Wavelet-Based Density Estimation in a Heteroscedastic Convolution Model

We consider a heteroscedastic convolution density model under the “ordinary smooth assumption.” We introduce a new adaptive wavelet estimator based on term-by-term hard thresholding rule. Its asymptotic properties are explored via the minimax approach under the mean integrated squared error over Besov balls. We prove that our estimator attains near optimal rates of convergence (lower bounds are determined). Simulation results are reported to support our theoretical findings.     

A Convolution Estimator for the Density of Nonlinear Regression Observations

Abstract. The problem of estimating an unknown density function has been widely studied. In this article, we present a convolution estimator for the density of the responses in a nonlinear heterogenous regression model. The rate of convergence for the mean square error of the convolution estimator is of order n ^?1 under certain regularity conditions. This is faster than the rate for the kernel density method. We derive explicit expressions for the asymptotic variance and the bias of the new estimator, and further a data‐driven bandwidth selector is proposed. We conduct simulation experiments to check the finite sample properties, and the convolution estimator performs substantially better than the kernel density estimator for well‐behaved noise densities.     

Orthogonal series density estimation in a disaggregation scheme

In this paper we consider long-memory processes obtained by aggregation of independent random parameter AR(1) processes. We propose an estimator of the density of the underlying random parameter. This estimator is based on the expansion of the density function on the basis of Gegenbauer polynomials. Rate of convergence to zero of the mean integrated square error (MISE) and of the uniform error are obtained. The results are illustrated by Monte-Carlo simulations.     

Nonparametric estimation of a two dimensional continuous–discrete density function by wavelets

We consider the estimation of a two dimensional continuous–discrete density function. A new methodology based on wavelets is proposed. We construct a linear wavelet estimator and a non-linear wavelet estimator based on a term-by-term thresholding. Their rates of convergence are established under the mean integrated squared error over Besov balls. In particular, we prove that our adaptive wavelet estimator attains a fast rate of convergence. A simulation study illustrates the usefulness of the proposed estimators.     

On the estimation of the marginal density of a moving average process

The authors present a new convolution‐type kernel estimator of the marginal density of an MA(1) process with general error distribution. They prove the √n; ‐consistency of the nonparametric estimator and give asymptotic expressions for the mean square and the integrated mean square error of some unobservable version of the estimator. An extension to MA(q) processes is presented in the case of the mean integrated square error. Finally, a simulation study shows the good practical behaviour of the estimator and the strong connection between the estimator and its unobservable version in terms of the choice of the bandwidth.     

A new biased estimator based on ridge estimation

In this paper we introduce a new biased estimator for the vector of parameters in a linear regression model and discuss its properties. We show that our new biased estimator is superior, in the mean square error(mse) sense, to the ordinary least squares (OLS) estimator, the ordinary ridge regression (ORR) estimator and the Liu estimator. We also compare the performance of our new biased estimator with two other special Liu-type estimators proposed in Liu (2003). We illustrate our findings with a numerical example based on the widely analysed dataset on Portland cement.     

A new estimator combining the ridge regression and the restricted least squares methods of estimation

It is well-known in the literature on multicollinearity that one of the major consequences of multicollinearity on the ordinary least squares estimator is that the estimator produces large sampling variances, which in turn might inappropriately lead to exclusion of otherwise significant coefficients from the model. To circumvent this problem, two accepted estimation procedures which are often suggested are the restricted least squares method and the ridge regression method. While the former leads to a reduction in the sampling variance of the estimator, the later ensures a smaller mean square error value for the estimator. In this paper we have proposed a new estimator which is based on a criterion that combines the ideas underlying these two estimators. The standard properties of this new estimator have been studied in the paper. It has also been shown that this estimator is superior to both the restricted least squares as well as the ordinary ridge regression estimators by the criterion of mean sauare error of the estimator of the regression coefficients when the restrictions are indeed correct. The conditions for superiority of this estimator over the other two have also been derived for the situation when the restrictions are not correct.     

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.     

Estimation of reliability for a multicomponent survival stress-strength model based on exponential distributions

Uniformly minimum variance unbiased estimator (UMVUE) of reliability in stress-strength model (known stress) is obtained for a multicomponent survival model based on exponential distributions for parallel system. The variance of this estimator is compared with Cramer-Rao lower bound (CRB) for the variance of unbiased estimator of reliability, and the mean square error (MSE) of maximum likelihood estimator of reliability in case of two component system.     

Iterative regularisation in nonparametric instrumental regression

This paper considers the nonparametric regression model with an additive error that is correlated with the explanatory variables. Motivated by empirical studies in epidemiology and economics, it also supposes that valid instrumental variables are observed. However, the estimation of a nonparametric regression function by instrumental variables is an ill-posed linear inverse problem with an unknown but estimable operator. We provide a new estimator of the regression function that is based on projection onto finite dimensional spaces and that includes an iterative regularisation method (the Landweber–Fridman method). The optimal number of iterations and the convergence of the mean square error of the resulting estimator are derived under both strong and weak source conditions. A Monte Carlo exercise shows the impact of some parameters on the estimator and concludes on the reasonable finite sample performance of the new estimator.     

An empirical investigation of different operating characteristics of several estimators of the intraclass correlation in the analysis of binary data

This paper is concerned with properties (bias, standard deviation, mean square error and efficiency) of twenty six estimators of the intraclass correlation in the analysis of binary data. Our main interest is to study these properties when data are generated from different distributions. For data generation we considered three over-dispersed binomial distributions, namely, the beta-binomial distribution, the probit normal binomial distribution and a mixture of two binomial distributions. The findings regarding bias, standard deviation and mean squared error of all these estimators, are that (a) in general, the distributions of biases of most of the estimators are negatively skewed. The biases are smallest when data are generated from the beta-binomial distribution and largest when data are generated from the mixture distribution; (b) the standard deviations are smallest when data are generated from the beta-binomial distribution; and (c) the mean squared errors are smallest when data are generated from the beta-binomial distribution and largest when data are generated from the mixture distribution. Of the 26, nine estimators including the maximum likelihood estimator, an estimator based on the optimal quadratic estimating equations of Crowder (1987), and an analysis of variance type estimator is found to have least amount of bias, standard deviation and mean squared error. Also, the distributions of the bias, standard deviation and mean squared error for each of these estimators are, in general, more symmetric than those of the other estimators. Our findings regarding efficiency are that the estimator based on the optimal quadratic estimating equations has consistently high efficiency and least variability in the efficiency results. In the important range in which the intraclass correlation is small (≤0 5), on the average, this estimator shows best efficiency performance. The analysis of variance type estimator seems to do well for larger values of the intraclass correlation. In general, the estimator based on the optimal quadratic estimating equations seems to show best efficiency performance for data from the beta-binomial distribution and the probit normal binomial distribution, and the analysis of variance type estimator seems to do well for data from the mixture distribution.     

Nonparametric regression in a statistical modified Helmholtz equation using the Fourier spectral regularization

In this paper, we deal with the Cauchy problem for the modified Helmholtz equation. We consider two models of data: the bounded variance model and the i.i.d. model. The trigonometric estimators of nonparametric regression is applied to solve the problem. In addition, the general forms of regularization parameter corresponding to the pointwise mean squared error and the mean integrated squared error are discussed in detail. The minimax rate convergence corresponding to the bounded variance model is also presented. In the i.i.d. model, we construct the asymptotic confidence interval for the solution of the problem. Finally, we give some numerical experiments and discuss the obtained results.     

A new non parametric estimator for Pdf based on inverse gamma distribution

ABSTRACT

The non parametric approach is considered to estimate probability density function (Pdf) which is supported on(0, ∞). This approach is the inverse gamma kernel. We show that it has same properties as gamma, reciprocal inverse Gaussian, and inverse Gaussian kernels such that it is free of the boundary bias, non negative, and it achieves the optimal rate of convergence for the mean integrated squared error. Also some properties of the estimator were established such as bias and variance. Comparison of the bandwidth selection methods for inverse gamma kernel estimation of Pdf is done.     

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.     

Bayes and Empirical Bayes Inference in Changepoint Problems

ABSTRACT

In the case of the random design nonparametric regression, the double smoothing technique is applied to estimate the multivariate regression function. The proposed estimator has desirable properties in both the finite sample and the asymptotic cases. In the finite sample case, it has bounded conditional (and unconditional) bias and variance. On the other hand, in the asymptotic case, it has the same mean square error as the local linear estimator in Fan (Design-Adaptive Nonparametric Regression. Journal of the American Statistical Association 1992, 87, 998–1004; Local Linear Regression Smoothers and Their Minimax Efficiencies. Annals of Statistics 1993, 21, 196–216). Simulation studies demonstrate that the proposed estimator is better than the local linear estimator, because it has a smaller sample mean integrated square error and gives smoother estimates.     

Simultaneous confidence intervals in the analysis of orthogonal saturated designs

We present the first known method of constructing exact simultaneous confidence intervals for the analysis of orthogonal, saturated factorial designs. Given m independent, normally distributed, unbiased estimators of treatment contrasts, if there is an independent chi-squared estimator of error variance, then simultaneous confidence intervals based on the Studentized maximum modulus distribution are exact under all parameter configurations. In this paper, an analogous method is developed for the case of an orthogonal saturated design, for which the treatment contrasts are independently estimable but there is no independent estimator of error variance. Lacking an independent estimator of the error variance, the smallest sums of squares of effect estimators are pooled. The simultaneous confidence intervals are based on a probability inequality, for which the simultaneous confidence coefficient is achieved in the null case.     

A minimum variance kernel estimator and a discrete frequency polygon estimator for ordinal contingency tables

This paper introduces two estimators, a boundary corrected minimum variance kernel estimator based on a uniform kernel and a discrete frequency polygon estimator, for the cell probabilities of ordinal contingency tables. Simulation results show that the minimum variance boundary kernel estimator has a smaller average sum of squared error than the existing boundary kernel estimators. The discrete frequency polygon estimator is simple and easy to interpret, and it is competitive with the minimum variance boundary kernel estimator. It is proved that both estimators have an optimal rate of convergence in terms of mean sum of squared error, The estimators are also defined for high-dimensional tables.     

The effects of the proxy information on the iterative Stein-rule estimator of the disturbance variance

In a regression model with proxy variables, we consider the iterative estimator of the disturbance variance to obtain more precise estimates. In the formula of the estimator of the disturbance variance, the estimator is obtained by using Stein-rule (SR) estimator instead of OLS (ordinary least squares) estimator is called Iterative estimator of the disturbance variance. It is shown that, in a regression model with proxy variables the mean square error (MSE) of the iterative estimator of the disturbance variance is greater than the MSE of the disturbance variance related to the OLS estimator under certain conditions.     

ON THE USE OF THE EMPIRICAL DISTRIBUTION AND CHARACTERISTIC FUNCTION TO ESTIMATE PARAMETERS OF REGULAR VARIATION

In this paper we present two new classes of estimators of parameters of regular variation, one based on the empirical distribution function and the other on the empirical characteristic function. They achieve the same rates of mean square error convergence as the estimators proposed by Hall (1982). The estimator based on the empirical characteristic function, unlike the other estimators, utilises the whole sample and not just a few extreme order statistics.