Effect of Violation of the Normal Assumption on MI and ML Estimators in the Analysis of Incomplete Data

Asymptotic distributions of normal-theory-based ML/MI estimators are studied in a simple regression model under general distributions with MAR missing data. The asymptotic variance of the ML/MI estimator of residuals’ variance is explicitly derived, from which it follows that the kurtosis of the error distribution primarily affects the asymptotic variance. Results of numerical simulations conducted to study finite sample properties of the estimators, conformed largely to the asymptotic results, and they also indicated interesting findings particularly for small samples, which do not follow from the asymptotic property. It is concluded that the ML estimators perform best in the situation studied here.     

Asymptotic optimality of estimators of a linear functional eeiation if the ratio of the error variances is known 2

For the problem of estimating a linear functional relation when the ratio of the error variances is known a general class of estimators is introduced. They include as special cases the instrumental variable and replication cases and some others. Conditions are given for consistency, asymptotic normality and asymptotic optimality within this class based on the variance of the limit distribution. Fisheb's lower bound for asymptotic variances is established, and under normality the asymptotically optimal estimators are shown to be best asymptotically normal. For an inhomogeneous linear relation only estimators which are invariant with respect to a translation of the origin are considered, and asymptotically optimal invariant and, under normality, best asymptotically normal invariant estimators are obtained. Several special cases are discussed.     

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.     

Sampling properties of estimators of the log-logistic distribution with application to Canadian precipitation data

We consider the probability-weighted moment and the maximum-likelihood estimators of two parameters in the log-logistic distribution. Quantile estimators are obtained using both methods. The distributional properties of these estimators are studied in large samples, via asymptotic theory, and in small and moderate samples, via Monte Carlo simulation. The distribution is shown to be appropriate for a wide variety of meteorological data.     

Least Squares estimation for the multivariate weibull model of bougaard based on accelerated life test of system and component

Accelerated life testing of products quickly yields information on life. In this article, we present a simple method to incorporate the information collected from accelerated life tests of both components and (series) systems. The multivariate Weibull distribution of Hougaard is applied to model lifetimes of components. Least-squares (LS) estimators of the model parameters and their joint asymptotic distribution are derived. The effects of the dependence parameter and the proportion of the system-data to the asymptotic relative efficiencies of the LS estimators are investigated.     

Large sample behavior of the Bernstein copula estimator

Bernstein polynomial estimators have been used as smooth estimators for density functions and distribution functions. The idea of using them for copula estimation has been given in Sancetta and Satchell (2004). In the present paper we study the asymptotic properties of this estimator: almost sure consistency rates and asymptotic normality. We also obtain explicit expressions for the asymptotic bias and asymptotic variance and show the improvement of the asymptotic mean squared error compared to that of the classical empirical copula estimator. A small simulation study illustrates this superior behavior in small samples.     

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.     

Asymptotic Normality of Extreme Quantile Estimators Based on the Peaks-Over-Threshold Approach

The POT (Peaks-Over-Threshold) approach consists of using the generalized Pareto distribution (GPD) to approximate the distribution of excesses over thresholds. In this article, we establish the asymptotic normality of the well-known extreme quantile estimators based on this POT method, under very general assumptions. As an illustration, from this result, we deduce the asymptotic normality of the POT extreme quantile estimators in the case where the maximum likelihood (ML) or the generalized probability-weighted moments (GPWM) methods are used. Simulations are provided in order to compare the efficiency of these estimators based on ML or GPWM methods with classical ones proposed in the literature.     

On the asymptotic efficiency of moment and maximum likelihood estimators in the three-parameter inverse gaussian distribution

The asymptotic distribution of estimators generated by the methods of moments and maximum likelihood are considered. Simple formulae are provided which enable comparisons of asymptotic relative efficiency to be effected.     

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.     

Robust estimation of AR coefficients under simultaneously influencing outliers and missing values

A family of robust estimators for coefficients of Gaussian AR(p) time series under simultaneously influencing distortions of two types: outliers and missing values, is proposed. The estimators are based on special properties of the Cauchy probability distribution; consistency and the asymptotic normality of these estimators are proven. An approximate solution of the problem of minimization of the asymptotic variance within the proposed family of estimators is found. Performance of the proposed estimators is illustrated for simulated time series and for real data sets.     

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.     

Edgeworth Expansion and Bootstrap Approximation for Studentized Product-Limit Estimator with Truncated and Censored Data

The Yule-Walker estimators of the AR coefficients of a causal multidimensional AR model are obtained by replacing the autocovariances with their estimators in the Yule-Walker equations. It is shown that only unbiased-type estimators of the autocovariances yield consistency of the Yule-Walker estimators. Also, the asymptotic joint distribution of the Yule-Walker estimators is presented.     

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.     

Closed-Form Estimators for the Gamma Distribution Derived From Likelihood Equations

It is well-known that maximum likelihood (ML) estimators of the two parameters in a gamma distribution do not have closed forms. This poses difficulties in some applications such as real-time signal processing using low-grade processors. The gamma distribution is a special case of a generalized gamma distribution. Surprisingly, two out of the three likelihood equations of the generalized gamma distribution can be used as estimating equations for the gamma distribution, based on which simple closed-form estimators for the two gamma parameters are available. Intuitively, performance of the new estimators based on likelihood equations should be close to the ML estimators. The study consolidates this conjecture by establishing the asymptotic behaviors of the new estimators. In addition, the closed-forms enable bias-corrections to these estimators. The bias-correction significantly improves the small-sample performance.     

A Pointwise Estimator for the k-Fold Convolution of a Distribution Function

ABSTRACT

This article is concerned with some parametric and nonparametric estimators for the k-fold convolution of a distribution function. An alternative estimator is proposed and its unbiasedness, asymptotic unbiasedness, and consistency properties are investigated. The asymptotic normality of this estimator is established. Some applications of the estimator are given in renewal processes. Finally, the computational procedures are described and the relative performance of these estimators for small sample sizes is investigated by a simulation study.     

Pitman closeness comparison of linear estimators:a canonical form

A general form is presented for the comparison of two linear estimators of a common parameter by means of the Pitman measure of closeness. Several asymptotic results are given. The case in which the estimators are linear combinations of the order statistics is discussed. The asymptotic comparison of the sample mean versus the sample median is derived for the Laplace distribution, and two other examples are given.     

Estimation using plug-in of the stationary distribution and Shannon entropy of continuous time Markov processes

A natural way to deal with the uncertainty of an ergodic finite state space Markov process is to investigate the entropy of its stationary distribution. When the process is observed, it becomes necessary to estimate this entropy.We estimate both the stationary distribution and its entropy by plug-in of the estimators of the infinitesimal generator. Three situations of observation are discussed: one long trajectory is observed, several independent short trajectories are observed, or the process is observed at discrete times. The good asymptotic behavior of the plug-in estimators is established. We also illustrate the behavior of the estimators through simulation.     

Estimating the conditional tail index by integrating a kernel conditional quantile estimator

This paper deals with the estimation of the tail index of a heavy-tailed distribution in the presence of covariates. A class of estimators is proposed in this context and its asymptotic normality established under mild regularity conditions. These estimators are functions of a kernel conditional quantile estimator depending on some tuning parameters. The finite sample properties of our estimators are illustrated on a small simulation study.     

Simultaneous confidence intervals on partial means of classes in the two-stage stratified sampling

Let’s consider a finite population of P units, each of them assumes a specific amount of the quantitative variable X. Moreover we assume that the range of values of X is subdivided into k classes and the sampling data come out from a two stage stratified sampling. The main purpose of the work is to determine the estimators, as well as their asymptotic distribution, of the partial means of classes, each of them is defined as a non linear function of the other parameters. Particularly, we are interested in determining the linear approximation estimators and, under convergence theorems, the asymptotic distribution. Afterwards we define the estimator of the vector of the partial means of classes and its asymptotic convergence to multivariate normal distribution is determined. These results are useful to develop simultaneous inferential procedures.