The aim of this note is to give an elegant proof of a result due to E. G. Olds which concerns the density distribution of the sum of independent uniform random variables non-identically distributed. The proof uses both analytical and combinatorial properties of Dirac distributions and their convolutions. The method is new and can apply to other situations.

The present article represents a review of the geometric generated family of distributions. Based on this family of distribution, several distributions are proposed. The family can be proposed by using the compounding concept of zero truncated geometric distribution with any other model or family of distributions. Here, we provide a complete survey on this family of distributions and also listed the contributory related research work, their sub-models, hazard rates, and utilized real datasets. We also address 10 power series distributions, 60 distributions based on the geometric family of distribution. These numbers show the importance of the geometric family of distribution.

Bayesian statistical methodology has become highly popular in a myriad of applications over the past several decades. In Bayesian statistics, it is often required to draw samples from intractable probability distributions. Markov chain Monte Carlo (MCMC) algorithms are common methods of obtaining samples from these distributions. When an MCMC algorithm is used, it is important to be able to obtain an answer to the question of how many iterations the chain must run before it is “close enough” to its target distribution to allow approximate sampling from this distribution. Several methods of approaching this question exist in the literature. Some rely on the output of the chain, and some are based on Markov chain theory. These techniques suffer from major practical limitations. This work provides a computational method of bounding the mixing time of a Metropolis–Hastings algorithm. This approach extends the work of Spade (Statistics and Computing 26 (2016) 761–781) and Spade (Markov Processes and Related Fields 26 (2020) 487–516) to general versions of the Metropolis–Hastings algorithm, while examining the convergence behavior of such samplers under symmetric and asymmetric proposal densities.

Nair, Sankaran and John (Metron 76 (2018) 133–153) have defined and studied the properties of reliability functions in terms of copulas. In the present paper, we investigate the utility of such functions in inferring the time-dependent association of bivariate distributions. We consider the Clayton measure of association for the study. A general expression for this measure in terms of the generator of Archimedean copulas is given, and a method of finding nature of association using the generators is provided. We derive the relationship of the association measure with the ageing property of the distribution, associated with the generator. We analyze how the hazard rate of survival copulas can be utilized in studying the association between two random variables. Applications of the results in real life situations are discussed.

We analyze the fluctuations of incomplete U-statistics over a triangular array of independent random variables. We give criteria for a Central Limit Theorem (CLT, for short) to hold in the sense that we prove that an appropriately scaled and centered version of the U-statistic converges to a normal random variable. Our method of proof relies on a martingale CLT. An application, a CLT for the hitting time for random walks on random graphs, will be presented in Löwe and Terveer (2020).

For modelling unbounded count data, Poisson distribution is a natural choice. However, count data arising in various fields of scientific research are often under-reported. In such situations, inference carried out on the basis of Poisson model will result in biased parameter estimates and suboptimal tests. A modified Poisson model is developed to accommodate the possible undercount. For model-identifiability a double sampling scheme of data collection has been adopted. The focus of this paper is to develop asymptotically optimal tests for the Poisson mean in presence of undercount. Simulation study is conducted to compare the performance of the tests with respect to level and power and also to investigate the impact of ignoring undercount on each of the tests. The findings are validated using real life data.

Assume we observe a finite number of inspection times together with information on whether a specific event has occurred before each of these times. Suppose replicated measurements are available on multiple event times. The set of inspection times, including the number of inspections, may be different for each event. This is known as mixed case interval censored data. We consider Bayesian estimation of the distribution function of the event time while assuming it is concave. We provide sufficient conditions on the prior such that the resulting procedure is consistent from the Bayesian point of view. We also provide computational methods for drawing from the posterior and illustrate the performance of the Bayesian method in both a simulation study and two real datasets.

In this paper, we consider a model for data analysis with measurement errors. The main objective of this work is to develop statistical inference tools, such as parameter estimation and hypothesis tests in a linear functional relationship with replicated observations. For this purpose, we use the maximum likelihood method in the presence of incidental parameters, and the unbiased estimating equations approach. Both approaches lead to explicit expressions for the asymptotic covariance matrices of the estimators of the model parameters. A simulation study is performed to assess the empirical behavior of estimators and of a Wald statistic. The methodology is illustrated with a real data set.

We compute higher derivatives of the Fréchet function on spheres with an absolutely continuous and rotationally symmetric probability distribution. Consequences include (i) a practical condition to test if the mode of the symmetric distribution is a local Fréchet mean; (ii) a central limit theorem on spheres with practical assumptions and an explicit limiting distribution; and (iii) an answer to the question of whether the smeary effect can occur on spheres with absolutely continuous and rotationally symmetric distributions: with the method presented here, it can in dimension at least 4.

We consider a version of continuum long-range percolation on finite boxes of ${\mathbb{R}}^{\mathit{d}}$ in which the vertex set is given by the points of a Poisson point process and each pair of two vertices at distance r is connected with probability proportional to ${\mathit{r}}^{-\mathit{s}}$ for a certain constant s. We explore the graph-theoretical distance in this model. The aim of this paper is to show that this random graph model undergoes phase transitions at values $\mathit{s}=\mathit{d}$ and $\mathit{s}=2\mathit{d}$ in analogy to classical long-range percolation on ${\mathbb{Z}}^{\mathit{d}}$, by using techniques which are based on an analysis of the underlying Poisson point process.

This is a study of the behavior under partition of the sample space of three multivariate distributions: multinomial, multinomial-Dirichlet, and Dirichlet. A general theorem is given, of which all three are special cases.

The class of generalized hyperbolic (GH) distributions is generated by a mean-variance mixture of a multivariate Gaussian with a generalized inverse Gaussian (GIG) distribution. This rich family of GH distributions includes some well-known heavy-tailed and symmetric multivariate distributions, including the Normal Inverse Gaussian and some members of the family of scale-mixture of skew-normal distributions. The class of GH distributions has received considerable attention in finance and signal processing applications. In this paper, we propose the likelihood ratio (LR) test to test hypotheses about the skewness parameter of a GH distribution. Due to the complexity of the likelihood function, the EM algorithm is used to find the maximum likelihood estimates both in the complete model and the reduced model. For comparative purposes and due to its simplicity, we also consider the Gradient (G) test. A simulation study shows that the LR and G tests are usually able to achieve the desired significance levels and the testing power increases as the asymmetry increases. The methodology developed in the paper is applied to two real datasets.