Inverse probability weighted estimators for single-index models with missing covariates

Abstract

In this article, we consider the inverse probability weighted estimators for a single-index model with missing covariates when the selection probabilities are known or unknown. It is shown that the estimator for the index parameter by using estimated selection probabilities has a smaller asymptotic variance than that with true selection probabilities, thus is more efficient. Therefore, the important Horvitz-Thompson property is verified for the index parameter in single index model. However, this difference disappears for the estimators of the link function. Some numerical examples and a real data application are also conducted to illustrate the performances of the estimators.     

An asymptotic equivalence of the cross-data and predictive estimators

Abstract

Estimators using multiplicative tuning parameters for maximum likelihood estimators in cross-validation are called cross-data estimators in this paper. Single-sample versions of the cross-data estimators have been called predictive estimators in literatures, which are given by maximizing the expected log-likelihood, where the two-fold expectations are taken over the distributions of future and current data using maximum likelihood estimators based on current data. An asymptotic equivalence of the cross-data and predictive estimators is shown, which guarantees an optimality of the predictive estimator when an unknown population parameter vector is replaced by the sample counterpart. Examples using typical statistical distributions are shown.     

Classical and Bayesian estimation of P(Y < X) for Kumaraswamy's distribution

The aim of this paper is to study the estimation of the reliability R=P(Y<X) when X and Y are independent random variables that follow Kumaraswamy's distribution with different parameters. If we assume that the first shape parameter is common and known, the maximum-likelihood estimator (MLE), the exact confidence interval and the uniformly minimum variance unbiased estimator of R are obtained. Moreover, when the first parameter is common but unknown, MLEs, Bayes estimators, asymptotic distributions and confidence intervals for R are derived. Furthermore, Bayes and empirical Bayes estimators for R are obtained when the first parameter is common and known. Finally, when all four parameters are different and unknown, the MLE of R is obtained. Monte Carlo simulations are performed to compare the different proposed methods and conclusions on the findings are given.     

Empirical Bayes estimation for the mean of the selected normal population when there are additional data

ABSTRACT

This paper is concerned with the problem of estimation for the mean of the selected population from two normal populations with unknown means and common known variance in a Bayesian framework. The empirical Bayes estimator, when there are available additional observations, is derived and its bias and risk function are computed. The expected bias and risk of the empirical Bayes estimator and the intuitive estimator are compared. It is shown that the empirical Bayes estimator is asymptotically optimal and especially dominates the intuitive estimator in terms of Bayes risk, with respect to any normal prior. Also, the Bayesian correlation between the mean of the selected population (random parameter) and some interested estimators are obtained and compared.     

Empirical likelihood dimension reduction inference for partially non-linear error-in-responses models with validation data

ABSTRACT

In this article, partially non linear models when the response variable is measured with error and explanatory variables are measured exactly are considered. Without specifying any error structure equation, a semiparametric dimension reduction technique is employed. Two estimators of unknown parameter in non linear function are obtained and asymptotic normality is proved. In addition, empirical likelihood method for parameter vector is provided. It is shown that the estimated empirical log-likelihood ratio has asymptotic Chi-square distribution. A simulation study indicates that, compared with normal approximation method, empirical likelihood method performs better in terms of coverage probabilities and average length of the confidence intervals.     

Seemingly Unrelated Ridge Regression in Semiparametric Models

This article is concerned with the problem of multicollinearity in the linear part of a seemingly unrelated semiparametric (SUS) model. It is also suspected that some additional non stochastic linear constraints hold on the whole parameter space. In the sequel, we propose semiparametric ridge and non ridge type estimators combining the restricted least squares methods in the model under study. For practical aspects, it is assumed that the covariance matrix of error terms is unknown and thus feasible estimators are proposed and their asymptotic distributional properties are derived. Also, necessary and sufficient conditions for the superiority of the ridge-type estimator over the non ridge type estimator for selecting the ridge parameter K are derived. Lastly, a Monte Carlo simulation study is conducted to estimate the parametric and nonparametric parts. In this regard, kernel smoothing and cross validation methods for estimating the nonparametric function are used.     

ASYMPTOTIC PROPERTIES FOR MAXIMUM LIKELIHOOD ESTIMATORS FOR RELIABILITY AND FAILURE RATES OF MARKOV CHAINS

ABSTRACT

In this paper, we shall study a homogeneous ergodic, finite state, Markov chain with unknown transition probability matrix. Starting from the well known maximum likelihood estimator of transition probability matrix, we define estimators of reliability and its measurements. Our aim is to show that these estimators are uniformly strongly consistent and converge in distribution to normal random variables. The construction of the confidence intervals for availability, reliability, and failure rates are also given. Finally we shall give a numerical example for illustration and comparing our results with the usual empirical estimator results.     

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.     

A family of estimators of population variance in two-occasion rotation patterns

Abstract

In this article, we have considered the problem of estimation of population variance on current (second) occasion in two occasion successive (rotation) sampling. A class of estimators of population variance has been proposed and its asymptotic properties have been discussed. The proposed class of estimators is compared with the sample variance estimator when there is no matching from the previous occasion and the Singh et al. (2013) estimator. Optimum replacement policy is discussed. It has been shown that the suggested estimator is more efficient than the Singh et al. (2013) estimator and a usual unbiased estimator when there is no matching. An empirical study is carried out in support of the present study.     

Minimum-Distance Estimator for Stable Exponent

Assume that X ₁, X ₂,…, X _n is a sequence of i.i.d. random variables with α-stable distribution (α ∈ (0,2], the stable exponent, is the unknown parameter). We construct minimum distance estimators for α by minimizing the Kolmogorov distance or the Cramér–von-Mises distance between the empirical distribution function G _n , and a class of distributions defined based on the sum-preserving property of stable random variables. The minimum distance estimators can also be obtained by minimizing a U-statistic estimate of an empirical distribution function involving the stable exponent. They share the same invariance property with the maximum likelihood estimates. In this article, we prove the strong consistency of the minimum distance estimators. We prove the asymptotic normality of our estimators. Simulation study shows that the new estimators are competitive to the existing ones and perform very closely even to the maximum likelihood estimator.     

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.     

On the asymptotic normality of the R-estimators of the slope parameters of simple linear regression models with associated errors

ABSTRACT

The purpose of this paper is to prove, under mild conditions, the asymptotic normality of the rank estimator of the slope parameter of a simple linear regression model with stationary associated errors. This result follows from a uniform linearity property for linear rank statistics that we establish under general conditions on the dependence of the errors. We prove also a tightness criterion for weighted empirical process constructed from associated triangular arrays. This criterion is needed for the proofs which are based on that of Koul [Behavior of robust estimators in the regression model with dependent errors. Ann Stat. 1977;5(4):681–699] and of Louhichi [Louhichi S. Weak convergence for empirical processes of associated sequences. Ann Inst Henri Poincaré Probabilités Statist. 2000;36(5):547–567].     

The LAD estimation of the change-point linear model with randomly censored data

Abstract

In this paper, a change-point linear model with randomly censored data is investigated. We propose the least absolute deviation estimation procedure for regression and change-point parameters simultaneously. The asymptotic properties of the change-point and regression parameter estimators are obtained. We show that the resulting regression parameter estimator is asymptotically normal, and the change-point estimator converges weakly to the minimizer of a given random process. The extensive simulation studies and the analysis of an acute myocardial infarction data set are conducted to illustrate the finite sample performance of the proposed method.     

Combining poisson means

The problem of estimating the Poisson mean is considered based on the two samples in the presence of uncertain prior information (not in the form of distribution) that two independent random samples taken from two possibly identical Poisson populations. The parameter of interest is λ₁ from population I. Three estimators, i.e. the unrestricted estimator, restricted estimator and preliminary test estimator are proposed. Their asymptotic mean squared errors are derived and compared; parameter regions have been found for which restricted and preliminary test estimators are always asymptotically more efficient than the classical estimator. The relative dominance picture of the estimators is presented. Maximum and minimum asymptotic efficiencies of the estimators relative to the classical estimator are tabulated. A max-min rule for the size of the preliminary test is also discussed. A Monte Carlo study is presented to compare the performance of the estimator with that of Kale and Bancroft (1967).     

Nonparametric kernel estimation of expected shortfall under negatively associated sequences

Abstract

In this article, Bahadur type expansions of a nonparametric kernel estimator for ES under NA sequences are given. The strong consistency and the uniformly asymptotic normality of the estimator are yielded from the Bahadur type expansions, while the convergence rates of the above asymptotic properties are also obtained. Moreover, the expectation, the variance and the mean squared error (MSE) of the estimator are given. Besides, the optimal bandwidth selection of this estimator is discussed. We point out that all above results are based on the NA sequences. Finally, we conduct numerical simulations and compare performances of some ES estimators.     

On cusp location estimation for perturbed dynamical systems

We consider the problem of parameter estimation in the case of observation of the trajectory of the diffusion process. We suppose that the drift coefficient has a singularity of cusp type and that the unknown parameter corresponds to the position of the point of the cusp. The asymptotic properties of the maximum likelihood estimator and Bayesian estimators are described in the asymptotic of small noise, that is, as the diffusion coefficient tends to zero. The consistency, limit distributions, and the convergence of moments of these estimators are established.     

Asymptotic Theory for the Multiscale Wavelet Density Derivative Estimator

Abstract

In this paper we consider the wavelet-based estimation of density derivatives. The multiscale density derivative estimator is proposed which is constructed by using a number of scaling functions. Asymptotic theory is developed in which asymptotic expressions for the bias, the variance and the mean integrated squared error are included. In addition, asymptotic normality of the proposed estimator is proved. Theoretical and numerical comparisons with the usual kernel-based estimators are also reported.     

Bayesian estimation for first-order autoregressive model with explanatory variables

In this article, we develop a Bayesian analysis in autoregressive model with explanatory variables. When σ² is known, we consider a normal prior and give the Bayesian estimator for the regression coefficients of the model. For the case σ² is unknown, another Bayesian estimator is given for all unknown parameters under a conjugate prior. Bayesian model selection problem is also being considered under the double-exponential priors. By the convergence of ρ-mixing sequence, the consistency and asymptotic normality of the Bayesian estimators of the regression coefficients are proved. Simulation results indicate that our Bayesian estimators are not strongly dependent on the priors, and are robust.     

Merging data from multiple sources: pretest and shrinkage perspectives

In this article, we have developed asymptotic theory for the simultaneous estimation of the k means of arbitrary populations under the common mean hypothesis and further assuming that corresponding population variances are unknown and unequal. The unrestricted estimator, the Graybill-Deal-type restricted estimator, the preliminary test, and the Stein-type shrinkage estimators are suggested. A large sample test statistic is also proposed as a pretest for testing the common mean hypothesis. Under the sequence of local alternatives and squared error loss, we have compared the asymptotic properties of the estimators by means of asymptotic distributional quadratic bias and risk. Comprehensive Monte-Carlo simulation experiments were conducted to study the relative risk performance of the estimators with reference to the unrestricted estimator in finite samples. Two real-data examples are also furnished to illustrate the application of the suggested estimation strategies.     

Estimating the common hazard rate of several exponential distributions

Abstract

In the present communication, we consider the estimation of the common hazard rate of several exponential distributions with unknown and unequal location parameters with a common scale parameter under a general class of bowl-shaped scale invariant loss functions. We have shown that the best affine equivariant estimator (BAEE) is inadmissible by deriving a non smooth improved estimator. Further, we have obtained a smooth estimator which improves upon the BAEE. As an application, we have obtained explicit expressions of improved estimators for special loss functions. Finally, a simulation study is carried out for numerically comparing the risk performance of various estimators.