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We first consider the sequential detection of transient signals by generalizing the moving average chart to exponential family and study the false detection probability (FDP) and power of detection (POD) in the steady state. Then windowed adjusted signed (or modified directed) likelihood ratio chart is studied by treating it as approximate normal random variable. In the multi-parameter exponential family, the detection of the transient change of one of the canonical parameters or a function of canonical parameters is considered by using the generalized adjusted signed likelihood ratio chart. Comparisons with window restricted CUSUM and Shiryayev-Roberts (S-R) procedures show that the generalized signed likelihood ratio chart performs quite well. Several important examples including the mean or variance change under normal model and a real-time example are used for illustration. [ABSTRACT FROM AUTHOR]

The process of discretization of continuous distributions creates and provides a large set of discrete probabilistic models used in various statistical applications. The most common way of doing so is by considering the probability distribution of the integral part of a continuous random variable. In this note we explore the following problem related to the latter discretization process and pose the following question: If the family of distributions that is discretized is an exponential family on the real line, when the (integral) resulting discrete probability model also generates an exponential family? We give a complete answer to this question and provide necessary and sufficient conditions under which the discretized version of an exponential family is also an exponential family. [ABSTRACT FROM AUTHOR]

We observe n pairs of independent (but not necessarily i.i.d.) random variables X 1 = (W 1 , Y 1) , ... , X n = (W n , Y n) and tackle the problem of estimating the conditional distributions Q i ⋆ (w i) of Y i given W i = w i for all i ∈ { 1 , ... , n }. Even though these might not be true, we base our estimator on the assumptions that the data are i.i.d. and the conditional distributions of Y i given W i = w i belong to a one parameter exponential family Q ¯ with parameter space given by an interval I. More precisely, we pretend that these conditional distributions take the form Q θ (w i) ∈ Q ¯ for some θ that belongs to a VC-class Θ ¯ of functions with values in I. For each i ∈ { 1 , ... , n } , we estimate Q i ⋆ (w i) by a distribution of the same form, i.e. Q θ ̂ (w i) ∈ Q ¯ , where θ ̂ = θ ̂ (X 1 , ... , X n) is a well-chosen estimator with values in Θ ¯. We establish non-asymptotic exponential inequalities for the upper deviations of a Hellinger-type distance between the true conditional distributions of the data and the estimated one based on the exponential family Q ¯ and the class of functions Θ ¯ we chose. We show that our estimation strategy is robust to model misspecification, contamination and the presence of outliers. Besides, when the data are truly i.i.d., the exponential family Q ¯ is suitably parametrized and the conditional distributions Q i ⋆ (w i) of the form Q θ ⋆ (w i) ∈ Q ¯ for some unknown Hölderian function θ ⋆ with values in I , we prove that the estimator θ ̂ of θ ⋆ is minimax (up to a logarithmic factor). Finally, we provide an algorithm for calculating θ ̂ when Θ ¯ is a VC-class of functions of low or moderate dimension and we carry out a simulation study to compare its performance to that of the MLE and median-based estimators. The proof of our main result relies on an upper bound, with explicit numerical constants, on the expectation of the supremum of an empirical process over a VC-subgraph class. This bound can be of independent interest. [ABSTRACT FROM AUTHOR]

The informational energy of Onicescu is a positive quantity that measures the amount of uncertainty of a random variable. However, contrary to Shannon's entropy, the informational energy is strictly convex and increases when randomness decreases. We report a closed-form formula for Onicescu's informational energy and its associated correlation coefficient when the probability distributions belong to an exponential family. We show how to instantiate the generic formula for several common exponential families. Finally, we discuss the characterization of valid thermodynamic process trajectories on a statistical manifold by enforcing that the entropy and the informational energy shall vary in opposite directions. [ABSTRACT FROM AUTHOR]

Every random variable (rv) X (or random vector) with finite moments generates a set of orthogonal polynomials, which can be used to obtain properties related to the distribution of X. This technique has been used in statistical inference, mainly connected to the exponential family of distributions. In this paper a review of some of its more relevant uses is provided. The first one deals with properties of expansions in terms of orthogonal polynomials for the Uniformly Minimum Variance Unbiased Estimator of a given parametric function, when sampling from a distribution in the Natural Exponential Family of distributions with Quadratic Variance Function. The second one compares two relevant methods, based on expansions in Laguerre polynomials, existing in the literature to approximate the distribution of linear combinations of independent chi‐square variables. [ABSTRACT FROM AUTHOR]

In this paper, a two-stage sequential estimation procedure is considered to construct fixed-accuracy confidence intervals of the reliability parameter R under the stress–strength model when the stress and strength are independent exponential random variables with different scale parameters. The exact distribution of the total sample size, explicit formulas for the expected value, and mean squared error of the maximum likelihood estimator of the reliability parameter under the stress–strength model are provided under the two-stage sequential procedure. Performances of the proposed methodology are investigated with the help of simulations. Finally, using three pairs of real datasets, the procedure is clearly illustrated. We used MATLAB in order to implement computations, simulation studies, and real data analysis. [ABSTRACT FROM AUTHOR]

Extending normal gamma and normal inverse Gaussian models, multivariate normal stable Tweedie (NST) models are composed by a fixed univariate stable Tweedie variable having a positive value domain, and the remaining random variables given the fixed one are real independent Gaussian variables with the same variance equal to the fixed component. Within the framework of multivariate exponential families, the NST models are recently classified by their covariance matrices V(m) depending on the mean vector m. In this paper, we prove the characterization of all the NST models through their determinants of V(m), also called generalized variance functions, which are power of only one component of m. This result is established under the NST assumptions of Monge-Ampère property and steepness. It completes the two special cases of NST, namely normal Poisson and normal gamma models. As a matter of fact, it provides explicit solutions of particular Monge-Ampère equations in differential geometry. [ABSTRACT FROM AUTHOR]

This paper proposes a new probability distribution, which belongs a member of the exponential family, defined on (0,1) unit interval. The new unit model has been defined by relation of a random variable defined on unbounded interval with respect to standard logistic function. Some basic statistical properties of newly defined distribution are derived and studied. The different estimation methods and some inferences for the model parameters have been derived. We assess the performance of the estimators of these estimation methods based on the three different simulation scenarios. The analysis of three real data examples which one is related to the coronavirus data, show better fit of proposed distribution than many known distributions on the unit interval under some comparing criteria. [ABSTRACT FROM AUTHOR]

A large number of useful distributions for data analysis are obtained by transforming different random variables. An example is the one-parameter unit-Lindley distribution, obtained by transforming a random variable which has a Lindley distribution. In this paper, we introduce a new one-parameter unit-Lindley distribution, useful for data analysis in the interval (0,1]. It follows some interesting properties such as having closed form expressions for the moments, belonging to the exponential family. We also analyze a practical application having covariates, by setting up a suitable regression and show that our model fits much better than both unit-Lindley and beta regressions. [ABSTRACT FROM AUTHOR]

Jammalamadaka and Mangalam introduced middle censoring which refers to data arising in situations, where the exact lifetime becomes unobservable if it falls within a random censoring interval. In the present article, we propose an additive risks regression model for a lifetime data subject to middle censoring, where the lifetimes are assumed to follow exponentiated exponential distribution. The regression parameters are estimated using the Expectation-Maximization algorithm. Asymptotic normality of the estimator is proposed. We report a simulation study to assess the finite sample properties of the estimator. We then analyze a real-life data on survival times of larynx cancer patients studied by Karduan. [ABSTRACT FROM AUTHOR]

We propose two new variants of chi-square-type statistics for testing the hypothesis on the shape of a distribution belonging to the one-parameter exponential family. Each variant assumes the use of unbiased estimators of the probabilities for the random variable to hit the atoms (intervals) of decomposition of its range. We establish the convergence of each statistics' distribution to the chi-square distribution with ( r − 1) degrees of freedom, where r is the number of decomposition atoms. [ABSTRACT FROM AUTHOR]

The sum of independent exponential random variables – the hypoexponential random variables – plays an important role of modeling in many domains. Khuong and Kong in (2006) were concerned in evaluating the performance of some diversity scheme, which deals with the problem of finding the probability density function of this hypoexponential random variable. They considered a particular case ofmindependent exponential random variable, whenlrandom variables have the same mean andm−lremaining random variables of different means and they found a closed expression of its probability density function. In this paper, we consider the general case of the hypoexponential random variable when the means do not have to be distinct. We find a more simple and general closed expression of its probability density function than that of Khuong and Kong. This expression is obtained using a new defined matrix called the Kad matrix, which is similar to the general Vandermonde matrix. Eventually, we present an application illustrating our work. [ABSTRACT FROM PUBLISHER]

Recently, a series of papers addressed the problem of decomposing the information of two random variables into shared information, unique information and synergistic information. Several measures were proposed, although still no consensus has been reached. Here, we compare these proposals with an older approach to define synergistic information based on the projections on exponential families containing only up to k-th order interactions. We show that these measures are not compatible with a decomposition into unique, shared and synergistic information if one requires that all terms are always non-negative (local positivity). We illustrate the difference between the two measures for multivariate Gaussians. [ABSTRACT FROM AUTHOR]

In this paper on the basis of unbiased estimates the problem of constructing asymptotic confidence interval for P( X < Y) is solved for the case where random variables X and Y have distributions, belonging to one-parameter exponential families. [ABSTRACT FROM AUTHOR]

Exponential CUSUM charts are used in monitoring the occurrence rate of rare events because the interarrival times of events for homogeneous Poisson processes are independent and identically distributed exponential random variables. In these applications, it is assumed that the exponential parameter, i.e. the mean, is known or has been accurately estimated. However, in practice, the in-control mean is typically unknown and must be estimated to construct the limits for the exponential CUSUM chart. In this article, we investigate the effect of parameter estimation on the run length properties of one-sided lower exponential CUSUM charts. In addition, analyzing conditional performance measures shows that the effect of estimation error can be significant, affecting both the in-control average run length and the quick detection of process deterioration. We also provide recommendations regarding phase I sample sizes. This sample size must be quite large for the in-control chart performance to be close to that for the known parameter case. Finally, we provide an industrial example to highlight the practical implications of estimation error, and to offer advice to practitioners when constructing/analyzing a phase I sample. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, we first propose a new estimator of entropy for continuous random variables. Our estimator is obtained by correcting the coefficients of Vasicek's [A test for normality based on sample entropy, J. R. Statist. Soc. Ser. B 38 (1976), pp. 54–59] entropy estimator. We prove the consistency of our estimator. Monte Carlo studies show that our estimator is better than the entropy estimators proposed by Vasicek, Ebrahimi et al. [Two measures of sample entropy, Stat. Probab. Lett. 20 (1994), pp. 225–234] and Correa [A new estimator of entropy, Commun. Stat. Theory Methods 24 (1995), pp. 2439–2449] in terms of root mean square error. We then derive the non-parametric distribution function corresponding to our proposed entropy estimator as a piece-wise uniform distribution. We also introduce goodness-of-fit tests for testing exponentiality and normality based on the said distribution and compare its performance with their leading competitors. [ABSTRACT FROM AUTHOR]

Abstract: An exponential inequality is established for a random variable with the finite Laplace transform. Using this inequality, we obtain an exponential inequality for identically distributed acceptable random variables (a class of random variables introduced in which includes negatively dependent random variables). Our result improves the corresponding ones in , , , , and . Our method is much simpler than those in the literature. [Copyright &y& Elsevier]

In this paper, we introduce a new class of mixture models based on the inverse Gaussian distribution, which is highly flexible and contains several well-known probability models. The new class of models is generated from symmetric distributions around zero by using the connection between the inverse Gaussian and standard normal distributions. We illustrate the obtained results by means of two real data sets through likelihood, goodness-of-fit and diagnostic methods. This illustration indicates the adequacy of the new model. [ABSTRACT FROM AUTHOR]

In this paper, a new generalized bimodal power exponential distribution is proposed. The Characterization of this distribution is investigated and special cases are considered. The problem of obtaining the maximum-likelihood estimates for the parameters of the distribution is studied and the information matrix is derived. A simulation study to assess the properties of the maximum likelihood estimators is carried out. Finally, two illustrative examples are presented. [ABSTRACT FROM AUTHOR]

Abstract: A sufficient condition for comparing convolutions of heterogeneous exponential random variables in terms of right spread order is established. As a consequence, it is shown that a convolution of heterogeneous independent exponential random variables is more skewed than that of homogeneous exponential random variables in the sense of NBUE order. This gives a new insight into the distribution theory of convolutions of independent random variables. A sufficient condition is also derived for comparing such convolutions in terms of Lorenz order. [Copyright &y& Elsevier]

The formalism of probabilistic graphical models provides a unifying framework for capturing complex dependencies among random variables, and building large-scale multivariate statistical models. Graphical models have become a focus of research in many statistical, computational and mathematical fields, including bioinformatics, communication theory, statistical physics, combinatorial optimization, signal and image processing, information retrieval and statistical machine learning. Many problems that arise in specific instances - including the key problems of computing marginals and modes of probability distributions - are best studied in the general setting. Working with exponential family representations, and exploiting the conjugate duality between the cumulant function and the entropy for exponential families, we develop general variational representations of the problems of computing likelihoods, marginal probabilities and most probable configurations. We describe how a wide variety of algorithms - among them sum-product, cluster variational methods, expectation-propagation, mean field methods, max-product and linear programming relaxation, as well as conic programming relaxations - can all be understood in terms of exact or approximate forms of these variational representations. The variational approach provides a complementary alternative to Markov chain Monte Carlo as a general source of approximation methods for inference in large-scale statistical models. [ABSTRACT FROM AUTHOR]

Boardman and Kendell (1970) considered the problem of estimation with respect to Type-I censoring when an item is subjected to only one of the two causes of failure assuming exponential model. Patel and Gajjar (1992) considered extension of the Boardman and Kendell's results in case of two-stage progressive censoring. Here we have considered geometric competing risk failure model with two independent causes of failures. Maximum likelihood estimation of the parameters is carried out using Type-I two-stage progressively censored and group censored samples. Asymptotic standard errors of the estimators are obtained for both the cases. Two illustrative examples are cited for ungroup and group competing risk models. [ABSTRACT FROM AUTHOR]

We introduce a new class of generalized linear mixed models based on the Tweedie exponential dispersion model distributions, accommodating a wide range of discrete, continuous and mixed data. Using the best linear unbiased predictor of random effects, we obtain an optimal estimating function for the regression parameters in the sense of Godambe, allowing an efficient common fitting algorithm for the whole class. Although allowing full parametric inference, our main results depend only on the first- and second-moment assumptions of unobserved random effects. In addition, we obtain consistent estimators for both regression and dispersion parameters. We illustrate the method by analysing the epilepsy data and cake baking data. Along with simulations and asymptotic justifications, this shows the usefulness of the method for analysis of clustered non-normal data. [ABSTRACT FROM AUTHOR]

In recent years, several attempts have been made to characterize the generalized Pareto distributions (GPD) based on the properties of order statistics and record values. In the present article, we give some characterization results on GPD based on order statistics and generalized order statistics. Some characterizations of uniform distribution based on expectation of some functions of order statistics are also given. [ABSTRACT FROM AUTHOR]

We consider a random walk { S n } with negative drift and heavy-tailed jumps. We study the asymptotic behavior at infinity of the distribution density of the supremum supn S n . [ABSTRACT FROM AUTHOR]

Let X 1 , X 2 ,..., X n be independent exponential random variables such that X i has hazard rate λ for i = 1,..., p and X j has hazard rate λ* for j = p + 1,..., n , where 1 ≤ p [ABSTRACT FROM AUTHOR]

Let Y[sub 1] , . . ., Y[sub n]denote independent random variables such that Y[sub j]has a one-parameter exponential family distribution with canonical parameter θ [sub j]=λ+&psgr; X[sub j]; here X[sub 1] , . . ., X[sub n]are known constants. Consider a test of the null hypothesis &psgr;=0. Under the null hypothesis, A=Σ Y[sub j]is sufficient for λ and, hence, a test of &psgr;=0 may be based on the conditional distribution of T=Σ X[sub j] Y[sub j]given A, which is independent of λ. In this paper, the effects of overdispersion due to a mixture model on the conditional distribution of Tgiven Aare considered. [ABSTRACT FROM AUTHOR]

In this paper a modification of the estimative prediction limit is derived, for which the coverage probability equals the target value to third-order accuracy, under ordinary repeated sampling. It provides an explicit approximate solution to the equation given by Barndorff-Nielsen & Cox (1996), which determines an upper α-prediction limit for an unobserved random variable. The corresponding predictive density is specified in a simple closed form and its main properties are emphasised. [ABSTRACT FROM AUTHOR]

• Sparsity in the PC expansion is promoted by the presence of multiplicative binary coefficients on the basis terms. • The chaos and binary coefficients are jointly estimated using Bayesian inference on a Hierarchical Bayesian model known as Relevance Vector Machine. • The marginal posteriors are estimated using Variational Inference. • The presence of exponential family priors enables the optimization of the KL divergence using a gradient ascent scheme. • Significant sparsity is achieved on a high dimensional deformation problem of a steel plate with random material property. The challenges for non-intrusive methods for Polynomial Chaos modeling lie in the computational efficiency and accuracy under a limited number of model simulations. These challenges can be addressed by enforcing sparsity in the series representation through retaining only the most important basis terms. In this work, we present a novel sparse Bayesian learning technique for obtaining sparse Polynomial Chaos expansions which is based on a Relevance Vector Machine model and is trained using Variational Inference. The methodology shows great potential in high-dimensional data-driven settings using relatively few data points and achieves user-controlled sparse levels that are comparable to other methods such as compressive sensing. The proposed approach is illustrated on two numerical examples, a synthetic response function that is explored for validation purposes and a low-carbon steel plate with random Young's modulus and random loading, which is modeled by stochastic finite element with 38 input random variables. [ABSTRACT FROM AUTHOR]

Let {\bf X} be a random vector in R^{n} with probability density f({\bf x}) = c_{n} \exp \{-({\bf x}\prime {\bf x})^{n/2}\} . We show that the random variable U = f({\bf X}) has uniform distribution on (0, c_{n}) . Our result generalizes previously derived special cases n = 1 (the Laplace distribution) and n = 2 (the uncorrelated bivariate normal distribution). [ABSTRACT FROM AUTHOR]