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Abstract: In the early stage of exploring a complex system, a preliminary experiment is used to capture the key characteristics of the model. Symmetrical global sensitivity analysis (SGSA) is one such experiment that explores the symmetrical structure of model by decomposing the model into independent symmetric functions. However, the existing experimental plans for SGSA rely on deterministic computational models that produce unique values of outputs when executed for specific values of inputs. In this paper, the problem of designing experiments for non-parametric SGSA is considered. Here the phrase “non-parametric” refers to model outputs containing random errors. The main result in the paper shows that a symmetrical design with certain constraints achieves A-optimum for the estimation of each output element function, and guarantees the superiority of the SGSA result. The statistical properties of non-parametric SGSA based on the optimal designs are further discussed, showing that the non-influential sensitivity indices can be estimated with low bias and volatility. Two explicit structures of the optimal designs are obtained. The optimality of the derived design is validated by simulation in the end. PubDate: 2023-02-01

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Abstract: This article investigates detecting the presence of significant predictors in marginal quantile regression. The main idea comes from the connection between the quantile correlation and the slope parameter of the marginal quantile regression, which is quite different from other methods. By introducing the local linear model and the plug-in empirical likelihood method, consistent asymptotic distribution and its adjusted version are obtained. We not only circumvent the non-regularity encountered by post-model-selected estimators but also make the results more concise. Two adaptive resampling test procedures are proposed in practice by comparing the t-statistics with a threshold to decide whether to use the traditional centered percentile bootstrap or otherwise adapt to the asymptotic distribution under the local model. Simulation studies compare these two resampling tests with other competing methods in several cases. Results show that the approaches proposed are more robust for each quantile level and can control type I error well. Two real datasets from Forbes magazine and the HIV drug resistance database are also applied to illustrate the new methods. PubDate: 2023-02-01

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Abstract: In this paper, we consider a coherent system composed of components whose lifetimes are independent and identically discretely distributed random variables. We study several aging and stochastic properties of the conditional residual lifetime of the system under the condition that some of its components have failed by time t. Moreover, we compare the conditional residual lifetimes of two coherent systems by using various stochastic orders. PubDate: 2023-02-01

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Abstract: Li et al. (Comm Statist Theory Methods 49: 924–941, 2020) introduced the concept of inverse Yates-order (IYO) designs, and obtained most of two-level IYO designs have general minimum lower-order confounding (GMC) property. For this reason, the paper extends two-level IYO designs to three-level cases. We first propose the definition of \(3^{n-m}\) IYO design \(D_q(n)\) from the saturated design \(H_q\) with three levels. Then, the formulas of lower-order confounding are obtained according to the factor number of \(3^{n-m}\) IYO design: (i) \(q<n<3^{q-1}\) , and (ii) \(3^{q-1}\le n\le (N-1)/2\) , where \(N=3^{n-m}\) . Under case (ii), we obtain the explicit expressions of lower-order confounding for four structure types of IYO designs. Some examples are given to illustrate the theoretical results. Compared with GMC designs, three-level IYO designs with 27- and 81-run are tabulated to show that some of them have GMC property through lower-order confounding. PubDate: 2023-02-01

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Abstract: The von Mises–Fisher distribution is one of the most widely used probability distributions to describe directional data. Finite mixtures of von Mises–Fisher distributions have found numerous applications. However, the likelihood function for the finite mixture of von Mises–Fisher distributions is unbounded and consequently the maximum likelihood estimation is not well defined. To address the problem of likelihood degeneracy, we consider a penalized maximum likelihood approach whereby a penalty function is incorporated. We prove strong consistency of the resulting estimator. An Expectation–Maximization algorithm for the penalized likelihood function is developed and experiments are performed to examine its performance. PubDate: 2023-02-01

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Abstract: In this article, we compare generalized order statistics in terms of increasing convex, mean residual life, hazard rate and reversed hazard rate orderings and establish some general results, including resolving an open problem. Some counterexamples are also provided for the cases wherein the results are not valid in general. Finally, some new applications of these results are demonstrated for sequential systems, k-record values and Pfeifer’s records. PubDate: 2023-01-01

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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: 2023-01-01

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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: 2023-01-01

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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: 2023-01-01

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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: 2023-01-01

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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: 2023-01-01

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Abstract: We study the Bahadur efficiency of several weighted L2-type goodness-of-fit tests based on the empirical characteristic function. The methods considered are for normality and exponentiality testing, and for testing goodness-of-fit to the logistic distribution. Our results are helpful in deciding which specific test a potential practitioner should apply. For the celebrated BHEP and energy tests for normality we obtain novel efficiency results, with some of them in the multivariate case, while in the case of the logistic distribution this is the first time that efficiencies are computed for any composite goodness-of-fit test. PubDate: 2022-12-26

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Abstract: The Poisson–Gamma model is obtained as a generalization of the Poisson model, when Gamma distributed block effects are assumed for Poisson count data. We show that optimal designs for estimating linear combinations of the model parameters coincide for the case of known and unknown parameters of the Gamma distribution. To obtain robust designs regarding parameter misspecification we determine standardized maximin D-optimal designs for a binary and a continuous design region. For standardized maximin c-optimality we show that the optimal designs for the Poisson–Gamma and Poisson model are equal and derive optimal designs for both models. PubDate: 2022-12-20

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Abstract: We consider parametric estimation and tests for multi-dimensional diffusion processes with a small dispersion parameter \(\varepsilon \) from discrete observations. For parametric estimation of diffusion processes, the main target is to estimate the drift parameter and the diffusion parameter. In this paper, we propose two types of adaptive estimators for both parameters and show their asymptotic properties under \(\varepsilon \rightarrow 0\) , \(n\rightarrow \infty \) and the balance condition that \((\varepsilon n^\rho )^{-1} =O(1)\) for some \(\rho >0\) . Using these adaptive estimators, we also introduce consistent adaptive testing methods and prove that test statistics for adaptive tests have asymptotic distributions under null hypothesis. In simulation studies, we examine and compare asymptotic behaviors of the two kinds of adaptive estimators and test statistics. Moreover, we treat the SIR model which describes a simple epidemic spread for a biological application. PubDate: 2022-12-16

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Abstract: Many a times a need may arise to estimate either a certain ratio or the sum of the shape parameters of two independent gamma populations. We try to tackle this problem through appropriate and novel two-stage sampling strategies. The first part of this paper deals with developing a two-stage methodology to estimate the ratio \(\alpha /(\alpha +\beta )\) coming from two independent gamma populations with parameters \((\alpha ,1)\) and \((\beta ,1)\) respectively. We assume a weighted squared error loss function and aim at controlling the associated risk function per unit cost by bounding it from above by a known constant \(\omega .\) We also establish first-order properties of our stopping rules. The second part of this paper deals with obtaining a two-stage sampling procedure to estimate the sum \(\alpha +\beta \) instead. We also provide extensive simulation analysis and real data analysis using data from cancer studies to show encouraging performances of our proposed stopping strategies. PubDate: 2022-12-04

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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: 2022-11-01 DOI: 10.1007/s00184-022-00858-1

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Abstract: In this paper, we propose a two-step test for parametric specification of volatility function based on high-frequency data with microstructure noise. The latent prices are first recovered at high precision under the assumption that the noise is a parametric function of observable trading information. An asymptotically distribution-free test is then built on the estimated latent prices using Khmaladze martingale transformation. We establish asymptotic theory associated with the test under both the null and alternative hypotheses. Moreover, an extension of the proposed method to incorporate intraday pattern is also formally discussed. Simulation results corroborate our theoretical findings demonstrating clear advantage of our method over an existing distribution-free method that does not take microstructure noise into account. We finally apply the test to the high-frequency data of Standard & Poor’s depository receipt (SPDR) that tracks the S&P 500 index. PubDate: 2022-11-01 DOI: 10.1007/s00184-021-00857-8

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Abstract: In this paper the problem of one dimensional parameter estimation is considered in the case where observations are coming from inhomogeneous Poisson processes. The method of moments estimation is studied and its stochastic expansion is obtained. This stochastic expansion is then used to obtain the expansion of the moments of the estimator and the expansion of the distribution function. The stochastic expansion, the expansion of the moments and the expansion of distribution function are non asymptotic in nature. Several examples are presented to illustrate the theoretical results. PubDate: 2022-11-01 DOI: 10.1007/s00184-021-00855-w

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Abstract: The interference model has been widely used and studied in block designs where the treatment in a particular plot effects on ones in its neighbor plots. There are many experiments specially in agricultre that block size (k) is greater than the number of treatment (t). If \(k>t\) , the universally optimal designs under the interference models are usually difficult to obtain. In this paper, we consider an interference model with equal left- and right-neighbor effects for uncorrelated errors when \(k>t\) . Based on Kushner’s method, we obtain the universally optimal designs on the class of circular block designs. We also present some methods to construct these designs. Then, we generalize our results for one-sided interference models. PubDate: 2022-11-01 DOI: 10.1007/s00184-022-00870-5

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Abstract: Finite Gamma mixture models are often used to describe randomness in income data, insurance data, and data in applications where the response values are intrinsically positive. The popular likelihood approach for model fitting, however, does not work for this model because its likelihood function is unbounded. Because of this, the maximum likelihood estimator is not well-defined. Other approaches have been developed to achieve consistent estimation of the mixing distribution, such as placing an upper bound on the shape parameter or adding a penalty to the log-likelihood function. In this paper, we show that if the shape parameter in the finite Gamma mixture model is structural, then the direct maximum likelihood estimator of the mixing distribution is well-defined and strongly consistent. We also present simulation results demonstrating the consistency of the estimator. We illustrate the application of the model with a structural shape parameter to household income data. The fitted mixture distribution leads to several possible subpopulation structures with regard to the level of disposable income. PubDate: 2022-11-01 DOI: 10.1007/s00184-021-00856-9