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Abstract: Nonparametric regression estimation with Gaussian measurement errors in predictors is a classical statistical problem. It is well known that the errors dramatically slow down the rate of regression estimation, and this paper complement that result by presenting a sharp constant. Then an interesting example of using this sharp constant to discover a new curse of dimensionality in functional nonparametric regression is presented, and analysis of real data complements the theory. PubDate: 2023-09-01
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Abstract: In this paper, we investigate multivariate doubly truncated moments for a class of multivariate location-scale mixture of elliptical (LSME) distributions. This rich family includes some well-known distributions, such as location-scale mixture of normal, location-scale mixture of Student- \(t\) , location-scale mixture of logistic and location-scale mixture of Laplace distributions, as special cases. We first present general formulae for computing the first two moments of the LSME distributions under the double truncation. We then consider a special case for cross moment. As an application, we present the results of multivariate tail conditional expectation (MTCE) for generalized hyperbolic (MGH) distribution. PubDate: 2023-09-01
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Abstract: In this paper, we study the information-generating (IG) measure of \(k\) -record values and examine some of its main properties. We establish some bounds for the IG measure of \(k\) -record values. In addition, we present some results related to the characterization of an exponential distribution by maximization (minimization) of the IG measure of record values under certain conditions. We also examine the relative information generating (RIG) measure between the distribution of record values and the corresponding underlying distribution and present some results in this regard. Several examples have been provided throughout the study to illustrate the results. We also consider the problem of estimation of the IG measure for a two-parameter Weibull distribution based on the upper \(k\) -record values. PubDate: 2023-09-01
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Abstract: The goal of this paper is to introduce an efficient method for solving problems formulated by stochastic mixed Volterra–Fredholm integral equations driven by space-time white noise. Two dimensional triangular functions and their operational matrix and stochastic operational matrix of integration are considered. This method has several benefits; in addition to validity and good degree of accuracy, arithmetic operations are carried out without the need to derivative or integration. Illustrative examples are included to demonstrate the efficiency and applicability of the operational matrices based on two dimensional triangular functions. PubDate: 2023-09-01
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Abstract: In this paper, we deal with the estimation problem for the extreme value parameters in the case of stationary \(\beta\) -mixing serials with heavy-tailed distributions. We first introduce two families of estimators generalizing the Hill’s estimator. And from those families, three asymptotically unbiased estimators of the extreme value index are established. Our reflection is based on the generalized Jackknife methodology which consists of taking any pair of three special cases of our family of estimators to cancel the bias term. The resulting estimators are also used to deduce three asymptotically unbiased estimators of the extreme quantiles. In a simulation survey, the performance of our proposed methods are compared to alternative estimators recently introduced in the literature. Finally, our methods are applied to high financial losses data in order to estimate the Value-at-Risk of the daily stock returns on the S&P500 index. PubDate: 2023-06-01 DOI: 10.3103/S1066530723020011
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Abstract: Families of distributions built from the fractional or continuous iteration of exponential-type functions are characterized by a wide range of tail-heaviness. The present paper aims to define classes of distributions supported on the whole real line based on the continuous iteration of the hyperbolic sine function sinh. This function has already been commonly employed in univariate transformations such as the Johnson’s \(S_{U}\) and sinh–arcsinh transforms. The tail versatility generated by a transformation based on the continuous iteration of sinh is highlighted based on an initial logistic distribution. It leads to the Hyperbolic Tetration distribution. The Double Hyperbolic Tetration distribution, defined from two successive hyperbolic transformations, is also introduced. It is among the first class of distributions with potential distinct tetration indices at plus and minus infinity. The distributions are applied to multiple data sets in hydrology. PubDate: 2023-06-01 DOI: 10.3103/S1066530723020023
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Abstract: Terrell [18] showed that the Pearson coefficient of correlation of an ordered pair from a random sample of size two is at most one-half, and the equality is attained only for rectangular (uniform over some interval) distributions. In the present note it is proved that the same is true for the discrete case, in the sense that the correlation coefficient attains its maximal value only for discrete rectangular (uniform over some finite lattice) distributions. PubDate: 2023-06-01 DOI: 10.3103/S1066530723020035
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Abstract: In [1], the authors introduced the geometric vitality function that explains the failure pattern of components or systems based on the geometric mean of the remaining lifetime. Recently, studies based on quantile function played an essential role in the research domain because of its unique properties than the distribution function method. Based on this, we define geometric vitality function in terms of quantile function and established monotone, ordering properties for the proposed measure. In addition to this, we also introduce the proposed measure in the context of order statistics. Some important properties were discussed for the proposed measure. Finally, we provide an application of the new measure for some distributions useful in lifetime data analysis. PubDate: 2023-03-01 DOI: 10.3103/S1066530723010040
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Abstract: Computation of reliability function of a coherent system is a difficult task especially when the system’s configuration is complex. Hence, sometimes, we should simply work with some approximations, as bounds reliability. The computation of these bounds has been widely studied in the case of coherent systems with independent and identically distributed (i.i.d) components. However, few results have been obtained in the case of heterogeneous (non i.d) dependent components. In this paper, we derive sharp bounds for the reliability of circular consecutive \(k\) -out-of- \(n:G\) systems consisting by dependent components with identical or arbitrary distribution functions. Also some stochastic comparisons are made. Illustrative examples are included. PubDate: 2023-03-01 DOI: 10.3103/S1066530723010052
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Abstract: In the present paper, we develop strong uniform consistency results for the generic kernel (including the kernel density estimator) on Riemannian manifolds with Riemann integrable kernels in order to accomplish these difficult tasks. The kernels of the Vapnik-Chervonenkis class that are commonly utilized in statistical problems are different to the isotropic kernels we address in this paper. Moreover, we show, in the same context, the uniform consistency for nonparametric inverse probability of censoring weighted (IPCW) estimators of the regression function under random censorship. As an application, we present the strong uniform consistency for estimators of the Nadaray-Watson type, which is of independent interest. PubDate: 2023-03-01 DOI: 10.3103/S1066530723010027
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Abstract: In this work we consider regularized Wasserstein barycenters (average in Wasserstein distance) in Fourier basis. We prove that random Fourier parameters of the barycenter converge to some Gaussian random vector in distribution. The convergence rate has been derived in finite-sample case with explicit dependence on measures count ( \(n\) ) and the dimension of parameters ( \(p\) ). PubDate: 2023-03-01 DOI: 10.3103/S1066530723010039
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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 DOI: 10.3103/S1066530722040044
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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 DOI: 10.3103/S1066530722040020
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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 DOI: 10.3103/S1066530722040032
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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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