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Abstract: This paper deals with extreme-value index estimation of a heavy-tailed distribution of a spatial dependent process. We are particularly interested in spatial rare events of a \(\beta\) -mixing process. Given a stationary real-valued multidimensional spatial process \(\left\{X_{\mathbf{i}},\mathbf{i}\in{\mathbb{Z}}^{N}\right\}\) , we investigate its heavy-tail index estimation. Asymptotic properties of the corresponding estimator are established under mild mixing conditions. The particularity of the tail proposed estimator is based on the spatial nature of the sample and its unbiased and reduced variance properties compared to well known tail index estimators. Extreme quantile estimation is also deduced. A numerical study on synthetic and real datasets is conducted to assess the finite-sample behaviour of the proposed estimators. PubDate: 2022-12-01
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Abstract: The covariance matrix of random variables \(X_{1},\dots,X_{n}\) is said to have an intraclass covariance structure if the variances of all the \(X_{i}\) ’s are the same and all the pairwise covariances of the \(X_{i}\) ’s are the same. We provide a possibly surprising characterization of such covariance matrices in the case when the \(X_{i}\) ’s are symmetric Bernoulli random variables. PubDate: 2022-12-01
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Abstract: Assessing dependence within co-movements of financial instruments has been of much interest in risk management. Typically, indices of tail dependence are used to quantify the strength of such dependence, although many of them underestimate the strength. Hence, we advocate the use of indices of maximal tail dependence, and for this reason we also develop a statistical procedure for estimating the indices. We illustrate the procedure using simulated and real data sets. PubDate: 2022-12-01
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Abstract: In the present work, we study the information generating (IG) function of record values and examine some main properties of it. We establish some comparison results associated with the IG measure of record values. We show that under equality of two given IG measures of upper record values, the corresponding parent distributions can be determined uniquely. We also present some bounds for the IG measure of upper record values based on upper records of a standard exponential distribution. Further, we provide some results associated with characterization of exponential distribution by maximization (minimization) of IG function of record values under some conditions. We also examine the relative information generating (RIG) measure between the distribution of records values and the corresponding underlying distribution and present some results in this regard. To illustrate the results, several examples have been presented through the paper. PubDate: 2022-09-01 DOI: 10.3103/S1066530722030036
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Abstract: In this paper, we study the asymptotic properties of the Maximum Likelihood Estimator (MLE) for a Zero-Inflated Bell regression model. Under some regularity conditions, we establish that the estimator is consistent and asymptotically normal. This lends a substantial support to the empirical findings that have already been obtained by some authors. Monte Carlo simulations are conducted to numerically illustrate the main results. The model is applied to a dataset of healthcare demand in USA. PubDate: 2022-09-01 DOI: 10.3103/S1066530722030012
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Abstract: In this work, we first establish exponential inequalities for the Robbins–Monro’s algorithm under \(\psi\) -mixing random errors. Then, we present a numerical application that uses the main result of this work to approximate the theoretical solution of the objective function. PubDate: 2022-09-01 DOI: 10.3103/S1066530722030024
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Abstract: In this paper, we define a multiple random good of order \(2\) denoted by \(X_{12}\) whose possible values are of a monetary nature. A two-risky asset portfolio is a multiple random good of order \(2\) . It is firstly possible to establish its expected return by using a linear and quadratic metric. We secondly establish the expected return on \(X_{12}\) denoted by \(\mathbf{P}(X_{12})\) by using a multilinear and quadratic metric. An extension of the notion of mathematical expectation of \(X_{12}\) is carried out by using the notion of \(\alpha\) -norm of an antisymmetric tensor of order \(2\) . An extension of the notion of variance of \(X_{12}\) denoted by \(\textrm{Var}(X_{12})\) is shown by using the notion of \(\alpha\) -norm of an antisymmetric tensor of order \(2\) based on changes of origin. An extension of the notion of expected utility connected with \(X_{12}\) is considered. An extension of Jensen’s inequality is shown as well. We focus on how the decision-maker maximizes the expected utility connected with multiple random goods of order \(2\) being chosen by her under conditions of uncertainty and riskiness. PubDate: 2022-06-01 DOI: 10.3103/S1066530722020028
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Abstract: We consider the order statistics based on independent identically distributed non-negative random variables. We determine sharp upper bounds on the expectations of arbitrary linear combinations of order statistics, expressed in the scale units being the \(p\) th roots of \(p\) th raw moments of original variables for various \(p\geq 1\) . The bounds are more precisely described for the single order statistics and spacings. The lower bounds are concluded from the upper ones. PubDate: 2022-06-01 DOI: 10.3103/S1066530722020041
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Abstract: In a variety of applicative fields the level of information in random quantities is commonly measured by means of the Shannon Entropy. In particular, in reliability theory and survival analysis, time-dependent generalizations of this measure of uncertainty have been considered to dynamically describe changes in the degree of information over time. The Residual Entropy and the Residual Varentropy, for example, have been considered in the specialized literature to measure the information and its variability in residual lifetimes. In a similar way, one can consider dynamic measures of information for past lifetimes, i.e., for random lifetimes of items when one assumes that their failures occur before a fixed inspection time. This paper provides a study of the Past Varentropy, defined as the dynamic measure of variability of information for past lifetimes. From this study emerges the interest on a particular family of lifetimes distributions, whose members satisfy the property to be the only ones having constant Past Varentropy. PubDate: 2022-06-01 DOI: 10.3103/S106653072202003X
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Abstract: The non-central Wishart and inverted Wishart distributions are studied in this work under elliptical models; some distributional results are based on some generalizations of the well-known Kummer relations, which leds us to determine that some moments have a polynomial representation. Then the non-central \(F\) and ‘‘studentized Wishart’’ distributions are derived in a general setting. After some generalizations, including the so called non-central generalized inverted Wishart distribution, the classical results based on Gaussian models are derived here as corollaries. PubDate: 2022-03-01 DOI: 10.3103/S1066530722010021
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Abstract: Mitscherlich’s function is a well-known three-parameter non-linear regression function that quantifies the relation between a stimulus or a time variable and a response. It has many applications, in particular in the field of measurement reliability. Optimal designs for estimation of this function have been constructed only for normally distributed responses with homoscedastic variances. In this paper we generalize this literature to D-optimal designs for discrete and continuous responses having their distribution function in the exponential family. We also demonstrate that our D-optimal designs can be identical to and different from optimal designs for variance weighted linear regression. PubDate: 2022-03-01 DOI: 10.3103/S1066530722010033
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Abstract: We consider the order statistics \(X_{1:n},\ldots,X_{n:n}\) based on independent identically symmetrically distributed random variables. We determine sharp upper bounds in the properly centered linear combinations of order statistics \(\sum_{i=1}^{n}c_{i}(X_{i:n}-\mu)\) , where \((c_{1},\ldots,c_{n})\) is an arbitrary vector of coefficients from the \(n\) -dimensional real space, and \(\mu\) is the symmetry center of the parent distribution, in various scale units. The scale units are constructed on the basis of absolute central moments of the parent distribution of various orders. The bounds are specified for single order statistics. The lower bounds are immediately concluded from the upper ones. PubDate: 2021-07-01 DOI: 10.3103/S1066530721030030
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Abstract: A novel Bayesian nonparametric test for assessing multivariate normal models is presented. Although there are extensive frequentist and graphical methods for testing multivariate normality, it is challenging to find Bayesian counterparts. The approach considered in this paper is based on the Dirichlet process and the squared radii of observations. Specifically, the squared radii are employed to transform the \(m\) -variate problem into a univariate problem by relying on the fact that if a random sample is coming from a multivariate normal distribution then the square radii follow a particular beta distribution. While the Dirichlet process is used as a prior on the distribution of the square radii, the concentration of the distribution of the Anderson–Darling distance between the posterior process and the beta distribution is compared to that between the prior process and beta distribution via a relative belief ratio. Key results of the approach are derived. The procedure is illustrated through several examples, in which it shows excellent performance. PubDate: 2021-07-01 DOI: 10.3103/S1066530721030029
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Abstract: We propose inferential results for a new integrated inequality curve, related to a new index of inequality and specifically designed for capturing significant shifts in the lower and upper tails of income distributions. In the last decades, indeed, substantial changes mainly occurred in the opposite sides of income distributions, raising serious concern to policy makers. These phenomena has been observed in countries like US, Germany, UK, and France. Properties of the index and curve have been investigated, and applications to real data disclosed a new way to look at inequality. First inferential results for the index have been published, as well. It seems natural, now, to be interested also in inferential results for the integrated curve. To fill this gap in the literature, we introduce two empirical estimators for the integrated curve, and show their asymptotical equivalence. Afterwards, we state their consistency. Finally, we prove the weak convergence in the space \(C[0,1]\) of the corresponding empirical process to a Gaussian process, which is a linear transformation of a Brownian bridge. An analysis of real data from the Bank of Italy Survey of Income and Wealth is also presented, on the base of the obtained inferential results. PubDate: 2021-01-01 DOI: 10.3103/S1066530721010026
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Abstract: In this paper, we revisit the concentration inequalities for the supremum of the cumulative distribution function (CDF) of a real-valued continuous distribution as established by Dvoretzky, Kiefer, Wolfowitz and revisited later by Massart in in two seminal papers. We focus on the concentration of the local supremum over a sub-interval, rather than on the full domain. That is, denoting \(U\) the CDF of the uniform distribution over \([0,1]\) and \(U_{n}\) its empirical version built from \(n\) samples, we study \(\mathbb{P}\Big{(}\sup_{u\in[\underline{u},\overline{u}]}U_{n}(u)-U(u)>\varepsilon\Big{)}\) for different values of \(\underline{u},\overline{u}\in[0,1]\) . Such local controls naturally appear for instance when studying estimation error of spectral risk-measures (such as the conditional value at risk), where \([\underline{u},\overline{u}]\) is typically \([0,\alpha]\) or \([1-\alpha,1]\) for a risk level \(\alpha\) , after reshaping the CDF \(F\) of the considered distribution into \(U\) by the general inverse transform \(F^{-1}\) . Extending a proof technique from Smirnov, we provide exact expressions of the local quantities \(\mathbb{P}\Big{(}\sup_{u\in[\underline{u},\overline{u}]}U_{n}(u)-U(u)>\varepsilon\Big{)}\) and \(\mathbb{P}\Big{(}\sup_{u\in[\underline{u},\overline{u}]}U(u)-U_{n}(u)>\varepsilon\Big{)}\) for each \(n,\varepsilon,\underline{u},\overline{u}\) . Interestingly these quantities, seen as a function of \(\varepsilon\) , can be easily inverted numerically into functions of the probability level \(\delta\) . Although not explicit, they can be computed and tabulated. We plot such expressions and compare them to the classical bound \(\sqrt{\frac{\ln(1/\delta)}{2n}}\) provided by Massart inequality. We then provide an application of such result to the control of generic functional of the CDF, motivated by the case of the conditional value at risk. Last, we extend the local concentration results holding individually for each \(n\) to time-uniform concentration inequalities holding simultaneously for all \(n\) , revisiting a reflection inequality by James, which is of independent interest for the study of sequential decision making strategies. PubDate: 2021-01-01 DOI: 10.3103/S1066530721010038
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Abstract: We consider here the use of the solution for the first multiplicity of types equation to compute exact probability distributions of statistical values and their exact approximations. We consider \({\Delta}\) -exact distributions as their exact approximations; \({\Delta}\) -exact distributions differ from exact distributions by no more than a predetermined, arbitrarily small value \({\Delta}\) . It is shown that the basis for the exact distribution computing method is an enumeration of search area elements for solution of a linear first multiplicity of type equation composed of multiplicity type vectors. Each element represents here the number of occurrences for elements of a certain type (any sign of an alphabet) in the considered sample. It is shown simultaneously, that the method for restricting the search area for solution of the first multiplicity of type equation is applied for calculating exact approximation. We give an expression defining the algorithmic complexity of exact distributions calculated using the first multiplicity solution method which is finite and allows for each value of alphabet power to determine the maximum sample size for which exact distributions can be calculated by the first multiplicity solution method using limited computing power. To estimate the algorithmic complexity of computing the exact approximations, we used the expression obtained for the first time for the number of first multiplicity equation’s solutions with limitation on the values of coordinates of solution vectors. An expression determining algorithmic complexity for computing the exact approximations using the solution method for the first multiplicity equation with the constraint on the values of solution vector coordinates was obtained. The statistic value of maximal frequency is used as a parameter for restricting the solution vector coordinates, the probability of its excess is less than a pre-determined, arbitrarily small value \({\Delta}\) . This permits to calculate the exact approximations of the distributions differing from their exact distribution values by no more than a chosen value \({\Delta}\) . Results for calculating the maximum sample sizes for which exact approximations can be computed are given. It is shown that the algorithmic complexity of computing exact distributions by many orders of magnitude exceeds the complexity of computing their exact approximations. It is shown that application of the first multiplicity method for computing exact approximations allows increasing the volume of samples by a factor of two or more for equal values of the alphabet power as compared to computing exact distributions. PubDate: 2020-10-01 DOI: 10.3103/S1066530720040031
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Abstract: There are many collaborative studies where the data are discrepant while uncertainty estimates reported in each study cannot be relied upon. The classical commonly used random effects model explains this phenomenon by additional noise with a constant heterogeneity variance. This assumption may be inadequate especially when the smallest uncertainty values correspond to the cases which are most deviant from the bulk of data. An augmented random effects model for meta-analysis of such studies is offered. It proposes to think about the data as consisting of different classes with the same heterogeneity variance only within each cluster. The choice of the classes is to be made on the basis of the classical or restricted likelihood. We discuss the properties of the corresponding procedures which indicate the studies whose heterogeneity effect is to be enlarged. The conditions for the convergence of several iterative algorithms are given. PubDate: 2020-10-01 DOI: 10.3103/S1066530720040043
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Abstract: In this paper, we consider the problem of censored Gamma regression when the censoring status is missing at random. Three estimation methods are investigated. They consist in solving a censored maximum likelihood estimating equation where missing data are replaced by values adjusted using either regression calibration or multiple imputation or inverse probability weights. We show that the resulting estimates are consistent and asymptotically normal. Moreover, while asymptotic variances in missing data problems are generally estimated empirically (using Rubin’s rules for example), we propose closed-form consistent variances estimates based on explicit formulas for the asymptotic variances of the proposed estimates. A simulation study is conducted to assess finite-sample properties of the proposed parameters and asymptotic variances estimates. PubDate: 2020-10-01 DOI: 10.3103/S106653072004002X
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Abstract: Recently, a time-dependent measure of divergence has been introduced by Mansourvar and Asadi (2020) to assess the discrepancy between the survival functions of two residual lifetime random variables. In this paper, we derive various time-dependent results on the proposed divergence measure in connection to other well-known measures in reliability engineering. The proposed criterion is also examined in mixture models and a general class of survival transformation models which results in some well-known models in the lifetime studies and survival analysis. In addition, the time-dependent measure is employed to evaluate the divergence between the lifetime distributions of \(k\) -out-of- \(n\) systems and also to assess the discrepancy between the distribution functions of the epoch times of a non-homogeneous Poisson process. PubDate: 2020-07-01 DOI: 10.3103/S1066530720030023
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Abstract: The concept of extended neighboring order statistics introduced in Asadi et al. (2001) is a general model containing models of ordered random variables that are included in the generalized order statistics. This model also includes several models of ordered random variables that are not included in the generalized order statistics and is a helpful tool in unifying characterization results from several models of ordered random variables. In this paper, some general classes of distributions with many applications in reliability analysis and engineering, such as negative exponential, inverse exponential, Pareto, negative Pareto, inverse Pareto, power function, negative power, beta of the first kind, rectangular, Cauchy, Raleigh, Lomax, etc., have been characterized by using the regression of extended neighboring order statistics and decreasingly ordered random variables. PubDate: 2020-07-01 DOI: 10.3103/S1066530720030035
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