In this paper, we derive the exact null distribution of Banerjee’s test [2] for assessing usual stochastic ordering in the two-sample problem. Critical values of the test are provided for some significance levels. Additionally, we obtain the power function under proportional hazards model alternatives.
The problem of nonparametric estimation of the ratio of two densities is a classical topic in discriminant analysis, machine learning and statistical classification. The aim of the paper is to develop the theory of efficient nonparametric estimation of the ratio under the mean integrated squared error (MISE) criterion. For the first time in the literature, a sharp lower bound for minimax MISE is developed for the oracle who knows data, an underlying Sobolev functional class of the ratios, and the reference (denominator) density. Then a data driven estimator is proposed that matches performance of the oracle. It is shown that the problem of ratio estimation may be dramatically more complicated than estimation of a density, and its complexity is captured by the coefficient of difficulty which is a special functional of the two densities defining the ratio.
Defining the notion of a multivariate quantile has been an open problem for more than half a century, motivating a plethora of possible solutions. Of these, the approach of Chakraborty and Chaudhuri [8] and Koltchinskii [26] leading to M-quantiles, is very appealing for its mathematical elegance, combining elements of convex analysis and probability theory. The key idea is the description of a convex function (the K-function) whose gradient (the K-transform) is in one-to-one correspondence between all of ℝ^d and the unit ball in ℝ^d . By analogy with the d=1 case where the K-transform is a cumulative distribution function like object (an M-distribution), the fact that its inverse is guaranteed to exist lends itself naturally to providing the basis for the definition of a quantile function for all d≥ 1 . The resulting M-quantiles have seen applications in a variety of fields, with outlier detection featuring prominently. We show that for odd d≥ 3 it is not the first derivative, but a poly-Laplacian of the K-function that is proportional to the density function. (For even values of d no relationship is apparent.) Two-dimensional examples from non-standard distributions reinforce this point, and illustrate a feature of the K-transform whereby high density regions of the support are mapped to larger volumes, thereby producing a magnification effect that moves inliers closer to the boundary than outliers. In conclusion, the K-transform should not be viewed as a distribution function in higher dimensions, and its inverse does not lead to a ‘‘useful’’ quantile function, in the sense that its level set contours do not plausibly delineate outlying regions.
This paper proposes a novel measure, the quantile-based reversed aging intensity (QRAI) function, which serves as a quantile-based analogue of the classical reversed aging intensity function traditionally defined within the distribution function framework. The QRAI function offers valuable characterizations for specific parametric lifetime models and provides new insights into aging properties when compared with existing aging classes. The paper also explores stochastic comparisons of random variables using the QRAI measure and establishes its relationship with well-known stochastic orderings. We have obtained the QRAI function for the series and parallel systems, and studied how component and system level QRAI functions are connected. Two non-parametric estimators of the QRAI function are proposed, and their performance is assessed through simulation studies under selected parametric models. Additionally, a real data application is presented to demonstrate the practical usefulness of the proposed estimators.
In many areas such as toxicology, chemistry and environmental sciences, data observed usually include left-censoring, that is, data below some practical limit of detection do not get observed. In this paper, we first provide a review of different non-parametric estimators for the cumulative distribution function under such a left-censoring situation. One of them is based on the chain rule and the other one is based on counting processes. We then propose a new estimator for the cumulative distribution function based on a non-parametric likelihood approach using reversed hazard rate. We then take on a classical non-parametric likelihood approach to derive the same estimator. We also propose another estimator for the cumulative distribution function using the relationship between the cumulative distribution function and the reversed hazard rate function. We then carry out an empirical comparison of all these estimators and make some comparative comments. Finally, we conclude with an application to real data.
In this paper, using known values of the expectations of upper record values E(R_2),…,E(R_m) , for some m∈{2,…,n-1} , we derive improved bounds for E(R_n) . Further improvements are obtained in the case of a symmetric underlying distribution. Numerical illustrations of the results are provided for the standard normal distribution. Finally, extensions of the results to k -record values are detailed.
Let X_1,…,X_n be unit gamma Gompertz (UGG) random variables with X_i∼ UGG(α_i,β_i,μ_i;G) for i=1,…,n and I_p_1,…,I_p_n are independent Bernoulli random variables, independent of the X_i ’s, with E(I_p_i)=p_i , i=1,…,n . Let Y_i=I_p_iX_i , for i=1,…,n . In actuarial science, Y_i corresponds to the claim amount in a portfolio of risks. In this paper, we establish usual stochastic order and reversed hazard rate order between the largest claim amounts, by using the concept of vector majorization and related orders, when claim severities are independent. We also discuss stochastic comparisons between the smallest claim amounts in the sense of the usual stochastic order when claim severities are dependent. Further, we apply the results for some special cases of the unit gamma Gompertz model with possibly different parameters to illustrate.
In this paper, we compare the weighted models proposed by Rao [9] and Bhattacharjee et al. [4] to assess the aging phenomena played by the two weighted frameworks. We examine the role of weight function in the two approaches using different stochastic orders that exist among the weighted and baseline random variables. We prove that the weighted failure rate model serves as a better option in capturing aging phenomena. We obtain some bounds for the difference between the aging functions of a weighted model and its baseline and study the importance of weighted failure rate model on closure properties in the formation of coherent system. We also propose a method for generating new probability distributions by a selective choice of weight function.
In this communication, quantile-based cumulative information generating functions with some of its properties have been addressed. It is shown that the quantile-based cumulative past and residual entropy generating functions can be deduced from this general quantile-based measure for particular values of the parameters. Quantile-based dynamic cumulative past information generating function and quantile-based dynamic cumulative residual information generating function have been studied. Relations between the proposed measures with some reliability measures have been proposed. Characterization results for the power and exponential distributions have been obtained. Quantile-based stochastic ordering results are also derived.
This paper investigates the theoretical properties of Dirichlet kernel density estimators for compositional data supported on simplices, for the first time addressing scenarios involving time-dependent observations characterized by strong mixing conditions. We establish rigorous results for the asymptotic normality and mean squared error of these estimators, extending previous findings from the independent and identically distributed (iid) context to the more general setting of strongly mixing processes. To demonstrate its practical utility, the estimator is applied to monthly market-share compositions of several Renault vehicle classes over a twelve-year period, with bandwidth selection performed via leave-one-out least squares cross-validation. Our findings underscore the reliability and strength of Dirichlet kernel techniques when applied to temporally dependent compositional data.
This paper analyzes the Gini coefficient estimator for zero-truncated Poisson populations, revealing the presence of bias, and provides a mathematical expression for the bias, along with a bias-corrected estimator, which is evaluated using Monte Carlo simulation methods.
We consider M-estimators and derive supremal-inequalities of exponential-or polynomial type according as a boundedness- or a moment-condition is fulfilled. This enables us to derive rates of r-complete convergence and also to show r-qick convergence in the sense of Strasser.
A linear m -consecutive- k -out-of- n:F(G) system with sparse d consists of n components arranged in a line. The system fails (works) if and if there are at least m non overlapping runs of k consecutive failed (working) components with sparse d . In this paper, we introduce a linear weighted m -consecutive- k -out-of- n:F(G) system with sparse d consisting of weighted components. Such a system can find many applications in practice. We consider the situation where the system components are non-homogeneous Markov-dependent, and we derive closed-form formulas for the system reliability, the marginal reliability importance, and the joint reliability importance using conditional probability generating function method. We present numerical examples to illustrate the use of formulas.
This paper explores the intricacies of the general regression functional where the explanatory variables are defined within a functional space. We are particularly interested in the Robbins-Monro-type estimator of this regression functional, especially when the data is drawn from an underlying weakly stationary process. To facilitate this study, we revisit the concept of weak dependence, initially introduced by [22] for real-valued random variables, and adapt it to accommodate functional data residing in a normed space. We also present several examples of functional processes that satisfy this weak dependence criterion. In our analysis, we rigorously establish the almost sure convergence of the estimator, along with its rate and the asymptotic distribution. These results are obtained under a set of relatively general conditions concerning the classes of functions and the distributions that underpin the data. The contributions of our research are twofold. Firstly, they provide deep insights that significantly enhance the existing statistical methodologies used in the analysis of functional data. Secondly, they lay the groundwork for further applications in various statistical functions. These applications include enhancing the understanding of regression functions, refining the estimation of conditional distribution functions. Through these applications, our findings have the potential to substantially advance the field of functional data analysis.
This paper deals with the problem of nonparametric regression when the response variable may be missing but not necessarily at random. Here, we propose a new approach to construct kernel-type estimators of an unknown regression function based on Horvitz–Thompson inverse weighting when the data suffers from missing response values. The proposed approach may be viewed as a two-step procedure: the first step involves constructing a family of kernel-type regression estimators based on inverse weighting where the members of this family are indexed by the unknown parameters of the missing probability mechanism (the selection probability). In the second step, a search will be carried out to find the member of a cover of this family that has the smallest mean-squared prediction error. Furthermore, we establish exponential performance bounds on the deviations of the proposed estimators from the true regression curve in general L_p norms; these bounds yield various strong convergence results. We also study the rates of convergence of these estimators. As an important application of our results, we consider the problem of statistical classification with incomplete data.
In this paper, we study a robust estimation method for a mixed nonlinear model. In this model, we consider that the distribution of the observations belongs to the elliptical family. For the parametric inference, we propose an estimator based on minimum density power divergence (MDPD). Under some regularity conditions, we establish the main properties of this estimator; that is, consistency and asymptotic normality. We illustrate the robustness of the MDPDE compared to the maximum likelihood using some Monte Carlo simulations. Finally, we provide the results of an application on real data.
In this paper, we investigate stochastic comparisons of extreme order statistics from dependent and heterogeneous observations from a generalization of Weibull distribution, which is called exponentiated additive Weibull distribution (EADDWD). The usual stochastic order is developed for the minimum and maximum of dependent observations coupled by Archimedean copulas. We establish some results when some parameters vary and the other ones are kept fixed, with the help of vector majorization. Finally, some of theoretical findings are illustrated by numerical examples. Also, we try to provide a brief review on the existing articles have been studied the stochastic comparisons of order statistics from dependent observation.
A distributional identity for generalized order statistics is presented in two versions along with a characterization of exponential and Pareto distributions, respectively, under mild conditions. In its multiplicative form, the identity extends previous random dilation results in models of ordered random variables. The relations can be utilized as a statistical method to predict future generalized order statistics or other quantities of interest within submodels.
