On the estimation of the parameters of the von mises distribution

The usual maximum likelihood estimators of the parameters of the von Mises distribution are shown to perform badly in small samples. In view of this and the fact that these estimators require a large amount of computation, alternative, simpler estimators are proposed. It is shown that these estimators are at least comparable to the traditional estimators and are, in many cases, superior to them. We also apply the procedure of jackknifing to the maximum likelihood estimator of the concentration parameter of the von Mises distribution and compare the properties of the jackknifed estimator with the other estimators considered in this paper.     

A New Estimation Method Based on Order Statistics in the Families of Symmetric Location-scale Distributions

In this study, new unbiased and nonlinear estimators based on order statistics are proposed for the family of symmetric location-scale distributions and these estimators can be computed from both uncensored and symmetric doubly Type II censored samples. In addition, other relevant unbiased estimators are proposed to estimate standard deviations of these new estimators. A simulation study has been performed to evaluate the performance of the new estimators compared to BLU estimators for small sample sizes. As a result of the simulation study, the new estimators proposed for the location-scale family in general performed nearly as good as BLU estimators. Furthermore, the computational advantage of the proposed estimators over BLU and ML estimators are worthy of notice. In addition, these new estimators have been applied to real data, and the estimation results obtained have been compatible with those of BLUE methods.     

Estimation of three-parameter exponentiated-Weibull distribution under type-II censoring

The present article obtains the point estimators of the exponentiated-Weibull parameters when all the three parameters of the distribution are unknown. Maximum likelihood estimator generalized maximum likelihood estimator and Bayes estimators are proposed for three-parameter exponentiated-Weibull distribution when available sample is type-II censored. Independent non-informative types of priors are considered for the unknown parameters to develop generalized maximum likelihood estimator and Bayes estimators. Although the proposed estimators cannot be expressed in nice closed forms, these can be easily obtained through the use of appropriate numerical techniques. The performances of these estimators are studied on the basis of their risks, computed separately under LINEX loss and squared error loss functions through Monte-Carlo simulation technique. An example is also considered to illustrate the estimators.     

Heteroscedasticity and Distributional Assumptions in the Censored Regression Model

Data censoring causes ordinary least-square estimators of linear models to be biased and inconsistent. The Tobit estimator yields consistent estimators in the presence of data censoring if the errors are normally distributed. However, nonnormality or heteroscedasticity results in the Tobit estimators being inconsistent. Various estimators have been proposed for circumventing the normality assumption. Some of these estimators include censored least absolute deviations (CLAD), symmetrically censored least-square (SCLS), and partially adaptive estimators. CLAD and SCLS will be consistent in the presence of heteroscedasticity; however, SCLS performs poorly in the presence of asymmetric errors. This article extends the partially adaptive estimation approach to accommodate possible heteroscedasticity as well as nonnormality. A simulation study is used to investigate the estimators’ relative performance in these settings. The partially adaptive censored regression estimators have little efficiency loss for censored normal errors and appear to outperform the Tobit and semiparametric estimators for nonnormal error distributions and be less sensitive to the presence of heteroscedasticity. An empirical example is considered, which supports these results.     

On invariant quadratic unbiased estimation of variance components

The present study deals with three different invarint quadratic unbiased estimators (IQUE) for variance components namely quadratic least squares estimators (QLSE), weighted quadratic least squares estimators (WQLSE) and Mitra type estimators (MTE). The variance and covariances of these three different estimators are presented for unbalanced one-way random model. The relative performances of these estimators are assessed based on different optimality criteria like, D-optimality, T-optimality and M-optimality together with variances of these estimators. As a result, it has been shown that MTE has optimal properties.     

Plug-in estimators for the mean value and variance functions in delayed renewal processes

Abstract

In this paper, we deal with the problem of estimating the delayed renewal and variance functions in delayed renewal processes. Two parametric plug-in estimators for these functions are proposed and their unbiasedness, asymptotic unbiasedness and consistency properties are investigated. The asymptotic normality of these estimators are established. Further, a method for the computation of the estimators is given. Finally, the performances of the estimators are evaluated for small sample sizes by a simulation study.     

Unbiased estimation of the MSE matrices of improved estimators in linear regression

Stein-rule and other improved estimators have scarcely been used in empirical work. One major reason is that it is not easy to obtain precision measures for these estimators. In this paper, we derive unbiased estimators for both the mean squared error (MSE) and the scaled MSE matrices of a class of Stein-type estimators. Our derivation provides the basis for measuring the estimators' precision and constructing confidence bands. Comparisons are made between these MSE estimators and the least squares covariance estimator. For illustration, the methodology is applied to data on energy consumption.     

Finite-sample properties of limited-iinformation estimators in misspecified simultaneous equation models

The two most common limited-information estimators in Simultaneous Equation Models are the two-stage least squares and limited-information maximum likelihood estimators. As both of these estimators are complicated functions of the underlying random variables, their exact distributions are difficult to derive. Consequently, their use was first justified on the basis of large sample criteria, such as consistency and asymptotic efficiency. However, in the early 1960s the analysis of the exact distributions and moments of these estimators began, and since this time substantial progress has been made. Although these estimators are asymptotically equivalent, recent research has shown that their finite-sample properties are substantially different. However, the majority of this research has simply concentrated on a correctly specified system of equations, even though, since typically in applied studies theory provides some guidance but falls short of specifying the precise form of structural relationship, the possibilities for misspecification in simultaneous equation models are numerous. The objective of this paper is to extend the finite-sample analysis of these two estimators to include various cases of misspecification.     

Estimators based on ranks for arma models

In this paper we introduce a new family of robust estimators for ARMA models. These estimators are defined by replacing the residual sample autocovariances in the least squares equations by autocovariances based on ranks. The asymptotic normality of the proposed estimators is provided. The efficiency and robustness properties of these estimators are studied. An adequate choice of the score functions gives estimators which have high efficiency under normality and robustness in the presence of outliers. The score functions can also be chosen so that the resulting estimators are asymptotically as efficient as the maximum likelihood estimators for a given distribution.     

关于均匀分布统计分析的一些思考

对于均匀分布给出了在全样本场合下参数的区间估计,在定数截尾场合下参数的点估计和区间估计,并通过大量Monte-Carlo模拟从均方差的角度比较各种点估计的优劣,考察各种区间估计方法的精度,通过实例说明这些点估计和区间估计方法的应用。     

On the nonparametric smooth estimation of the reversed Hazard Rate function

The Reversed Hazard Rate (RHR) function is an important measure as a tool in the analysis of the reliability of both natural and man-made systems. In this paper, we present several new estimators of the RHR function using nonparametric techniques. These estimators are obtained by incorporating different binning techniques with fixed design local polynomial regression. We show that these estimators are asymptotically unbiased and consistent and, to determine the bandwidth, we propose two simple yet efficient plug-in bandwidth selection methods for even and odd order local polynomial estimators. Simulated and real life data are subsequently used to evaluate the performances of these estimators.     

Separate Ratio Estimators for the Population Variance in Stratified Random Sampling

We propose separate ratio estimators for population variance in stratified random sampling. We obtain mean square error equations and compare proposed estimators about efficiency with each other. By these comparisons, we find the conditions which make proposed estimators more efficient than others. It has been shown that proposed classes of estimators are more efficient than usual unbiased estimator. We find that separate ratio estimators are more efficient than combined ratio estimators for population variance. The theoretical results are supported by a numerical illustration with original data. A simulation study is also carried out to investigate empirical performance of estimators.     

Reliability estimation for the exponential distribution

This paper is concerned with classical statistical estimation of the reliability function for the exponential density with unknown mean failure time θ, and with a known and fixed mission time τ. The minimum variance unbiased (MVU) estimator and the maximum likelihood (ML) estimator are reviewed and their mean square errors compared for different sample sizes. These comparisons serve also to extend previous work, and reinforce further the nonexistence of a uniformly best estimator. A class of shrunken estimators is then defined, and it produces a shrunken quasi-estimator and a shrunken estimator. The mean square errors for both these estimators are compared to the mean square errors of the MVU and ML estimators, and the new estimators are found to perform very well. Unfortunately, these estimators are difficult to compute for practical applications. A second class of estimators, which is easy to compute is also developed. Its mean square error properties are compared to the other estimators, and it outperforms all the contending estimators over the high and low reliability parameter space. Since, for all the estimators, analytical mean square error comparisons are not tractable, extensive numerical analyses are done in obtaining both the exact small sample and large sample results.     

Mean Estimation Under Successive Sampling with Calibration Estimators

The estimation of the finite population mean in successive occasions is investigated with calibration estimators in this article. We propose several estimators based on calibration techniques with arbitrary sampling design in each of the occasions. Asymptotic variance formulaes are derived for the proposed estimators. The properties of these estimators are studied via a simulation study and using natural populations.     

Asymptotic Theory of Taguchi’s Natural Estimators of the Signal to Noise Ratio for Dynamic Robust Parameter Design

This article discusses the asymptotic theory of Taguchi’s natural estimators of the signal to noise ratio (SNR) for dynamic robust parameter design. Three asymptotic properties are shown. First, two natural estimators of the population SNR are asymptotically equivalent. Second, both of these estimators are consistent. Finally, both of these estimators are asymptotically normally distributed.     

Modified ratio estimators using robust regression methods

When there is an outlier in the data set, the efficiency of traditional methods decreases. In order to solve this problem, Kadilar et al. (2007) adapted Huber-M method which is only one of robust regression methods to ratio-type estimators and decreased the effect of outlier problem. In this study, new ratio-type estimators are proposed by considering Tukey-M, Hampel M, Huber MM, LTS, LMS and LAD robust methods based on the Kadilar et al. (2007). Theoretically, we obtain the mean square error (MSE) for these estimators. We compared with MSE values of proposed estimators and MSE values of estimators based on Huber-M and OLS methods. As a result of these comparisons, we observed that our proposed estimators give more efficient results than both Huber M approach which was proposed by Kadilar et al. (2007) and OLS approach. Also, under all conditions, all of the other proposed estimators except Lad method are more efficient than robust estimators proposed by Kadilar et al. (2007). And, these theoretical results are supported with the aid of a numerical example and simulation by basing on data that includes an outlier.     

Discriminant analysis of survey data

We consider the problem of the effect of sample designs on discriminant analysis. The selection of the learning sample is assumed to depend on the population values of auxiliary variables. Under a superpopulation model with a multivariate normal distribution, unbiasedness and consistency are examined for the conventional estimators (derived under the assumptions of simple random sampling), maximum likelihood estimators, probability-weighted estimators and conditionally unbiased estimators of parameters. Four corresponding sampled linear discriminant functions are examined. The rates of misclassification of these four discriminant functions and the effect of sample design on these four rates of misclassification are discussed. The performances of these four discriminant functions are assessed in a simulation study.     

Inference under Heteroscedasticity of Unknown Form Using an Adaptive Estimator

The heteroscedasticity consistent covariance matrix estimators are commonly used for the testing of regression coefficients when error terms of regression model are heteroscedastic. These estimators are based on the residuals obtained from the method of ordinary least squares and this method yields inefficient estimators in the presence of heteroscedasticity. It is usual practice to use estimated weighted least squares method or some adaptive methods to find efficient estimates of the regression parameters when the form of heteroscedasticity is unknown. But HCCM estimators are seldom derived from such efficient estimators for testing purposes in the available literature. The current article addresses the same concern and presents the weighted versions of HCCM estimators. Our numerical work uncovers the performance of these estimators and their finite sample properties in terms of interval estimation and null rejection rate.     

Delay Estimation for Some Stationary Diffusion-type Processes

In this paper the asymptotic behaviour of the maximum likelihood and Bayesian estimators of a delay parameter is studied. The observed process is supposed to be the solution of a linear stochastic differential equation with one time delay term. It is shown that these estimators are consistent and their limit distributions are described. The behaviour of the estimators is similar to the behaviour of corresponding estimators in change-point problems. The question of asymptotical efficiency is also discussed.     

Improving Ratio Estimators of Second Order Point Process Characteristics

Ripley's K function, the L function and the pair correlation function are important second order characteristics of spatial point processes. These functions are usually estimated by ratio estimators, where the numerators are Horvitz–Thompson edge corrected estimators and the denominators estimate the intensity or its square. It is possible to improve these estimators with respect to bias and estimation variance by means of adapted distance dependent intensity estimators. Further improvement is possible by means of refined estimators of the square of intensity. All this is shown by statistical analysis of simulated Poisson, cluster and hard core processes.