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Abstract: Abstract We consider here two finite (arithmetic) mixture models (FMMs) with general parametric family of distributions. Sufficient conditions for the usual stochastic order and hazard rate order are then established under the assumption that the model parameter vectors are connected in p-larger order, reciprocal majorization order and weak super/sub majorization order. Furthermore, we establish hazard rate order and reversed hazard rate order between two mixture random variables (MRVs) when a matrix of model parameters and mixing proportions changes to another matrix in some mathematical sense. We have also considered scale family of distributions to establish some sufficient conditions under which the MRVs have hazard rate order. Several examples are presented to illustrate and clarify all the results established here. PubDate: 2023-11-20
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Abstract: Abstract To capture the bivariate count time series showing piecewise phenomena, we introduce a first-order bivariate threshold Poisson integer-valued autoregressive process. Basic probabilistic and statistical properties of the model are discussed. Conditional least squares and conditional maximum likelihood estimators, as well as their asymptotic properties, are obtained for both the cases that the threshold parameter is known or not. A new algorithm to estimate the threshold parameter of the model is also provided. Moreover, the nonlinearity test and forecasting problems are also addressed. Finally, some numerical results of the estimates and a real data example are presented. PubDate: 2023-11-01
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Abstract: Abstract In the canonical setting of the general linear model, we are concerned with estimating the loss of a point estimator when sampling from a spherically symmetric distribution. More precisely, from an observable (X, U) in \({\mathbb {R}}^p \times {\mathbb {R}}^k\) having a density of the form \(1 / \sigma ^{p+k} \, f \! \left( \big ( \Vert {\textbf{x}}- \varvec{\theta }\Vert ^2 + \Vert {\textbf{u}}\Vert ^2 / \sigma ^2 \big ) \right) \) where \(\varvec{\theta }\) and \(\sigma \) are both unknown, we consider general estimators \( \varphi (X,\Vert U\Vert ^2) \) of \(\varvec{\theta }\) under two losses: the usual quadratic loss \(\Vert \varphi (X,\Vert U\Vert ^2) - \varvec{\theta }\Vert ^2\) and the data-based loss \(\Vert \varphi (X,\Vert U\Vert ^2) - \varvec{\theta }\Vert ^2 / \Vert U\Vert ^2\) . Then, for each loss, we compare, through a squared error risk, their unbiased loss estimator \(\delta _0(X,\Vert U\Vert ^2)\) with a general alternative loss estimator \(\delta (X,\Vert U\Vert ^2)\) . Thanks to the new Stein type identity in Fourdrinier and Strawderman (Metrika 78(4):461–484, 2015), we provide an unbiased estimator of the risk difference between \(\delta (X,\Vert U\Vert ^2)\) and \(\delta _0(X,\Vert U\Vert ^2)\) , which gives rise to a sufficient domination condition of \(\delta (X,\Vert U\Vert ^2)\) over \(\delta _0(X,\Vert U\Vert ^2)\) . Minimax estimators of Baranchik form illustrate the theory. It is found that the distributional assumptions and dimensional conditions on the residual vector U are weaker when the databased loss is used. PubDate: 2023-11-01
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Abstract: Abstract The paper deal with the robust equivariant nonparametric regression when the covariates are functional and the response variables are missing at random (MAR). Under some mild conditions, the almost complete convergence rate of the proposed estimators for both cases known and unknown scale parameter are established. Some simulations study are drawing, and real data analysis are given to illustrate the higher predictive performances of our proposed method. PubDate: 2023-11-01
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Abstract: Abstract Functional linear concurrent regression model is an important model in functional regression. It is usually assumed that realizations of functional covariate are independent and observed precisely. But in practice, the dependence across different functional sample curves often exists. Moreover, each realization of functional covariate may be contaminated with noise. To address this issue, we propose a novel estimation method, which makes full use of dependence information and filters out the impact of measured noise. Then, we extend the proposed method to partially observed functional data. Under some regular conditions, we establish asymptotic properties of the estimators of the model. Finite-sample performance of our estimation is illustrated by Monte Carlo simulation studies and a real data example. Numerical results reveal that the proposed method exhibits superior performance compared with the existing methods. PubDate: 2023-11-01
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Abstract: Abstract The knowledge of the Fisher information is a fundamental tool to judge the quality of an experiment. Unlike in linear and generalized linear models without random effects, there is no closed form for the Fisher information in the situation of generalized linear mixed models, in general. To circumvent this problem, we make use of the quasi-information in this paper as an approximation to the Fisher information. We derive optimal designs based on the V-criterion, which aims to minimize the average variance of prediction of the mean response. For this criterion, we obtain locally optimal designs in two specific cases of a Poisson straight line regression model with either random intercepts or random slopes. PubDate: 2023-11-01
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Abstract: Abstract The analysis of multivariate longitudinal data may often encounter a difficult task, particularly in the presence of censored measurements induced by detection limits and intermittently missing values arising when subjects do not respond to a part of outcomes during scheduled visits. The multivariate nonlinear mixed model (MNLMM) has emerged as a promising analytical tool for multi-outcome longitudinal data following arbitrarily nonlinear profiles with random phenomena. This article presents a generalization of the MNLMM, called MNLMM-CM, designed to simultaneously accommodate the effects of censorship and missingness within a Bayesian framework. Specifically, we develop a Markov chain Monte Carlo procedure that combines a Gibbs sampler with the Metropolis–Hastings algorithm. This hybrid approach facilitates Bayesian estimation of essential model parameters and imputation of non-responses under the missing at random mechanism. The issue of posterior predictive inference for the censored and missing outcomes is also addressed. The effectiveness and performance of the proposed methodology are demonstrated through the analysis of simulated data and a real example from an AIDS clinical study. PubDate: 2023-10-26
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Abstract: Abstract In this paper, we obtain the asymptotic form of the joint distribution of maxima and minima of independent observations, when the sample size is a random variable. We also discuss the asymptotic distribution of the range. PubDate: 2023-10-17
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Abstract: Abstract In case of multicollinearity, PRESS statistics has been proposed to be used in the selection of the ridge biasing parameter of the ridge estimator which is introduced as an alternative to BLUE. This newly proposed PRESS statistic for the ridge estimator, \(\textit{CPRESS}_{k}\) , depends on the conditional ridge residual and can be computed once at a time by fitting the linear mixed model with all the observations. We also define \(R^2_{RidPred}\) statistic to evaluate the predictive ability of the ridge fit. Since the PRESS statistic for the BLUE is a special \(\textit{CPRESS}_{k}\) statistic, we indirectly also give closed form solution of the PRESS statistic for the BLUE. Then, we compared the predictive performance of the linear mixed model via the statistics \( \textit{CPRESS}_{k}\) , \(GCV_{k}\) and \(C_{p}\) by considering a real data analysis and a simulation study where the optimal ridge biaisng parameter is obtained by minimizing each statistic. The study shows that the ridge predictors improve the predictive performance of a linear mixed model over BLUE in the presence of multicollinearity and each statistic gives a different optimum ridge biasing value and they show the best predictive performance at their optimum ridge biasing values. In addition, the simulation study has shown that the intensity of variance and multicollinearity is effective in determining the optimum ridge biasing value and this optimum ridge biasing value is effective on the superiority of the predictive performance of ridge estimator over BLUE. PubDate: 2023-10-05
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Abstract: Abstract In this work, we introduce the random weighting method to the nonlinear regression model and study the asymptotic properties for the randomly weighted least squares estimator with dependent errors. The results reveal that this new estimator is consistent. Moreover, some simulations are also carried out to show the performance of the proposed estimator. PubDate: 2023-10-04
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Abstract: Abstract We consider two negative exponential (NE) populations with unknown location parameters and unknown but unequal scale parameters. We develop fixed-width confidence interval (FWCI) estimation strategies for comparing the location parameters. First, we formulate a unified multi-stage estimation strategy and derive the theory for an asymptotic second-order (s.o.) expansion of the coverage probability as well as s.o. efficiency. Next, we successively specialize by providing a range of asymptotics associated with (i) purely sequential, (ii) parallel piecewise sequential, (iii) accelerated sequential, and (iv) three-stage strategies. Theoretical findings are supplemented by simulations and real data illustrations from head and neck cancer research. PubDate: 2023-10-01
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Abstract: 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: 2023-10-01 DOI: 10.1007/s00184-022-00891-0
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Abstract: Abstract This paper considers the modeling problem of the weekly number of districts with new cases of cryptosporidiosis infection, and proposes a covariate-driven beta-binomial integer-valued GARCH model with a logit transformation to illustrate such bounded integer-valued time series data with extra-binomial variation and high volatility. We establish the existence of the stationary and ergodic solution by imposing a contraction condition on its conditional mean process and a Markov structure on the incorporated covariate process, consider the conditional maximum likelihood (CML) estimator for the parameter vector and discuss its asymptotic properties, conduct a simulation study to examine the finite sample performance of the CML estimator for the proposed model with three data generating mechanisms of the covariate process. Finally, an application to the weekly number of districts with new cases of cryptosporidiosis infection is considered to illustrate the superior performance of the proposed model. PubDate: 2023-10-01 DOI: 10.1007/s00184-023-00894-5
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Abstract: Abstract In this paper, we prove a local limit theorem and a refined continuity correction for the negative binomial distribution. We present two applications of the results. First, we find the asymptotics of the median for a \(\textrm{NegativeBinomial}(r,p)\) random variable jittered by a \(\textrm{Uniform}(0,1)\) , which answers a problem left open in Coeurjolly and Trépanier (Metrika 83(7):837–851, 2020). This is used to construct a simple, robust and consistent estimator of the parameter p, when \(r > 0\) is known. The case where r is unknown is also briefly covered. Second, we find an upper bound on the Le Cam distance between negative binomial and normal experiments. PubDate: 2023-10-01 DOI: 10.1007/s00184-023-00897-2
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Abstract: Abstract The aliased effect-number pattern, proposed by Zhang et al. (Stat Sin 18:1689–1705, 2008), is often used to express the overall confounding between factorial effects in two-level regular designs. The confounding relationships of main effects and two-factor interactions have been well revealed in literature, but little is known about three and higher-order interactions. To fill the gaps, this paper aims to study the confounding of three-order interactions in any two-level design. When the factor number of the design is larger, we derive the confounding among lower-order and three-order interactions via the complementary method. Further, confounding formulas can be obtained owing to the consulting sets of the structured designs. These two approaches can cover all designs in the sense of isomorphism. As an application of the previous results, the confounding numbers of three-order interactions are tabulated for 16, 32, and 64-run optimal designs under the general minimum lower-order confounding criterion. PubDate: 2023-10-01 DOI: 10.1007/s00184-023-00892-7
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Abstract: Abstract Much of classical optimal design theory relies on specifying a model with only a small number of parameters. In many applications, such models will give reasonable approximations. However, they will often be found not to be entirely correct when enough data are at hand. A property of classical optimal design methodology is that the amount of data does not influence the design when a fixed model is used. However, it is reasonable that a low dimensional model is satisfactory only if limited data is available. With more data available, more aspects of the underlying relationship can be assessed. We consider a simple model that is not thought to be fully correct. The model misspecification, that is, the difference between the true mean and the simple model, is explicitly modeled with a stochastic process. This gives a unified approach to handle situations with both limited and rich data. Our objective is to estimate the combined model, which is the sum of the simple model and the assumed misspecification process. In our situation, the low-dimensional model can be viewed as a fixed effect and the misspecification term as a random effect in a mixed-effects model. Our aim is to predict within this model. We describe how we minimize the prediction error using an optimal design. We compute optimal designs for the full model in different cases. The results confirm that the optimal design depends strongly on the sample size. In low-information situations, traditional optimal designs for models with a small number of parameters are sufficient, while the inclusion of the misspecification term lead to very different designs in data-rich cases. PubDate: 2023-10-01 DOI: 10.1007/s00184-023-00893-6
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Abstract: Abstract In an effort to estimate the number of true nulls in large scale multiplicity problems (the normal means problem), we generalize the current Fourier transform based oracle estimator with a Laplace transform based estimator. Our interest in this problem stems from the application of r-power which requires knowledge of the number of nulls (Dasgupta et al. in Sankhya B 78(1):96–118, 2016). We analytically show that our method is consistent and theoretically has lower mean squared error than the existing competitor (Jin in J R Stat Soc Ser B (Stat Methodol) 70(3):461–493, 2008). We follow up by a numerical example and a simulation study that ratifies our theoretical results. PubDate: 2023-09-16
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Abstract: Abstract Even though the proportional hazards model has been used extensively in reliability and survival analysis, it often fails to satisfy the basic assumptions wherein additive hazard model is a good alternative. Unlike the distribution function, quantile function are of efficient alternatives in the modelling and analysis of lifetime data. Motivated by these, in the present study we introduce an additive hazards quantile model and study its various properties and applications. The proposed model possess some interesting properties that are not shared by its distribution function counterpart. We also present a class of distributions with quadratic hazard quantile function and examine its usefulness through real-life examples. PubDate: 2023-09-11
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Abstract: Abstract This paper considers the problem of kernel regression and classification with possibly unobservable response variables in the data, where the mechanism that causes the absence of information can depend on both predictors and the response variables. Our proposed approach involves two steps: First we construct a family of models (possibly infinite dimensional) indexed by the unknown parameter of the missing probability mechanism. In the second step, a search is carried out to find the empirically optimal member of an appropriate cover (or subclass) of the underlying family in the sense of minimizing the mean squared prediction error. The main focus of the paper is to look into some of the theoretical properties of these estimators. The issue of identifiability is also addressed. Our methods use a data-splitting approach which is quite easy to implement. We also derive exponential bounds on the performance of the resulting estimators in terms of their deviations from the true regression curve in general \(L_p\) norms, where we allow the size of the cover or subclass to diverge as the sample size n increases. These bounds immediately yield various strong convergence results for the proposed estimators. As an application of our findings, we consider the problem of statistical classification based on the proposed regression estimators and also look into their rates of convergence under different settings. Although this work is mainly stated for kernel-type estimators, it can also be extended to other popular local-averaging methods such as nearest-neighbor and histogram estimators. PubDate: 2023-09-10
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Abstract: Abstract The non-null distribution of the sample correlation coefficient under bivariate normality is derived when each of the associated two sample variances is subject to stripe truncation including usual single and double truncation as special cases. The probability density function is obtained using series expressions as in the untruncated case with new definitions of weighted hypergeometric functions. Formulas of the moments of arbitrary orders are given using the weighted hypergeometric functions. It is shown that the null joint distribution of the sample correlation coefficients under multivariate untruncated normality holds also in the variance-truncated cases. Some numerical illustrations are shown. PubDate: 2023-08-25 DOI: 10.1007/s00184-023-00918-0
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