Consistency of least-squares estimator and its jackknife variance estimator in nonlinear models

The purpose of this paper is twofold: (1) We establish the consistency of the least-squares estimator in a nonlinear modely_i = f(x_i,θ) +σ_ie_i where the range of the parameter θ is noncompact, the regression function is unbounded, and the σ_i,'s are not necessarily equal. This extends the results in Jennrich (1969) and Wu (1981). (2) Under the same model, the jackknife estimator of the asymptotic covariance matrix of the least-squares estimator is shown to be consistent, which provides a theoretical justification of the empirical results in Duncan (1978) and the use of the jackknife method in large-sample inferences.     

Jackknifing the Mantel–Haenszel Estimator in Sparse Contingency Tables

This article investigates the performance of two jackknife techniques under an asymptotic model in which the number of 2 × 2 tables increases but the possible marginal configurations remain fixed. These approaches are applied to the Mantel–Haenszel estimator, or transformed versions of this estimator, respectively. The resulting jackknife estimators are shown to be consistent for the common odds ratio. Their asymptotic distributions are derived; they can be used for constructing appropriate sparse-data confidence intervals.     

On estimating a function of the mean of a multivariate normal distribution

The unique minimum variance of unbiased estimator is obtained for analysis functions of the mean of a multivariate normal distribution with either unknown covariance matrix or with covariance matrix of the form σ²v where σ² is unknown.     

Jackknifing L-estimates

Linear functions of order statistics (“L-estimates”) of the form T_n =under jackknifing are investigated. This paper proves that with suitable conditions on the function J, the jackknifed version T_n of the L-estimate T_n has the same limit distribution as T_n. It is also shown that the jackknife estimate of the asymptotic variance of n^1/2 is consistent. Furthermore, the Berry-Esséen rate associated with asymptotic normality, and a law of the iterated logarithm of a class of jackknife L-estimates, are characterized.     

VARIANCE ESTIMATION FOR SAMPLE AUTOCOVARIANCES: DIRECT AND RESAMPLING APPROACHES

The usual covariance estimates for data _n-1 from a stationary zero-mean stochastic process {X_t} are the sample covariances Both direct and resampling approaches are used to estimate the variance of the sample covariances. This paper compares the performance of these variance estimates. Using a direct approach, we show that a consistent windowed periodogram estimate for the spectrum is more effective than using the periodogram itself. A frequency domain bootstrap for time series is proposed and analyzed, and we introduce a frequency domain version of the jackknife that is shown to be asymptotically unbiased and consistent for Gaussian processes. Monte Carlo techniques show that the time domain jackknife and subseries method cannot be recommended. For a Gaussian underlying series a direct approach using a smoothed periodogram is best; for a non-Gaussian series the frequency domain bootstrap appears preferable. For small samples, the bootstraps are dangerous: both the direct approach and frequency domain jackknife are better.     

An asymptotic representation of a ratio of two statistics and its applications

Some statistics in common use take a form of a ratio of two statistics.In this paper, we will discuss asymptotic properties of the ratio statistic.We obtain an asymptotic representation of the ratio with remainder term o _p(n ^-1) and a Edgeworth expansion with remainder term o(n ^-1/2) And as example, the asymptotic representation and the Edgeworth expansion of the jackknife skewness estimator for U-statistics are established and we discuss the biases of the skewness estimator theoretically.We also apply the result to an estimator of Pearson’s coefficient of variation and the sample correlation coefficient.     

On a Berry-Esséen theorem for a Studentized jackknife L-estimate

Consider a linear function of order statistics (“L-estimate”) which can be expressed as a statistical function T(F_n) based on the sample cumulative distribution function F_n. Let T*(F_n) be the corresponding jackknifed version of T(F_n), and let V²_n be the jackknife estimate of the asymptotic variance of n ^1/2T(F_n) or n ^1/2T*(F_n). In this paper, we provide a Berry-Esséen rate of the normal approximation for a Studentized jackknife L-estimate n^1/2[T*(F_n) - T(F)]/V_n, where T(F) is the basic functional associated with the L-estimate.     

Statistical inference for the difference of two Lorenz curves

Testing the Lorenz dominance is of importance in economic and social sciences. In this article, we propose new tools to do inferences for the difference of two Lorenz curves. The asymptotic normality of the proposed smoothed nonparametric estimator is proved. We also propose a smoothed jackknife empirical likelihood (JEL) method which avoids to estimate the complicate asymptotic variance. It is proved that the proposed JEL ratio statistics converge to the standard chi-square distribution. Simulation studies and real data analysis are also conducted, and show encouraging finite-sample performance.     

Asymptotic efficiency of model selection criteria: the nonzero mean gaussian ar(∞) case

Motivated by Shibata’s (1980) asymptotic efficiency results this paper dis-cusses the asymptotic efficiency of the order selected by a selection procedure for an infinite order autoregressive process with nonzero mean and unob servable errors that constitute a sequence of independent Gaussian random variables with mean zero and variance σ² The asymptotic efficiency is established for AIC–type selection criteria such as AIC’, FPE, and S_n(k). In addition, some asymptotic results about the estimators of the parameters of the process and the error–sequence are presented.     

Bias evaluation in the proportional hazards model

We consider two approaches for bias evaluation and reduction in the proportional hazards model proposed by Cox. The first one is an analytical approach in which we derive the n^-1 bias term of the maximum partial likelihood estimator. The second approach consists of resampling methods, namely the jackknife and the bootstrap. We compare all methods through a comprehensive set of Monte Carlo simulations. The results suggest that bias-corrected estimators have better finite-sample performance than the standard maximum partial likelihood estimator. There is some evidence oithe bootstrap-correction superiority over the jackknife-correction but its performance is similar to the analytical estimator. Finaily an application iliustrates the proposed approaches.     

Estimators of Certain Functions of the Variance of a Normal Distribution

Estimators of σ^aand log σ which are functions of Σ(x?x)²/d are considered. Besides the usual sampling theory estimators, Bayesian point estimators which are the usual measures of location of the posterior distribution are given, and in each case an exact or asymptotic expression for the divisor d is stated.     

Small sample performance of jackknife confidence intervals for the james-stein estimator

The primary goal of this paper is to examine the small sample coverage probability and size of jackknife confidence intervals centered at a Stein-rule estimator. A Monte Carlo experiment is used to explore the coverage probabilities and lengths of nominal 90% and 95% delete-one and infinitesimal jackknife confidence intervals centered at the Stein-rule estimator; these are compared to those obtained using a bootstrap procedure.     

Jackknife empirical likelihood method for testing the equality of two variances

In this paper, we propose a nonparametric method based on jackknife empirical likelihood ratio to test the equality of two variances. The asymptotic distribution of the test statistic has been shown to follow χ² distribution with the degree of freedom 1. Simulations have been conducted to show the type I error and the power compared to Levene's test and F test under different distribution settings. The proposed method has been applied to a real data set to illustrate the testing procedure.     

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.     

SMALL SAMPLE VARIANCE ESTIMATORS FOR U-STATISTICS

New unbiased estimators are presented for the dominant and lower-order terms of the variance expansion for U-statistics. In small samples these provide important corrections to the usual estimate of asymptotic standard error which is based on the leading term in the expansion. The new estimators for the first term cannot be recommended. The ordinary jackknife estimator is found to be more effective than the direct estimates of the separate terms.     

Selecting the normal population with the smallest variance: A restricted subset selection rule

Consider k( ? 2) normal populations whose means are all known or unknown and whose variances are unknown. Let σ²_[1] ? ??? ? σ_[k]² denote the ordered variances. Our goal is to select a non empty subset of the k populations whose size is at most m(1 ? m ? k ? 1) so that the population associated with the smallest variance (called the best population) is included in the selected subset with a guaranteed minimum probability P* whenever σ²_[2]/σ_[1]² ? δ* > 1, where P* and δ* are specified in advance of the experiment. Based on samples of size n from each of the populations, we propose and investigate a procedure called R_BCP. We also derive some asymptotic results for our procedure. Some comparisons with an earlier available procedure are presented in terms of the average subset sizes for selected slippage configurations based on simulations. The results are illustrated by an example.     

Higher than second order approximations for sequential point estimation by a two-stage procedure

We consider the sequential point estimation problem of the mean of a normal distribution N(μ, σ²) when the loss function is squared error plus linear cost. It is shown that a two-stage procedure has the asymptotic efficiency of which the order is higher than second order, provided the standard deviation has a known lower bound. We also give a higher than second-order approximation to the risk.     

On Nonsmooth Estimating Functions via Jackknife Empirical Likelihood

In many applications, the parameters of interest are estimated by solving non‐smooth estimating functions with U‐statistic structure. Because the asymptotic covariances matrix of the estimator generally involves the underlying density function, resampling methods are often used to bypass the difficulty of non‐parametric density estimation. Despite its simplicity, the resultant‐covariance matrix estimator depends on the nature of resampling, and the method can be time‐consuming when the number of replications is large. Furthermore, the inferences are based on the normal approximation that may not be accurate for practical sample sizes. In this paper, we propose a jackknife empirical likelihood‐based inferential procedure for non‐smooth estimating functions. Standard chi‐square distributions are used to calculate the p‐value and to construct confidence intervals. Extensive simulation studies and two real examples are provided to illustrate its practical utilities.     

On selecting a subset which contains all populations better than a control

Let π₁…, π_k denote k(≥ 2) populations with unknown means μ₁ , …, μ_k and variances σ₁ ² , …, σ_k ² , respectively and let π_o denote the control population having mean μ_o and variance σ_o ² . It is assumed that these populations are normally distributed with correlation matrix {ρ_ij}. The goal is to select a subset, of populations of π₁ , …, π_k which contains all the populations with means larger than or equal to the mean of the control one. Procedures are given for selecting such a subset so that the probability that all the populations with means larger than or equal to the mean of the control one are included in the selected subset is at least equal to a predetermined value P?(l/k < P? < 1). The goal treated here is a first step screening procedure that allows the experimenter to choose a subset and withhold judgement about which one has the largest mean. Then, if the one with the largest mean is desired it can be chosen from the selected subset on the basis of cost and other considerations. Percentage points are also included.