Estimation of the SURE model

The paper presents the essentials of the SURE model and the estimation of its parameters β and ω. Two alternative compact representations of the model are being used. The parameter β is estimated by least squares (LS), generalized least squares (GLS) and maximum likelihood (ML) (under normality). For ω two estimators are being considered, viz an LS-related estimator and a maximum likelihood estimator (under normality). Attention is being given to the study of asymptotic properties of all estimators examined. It turns out that the LS-related and ML estimators of ω follow the same asymptotic (normal) distribution. Efficiency comparisons for the various estimators of β conclude the paper.     

Optimal step-stress test under type I progressive group-censoring with random removals

Some traditional life tests result in no or very few failures by the end of test. In such cases, one approach is to do life testing at higher-than-usual stress conditions in order to obtain failures quickly. This paper discusses a k-level step-stress accelerated life test under type I progressive group-censoring with random removals. An exponential failure time distribution with mean life that is a log-linear function of stress and a cumulative exposure model are considered. We derive the maximum likelihood estimators of the model parameters and establish the asymptotic properties of the estimators. We investigate four selection criteria which enable us to obtain the optimum test plans. One is to minimize the asymptotic variance of the maximum likelihood estimator of the logarithm of the mean lifetime at use-condition, and the other three criteria are to maximize the determinant, trace and the smallest eigenvalue of Fisher's information matrix. Some numerical studies are discussed to illustrate the proposed criteria.     

Asymptotic theory for bent-cable regression—the basic case

We use what we call the bent-cable model to describe potential change-point phenomena. The class of bent cables includes the commonly used broken stick (a bent cable without a bend segment). Theory for least-squares (LS) estimation is developed for the basic bent cable, whose incoming and outgoing linear phases have slopes 0 and 1, respectively, and are joined smoothly by a quadratic bend. Conditions on the design are given to ensure regularity of the estimation problem, despite non-differentiability of the model's first partial derivatives (with respect to the covariate and model parameters). Under such conditions, we show that the LS estimators (i) are consistent, regardless of a zero or positive true bend width; and (ii) asymptotically follow a bivariate normal distribution, if the underlying cable has all three segments. In the latter case, we show that the deviance statistic has an asymptotic chi-squared distribution with two degrees of freedom.     

A comparison of LS/ML and GMM estimation in a simple AR(1) model

This paper compares least squares (LS)/maximum likelihood (ML) and generalised method of moments (GMM) estimation in a simple. Gaussian autoregressive of order one (AR(1)) model. First, we show that the usual LS/ML estimator is a corner solution to a general minimisation problem that involves two moment conditions, while the new GMM we devise is not. Secondly, we examine asymptotic and finite sample properties of the new GMM estimator in comparison to the usual LS/ML estimator in a simple AR(1) model. For both stable and unstable (unit root) specifications, we show asymptotic equivalence of the distributions of the two estimators. However, in finite samples, the new GMM estimator performs better.     

Stochastic analysis of covariance when the error distribution is long-tailed symmetric

In this study, we consider stochastic one-way analysis of covariance model when the distribution of the error terms is long-tailed symmetric. Estimators of the unknown model parameters are obtained by using the maximum likelihood (ML) methodology. Iteratively reweighting algorithm is used to compute the ML estimates of the parameters. We also propose new test statistic based on ML estimators for testing the linear contrasts of the treatment effects. In the simulation study, we compare the efficiencies of the traditional least-squares (LS) estimators of the model parameters with the corresponding ML estimators. We also compare the power of the test statistics based on LS and ML estimators, respectively. A real-life example is given at the end of the study.     

ASYMPTOTIC PROPERTIES OF RESTRICTED MAXIMUM LIKELIHOOD (REML) ESTIMATES FOR HIERARCHICAL MIXED LINEAR MODELS

This paper explores the asymptotic distribution of the restricted maximum likelihood estimator of the variance components in a general mixed model. Restricting attention to hierarchical models, central limit theorems are obtained using elementary arguments with only mild conditions on the covariates in the fixed part of the model and without having to assume that the data are either normally or spherically symmetrically distributed. Further, the REML and maximum likelihood estimators are shown to be asymptotically equivalent in this general framework, and the asymptotic distribution of the weighted least squares estimator (based on the REML estimator) of the fixed effect parameters is derived.     

Estimation in a truncated bivariate poisson distribution

Moment estimators for parameters in a truncated bivariate Poisson distribution are derived in Hamdan (1972) for the special case of λ₁ = λ₂, Where λ₁, λ₂ are the marginal means. Here we derive the maximum likelihood estimators for this special case. The information matrix is also obtained which provides asymptotic covariance matrix of the maximum likelihood estimators. The asymptotic covariance matrix of moment estimators is also derived. The asymptotic efficiency of moment estimators is computed and found to be very low.     

On the estimation of reliabilty function in a Weibull lifetime distribution

The estimation of the reliability function of the Weibull lifetime model is considered in the presence of uncertain prior information (not in the form of prior distribution) on the parameter of interest. This information is assumed to be available in some sort of a realistic conjecture. In this article, we focus on how to combine sample and non-sample information together in order to achieve improved estimation performance. Three classes of point estimatiors, namely, the unrestricted estimator, the shrinkage estimator and shrinkage preliminary test estimator (SPTE) are proposed. Their asymptotic biases and mean-squared errors are derived and compared. The relative dominance picture of the estimators is presented. Interestingly, the proposed SPTE dominates the unrestricted estimator in a range that is wider than that of the usual preliminary test estimator. A small-scale simulation experiment is used to examine the small sample properties of the proposed estimators. Our simulation investigations have provided strong evidence that corroborates with asymptotic theory. The suggested estimation methods are applied to a published data set to illustrate the performance of the estimators in a real-life situation.     

On the comparison of Fisher information of the Weibull and GE distributions

In this paper, we consider the Fisher information matrices of the generalized exponential (GE) and Weibull distributions for complete and Type-I censored observations. Fisher information matrix can be used to compute asymptotic variances of the different estimators. Although both distributions may provide similar data fit but the corresponding Fisher information matrices can be quite different. Moreover, the percentage loss of information due to truncation of the Weibull distribution is much more than the GE distribution. We compute the total information of the Weibull and GE distributions for different parameter ranges. We compare the asymptotic variances of the median estimators and the average asymptotic variances of all the percentile estimators for complete and Type-I censored observations. One data analysis has been preformed for illustrative purposes. When two fitted distributions are very close to each other and very difficult to discriminate otherwise, the Fisher information or the above mentioned asymptotic variances may be used for discrimination purposes.     

Optimality of quasi-score in the multivariate mean–variance model with an application to the zero-inflated Poisson model with measurement errors

In a multivariate mean–variance model, the class of linear score (LS) estimators based on an unbiased linear estimating function is introduced. A special member of this class is the (extended) quasi-score (QS) estimator. It is ‘extended’ in the sense that it comprises the parameters describing the distribution of the regressor variables. It is shown that QS is (asymptotically) most efficient within the class of LS estimators. An application is the multivariate measurement error model, where the parameters describing the regressor distribution are nuisance parameters. A special case is the zero-inflated Poisson model with measurement errors, which can be treated within this framework.     

Model averaging procedure for partially linear single-index models

This paper is concerned with model selection and model averaging procedures for partially linear single-index models. The profile least squares procedure is employed to estimate regression coefficients for the full model and submodels. We show that the estimators for submodels are asymptotically normal. Based on the asymptotic distribution of the estimators, we derive the focused information criterion (FIC), formulate the frequentist model average (FMA) estimators and construct proper confidence intervals for FMA estimators and FIC estimator, a special case of FMA estimators. Monte Carlo studies are performed to demonstrate the superiority of the proposed method over the full model, and over models chosen by AIC or BIC in terms of coverage probability and mean squared error. Our approach is further applied to real data from a male fertility study to explore potential factors related to sperm concentration and estimate the relationship between sperm concentration and monobutyl phthalate.     

Estimation of the linear-linear segmented regression model in the presence of measurement error

A simple segmented regression model in which the independent variable is measured with error is considered. The method of moments is used to obtain parameter estimates and the joint asymptotic distribution of the estimators is presented. The small sample properties of the inference procedures based on the asymptotic distribution of the estimators are studied numerically.     

A fiducial-based approach to the one-way ANOVA in the presence of nonnormality and heterogeneous error variances

In this study, we propose a new test for testing the equality of the treatment means in one-way ANOVA when the usual normality and the homogeneity of variances assumptions are not met. In developing the proposed test, we benefit from the Fisher's fiducial inference [1–3]. Distribution of the error terms is assumed to be long-tailed symmetric (LTS) which includes the normal distribution as a limiting case. Modified maximum likelihood (MML) estimators are used in the test statistics rather than the traditional least squares (LS) estimators, since LS estimators have very low efficiencies under nonnormal distributions, see Tiku [4] for the details of MML methodology. An extensive Monte Carlo simulation study is done to compare the efficiency of the proposed test with the corresponding test based on normal theory, see Li et al. [5]. Finally, we give a real life example to show the applicability of the proposed methodology.     

Nonparametric smooth estimation of the expected inactivity time function

The expected inactivity time (EIT) function (also known as the mean past lifetime function) is a well known reliability function which has application in many disciplines such as survival analysis, actuarial studies and forensic science, to name but a few. In this paper, we use a fixed design local polynomial fitting technique to obtain estimators for the EIT function when the lifetime random variable has an unknown distribution. It will be shown that the proposed estimators are asymptotically unbiased, consistent and also, when standardized, has an asymptotic normal distribution. An optimal bandwidth, which minimizes the AMISE (asymptotic mean integrated squared error) of the estimator, is derived. Numerical examples based on simulated samples from various lifetime distributions common in reliability studies will be presented to evaluate the performances of these estimators. Finally, three real life applications will also be presented to further illustrate the wide applicability of these estimators.     

Nonparametric estimation with recurrent competing risks data

Nonparametric estimators of component and system life distributions are developed and presented for situations where recurrent competing risks data from series systems are available. The use of recurrences of components’ failures leads to improved efficiencies in statistical inference, thereby leading to resource-efficient experimental or study designs or improved inferences about the distributions governing the event times. Finite and asymptotic properties of the estimators are obtained through simulation studies and analytically. The detrimental impact of parametric model misspecification is also vividly demonstrated, lending credence to the virtue of adopting nonparametric or semiparametric models, especially in biomedical settings. The estimators are illustrated by applying them to a data set pertaining to car repairs for vehicles that were under warranty.     

Statistic inference for a single-index ARCH-M model

For a single-index autoregressive conditional heteroscedastic (ARCH-M) model, estimators of the parametric and non parametric components are proposed by the profile likelihood method. The research results had shown that all the estimators have consistency and the parametric estimators have asymptotic normality. We extend this line of research by deriving the asymptotic normality of the non parametric estimator. Based on the asymptotic properties, we propose Wald statistic and generalized likelihood ratio statistic to investigate the testing problems for ARCH effect and goodness of fit, respectively. A simulation study is conducted to evaluate the finite-sample performance of the proposed estimation methodology and testing procedure.     

Some computational aspects of the generalized von Mises distribution

This article deals with some important computational aspects of the generalized von Mises distribution in relation with parameter estimation, model selection and simulation. The generalized von Mises distribution provides a flexible model for circular data allowing for symmetry, asymmetry, unimodality and bimodality. For this model, we show the equivalence between the trigonometric method of moments and the maximum likelihood estimators, we give their asymptotic distribution, we provide bias-corrected estimators of the entropy, the Akaike information criterion and the measured entropy for model selection, and we implement the ratio-of-uniforms method of simulation.     

Estimation in step-stress partially accelerated life tests for the Burr type XII distribution using type I censoring

In this paper, step-stress partially accelerated life tests are considered when the lifetime of a product follows a Burr type XII distribution. Based on type I censoring, the maximum likelihood estimates (MLEs) are obtained for the distribution parameters and acceleration factor. In addition, asymptotic variance and covariance matrix of the estimators are given. An iterative procedure is used to obtain the estimators numerically using Mathcad (2001). Furthermore, confidence intervals of the estimators are presented. Simulation results are carried out to study the precision of the MLEs for the parameters involved.     

Information attainable in some randomly incomplete data models

The Fisher information is intricately linked to the asymptotic (first-order) optimality of maximum likelihood estimators for parametric complete-data models. When data are missing completely at random in a multivariate setup, it is shown that information in a single observation is well-defined and it plays the same role as in the complete-data model in characterizing the first-order asymptotic optimality properties of associated maximum likelihood estimators; computational aspects are also thoroughly appraised. As an illustration, the logistic regression model with incomplete binary responses and an incomplete categorical covariate is worked out.     

Estimation for Location-Scale Models with Censored Data

This article considers a class of estimators for the location and scale parameters in the location-scale model based on ‘synthetic data’ when the observations are randomly censored on the right. The asymptotic normality of the estimators is established using counting process and martingale techniques when the censoring distribution is known and unknown, respectively. In the case when the censoring distribution is known, we show that the asymptotic variances of this class of estimators depend on the data transformation and have a lower bound which is not achievable by this class of estimators. However, in the case that the censoring distribution is unknown and estimated by the Kaplan–Meier estimator, this class of estimators has the same asymptotic variance and attains the lower bound for variance for the case of known censoring distribution. This is different from censored regression analysis, where asymptotic variances depend on the data transformation. Our method has three valuable advantages over the method of maximum likelihood estimation. First, our estimators are available in a closed form and do not require an iterative algorithm. Second, simulation studies show that our estimators being moment-based are comparable to maximum likelihood estimators and outperform them when sample size is small and censoring rate is high. Third, our estimators are more robust to model misspecification than maximum likelihood estimators. Therefore, our method can serve as a competitive alternative to the method of maximum likelihood in estimation for location-scale models with censored data. A numerical example is presented to illustrate the proposed method.