Bias Correction in the Type I Generalized Logistic Distribution

Four strategies for bias correction of the maximum likelihood estimator of the parameters in the Type I generalized logistic distribution are studied. First, we consider an analytic bias-corrected estimator, which is obtained by deriving an analytic expression for the bias to order n ^?1; second, a method based on modifying the likelihood equations; third, we consider the jackknife bias-corrected estimator; and fourth, we consider two bootstrap bias-corrected estimators. All bias correction estimators are compared by simulation. Finally, an example with a real data set is also presented.     

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.     

On improving convergence rate of Bernstein polynomial density estimator

This paper is concerned with the Bernstein estimator [Vitale, R.A. (1975), ‘A Bernstein Polynomial Approach to Density Function Estimation’, in Statistical Inference and Related Topics, ed. M.L. Puri, 2, New York: Academic Press, pp. 87–99] to estimate a density with support [0, 1]. One of the major contributions of this paper is an application of a multiplicative bias correction [Terrell, G.R., and Scott, D.W. (1980), ‘On Improving Convergence Rates for Nonnegative Kernel Density Estimators’, The Annals of Statistics, 8, 1160–1163], which was originally developed for the standard kernel estimator. Moreover, the renormalised multiplicative bias corrected Bernstein estimator is studied rigorously. The mean squared error (MSE) in the interior and mean integrated squared error of the resulting bias corrected Bernstein estimators as well as the additive bias corrected Bernstein estimator [Leblanc, A. (2010), ‘A Bias-reduced Approach to Density Estimation Using Bernstein Polynomials’, Journal of Nonparametric Statistics, 22, 459–475] are shown to be O(n^?8/9) when the underlying density has a fourth-order derivative, where n is the sample size. The condition under which the MSE near the boundary is O(n^?8/9) is also discussed. Finally, numerical studies based on both simulated and real data sets are presented.     

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.     

Estimation and Inference for Linear Panel Data Models Under Misspecification When Both n and T are Large

This article considers fixed effects (FE) estimation for linear panel data models under possible model misspecification when both the number of individuals, n, and the number of time periods, T, are large. We first clarify the probability limit of the FE estimator and argue that this probability limit can be regarded as a pseudo-true parameter. We then establish the asymptotic distributional properties of the FE estimator around the pseudo-true parameter when n and T jointly go to infinity. Notably, we show that the FE estimator suffers from the incidental parameters bias of which the top order is O(T^{? 1}), and even after the incidental parameters bias is completely removed, the rate of convergence of the FE estimator depends on the degree of model misspecification and is either (nT)^{? 1/2} or n^{? 1/2}. Second, we establish asymptotically valid inference on the (pseudo-true) parameter. Specifically, we derive the asymptotic properties of the clustered covariance matrix (CCM) estimator and the cross-section bootstrap, and show that they are robust to model misspecification. This establishes a rigorous theoretical ground for the use of the CCM estimator and the cross-section bootstrap when model misspecification and the incidental parameters bias (in the coefficient estimate) are present. We conduct Monte Carlo simulations to evaluate the finite sample performance of the estimators and inference methods, together with a simple application to the unemployment dynamics in the U.S.     

Bias reduction for the maximum likelihood estimator of the doubly-truncated Poisson distribution

ABSTRACT

We derive an analytic expression for the bias of the maximum likelihood estimator of the parameter in a doubly-truncated Poisson distribution, which proves highly effective as a means of bias correction. For smaller sample sizes, our method outperforms the alternative of bias correction via the parametric bootstrap. Bias is of little concern in the positive Poisson distribution, the most common form of truncation in the applied literature. Bias appears to be the most severe in the doubly-truncated Poisson distribution, when the mean of the distribution is close to the right (upper) truncation.     

A note on bias reduction in variable-kernel density estimates

It is well known that the inverse-square-root rule of Abramson (1982) for the bandwidth h of a variable-kernel density estimator achieves a reduction in bias from the fixed-bandwidth estimator, even when a nonnegative kernel is used. Without some form of “clipping” device similar to that of Abramson, the asymptotic bias can be much greater than O(h⁴) for target densities like the normal (Terrell and Scott 1992) or even compactly supported densities. However, Abramson used a nonsmooth clipping procedure intended for pointwise estimation. Instead, we propose a smoothly clipped estimator and establish a globally valid, uniformly convergent bias expansion for densities with uniformly continuous fourth derivatives. The main result extends Hall's (1990) formula (see also Terrell and Scott 1992) to several dimensions, and actually to a very general class of estimators. By allowing a clipping parameter to vary with the bandwidth, the usual O(h⁴) bias expression holds uniformly on any set where the target density is bounded away from zero.     

Image estimators based on marked bins

The problem of approximating an ‘image’ S?? ^d from a random sample of points is considered. If S is included in a grid of square bins, a plausible estimator of S is defined as the union of the ‘marked’ bins (those containing a sample point). We obtain convergence rates for this estimator and study its performance in the approximation of the border of S. The practical aspects of implementation are discussed, including some technical improvements on the estimator, whose performance is checked through a real data example.     

Efficient bias-Robust M-Estimators of location in Kolmogorov normal neighbourhoods

We consider minimax-bias M-estimation of a location parameter in a Kolmogorov neighbourhood K() of a normal distribution. The maximum asymptotic bias of M-estimators for the Kolmogorov normal neighbourhood is derived, and its relation with the gross-error sensitivity of the estimator at the nominal model (the Gaussian case) is found. In addition, efficient bias-robust M-estimators T_i are constructed. Numerical results are also obtained to show the percentage of increase in maximum asymptotic bias and the efficiency we can achieve for some well-known -functions.     

On a class of nonparametric tests for the treatment vs control problem

The classical problem of testing treatment versus control is revisited by considering a class of test statistics based on a kernel that depends on a constant ‘a’. The proposed class includes the celebrated Wilcoxon-Mann-Whitnet statistics as a special case when ‘a’=1. It is shown that, with optimal choice of ‘a’ depending on the underlying distribution, the optimal member performs better (in terms of Pitman efficiency) than the Wilcoxon-Mann-Whitney and the Median tests for a wide range of underlying distributions. An extended Hodges-Lehmann type point estimator of the shift prameter corresponding to the proposed ‘optimal’ test statistic is also derived.     

Adaptive nonparametric estimation in the presence of dependence

We consider nonparametric estimation problems in the presence of dependent data, notably nonparametric regression with random design and nonparametric density estimation. The proposed estimation procedure is based on a dimension reduction. The minimax optimal rate of convergence of the estimator is derived assuming a sufficiently weak dependence characterised by fast decreasing mixing coefficients. We illustrate these results by considering classical smoothness assumptions. However, the proposed estimator requires an optimal choice of a dimension parameter depending on certain characteristics of the function of interest, which are not known in practice. The main issue addressed in our work is an adaptive choice of this dimension parameter combining model selection and Lepski's method. It is inspired by the recent work of Goldenshluger and Lepski [(2011), ‘Bandwidth Selection in Kernel Density Estimation: Oracle Inequalities and Adaptive Minimax Optimality’, The Annals of Statistics, 39, 1608–1632]. We show that this data-driven estimator can attain the lower risk bound up to a constant provided a fast decay of the mixing coefficients.     

Exploring the distribution for the estimator of Rosenthal's ‘fail-safe’ number of unpublished studies in meta-analysis

The present article discusses the statistical distribution for the estimator of Rosenthal's ‘file-drawer’ number N_R, which is an estimator of unpublished studies in meta-analysis. We calculate the probability distribution function of N_R. This is achieved based on the central limit theorem and the proposition that certain components of the estimator N_R follow a half-normal distribution, derived from the standard normal distribution. Our proposed distributions are supported by simulations and investigation of convergence.     

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.     

Bias reduction of the maximum-likelihood estimator for a conditional Gaussian MA(1) model

In this paper, we consider an estimation for the unknown parameters of a conditional Gaussian MA(1) model. In the majority of cases, a maximum-likelihood estimator is chosen because the estimator is consistent. However, for small sample sizes the error is large, because the estimator has a bias of O(n^{? 1}). Therefore, we provide a bias of O(n^{? 1}) for the maximum-likelihood estimator for the conditional Gaussian MA(1) model. Moreover, we propose new estimators for the unknown parameters of the conditional Gaussian MA(1) model based on the bias of O(n^{? 1}). We investigate the properties of the bias, as well as the asymptotical variance of the maximum-likelihood estimators for the unknown parameters, by performing some simulations. Finally, we demonstrate the validity of the new estimators through this simulation study.     

Kernel Method Starting with Half-Normal Detection Function for Line Transect Density Estimation

In this article, we introduce the nonparametric kernel method starting with half-normal detection function using line transect sampling. The new method improves bias from O(h ²), as the smoothing parameter h → 0, to O(h ³) and in some cases to O(h ⁴). Properties of the proposed estimator are derived and an expression for the asymptotic mean square error (AMSE) of the estimator is given. Minimization of the AMSE leads to an explicit formula for an optimal choice of the smoothing parameter. Small-sample properties of the estimator are investigated and compared with the traditional kernel estimator by using simulation technique. A numerical results show that improvements over the traditional kernel estimator often can be realized even when the true detection function is far from the half-normal detection function.     

Finite-sample properties of the maximum likelihood estimator for the binary logit model with random covariates

We examine the finite sample properties of the maximum likelihood estimator for the binary logit model with random covariates. Previous studies have either relied on large-sample asymptotics or have assumed non-random covariates. Analytic expressions for the first-order bias and second-order mean squared error function for the maximum likelihood estimator in this model are derived, and we undertake numerical evaluations to illustrate these analytic results for the single covariate case. For various data distributions, the bias of the estimator is signed the same as the covariate’s coefficient, and both the absolute bias and the mean squared errors increase symmetrically with the absolute value of that parameter. The behaviour of a bias-adjusted maximum likelihood estimator, constructed by subtracting the (maximum likelihood) estimator of the first-order bias from the original estimator, is examined in a Monte Carlo experiment. This bias-correction is effective in all of the cases considered, and is recommended for use when this logit model is estimated by maximum likelihood using small samples.     

Investigation of the performance of trimmed estimators of life time distributions with censoring

For the lifetime (or negative) exponential distribution, the trimmed likelihood estimator has been shown to be explicit in the form of a β‐trimmed mean which is representable as an estimating functional that is both weakly continuous and Fréchet differentiable and hence qualitatively robust at the parametric model. It also has high efficiency at the model. The robustness is in contrast to the maximum likelihood estimator (MLE) involving the usual mean which is not robust to contamination in the upper tail of the distribution. When there is known right censoring, it may be perceived that the MLE which is the most asymptotically efficient estimator may be protected from the effects of ‘outliers’ due to censoring. We demonstrate that this is not the case generally, and in fact, based on the functional form of the estimators, suggest a hybrid defined estimator that incorporates the best features of both the MLE and the β‐trimmed mean. Additionally, we study the pure trimmed likelihood estimator for censored data and show that it can be easily calculated and that the censored observations are not always trimmed. The different trimmed estimators are compared by a modest simulation study.     

Estimation for a scale parameter with known coefficient of variation

A loss function proposed by Wasan (1970) is well-fitted for a measure of inaccuracy for an estimator of a scale parameter of a distribution defined onR ⁺=(0, ∞). We refer to this loss function as the K-loss function. A relationship between the K-loss and squared error loss functions is discussed. And an optimal estimator for a scale parameter with known coefficient of variation under the K-loss function is presented.     

Robust M-estimators of scale: Minimax bias versus maximal variance

In the location-scale estimation problem, we study robustness properties of M-estimators of the scale parameter under unknown ?-contamination of a fixed symmetric unimodal error distribution F₀. Within a general class of M-estimators, the estimator with minimax asymptotic bias is shown to lie within the subclass of α-interquantile ranges of the empirical distribution symmetrized about the sample median. Our main result is that as ? → 0, the limiting minimax asymptotic bias estimator is sometimes (e.g., when F_o is Cauchy), but not always, the median absolute deviation about the median. It is also shown that contamination in the neighbourhood of a discontinuity of the influence function of a minimax bias estimator can sometimes inflate the asymptotic variance beyond that achieved by placing all the ?-contamination at infinity. This effect is quantified by a new notion of asymptotic efficiency that takes into account the effect of infinitesimal contamination of the parametric model for the error distribution.     

Negative moments of a modified power series distribution and bias of the maximum likelihood estimator

The class of Modified Power Series distributions (MPSD) containing Lagrangian Poisson (LPD) (Consul and Jain, 1973) and Lagrangian binomial distributions (LBD) (Jain and Consul, 1971) was studied by Gupta (1974). We investigate the problem of finding the negative momentsE[X^-r ], of displaced and decapitated Modified Power Series Distributions. We derive the relationship between rand (r-1) negative moments. The negative moments of the decapitated and displaced LPD are obtained. These results are, then, used to find the exact amount of bias in the ML estimators of the parameters in the LPD and the LBD. We have also given the variances of the ML estimator and the minimum variance unbiased estimator of the parameter in the LPD.