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Abstract: The object of the present paper is the study of the joint lifetime of d components subject to a common stressful external environment. Out of the stressing environment, the components are independent and the lifetime of each component is characterized by its failure (hazard) rate function. The impact of the external environment is modelled through an increase in the individual failure rates of the components. The failure rate increments due to the environment increase over time and they are dependent among components. The evolution of the joint failure rate increments is modelled by a non negative multivariate additive process, which include Lévy processes and non-homogeneous compound Poisson processes, hence encompassing several models from the previous literature. A full form expression is provided for the multivariate survival function with respect to the intensity measure of a general additive process, using the construction of an additive process from a Poisson random measure (or Poisson point process). The results are next specialized to Lévy processes and other additive processes (time-scaled Lévy processes, extended Lévy processes and shock models), thus providing simple and easily computable expressions. All results are provided under the assumption that the additive process has bounded variations, but it is possible to relax this assumption by means of approximation procedures, as is shown for the last model of this paper. PubDate: 2022-05-09

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Abstract: Abstract We introduce and discuss a multivariate version of the classical median that is based on an equipartition property with respect to quarter spaces. These arise as pairwise intersections of the half-spaces associated with the coordinate hyperplanes of an orthogonal basis. We obtain results on existence, equivariance, and asymptotic normality. PubDate: 2022-05-01

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Abstract: Abstract We extended the application of uniformly most powerful tests to sequential tests with different stage-specific sample sizes and critical regions. In the one parameter exponential family, likelihood ratio sequential tests are shown to be uniformly most powerful for any predetermined \(\alpha \) -spending function and stage-specific sample sizes. To obtain this result, the probability measure of a group sequential design is constructed with support for all possible outcome events, as is useful for designing an experiment prior to having data. This construction identifies impossible events that are not part of the support. The overall probability distribution is dissected into components determined by the stopping stage. These components are the sub-densities of interim test statistics first described by Armitage et al. (J R Stat Soc: Ser A 132:235–244, 1969) that are commonly used to create stopping boundaries given an \(\alpha \) -spending function and a set of interim analysis times. Likelihood expressions conditional on reaching a stage are given to connect pieces of the probability anatomy together. The reduction of support caused by the adoption of an early stopping rule induces sequential truncation (not nesting) in the probability distributions of possible events. Multiple testing induces mixtures on the adapted support. Even asymptotic distributions of inferential statistics that are typically normal, are not. Rather they are derived from mixtures of truncated multivariate normal distributions. PubDate: 2022-05-01

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Abstract: Abstract Regional prevalence estimation requires the use of suitable statistical methods on epidemiologic data with substantial local detail. Small area estimation with medical treatment records as covariates marks a promising combination for this purpose. However, medical routine data often has strong internal correlation due to diagnosis-related grouping in the records. Depending on the strength of the correlation, the space spanned by the covariates can become rank-deficient. In this case, prevalence estimates suffer from unacceptable uncertainty as the individual contributions of the covariates to the model cannot be identified properly. We propose an area-level logit mixed model for regional prevalence estimation with a new fitting algorithm to solve this problem. We extend the Laplace approximation to the log-likelihood by an \(\ell _2\) -penalty in order to stabilize the estimation process in the presence of covariate rank-deficiency. Empirical best predictors under the model and a parametric bootstrap for mean squared error estimation are presented. A Monte Carlo simulation study is conducted to evaluate the properties of our methodology in a controlled environment. We further provide an empirical application where the district-level prevalence of multiple sclerosis in Germany is estimated using health insurance records. PubDate: 2022-05-01

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Abstract: Abstract Maximin distance designs and orthogonal designs have become increasingly popular in computer and physical experiments. The construction of such designs is challenging, especially under the maximin distance criterion. This paper studies a class of Latin hypercube designs by calculating the minimum distances between design points. We derive a general formula for the minimum intersite distance of this kind of design. The row pairs with the minimum intersite distance are also specified. The results show that such kind of Latin hypercube design is asymptotically optimal under both the maximin distance criterion and the orthogonality criterion. PubDate: 2022-05-01

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Abstract: Abstract In this paper, we consider prediction for the exponential distribution with unknown location. For the most part, we treat the one-dimensional case and assume that the location parameter is restricted to an interval. The Bayesian predictive densities with respect to prior densities supported on the real line and the restricted space are compared under the Kullback–Leibler divergence. We first consider the case where the scale parameter is known. We obtain general dominance conditions and also minimaxity and admissibility results. Next, we treat the case of unknown scale. In this case, the location parameter is assumed to be less than a known constant and sufficient conditions for domination are obtained. Finally, we treat a multidimensional problem with known scale where the location parameter is restricted to a convex set. The performance of several Bayesian predictive densities is investigated through simulation. Some of the prediction methods are applied to real data. PubDate: 2022-05-01

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Abstract: Abstract We study the problem of estimating conditional distribution functions from data containing additional errors. The only assumption on these errors is that a weighted sum of the absolute errors tends to zero with probability one for sample size tending to infinity. We prove sufficient conditions on the weights (e.g. fulfilled by kernel weights) of a local averaging estimate of the codf, based on data with errors, which ensure strong pointwise consistency. We show that two of the three sufficient conditions on the weights and a weaker version of the third one are also necessary for the spc. We also give sufficient conditions on the weights, which ensure a certain rate of convergence. As an application we estimate the codf of the number of cycles until failure based on data from experimental fatigue tests and use it as objective function in a shape optimization of a component. PubDate: 2022-04-01 DOI: 10.1007/s00184-021-00831-4

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Abstract: Abstract We consider together the retrospective and the sequential change-point detection in a general class of integer-valued time series. The conditional mean of the process depends on a parameter \(\theta ^*\) which may change over time. We propose procedures which are based on the Poisson quasi-maximum likelihood estimator of the parameter, and where the updated estimator is computed without the historical observations in the sequential framework. For both the retrospective and the sequential detection, the test statistics converge to some distributions obtained from the standard Brownian motion under the null hypothesis of no change and diverge to infinity under the alternative; that is, these procedures are consistent. Some results of simulations as well as real data application are provided. PubDate: 2022-04-01 DOI: 10.1007/s00184-021-00834-1

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Abstract: Abstract This paper proposes a target specialized meta-heuristic optimization algorithm, called Mixture Design Threshold Accepting (MDTA) algorithm, which applies the idea of the Threshold Accepting to generate the corresponding approximate D-optimal designs for a wide range of mixture models, with or without constraints imposed on the components. The MDTA algorithm is tested by many of common mixture models, among which some even have no solutions of the D-optimal design available in the literature. Other tests include 5 models with specific upper bound constraints. These results prove that the MDTA algorithm is very efficient in finding D-optimal designs for mixture models. In some scenarios it even outperforms the state-of-art algorithms, such as the ProjPSO algorithm and the REX algorithm. The source codes of the MDTA algorithm are freely available by writing to the first author. PubDate: 2022-04-01 DOI: 10.1007/s00184-021-00832-3

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Abstract: Abstract Let X, Y, W, \(\delta \) and \(\varepsilon \) be continuous univariate random variables defined on a probability space such that \(Y = X+\varepsilon \) and \(W = X + \delta \) . Herein X, \(\delta \) and \(\varepsilon \) are assumed to be mutually independent. The variables \(\varepsilon \) and \(\delta \) are called classical and Berkson errors, respectively. Their distributions are known exactly. Suppose we only observe a random sample \(Y_1, \ldots , Y_n\) from the distribution of Y. This paper is devoted to a nonparametric estimation of the unknown cumulative distribution function \(F_W\) of W based on the observations as well as on the distributions of \(\varepsilon \) , \(\delta \) . An estimator for \(F_W\) depending on a smoothing parameter is suggested. It is shown to be consistent with respect to the mean squared error. Under certain regularity assumptions on the densities of X, \(\delta \) and \(\varepsilon \) , we establish some upper and lower bounds on the convergence rate of the proposed estimator. Finally, we perform some numerical examples to illustrate our theoretical results. PubDate: 2022-04-01 DOI: 10.1007/s00184-021-00830-5

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Abstract: Abstract 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. PubDate: 2022-03-20

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Abstract: Abstract Nonresponse data are common in achievement tests or questionnaires. Chang et al. (Br J Math Stat Psychol 74:487–512, 2021) proposed an Item Response tree model, namely TR4, for modeling some potential mechanisms underlying nonresponses so that the estimates of parameters of interest would not be biased due to missing not at random (Rubin in Biometrika 63:581–592, 1976). TR4 has two notable degenerate cases, both with insightful practical meanings. When TR4 is fitted to data originated from some degenerate cases, there exist model identifiability issues so that the existing frequentist inference for the TR4 model is not suitable. In the current study, we propose a Bayesian estimation procedure that incorporates the Markov chain Monte Carlo technique for estimating the TR4 model. We conducted simulation studies to demonstrate the effectiveness of the Bayesian estimation procedure in solving the model unidentifiability issue. In addition, the TR4 model is further extended in the present study to effectively accommodate the complexity underlying some real data. The advantage of the extended models over TR4 is demonstrated in the real data analysis where we apply our method to the data of a geography test for college admission in Taiwan. PubDate: 2022-03-19

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Abstract: Abstract Consider an n-components coherent system monitored at one or two inspection times, and some information about the system and its components is obtained. Under these conditions, some variants of mean residual lifetimes can be defined. Also, the dual concept of the residual lifetime, i.e., inactivity time is defined for a failed system under different conditions. This article is concerned with the study of mean residual lives and mean inactivity times for a coherent system made of multiple types of dependent components. The dependency structure is modeled by a survival copula. The notion of survival signature is employed to represent the system’s reliability function and subsequently its mean residual lives and mean inactivity times under different events at the monitoring time. These dynamic measures are used frequently to study the reliability characteristics of a system. Also, they provide helpful tools for designing the optimal maintenance policies to preserving the system from sudden and costly failures. Here, we extend some maintenance strategies for a coherent system consists of multiple dependent components. Some illustrative examples are provided. PubDate: 2022-03-06 DOI: 10.1007/s00184-022-00862-5

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Abstract: Abstract This paper discusses a test of goodness-of-fit of a known error density in a two phase linear regression model in the case jump size at the phase transition point is fixed or tends to zero with the increasing sample size. The proposed test is based on an integrated square difference between a nonparametric error density estimator obtained from the residuals and its expected value under the null error density when the underlying regression parameters are known. The paper establishes the asymptotic normality of the proposed test statistic under the null hypothesis and under certain global \(L_2\) alternatives. The asymptotic null distribution of the test statistic is the same as in the case of the known regression parameters. Under the chosen alternatives, unlike in the linear autoregressive time series models with known intercept, it depends on the parameters and their estimates in general. We also describe the analogous results for the self-exciting threshold autoregressive time series model of order 1. PubDate: 2022-02-23 DOI: 10.1007/s00184-022-00861-6

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Abstract: Abstract This paper investigates the Lasso method for sparse linear models with exponential \(\varphi \) -mixing errors under a fixed design, where the number of covariates p is large, or even much larger than the sample size n. The non-asymptotic concentration inequalities for the estimation and prediction errors of the Lasso estimators are given when the errors follow the Gaussian distribution and the sub-exponential distribution, respectively. The prediction and variable selection performance of Lasso estimators are further illustrated through numerical simulations. Finally, the results of the empirical application show that the Index Tracking Fund based on the sparse selection of Lasso can closely track the trend of the target index, and thus provide some useful guidance for the investors. PubDate: 2022-02-18 DOI: 10.1007/s00184-022-00860-7

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Abstract: Abstract The Gumbel–Barnett family of bivariate distributions with given marginals, is frequently used in theory and applications. This family has been generalized in several ways. We propose and study a broad generalization by using two differentiable functions. We obtain some properties and describe particular cases. PubDate: 2022-02-17 DOI: 10.1007/s00184-022-00859-0

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Abstract: Abstract Minimax robust designs for regression models with heteroscedastic errors are studied and constructed. These designs are robust against possible misspecification of the error variance in the model. We propose a flexible assumption for the error variance and use a minimax approach to define robust designs. As usual it is hard to find robust designs analytically, since the associated design problem is not a convex optimization problem. However, we can show that the objective function of the minimax robust design problem is a difference of two convex functions. An effective algorithm is developed to compute minimax robust designs under the least squares estimator and generalized least squares estimator. The algorithm can be applied to construct minimax robust designs for any linear or nonlinear regression model with heteroscedastic errors. In addition, several theoretical results are obtained for the minimax robust designs. PubDate: 2022-02-01 DOI: 10.1007/s00184-021-00827-0

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Abstract: Abstract In life-testing experiments, it is often of interest to predict unobserved future failure times based on observed early failure times. A point best linear unbiased predictor (BLUP) has been developed in this context by Kaminsky and Nelson (J Am Stat Assoc 70:145–150, 1975). In this article, we develop joint BLUPs of two future failure times based on early failure times by minimizing the determinant of the variance–covariance matrix of the predictors. The advantage of applying joint prediction is demonstrated by using two real data sets. The non-existence of joint BLUPs in certain setups is also discussed. PubDate: 2022-02-01 DOI: 10.1007/s00184-021-00835-0

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Abstract: Abstract In this paper, we study partially varying-coefficient single-index model where both the response and predictors are observed with multiplicative distortions which depend on a commonly observable confounding variable. Due to the existence of measurement errors, the existing methods cannot be directly applied, so we recommend using the nonparametric regression to estimate the distortion functions and obtain the calibrated variables accordingly. With these corrected variables, the initial estimators of unknown coefficient and link functions are estimated by assuming that the parameter vector \(\beta \) is known. Furthermore, we can obtain the least square estimators of unknown parameters. Moreover, we establish the asymptotic properties of the proposed estimators. Simulation studies and real data analysis are given to illustrate the advantage of our proposed method. PubDate: 2022-02-01 DOI: 10.1007/s00184-021-00823-4

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Abstract: Abstract Empirical and kernel estimators are considered for the distribution of positive length biased data. Their asymptotic bias, variance and limiting distribution are obtained. For the kernel estimator, the asymptotically optimal bandwidth is calculated and rule of thumb bandwidths are proposed. At any point below the median, the asymptotic mean squared error of the kernel estimator is smaller than that of the empirical estimator. A suitably truncated kernel estimator is positive and we prove the strong uniform, and \(L_2\) consistency of this estimator. Simulations reveal the improved performance of the truncated kernel estimator in estimating tail probabilities based on length biased data. PubDate: 2022-01-13 DOI: 10.1007/s00184-021-00824-3