Moments of the sampled autocovariances and autocorrelations for a Gaussian white-noise process

Cumulants, moments about zero, and central moments are obtained for the mean-corrected serial covariances and serial correlations for series realizations of length n from a white-noise Gaussian process. All first and second moments (and some third, fourth, and higher moments) are given explicitly for the serial covariances; and the corresponding moments for the serial correlations are derived either explicitly or implicitly.     

Alternative moment approximations for berkson's minimum logit chi-squared estimator

Expressions are derived for the bias to order J^-1 , the variance to order J^-2 and the mean squared error to order J^-2 of Berkson's minimum logit chi-squared estimator where J is the number of distinct design points. These moment approximations are numerically compared to Monte Carlo estimates of the true moments and the moment approximations of Amemiya (1980) which are appropriate when the “average” number of observations per design point is large. They are used to compare the mean squared error of the minimum logit chi-squared estimator to that of the maximum likelihood estimator and to investigate the effect of bias on confidence intenrals constructed using the minimum logit chi-squared estimator.     

On the minimum bias response surface designs of Box and Draper

The minimum bias design criterion is an important part of response surface methodology. In this paper, necessary and sufficient conditions for minimum bias designs are derived in terms of the design moments. It is shown that the sufficient conditions of Box and Draper (J. Amer. Statist. Assoc. 54 (1959) 622) are sometimes, but not always, necessary depending on the order of the polynomial model and the number of factors.     

Adjusted Pearson residuals in beta regression models

In this paper, matrix formulae of order n^?1, where n is the sample size, for the first two moments of Pearson residuals are obtained in beta regression models. Adjusted Pearson residuals are also obtained, having, to this order, expected value zero and variance one. Monte Carlo simulation results are presented illustrating the behaviour of both adjusted and unadjusted residuals.     

Panel data unit roots tests: The role of serial correlation and the time dimension

We investigate the influence of residual serial correlation and of the time dimension on statistical inference for a unit root in dynamic longitudinal data, known as panel data in econometrics. To this end, we introduce two test statistics based on method of moments estimators. The first is based on the generalized method of moments estimators, while the second is based on the instrumental variables estimator. Analytical results for the Instrumental Variables (IV) based test in a simplified setting show that (i) large time dimension panel unit root tests will suffer from serious size distortions in finite samples, even for samples that would normally be considered large in practice, and (ii) negative serial correlation in the error terms of the panel reduces the power of the unit root tests, possibly up to a point where the test becomes biased. However, near the unit root the test is shown to have power against a wide range of alternatives. These findings are confirmed in a more general set-up through a series of Monte Carlo experiments.     

The Mean Squared Error of the Instrumental Variables Estimator When the Disturbance Has an Elliptical Distribution

This paper generalizes Nagar's (1959) approximation to the finite sample mean squared error (MSE) of the instrumental variables (IV) estimator to the case in which the errors possess an elliptical distribution whose moments exist up to infinite order. This allows for types of excess kurtosis exhibited by some financial data series. This approximation is compared numerically to Knight's (1985) formulae for the exact moments of the IV estimator under nonnormality. We use the results to explore two questions on instrument selection. First, we complement Buse's (1992) analysis by considering the impact of additional instruments on both bias and MSE. Second, we evaluate the properties of Andrews's (1999) selection method in terms of the bias and MSE of the resulting IV estimator.     

On intra-class correlation coefficient estimation

Consider the problem of estimating the intra-class correlation coefficient of a symmetric normal distribution. In a recent article (Pal and Lim (1999)) it has been shown that the three popular estimators, namely—the maximum likelihood estimator (MLE), the method of moments estimator (MME) and the unique minimum variance unbiased estimator (UMVUE), are second order admissible under the squared error loss function. In this paper we study the performance of the above mentioned estimators in terms of Pitman Nearness Criterion (PNC) as well as Stochastic Domination Criterion (SDC). We then apply the aforementioned estimators to two real life data sets with moderate to large sample sizes, and bootstrap bias as well as mean squared errors are computed to compare the estimators. In terms of overall performance the MME seems most appealing among the three estimators considered here and this is the main contribution of our paper. Formerly University of Southewestern Louisisna     

A discrete distribution associated with a pure birth process

A discrete distribution associated with a pure birth process starting with no individuals, with birth rates λ ⁿ =λ forn=0, 2, …,m−1 and λ ⁿ forn≥m is considered in this paper. The probability mass function is expressed in terms of an integral that is very convenient for computing probabilities, moments, generating functions and others. Using this representation, the mean and the k-th factorial moments of the distribution are obtained. Some nice characterizations of this distribution are also given.     

A general class of chain estimators for ratio and product of two means of a finite population

This paper proposes a class of estimators for estimating ratio and product of two means of a finite population using information on two auxiliary characters. Asymptotic expression to terms of order 0(n^-1) for bias and mean square error (MSE) of the proposed class of estimators are derived. Optimum conditions are obtained under which the proposed class of estimators has the minimum MSE. An empirical study is carried out to compare the performance of various estimators of ratio with the conventional estimators.     

First‐order bias correction for fractionally integrated time series

Most of the long memory estimators for stationary fractionally integrated time series models are known to experience non‐negligible bias in small and finite samples. Simple moment estimators are also vulnerable to such bias, but can easily be corrected. In this article, the authors propose bias reduction methods for a lag‐one sample autocorrelation‐based moment estimator. In order to reduce the bias of the moment estimator, the authors explicitly obtain the exact bias of lag‐one sample autocorrelation up to the order n⁻¹. An example where the exact first‐order bias can be noticeably more accurate than its asymptotic counterpart, even for large samples, is presented. The authors show via a simulation study that the proposed methods are promising and effective in reducing the bias of the moment estimator with minimal variance inflation. The proposed methods are applied to the northern hemisphere data. The Canadian Journal of Statistics 37: 476–493; 2009 © 2009 Statistical Society of Canada     

Moments of order statistics from one truncation parameter densities

Analogs of some classical recurrence relations for moments of order statistics for a one truncation parameter densities are derived. These relate the k^thorder moment of the r^thorder statistic X_r:nto the k^thorder moment of X_l:ior X_r:rvia an integral operator. Similar results are obtained for product moments. An application is also given.     

Higher-order Bartlett-type adjustment

This paper deals with Bartlett-type adjustment which makes all the terms up to order n^−k in the asymptotic expansion vanish, where k is an integer k ⩾ 1 and n depends on the sample size. Extending Cordeiro and Ferrari (1991, Biometrika, 78, 573–582) for the case of k = 1, we derive a general formula of the kth-order Bartlett-type adjustment for the test statistic whose kth-order asymptotic expansion of the distribution is given by a finite linear combination of chi-squared distribution with suitable degrees of freedom. Two examples of the second-order Bartlett-type adjustment are given. We also elucidate the connection between Bartlett-type adjustment and Cornish-Fisher expansion.     

Control variates and importance sampling for efficient bootstrap simulations

Importance sampling and control variates have been used as variance reduction techniques for estimating bootstrap tail quantiles and moments, respectively. We adapt each method to apply to both quantiles and moments, and combine the methods to obtain variance reductions by factors from 4 to 30 in simulation examples.We use two innovations in control variates—interpreting control variates as a re-weighting method, and the implementation of control variates using the saddlepoint; the combination requires only the linear saddlepoint but applies to general statistics, and produces estimates with accuracy of order n ^-1/2 B ^-1, where n is the sample size and B is the bootstrap sample size.We discuss two modifications to classical importance sampling—a weighted average estimate and a mixture design distribution. These modifications make importance sampling robust and allow moments to be estimated from the same bootstrap simulation used to estimate quantiles.     

Jackknifing two-sample statistics

In this paper, a new simple method for jackknifing two-sample statistics is proposed. The method is based on a two-step procedure. In the first step, the point estimator is calculated by leaving one X (or Y) out at a time. At the second step, the point estimator obtained in the first step is further jackknifed, leaving one Y (or X) out at a time, resulting in a simple formula for the proposed point estimator. It is shown that by using the two-step procedure, the bias of the point estimator is reduced in terms of asymptotic order, from O(n⁻¹) up to O(n⁻²), under certain regularity conditions. This conclusion is also confirmed empirically in terms of finite sample numerical examples via a small-scale simulation study. We also discuss the idea of asymptotic bias to obtain parallel results without imposing some conditions that may be difficult to check or too restrictive in practice.     

Note on the bias in the estimation of the serial correlation coefficient of AR(1) processes

We derive approximating formulas for the mean and the variance of an autocorrelation estimator which are of practical use over the entire range of the autocorrelation coefficient ρ. The least-squares estimator ∑ ^{n −1} _{i =1}ε _i ε _{i +1} / ∑ ^{n −1} _{i =1}ε² _i is studied for a stationary AR(1) process with known mean. We use the second order Taylor expansion of a ratio, and employ the arithmetic-geometric series instead of replacing partial Cesàro sums. In case of the mean we derive Marriott and Pope's (1954) formula, with (n− 1)⁻¹ instead of (n)⁻¹, and an additional term α (n− 1)⁻². This new formula produces the expected decline to zero negative bias as ρ approaches unity. In case of the variance Bartlett's (1946) formula results, with (n− 1)⁻¹ instead of (n)⁻¹. The theoretical expressions are corroborated with a simulation experiment. A comparison shows that our formula for the mean is more accurate than the higher-order approximation of White (1961), for |ρ| > 0.88 and n≥ 20. In principal, the presented method can be used to derive approximating formulas for other estimators and processes. Received: November 30, 1999; revised version: July 3, 2000     

Moments of the Product-Limit Estimator Under Left-Truncation and Right-Censoring

ABSTRACT

The product-limit estimator (PLE) is a well-known nonparametric estimator for the distribution function of the lifetime when data are left-truncated and right-censored. Much work has focused on developing its asymptotic properties. Finite sample results have been difficult to obtain. This article is concerned about finite moments of the PLE. The moments of the PLE can be represented as a power series in n ^?1. In addition, through the U-statistic mechanism, we obtain also computable formulas for the first, second, third, and fourth of the PLE up to o(n ^?2). Finally, a numerical example is presented.     

Testing for pairwise serial independence via the empirical distribution function

Built on Skaug and Tjøstheim's approach, this paper proposes a new test for serial independence by comparing the pairwise empirical distribution functions of a time series with the products of its marginals for various lags, where the number of lags increases with the sample size and different lags are assigned different weights. Typically, the more recent information receives a larger weight. The test has some appealing attributes. It is consistent against all pairwise dependences and is powerful against alternatives whose dependence decays to zero as the lag increases. Although the test statistic is a weighted sum of degenerate Cramér–von Mises statistics, it has a null asymptotic N (0, 1) distribution. The test statistic and its limit distribution are invariant to any order preserving transformation. The test applies to time series whose distributions can be discrete or continuous, with possibly infinite moments. Finally, the test statistic only involves ranking the observations and is computationally simple. It has the advantage of avoiding smoothed nonparametric estimation. A simulation experiment is conducted to study the finite sample performance of the proposed test in comparison with some related tests.     

Asymptotic convergence in distribution for spearman's rank correlation coefficient

The Edgeworth expansion for the distribution function of Spearman's rank correlation coefficient may be used to show that the rates of convergence for the normal and Pearson type II approximations are l/nand l/n² respectively. Using the Edgeworth expansion up to terms involving the sixth moment of the exact distribution allows an approximation with an error of order l/n³.     

Matching Three Moments with Minimal Acyclic Phase Type Distributions

Abstract

A number of approximate analysis techniques are based on matching moments of continuous time phase type (PH) distributions. This paper presents an explicit method to compose minimal order continuous time acyclic phase type (APH) distributions with a given first three moments. To this end we also evaluate the bounds for the first three moments of order n APH distributions (APH(n)). The investigations of these properties are based on a basic transformation, which extends the APH(n ? 1) class with an additional phase in order to describe the APH(n) class.     

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.