A generalization of the growth curve model which allows missing data

This study presents methods for estimating and testing hypotheses about linear functions of the unknown parameters in a generalization of the growth curve model which allows missing data. The estimators proposed are best asymptotically normal (BAN). A testing method for large samples is described which uses a test criterion given in general form by Wald. The asymptotic null distribution of the test statistic is a central chi-square variable. A BAN estimator of a linear vector function of the unknown parameters of the expectation model and consistent estimators of the variance-covariance parameters are required for computation.     

Asymptotic properties of estimators of interclass correlation from familial data

Summary Asymptotic properties of several estimators of interclass correlation from familial data are examined in the case of a variable number of siblings per family. After showing that the usual sib-mean estimator is not consistent, a modified sib-mean estimator is proposed. Asymptotic distributions of estimators are derived and a test procedure is provided for a certain testing problem concerning interclass correlation. Several estimators are compared in the various mean number of siblings per family, using asymptotic mean square errors. The Institute of Statistical Mathematics     

Consistent estimation and testing in heteroscedastic polynomial errors-in-variables models

The paper concentrates on consistent estimation and testing in functional polynomial measurement errors models with known heterogeneous variances. We rest on the corrected score methodology which allows the derivation of consistent and asymptotically normal estimators for line parameters and also consistent estimators for the asymptotic covariance matrix. Hence, Wald and score type statistics can be proposed for testing the hypothesis of a reduced linear relationship, for example, with asymptotic chi-square distribution which guarantees correct asymptotic significance levels. Results of small scale simulation studies are reported to illustrate the agreement between theoretical and empirical distributions of the test statistics studied. An application to a real data set is also presented.     

Confidence intervals and ellipsoids for estimable functions and estimable vectors in models with orthogonal block structure

Sub-models of mixed linear models are considered. The independence of these sub-models leads to sufficient statistics for the parameters relevant for their densities. Using pivot variables, confidence regions are obtained as well hypothesis testing for variance components, estimable functions, and estimable vectors. In addition, to compare the estimators and the models, we present the histograms with the empirical joint densities for positive and negative parts of the estimators. The figures, for the two-dimensional charts, contain the corresponding UMVUE and are all unimodal with the UMVUE near the mode. The nearness of the estimators and the modes validates the presented methodology and allows the safe use of induced densities. A numerical example applied to real data is presented.     

Estimation under failure-censored step-stress life test for the generalized exponential distribution parameters

Accelerated life testing of materials is used to collect failure data quickly when the lifetime of a specimen under use condition is too long. This article considers estimates of the generalized exponential distribution parameters under step-stress partially accelerated life testing with Type-II censoring. The maximum likelihood approach is applied to derive point and asymptotic confidence interval estimations of the model parameters. The performance of the estimators is evaluated numerically for different parameter values and different sample sizes via their mean square error. Also, the average confidence intervals lengths and the associated coverage probabilities are obtained. A simulation study is conducted for illustration.     

Degrees of freedom in low rank matrix estimation

The objective of this paper is to quantify the complexity of rank and nuclear norm constrained methods for low rank matrix estimation problems. Specifically, we derive analytic forms of the degrees of freedom for these types of estimators in several common settings. These results provide efficient ways of comparing different estimators and eliciting tuning parameters. Moreover, our analyses reveal new insights on the behavior of these low rank matrix estimators. These observations are of great theoretical and practical importance. In particular, they suggest that, contrary to conventional wisdom, for rank constrained estimators the total number of free parameters underestimates the degrees of freedom, whereas for nuclear norm penalization, it overestimates the degrees of freedom. In addition, when using most model selection criteria to choose the tuning parameter for nuclear norm penalization, it oftentimes suffices to entertain a finite number of candidates as opposed to a continuum of choices. Numerical examples are also presented to illustrate the practical implications of our results.     

Estimation in the presence of many nuisance parameters: Composite likelihood and plug-in likelihood

We consider the incidental parameters problem in this paper, i.e. the estimation for a small number of parameters of interest in the presence of a large number of nuisance parameters. By assuming that the observations are taken from a multiple strictly stationary process, the two estimation methods, namely the maximum composite quasi-likelihood estimation (MCQLE) and the maximum plug-in quasi-likelihood estimation (MPQLE) are considered. For the MCQLE, we profile out nuisance parameters based on lower-dimensional marginal likelihoods, while the MPQLE is based on some initial estimators for nuisance parameters. The asymptotic normality for both the MCQLE and the MPQLE is established under the assumption that the number of nuisance parameters and the number of observations go to infinity together, and both the estimators for the parameters of interest enjoy the standard root-n$n$ convergence rate. Simulation with a spatial–temporal model illustrates the finite sample properties of the two estimation methods.     

The application of spectral distribution of product of two random matrices in the factor analysis

In the factor analysis model with large cross-section and time-series dimensions,we pro- pose a new method to estimate the number of factors.Specially if the idiosyncratic terms satisfy a linear time series model,the estimators of the parameters can be obtained in the time series model. The theoretical properties of the estimators are also explored.A simulation study and an empirical analysis are conducted.     

Second derivative estimation using harmonic analysis

Simulation sensitivity analysis is an important problem for simulation practitioners analyzing complex systems. The significance of this problem has resulted in the development of various gradient estimators that can be used to address this issue. Although higher derivative estimators have been discussed concurrently, less attention has been given to assess the efficiency and feasibility of computing such estimators. In this paper, two second derivative estimators are presented. The first estimators, called the HFD estimators, combine harmonic gradient estimators with finite differences second derivative estimators. The resulting hybrid estimators requireO(p) fewer simulation runs to implement compared to the straightforward finite differences approach, wherep is the number of input parameters in the simulation model. The second estimators, called the HA estimators, incorporate harmonic analysis directly, requiring one or two simulation runs to implement, depending on whether a control variate simulation run is made. Expressions for the bias and the variance of the HFD and the HA estimators (with and without variance reduction techniques) are derived. Optimal mean squared error convergence rates are also discussed. In particular, the convergence rates for both these estimators are shown to be the same, though the computational performance of the HFD estimators is better than that for the HA estimators on anM/M/1 queue simulation model. Computational results for the HFD estimators on an (s, S) inventory system simulation model are also included.     

The estimation of M4 processes with geometric moving patterns

There are many parameters in multivariate maxima of moving maxima processes—or M4 processes. However, the more parameters there are, the more difficult it is to estimate them. It is not just an issue of numerical stability, of course. The statistical precision of the estimates will be poor if the number of parameters is too large. We consider asymmetric geometric structures which correspond to special moving patterns of extreme observations in observed time series. We study the model identifiability and propose parameter estimators. All proposed estimators are shown to be consistent and asymptotically joint normal. Simulation study and real data modeling of North Sea wave height data are illustrated.     

A study of the number of solutions of the system of the log-likelihood equations for the 3-parameter Weibull distribution

The maximum likelihood estimators of the parameters for the 3-parameter Weibull distribution do not always exist. Furthermore, computationally it is difficult to find all the solutions. Thus, the case of missing some solutions and among them the maximum likelihood estimators cannot be excluded. In this paper we provide a simple rule with help of which we are able to know if the system of the log-likelihood equations has even or odd number of solutions. It is a useful tool for the detection of all the solutions of the system.     

Estimating parameters in one-way analysis of covariance model with short-tailed symmetric error distributions

We consider one-way analysis of covariance (ANCOVA) model with a single covariate when the distribution of error terms are short-tailed symmetric. The maximum likelihood (ML) estimators of the parameters are intractable. We, therefore, employ a simple method known as modified maximum likelihood (MML) to derive the estimators of the model parameters. The method is based on linearization of the intractable terms in likelihood equations. Incorporating these linearizations in the maximum likelihood, we get the modified likelihood equations. Then the MML estimators which are the solutions of these modified equations are obtained. Computer simulations were performed to investigate the efficiencies of the proposed estimators. The simulation results show that the proposed estimators are remarkably efficient compared with the conventional least squares (LS) estimators.     

Multiple imputation and other resampling schemes for imputing missing observations

The problem of imputing missing observations under the linear regression model is considered. It is assumed that observations are missing at random and all the observations on the auxiliary or independent variables are available. Estimates of the regression parameters based on singly and multiply imputed values are given. Jackknife as well as bootstrap estimates of the variance of the singly imputed estimator of the regression parameters are given. These estimators are shown to be consistent estimators. The asymptotic distributions of the imputed estimators are also given to obtain interval estimates of the parameters of interest. These interval estimates are then compared with the interval estimates obtained from multiple imputation. It is shown that singly imputed estimators perform at least as good as multiply imputed estimators. A new nonparametric multiply imputed estimator is proposed and shown to perform as good as a multiply imputed estimator under normality. The singly imputed estimator, however, still remains at least as good as a multiply imputed estimator.     

Inference for 2-D GARCH models

The purpose of this paper is, in the first step, to consider a class of GMM estimators with interesting asymptotic properties and a reasonable number of computations for two dimensionally indexed Generalized Autoregressive Conditional Heteroscedasticity (GARCH) model. In the second step, we use the central limit theorem of Huang (1992) for spatial martingale differences to establish the LAN property for general two-dimensional discrete models on a regular grid with Gaussian errors. We then apply this result to the spatial GARCH model and derive the limit distribution of the maximum likelihood estimators of the parameters. Results of numerical simulations are presented.     

Asymptotic expansions for the ability estimator in item response theory

Asymptotic approximations to the distributions of the ability estimator and its transformations in item response theory are derived beyond the usual normal one when associated item parameters are given as in tailored testing. For the approximations, the asymptotic cumulants of the estimators up to the fourth order with the higher-order asymptotic variances are obtained under possible model misspecification. For testing and interval estimation of abilities, the asymptotic cumulants of the pivots studentized in four ways are derived. Numerical examples with simulations including those for confidence intervals for abilities are given using the three-parameter logistic model.     

Asymptotic inference for stochastic processes

This is a survey of some aspects of large-sample inference for stochastic processes. A unified framework is used to study the asymptotic properties of tests and estimators parameters in discrete-time, continuous-time jump-type, and diffusion processes. Two broad families of processes, viz, ergodic and non-ergodic type are introduced and the qualitative differences in the asymptotic results for the two families are discussed and illustrated with several examples. Some results on estimation and testing via Bayesian, nonparametric, and sequential methods are also surveyed briefly.     

Estimation and construction of confidence regions in regression models

In the present paper, linear and nonlinear regression models with a growing number of unknown parameters are considered. Conditions sufficient for the least squares estimators to be consistent are formulated. Estimates for the covariance matrix of least squares estimators, which make it possible to construct confidence regions for the regression function, are given. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 142–150, Perm, 1990.     

A NEW WAY TO ESTIMATE THE PARAMETERS IN THE PROGRESSIVE STRESS ACCELERATED LIFE TESTING

Among the three types of accelerated life testing, that is, the constant stress accelerated life testing, the step stress accelerated life testing, and the progressive stress accelerated life testing, the last one is the most effective and economical way to get the failure times of products in a short period of time. In this artiele a deficiency inthe classical approach is pointed out and a new method for the statisticul analysis of lifedata, especially for electronic products, under progressive stress (voltage) V(t)=Kt is given. Using the Bayesian method, for certain choices of the prior distribution, several forms of estimators of the parameters in the noraml stress We|bull distribution and the htverse power law model are derived. In particular, when the new shape parameter computers. The formulations are given for the general case and are illustrated for a special sitnpio case. comparison with the classical approach using pseudorandom dataas well as real data on solid tantatum electrolytic capacitors shows that the proposed method is better and more effective.     

用于指数可靠性增长模型的一类新的先验分布

该文提出了可用于指数分布产品四种可靠性增长试验方案的一类新的先验分布. 这类先验分布以条件分布形式给出, 它适合可靠性增长试验中的各种情况. 各阶段的条件均值和条件方差的表达式被获得, 先验分布的形式与它们的参数间的关系被讨论. 这些结果有助于与专家意见相结合.本文还给出试验末尾产品可靠性的后验密度, Bayesian估计和Bayesian下限.     

Asymptotic properties of pseudo maximum likelihood estimators and test in semi-parametric copula models with multiple change points

The main purpose of the present paper is to establish the asymptotic properties of pseudo maximum likelihood estimators of the parameters of a multiple change-point model in the multivariate copula models when marginal distributions are unspecified but the copula function is parametrized. A pseudo likelihood ratio-type statistic is proposed for testing a sequence of observations for no change in the copula parameter against possible changes. Finally, a weighted bootstrap procedure that aims at evaluating the limiting distributions is examined.