Bootstrap for finite populations

The bootstrap method is compared with the classical (linearization) and jackknife procedures for estimating the mean square errors (MSEs) of the ratio estimator and the combined ratio estimator. The initial samples are considered to be selected without replacement, and different procedures for selecting the bootstrap samples with or without replacement from them are examined. The biases, stabilities, coverage probabilities and confidence widths of all the procedures are compared.     

On the estimation of the true probability of a correct selection

For ranking and selection problems, the true probabiIity of a correct selection P(CS) is unknown even if a selection is made under the indifference-zone approach. Thus to estimate the true P(CS) some Bayes estimators and a bootstrap estimator are proposed for two normcal populations with common known variance. Also a bootstrap estimator and a bootstrap confidence interval are proposed for normal populations with common unknown variance. Some comparisons between proposed estimators and some other known estimators are made via Monte Carlo simulations.     

Finite sample moments of a bootstrap estimator of the james-stein rule

The finite sample moments of the bootstrap estimator of the James-Stein rule are derived and shown to be biased. Analytical results shed some light upon the source of bias and suggest that the bootstrap will be biased in other settings where the moments of the statistic of interest depends on nonlinear functions of the parameters of its distribution.     

Finite sample moments of a bootstrap estimator of the james-stein rule

The finite sample moments of the bootstrap estimator of the James-Stein rule are derived and shown to be biased. Analytical results shed some light upon the source of bias and suggest that the bootstrap will be biased in other settings where the moments of the statistic of interest depends on nonlinear functions of the parameters of its distribution.     

Some properties of generalized ridge estimators

Exact expressions, in the form of infinite series expansions, are given for the first and second moments of two well known generalized ridge estimators. These series expansions are then evaluated using recursive formulas and computations are verified using approximations. Results are presented for the relative mean square error and bias of these estimators as well as their relative efficiency with respect to least squares.     

Bias Reduction for the Maximum Likelihood Estimators of the Parameters in the Half-Logistic Distribution

We derive analytic expressions for the biases of the maximum likelihood estimators of the scale parameter in the half-logistic distribution with known location, and of the location parameter when the latter is unknown. Using these expressions to bias-correct the estimators is highly effective, without adverse consequences for estimation mean squared error. The overall performance of the first of these bias-corrected estimators is slightly better than that of a bootstrap bias-corrected estimator. The bias-corrected estimator of the location parameter significantly out-performs its bootstrapped-based counterpart. Taking computational costs into account, the analytic bias corrections clearly dominate the use of the bootstrap.     

Simultaneous bootstrap for all three parameters in random coefficient autoregressive models

In this paper we consider autoregressive processes with random coefficients and develop bootstrap approaches that asymptotically work for the distribution of estimated autoregressive parameter as well as for the distribution of estimated variances of the innovation noise and the disturbance noise. We discuss how to obtain approximative residuals of the process and how to separate between the innovation and the disturbance noise in order to be able to extend the classical residual bootstrap for autoregressive processes to the situation considered in this paper. Thereafter, we propose a wild bootstrap procedure as a variation of the residual bootstrap that uses estimated densities of the innovation and the disturbance noise to generate bootstrap replicates of the data generating process. The consistency of the bootstrap approaches is established and their performance is illustrated by a simulation study.     

Spectral Density Based Goodness-of-Fit Tests for Time Series Models

A new goodness-of-fit test for time series models is proposed. The test statistic is based on the distance between a kernel estimator of the ratio between the true and the hypothesized spectral density and the expected value of the estimator under the null. It provides a quantification of how well a parametric spectral density model fits the sample spectral density (periodogram). The asymptotic distribution of the statistic proposed is derived and its power properties are discussed. To improve upon the large sample (Gaussian) approximation of the distribution of the test statistic under the null, a bootstrap procedure is presented and justified theoretically. The finite sample performance of the test is investigated through a simulation experiment and applications to real data sets are given.     

Estimation of parameters and the mean life of a mixed failure time distribution

This paper deals with estimation of parameters and the mean life of a mixed failure time distribution that has a discrete probability mass at zero and an exponential distribution with mean O for positive values. A new sampling scheme similar to Jayade and Prasad (1990) is proposed for estimation of parameters. We derive expressions for biases and mean square errors (MSEs) of the maximum likelihood estimators (MLEs). We also obtain the uniformly minimum variance unbiased estimators (UMVUEs) of the parameters. We compare the estimator of O and mean life fj based on the proposed sampling scheme with the estimators obtained by using the sampling scheme of Jayade and Prasad (1990).     

Bias-corrected maximum likelihood estimation of the parameters of the generalized Pareto distribution

ABSTRACT

We derive analytic expressions for the biases, to O(n^?1), of the maximum likelihood estimators of the parameters of the generalized Pareto distribution. Using these expressions to bias-correct the estimators in a selective manner is found to be extremely effective in terms of bias reduction, and can also result in a small reduction in relative mean squared error (MSE). In terms of remaining relative bias, the analytic bias-corrected estimators are somewhat less effective than their counterparts obtained by using a parametric bootstrap bias correction. However, the analytic correction out-performs the bootstrap correction in terms of remaining %MSE. It also performs credibly relative to other recently proposed estimators for this distribution. Taking into account the relative computational costs, this leads us to recommend the selective use of the analytic bias adjustment for most practical situations.     

Bias Reduction for the Maximum Likelihood Estimator of the Parameters of the Generalized Rayleigh Family of Distributions

We derive analytic expressions for the biases, to O(n^{? 1}), of the maximum likelihood estimators of the parameters of the generalized Rayleigh distribution family. Using these expressions to bias-correct the estimators is found to be extremely effective in terms of bias reduction, and generally results in a small reduction in relative mean squared error. In general, the analytic bias-corrected estimators are also found to be superior to the alternative of bias-correction via the bootstrap.     

Asymptotic behaviour of M‐estimators in AR(p) models under nonstandard conditions

The authors derive the limiting distribution of M‐estimators in AR(p) models under nonstandard conditions, allowing for discontinuities in score and density functions. Unlike usual regularity assumptions, these conditions are satisfied in the context of L₁‐estimation and autoregression quantiles. The asymptotic distributions of the resulting estimators, however, are not generally Gaussian. Moreover, their bootstrap approximations are consistent along very specific sequences of bootstrap sample sizes only.     

Bias-Corrected maximum likelihood estimation of the parameters of the weighted Lindley distribution

The two-parameter weighted Lindley distribution is useful for modeling survival data, whereas its maximum likelihood estimators (MLEs) are biased in finite samples. This motivates us to construct nearly unbiased estimators for the unknown parameters. We adopt a “corrective” approach to derive modified MLEs that are bias-free to second order. We also consider an alternative bias-correction mechanism based on Efron’s bootstrap resampling. Monte Carlo simulations are conducted to compare the performance between the proposed and two previous methods in the literature. The numerical evidence shows that the bias-corrected estimators are extremely accurate even for very small sample sizes and are superior than the previous estimators in terms of biases and root mean squared errors. Finally, applications to two real datasets are presented for illustrative purposes.     

The relative performances of improved ridge estimators and an empirical bayes estimator: some monte carlo results

The relative 'performances of improved ridge estimators and an empirical Bayes estimator are studied by means of Monte Carlo simulations. The empirical Bayes method is seen to perform consistently better in terms of smaller MSE and more accurate empirical coverage than any of the estimators considered here. A bootstrap method is proposed to obtain more reliable estimates of the MSE of ridge esimators. Some theorems on the bootstrap for the ridge estimators are also given and they are used to provide an analytical understanding of the proposed bootstrap procedure. Empirical coverages of the ridge estimators based on the proposed procedure are generally closer to the nominal coverage when compared to their earlier counterparts. In general, except for a few cases, these coverages are still less accurate than the empirical coverages of the empirical Bayes estimator.     

Outlier Robust Small‐Area Estimation Under Spatial Correlation

Modern systems of official statistics require the estimation and publication of business statistics for disaggregated domains, for example, industry domains and geographical regions. Outlier robust methods have proven to be useful for small‐area estimation. Recently proposed outlier robust model‐based small‐area methods assume, however, uncorrelated random effects. Spatial dependencies, resulting from similar industry domains or geographic regions, often occur. In this paper, we propose an outlier robust small‐area methodology that allows for the presence of spatial correlation in the data. In particular, we present a robust predictive methodology that incorporates the potential spatial impact from other areas (domains) on the small area (domain) of interest. We further propose two parametric bootstrap methods for estimating the mean‐squared error. Simulations indicate that the proposed methodology may lead to efficiency gains. The paper concludes with an illustrative application by using business data for estimating average labour costs in Italian provinces.     

On the construction of nonparametric density function estimators using the bootstrap

The derivation of a new class of nonparametric density function estimators, the so-called bootstrap functional estimators (BFE's), is given. These estimators are shown to be strongly consistent under fairly nonrestrictive conditions. Some small-sample properties are discussed and a number of graphs are presented.     

Maximum likelihood estimators of the location and scale parameters of the exponential distribution from a censored sample

In this note explicit expressions are given for the maximum likelihood estimators of the parameters of the two-parameter exponential distribution, when a doubly censored sample is available.     

Smooth Residual Bootstrap for Empirical Processes of Non-parametric Regression Residuals

Abstract.  The aim of this paper is to prove the validity of smooth residual bootstrap versions of procedures that are based on the empirical process of residuals estimated from a non-parametric regression model. From this result, consistency of various model tests in non-parametric regression is deduced, such as goodness-of-fit tests for the regression and variance function, tests for equality of regression functions and tests concerning the error distribution.     

Bootstrap test of goodness of fit to a linear model when errors are correlated

Given the regression model Y_i = m(x_i) +ε_i (x_i ε C, i = l,…,n, C a compact set in R) where m is unknown and the random errors {ε_i} present an ARMA structure, we design a bootstrap method for testing the hypothesis that the regression function follows a general linear model: H_o : m ε {mθ(.) = A^t(.)θ : θ ε ? ? R^q} with A a functional from R to R^q. The criterion of the test derives from a Cramer-von-Mises type functional distance D = d²([mcirc]_n, A^t(.)θ_n), between [mcirc]_n, a Gasser-Miiller non-parametric estimator of m, and the member of the class defined in H_o that is closest to m_n in terms of this distance. The consistency of the bootstrap distribution of D and θ_n is obtained under general conditions. Finally, simulations show the good behavior of the bootstrap approximation with respect to the asymptotic distribution of D = d².     

Quantile estimation and comparing two independent groups with an approach based on percentile bootstrap

Although the most common approach for comparing two independent groups is on the basis of some measure of location, determination of the differences in the tails of the groups is often of interest. In this study, Harrell–Davis estimator, Sfakianakis–Verginis estimators and default quantile estimator of R are used in conjunction with a percentile bootstrap method with the aim of comparing two independent groups via the quantiles, and the relative efficiencies of Harrell–Davis and Sfakianakis–Verginis estimators are compared. General performance of Sfakianakis–Verginis estimators was much better than Harrell–Davis estimator in terms of both saving actual type I error and relative efficiency.