Improved estimators for the location of double exponential distribution

In this paper, attention is focused on estimation of the location parameter in the double exponential case using a weighted linear combination of the sample median and pairs of order statistics, with symmetric distance to both sides from the sample median. Minimizing with respect to weights and distances we get smaller asymptotic variance in the second order. If the number of pairs is taken as infinite and the distances as null we attain the least asymptotic variance in this class of estimators. The Pitman estimator is also noted. Similarly improved estimators are scanned over their probability of concentration to investigate its bound. Numerical comparison of the estimators is shown.     

The distribution of a moment estimator for a parameter of the generalized poisson distribution

The ratio of the sample variance to the sample mean estimates a simple function of the parameter which measures the departure of the Poisson-Poisson from the Poisson distribution. Moment series to order n?²⁴ are given for related estimators. In one case, exact integral formulations are given for the first two moments, enabling a comparison to be made between their asymptotic developments and a computer-oriented extended Taylor series (COETS) algorithm. The integral approach using generating functions is sketched out for the third and fourth moments. Levin's summation algorithm is used on the divergent series and comparative simulation assessments are given.     

Estimation of the Weibull tail-coefficient with linear combination of upper order statistics

We present a new family of estimators of the Weibull tail-coefficient. The Weibull tail-coefficient is defined as the regular variation coefficient of the inverse failure rate function. Our estimators are based on a linear combination of log-spacings of the upper order statistics. Their asymptotic normality is established and illustrated for two particular cases of estimators in this family. Their finite sample performances are presented on a simulation study.     

Replicated measurement error model under exact linear restrictions

We consider a replicated ultrastructural measurement error regression model where predictor variables are observed with error. It is assumed that some prior information regarding the regression coefficients is available in the form of exact linear restrictions. Three classes of estimators of regression coefficients are proposed. These estimators are shown to be consistent as well as satisfying the given restrictions. The asymptotic properties of unrestricted as well as restricted estimators are studied without imposing any distributional assumption on any random component of the model. A Monte Carlo simulations study is performed to assess the effect of sample size, replicates and non-normality on the estimators.     

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.     

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.     

Efficient estimation in partially linear single‐index models for longitudinal data

In this paper, we consider the estimation of both the parameters and the nonparametric link function in partially linear single‐index models for longitudinal data that may be unbalanced. In particular, a new three‐stage approach is proposed to estimate the nonparametric link function using marginal kernel regression and the parametric components with generalized estimating equations. The resulting estimators properly account for the within‐subject correlation. We show that the parameter estimators are asymptotically semiparametrically efficient. We also show that the asymptotic variance of the link function estimator is minimized when the working error covariance matrices are correctly specified. The new estimators are more efficient than estimators in the existing literature. These asymptotic results are obtained without assuming normality. The finite‐sample performance of the proposed method is demonstrated by simulation studies. In addition, two real‐data examples are analyzed to illustrate the methodology.     

Relative error prediction via kernel regression smoothers

In this article, we introduce and study local constant and local linear nonparametric regression estimators when it is appropriate to assess performance in terms of mean squared relative error of prediction. We give asymptotic results for both boundary and non-boundary cases. These are special cases of more general asymptotic results that we provide concerning the estimation of the ratio of conditional expectations of two functions of the response variable. We also provide a good bandwidth selection method for the estimators. Examples of application, limited simulation results and discussion of related problems and approaches are also given.     

Outlier-detection tests and robust estimators based on signs of residuals

The asymptotic structure of a vector of weighted sums of signs of residuals, in the general linear model, is studied. The vector can be used as a basis for outlier-detection tests, or alternatively, setting the vector to zero and solving for the parameter yields a class of robust estimators which are analogues of the sample median. Asymptotic results for both estimates and tests are obtained. The question of optimal weights is investigated, and the optimal estimators in the case of simple linear regression are found to coincide with estimators introduced by Adichie.     

Estimation in the presence of incidental parameters

This paper considers the estimation of “structural” parameters when the number of unknown parameters increases with the sample size. Neyman and Scott (1948) had demonstrated that maximum likelihood estimators (MLE) of structural parameters may be inconsistent in this case. Patefield (1977) further observed that the asymptotic covariance matrix of the MLE is not equal to the inverse of the information matrix. In this paper we establish asymptotic properties of estimators (which include in particular the MLE) obtained via the usual likelihood approach when the incidental parameters are first replaced by their estimates (which are allowed to depend on the structural parameters). Conditions for consistency and asymptotic normality together with a proper formula for the asymptotic covariance matrix are given. The results are illustrated and applied to the problem of estimating linear functional relationships, and mild conditions on the incidental parameters for the MLE (or an adjusted MLE) to be consistent and asymptotically normal are obtained. These conditions are weaker than those imposed by previous authors.     

On the mean squared error of nonparametric quantile estimators under random right-censorship

For randomly right-censored data, new asymptotic expressions for the mean squared errors of the product-limit quantile estimator and a kernel-type quantile estimator are presented in this paper. From these results a comparison of the two quantile estimators with respect to their mean squared errors is given.     

Setting confidence intervals by ratio estimator

For the survey population total of a variable y when values of an auxiliary variable x are available a popular procedure is to employ the ratio estimator on drawing a simple random sample without replacement (SRSWOR) especially when the size of the sample is large. To set up a confidence interval for the total, various variance estimators are available to pair with the ratio estimator. We add a few more variance estimators studded with asymptotic design-cum-model properties. The ratio estimator is traditionally known to be appropriate when the regression of y on x is linear through the origin and the conditional variance of y given x is proportional to x. But through a numerical exercise by simulation we find the confidence intervals to fare better if the regression line deviates from the origin or if the conditional variance is disproportionate with x. Also, comparing the confidence intervals using alternative variance estimators we find our newly proposed variance estimators to yield favourably competitive results.     

Comparison of improved regression estimators with and without moments

For estimating the coefficients in a linear regression model, the double k–class estimators are considered and the small disturbance asymptotic approximation for their density function is obtained. Then employing the criterion of concentration probability around the true parameter values, a comparison is made between the estimators possessing finite moments and the estimators having no finite moments.     

Estimating and testing group effects from type i censored normal samples in experimental design

The maximum likelihood (ML) equations calculated from censored normal samples do not admit explicit solutions. A principle of modification is given and modified maximum likelihood (MML) equations, which admit explicit solutions, are defined. This approach makes it possible to tackle the hitherto unresolved problem of estimating and testing hypotheses about group-effects in one-way classification experimental designs based on Type I censored normal samples. The MML estimators of group-effects are obtained as explicit functions of sample observations and shown to be asymptotically identical with the ML estimators and hence BAN (best asymptotic normal) estimators. A statistic t is defined to test a linear contrast of group-effects and shown to be asymptotically normally distributed. A numerical example is presented which illustrates the procedure.     

Estimation in two sample randomly truncated scale model

This paper studies a class of Cramér–von Mises type minimum distance estimators of the scale parameter in the two sample randomly left truncated scale models. The proposed class of estimators includes an analogue of the well-known Hodges–Lehmann estimator. The paper proves the asymptotic normality of these estimators under mild conditions. It also contains a real data application and a simulation study making a comparison of some of the estimators in the class with the ratio of the two means.     

Goodness-of-fit testing for Weibull-type behavior

In the process of analyzing data, testing the fit of a model under consideration is a prerequisite for performing inference about the model parameters. In this paper we examine the goodness-of-fit testing problem for assessing whether a sample is consistent with the Weibull-type model. Inspired by the Jackson and the Lewis test statistics, originally proposed as goodness-of-fit tests for the exponential distribution, we introduce two new statistics for testing Weibull-type behavior, and study their asymptotic properties. Moreover, given that the statistics are ratios of estimators for the Weibull-tail coefficient, we obtain new estimators for the latter, and establish their consistency and asymptotic normality. The small sample behavior of our statistics and estimators is evaluated on the basis of a simulation study.     

Estimation on Varying-Coefficient Partially Linear Model With Different Smoothing Variables

The authors give the estimation on the varying-coefficient partially linear regression model with different smoothing variables. The efficient estimators of the intercept function and the coefficient functions are obtained by a one-step back-fitting technique based on their initial estimators given by local linear technique and the averaged method. Furthermore, their asymptotic normalities are given. Some simulation studies are used to illustrate the performances of the estimation.     

Asymptotic efficiency of estimators in the multivariate linear model

Conditions are given under which the least squares estimator and a certain two-step estimator in the multivariate linear model are best asymptotically normal. Normality turns out to be necessary. Under normality the asymptotic efficiency in the sense of RAO of these two estimators is derived.     

Statistical inference in the partial linear models with the inverse gaussian kernel

Symmetric kernel smoothing is commonly used in estimating the nonparametric component in the partial linear regression models. In this article, we propose a new estimation method for the partial linear regression models using the inverse Gaussian kernel when the explanatory variable of the nonparametric component is non-negatively supported. As an asymmetric kernel function, the inverse Gaussian kernel is also supported on the non-negative half line. The asymptotic properties, including the asymptotic normality, uniform almost sure convergence, and the iterated logarithm laws, of the proposed estimators are thoroughly discussed for both homoscedastic and heteroscedastic cases. The simulation study is conducted to evaluate the finite sample performance of the proposed estimators.     

Model selection and post estimation based on a pretest for logistic regression models

ABSTRACT

This article addresses the problem of parameter estimation of the logistic regression model under subspace information via linear shrinkage, pretest, and shrinkage pretest estimators along with the traditional unrestricted maximum likelihood estimator and restricted estimator. We developed an asymptotic theory for the linear shrinkage and pretest estimators and compared their relative performance using the notion of asymptotic distributional bias and asymptotic quadratic risk. The analytical results demonstrated that the proposed estimation strategies outperformed the classical estimation strategies in a meaningful parameter space. Detailed Monte-Carlo simulation studies were conducted for different combinations and the performance of each estimation method was evaluated in terms of simulated relative efficiency. The results of the simulation study were in strong agreement with the asymptotic analytical findings. Two real-data examples are also given to appraise the performance of the estimators.