] and clearly exposed by DeGroot [Probability and Statistics (1975) Addison-Wesley], with the aim of developing an approach that is attractive to all schools of statistics, resulting in a procedure better suited to the needs of science. The essential idea is to base testing statistical hypotheses on minimizing a weighted sum of type I and type II error probabilities instead of the prevailing paradigm, which is fixing type I error probability and minimizing type II error probability. For simple vs simple hypotheses, the optimal criterion is to reject the null using the likelihood ratio as the evidence (ordering) statistic, with a fixed threshold value instead of a fixed tail probability. By defining expected type I and type II error probabilities, we generalize the weighting approach and find that the optimal region is defined by the evidence ratio, that is, a ratio of averaged likelihoods (with respect to a prior measure) and a fixed threshold. This approach yields an optimal theory in complete generality, which the classical theory of testing does not. This can be seen as a Bayesian/non-Bayesian compromise: using a weighted sum of type I and type II error probabilities is Frequentist, but basing the test criterion on a ratio of marginalized likelihoods is Bayesian. We give arguments to push the theory still further, so that the weighting measures (priors) of the likelihoods do not have to be proper and highly informative, but just “well calibrated.” That is, priors that give rise to the same evidence (marginal likelihoods) using minimal (smallest) training samples.

The theory that emerges, similar to the theories based on objective Bayesian approaches, is a powerful response to criticisms of the prevailing approach of hypothesis testing. For criticisms see, for example, Ioannidis [PLoS Medicine 2 (2005) e124] and Siegfried [Science News 177 (2010) 26–29], among many others.

In many practical situations, it is desired to compare several populations, find the best one and estimate some parametric functions associated with the selected population. This has been recognized as an important problem that arises in various applications in agricultural, industrial and medical studies. This paper concerns unbiased estimation of a general parametric function, say $\gamma(\theta)$, of selected populations under the squared error loss (SEL) function. Examples of $\gamma(\cdot)$ include reliability function, odds ratio and variance, among others. Also, we obtain the uniformly minimum risk unbiased estimators of the parameters of selected populations under some general class of loss functions other than the commonly used SEL function. Furthermore, we characterize some loss functions for which the risk unbiased estimators of parameters of selected populations do not exist. Theoretical results are augmented with various illustrations and examples.

In this paper, we first introduce a new thinning operator and derive some of its properties. Then, by using the thinning operator we define a new stationary time series with discrete Laplace marginal distributions with either positive or negative lag-one autocorrelation. We show that this time series is distributed as the difference of two independent $\operatorname{NGINAR} (1)$ time series and, using this fact, we discuss some of its properties. The Yule–Walker estimators for the unknown parameters are derived and their asymptotic properties are discussed.

Given is the outcome $s$ of $S\sim{\mathrm{B}}(n,p)$ ($n$ known, $p$ fully unknown) and two numbers $0<a\leq b<1$. Required are probabilities $\alpha_{<}(s)$, $\alpha_{a,b}(s)$, and $\alpha_{>}(s)$ of the hypotheses $\mathrm{H}_{<}$: $p<a$, $\mathrm{H}_{a,b}$: $a\leq p\leq b$, and $\mathrm{H}_{>}$: $p>b$, such that their sum is equal to 1. The degenerate case $a=b(=c)$ is of special interest. A method, optimal with respect to a class of functions, is derived under Neyman–Pearsonian restrictions, and applied to a case from medicine.

This paper describes a random walk on the $p$-adic integers, which turns out to be recurrent.