Lifetime distribution and estimation problems of consecutive-k-out-of-n:F systems

Explicit formula is given for the lifetime distribution of a consecutive-k-out-of-n:F system. It is given as a linear combination of distributions of order statistics of the lifetimes of n components. We assume that the lifetimes are independent and identically distributed. The results should make it possible to treat the parametric estimation problems based on the observations of the lifetimes of the system. In fact, we take up, as some examples, the cases where the lifetimes of the components follow the exponential, the Weibull, and the Pareto distributions, and obtain feasible estimators by moment method. In particular, it is shown that the moment estimator is quite good for the exponential case in the sense that the asymptotic efficiency is close to one.This research was partially supported by the ISM Cooperative Research Program (94-ISM-CRP-5).     

On stable laws for estimating functions and derived estimators

Summary Stable laws forM-estimators, maximum likelihood and other estimators and obtained through parallel results for the estimating functions and relative compactness of some related estimating functional processes. Work supported by the Office of Naval Research, Contract No. N00014-83-K-0387.     

Consistent nonparametric regression from recursive partitioning schemes

We here extend our results on asymptotically Bayes risk efficient classification to the general regression scenario. More precisely, we find L^p consistent estimators for an arbitrary regression function provided only that the dependent variable has a finite absolute pth moment. The estimators are truncated and untruncated local means derived from recursive partitioning schemes.     

Comparisons among some estimators in misspecified linear models with multicollinearity

In this paper we deal with comparisons among several estimators available in situations of multicollinearity (e.g., the r-k class estimator proposed by Baye and Parker, the ordinary ridge regression (ORR) estimator, the principal components regression (PCR) estimator and also the ordinary least squares (OLS) estimator) for a misspecified linear model where misspecification is due to omission of some relevant explanatory variables. These comparisons are made in terms of the mean square error (mse) of the estimators of regression coefficients as well as of the predictor of the conditional mean of the dependent variable. It is found that under the same conditions as in the true model, the superiority of the r-k class estimator over the ORR, PCR and OLS estimators and those of the ORR and PCR estimators over the OLS estimator remain unchanged in the misspecified model. Only in the case of comparison between the ORR and PCR estimators, no definite conclusion regarding the mse dominance of one over the other in the misspecified model can be drawn.     

Universally consistent conditionalU-statistics for absolutely regular processes and its applications for hidden Markov models

A general class of conditionalU-statistics was introduced by W. Stute as a generalization of the Nadaraya-Watson estimates of a regression function. It was shown that such statistics are universally consistent. Also, universal consistentcies of the window andk _n -nearest neighbor estimators (as two special cases of the conditionalU-statistics) were proved. In this paper, we extend these results from the independent case to dependent case. The result is applied to verify the Bayes risk consistency of the corresponding discrimination rules. Research supported by the Office of Naval Research Contract N00014-91-J-1020.     

On the Waiting Time for the First Success Run

Let k and m are positive integers with k ≥ m. The probability generating function of the waiting time for the first occurrence of consecutive k successes in a sequence of m-th order Markov dependent trials is given as a function of the conditional probability generating functions of the waiting time for the first occurrence of consecutive m successes. This provides an efficient algorithm for obtaining the probability generating function when k is large. In particular, in the case of independent trials a simple relationship between the geometric distribution of order k and the geometric distribution of order k−1 is obtained. This research was partially supported by the ISM Cooperative Research Program(2004-ISM-CRP-2006) and by a Grant-in-Aid for Scientific Research (C) of the JSPI (Grant Number 16500183)     

Asymptotics of R-, MD- and LAD-estimators in linear regression models with long range dependent errors

Summary This paper establishes the uniform closeness of a weighted residual empirical process to its natural estimate in the linear regression setting when the errors are Gaussian, or a function of Gaussian random variables, that are strictly stationary and long range dependent. This result is used to yield the asymptotic uniform linearity of a class of rank statistics in linear regression models with long range dependent errors. The latter result, in turn, yields the asymptotic distribution of the Jaeckel (1972) rank estimators. The paper also studies the least absolute deviation and a class of certain minimum distance estimators of regression parameters and the kernel type density estimators of the marginal error density when the errors are long range dependent.Research of this author was partly supported by the NSF grant: DMS-9102041     

Mixed moment estimator and location invariant alternatives

A new class of estimators of the extreme value index is developed. It has a simple form and is asymptotically very close to the maximum likelihood estimator for a wide class of heavy-tailed models. We also propose an alternative class of estimators, dependent on a tuning parameter p ∈ (0,1) and invariant for changes in both scale and/or location. Such a tuning parameter can help us to choose the number of top order statistics to be used in the estimation of extreme parameters. Research partially supported by FCT / POCTI, POCI, PCDT and PPCDT / FEDER.     

Minimum <Emphasis Type="Italic">Φ</Emphasis>-Divergence Estimator and <Emphasis Type="Italic">Φ</Emphasis>-Divergence Statistics in Generalized Linear Models with Binary Data

In this paper, we assume that the data are distributed according to a binomial distribution whose probabilities follow a generalized linear model. To fit the data the minimum φ-divergence estimator is studied as a generalization of the maximum likelihood estimator. We use the minimum φ-divergence estimator, which is the basis of some new statistics, for solving the problems of testing in a generalized linear model with binary data. A wide simulation study is carried out for studying the behavior of the new family of estimators as well as of the new family of test statistics. This work was partially supported by Grant MTM2006-06872 and UCM2006-910707.     

Testing a sub-hypothesis in linear regression models with long memory covariates and errors

This paper considers the problem of testing a sub-hypothesis in homoscedastic linear regression models when the covariate and error processes form independent long memory moving averages. The asymptotic null distribution of the likelihood ratio type test based on Whittle quadratic forms is shown to be a chi-square distribution. Additionally, the estimators of the slope parameters obtained by minimizing the Whittle dispersion is seen to be n ^1/2-consistent for all values of the long memory parameters of the design and error processes. Research of the first author was partly supported by the NSF DMS Grant 0701430. Research of the second author was partly supported by the bilateral France-Lithuania scientific project Gilibert and the Lithuanian State Science and Studies Foundation grant T-15/07.     

Semiparametric quantile regression with random censoring

This paper considers estimation and inference in semiparametric quantile regression models when the response variable is subject to random censoring. The paper considers both the cases of independent and dependent censoring and proposes three iterative estimators based on inverse probability weighting, where the weights are estimated from the censoring distribution using the Kaplan–Meier, a fully parametric and the conditional Kaplan–Meier estimators. The paper proposes a computationally simple resampling technique that can be used to approximate the finite sample distribution of the parametric estimator. The paper also considers inference for both the parametric and nonparametric components of the quantile regression model. Monte Carlo simulations show that the proposed estimators and test statistics have good finite sample properties. Finally, the paper contains a real data application, which illustrates the usefulness of the proposed methods.

     

Asymptotic cumulants of the parameter estimators in item response theory

The asymptotic cumulants of the parameter estimators for the three-parameter logistic model in item response theory are derived up to the fourth order with the higher-order added asymptotic variances. The asymptotic cumulants of the corresponding Studentized estimators up to the third order are also given. The estimators are obtained by marginal maximum likelihood using the standard normal distribution for the latent variable with and without model misspecification. Numerical examples with fixed guessing parameters show advantages of the asymptotic expansions over the usual normal approximation. This work was partially supported by Grant-in-Aid for Scientific Research from the Japanese Ministry of Education, Culture, Sports, Science and Technology.     

Retrial queues with collision arising from unslottedCSMA/CD protocol

We consider a retrial queueing model with collision arising from the specific communication protocolCSMAICD. Under the retrial control policy in which the retrial rate is inversely proportional to the number of customers in the retrial group, we derive the generating function of the limiting distribution of the number of customers in the retrial group at the moment when the channel is free. Using the theory of Markov regenerative processes, we also obtain the limiting distribution of the number of customers in the system at arbitrary time points.This paper was supported in part by the Non-Directed Research Fund, Korea Research Foundation, 1990.     

Inference for ordered parameters in multinomial distributions

This paper discusses inference for ordered parameters of multinomial distributions. We first show that the asymptotic distributions of their maximum likelihood estimators (MLEs) are not always normal and the bootstrap distribution estimators of the MLEs can be inconsistent. Then a class of weighted sum estimators (WSEs) of the ordered parameters is proposed. Properties of the WSEs are studied, including their asymptotic normality. Based on those results, large sample inferences for smooth functions of the ordered parameters can be made. Especially, the confidence intervals of the maximum cell probabilities are constructed. Simulation results indicate that this interval estimation performs much better than the bootstrap approaches in the literature. Finally, the above results for ordered parameters of multinomial distributions are extended to more general distribution models. This work was supported by National Natural Science Foundation of China (Grant No. 10371126)     

Optimal designs with respect to Elfving's partial minimax criterion in polynomial regression

For the polynomial regression model on the interval [a, b] the optimal design problem with respect to Elfving's minimax criterion is considered. It is shown that the minimax problem is related to the problem of determining optimal designs for the estimation of the individual parameters. Sufficient conditions are given guaranteeing that an optimal design for an individual parameter in the polynomial regression is also minimax optimal for a subset of the parameters. The results are applied to polynomial regression on symmetric intervals [–b, b] (b1) and on nonnegative or nonpositive intervals where the conditions reduce to very simple inequalities, involving the degree of the underlying regression and the index of the maximum of the absolute coefficients of the Chebyshev polynomial of the first kind on the given interval. In the most cases the minimax optimal design can be found explicitly.Research supported in part by the Deutsche Forschungsgemeinschaft.Research supported in part by NSF Grant DMS 9101730.     

Minimum f-divergence estimators and quasi-likelihood functions

Maximum quasi-likelihood estimators have several nice asymptotic properties. We show that, in many situations, a family of estimators, called the minimum f-divergence estimators, can be defined such that each estimator has the same asymptotic properties as the maximum quasi-likelihood estimator. The family of minimum f-divergence estimators include the maximum quasi-likelihood estimators as a special case. When a quasi-likelihood is the log likelihood from some exponential family, Amari's dual geometries can be used to study the maximum likelihood estimator. A dual geometric structure can also be defined for more general quasi-likelihood functions as well as for the larger family of minimum f-divergence estimators. The relationship between the f-divergence and the quasi-likelihood function and the relationship between the f-divergence and the power divergence is discussed.This work was supported by National Science Foundation grant DMS 88-03584.     

Variable selection via quantile regression with the process of Ornstein-Uhlenbeck type

Based on the data-cutoff method,we study quantile regression in linear models,where the noise process is of Ornstein-Uhlenbeck type with possible jumps.In single-level quantile regression,we allow the noise process to be heteroscedastic,while in composite quantile regression,we require that the noise process be homoscedastic so that the slopes are invariant across quantiles.Similar to the independent noise case,the proposed quantile estimators are root-n consistent and asymptotic normal.Furthermore,the adaptive least absolute shrinkage and selection operator(LASSO)is applied for the purpose of variable selection.As a result,the quantile estimators are consistent in variable selection,and the nonzero coefficient estimators enjoy the same asymptotic distribution as their counterparts under the true model.Extensive numerical simulations are conducted to evaluate the performance of the proposed approaches and foreign exchange rate data are analyzed for the illustration purpose.     

Component risk in multiparameter estimation

Summary For estimating the mean of ap-variate normal distribution under a quadratic loss, a class of estimators, known as Stein's estimators, is known to dominate the maximum likelihood estimator (MLE) forp≧3. But, whereas the risk of the MLE has the same value, equal to a constant, for each component, the maximum component risk of Stein's estimator is large for large values ofp. Certain modification of Stein's rule has been proposed in the literature for reducing the maximum component risk. In this paper, a new rule is given for reducing the maximum component risk. The new rule yields larger reduction in the maximum component risk, compared to its competitor.     

Efficiencies of tests and estimators forp-order autoregressive processes when the error distribution is nonnormal

Summary We considerpth order autoregressive time series where the shocks need not be normal. By employing the concept of contiguity, we obtain the sysmptotic power for tests of hypothesis concerning the autoregressive parameters. Our approach allows consideration of the double exponential and other thicker-tailed distributions for the shocks. We derive a new result in the contiguity framework that leads directly to an expression for the Pitman efficiencies of tests as well as estimators. The numerical values of the efficiencies suggest a lack of robustness for the normal theory least squares estimators when the shock distribution is thick tailed or an outlier prone mixed normal. An important alternative test statistic is proposed that competes with the normal theory tests. This research was supported by the Office of Naval Research under Grant No. N00014-78-C-0722 and by the Army Research Office.     

SEM modeling with singular moment matrices Part III: GLS estimation

We discuss generalized least squares (GLS) and maximum likelihood (ML) estimation for structural equations models (SEM), when the sample moment matrices are possibly singular. This occurs in several instances, for example, for panel data when there are more panel waves than independent replications or for time series data where the number of time points is large, but only one unit is observed. In previous articles, it was shown that ML estimation of the SEM is possible by using a correct Gaussian likelihood function. In this article, the usual GLS fit function is modified so that it is also defined for singular sample moment matrices S. In large samples, GLS and ML estimation perform similarly, and the modified GLS approach is a good alternative when S becomes nearly singular. Both GLS approaches do not work for N = 1, since here S = 0 and the modified GLS approach yields biased estimates. In conclusion, ML estimation (and pseudo ML under misspecification) is recommended for all sample sizes including N = 1.