Umvu estimators for the population mean and variance based on random effects models for lognormal data

Cross-classified data are often obtained in controlled experimental situations and in epidemiologic studies. As an example of the latter, occupational health studies sometimes require personal exposure measurements on a random sample of workers from one or more job groups, in one or more plant locations, on several different sampling dates. Because the marginal distributions of exposure data from such studies are generally right-skewed and well-approximated as lognormal, researchers in this area often consider the use of ANOVA models after a logarithmic transformation. While it is then of interest to estimate original-scale population parameters (e.g., the overall mean and variance), standard candidates such as maximum likelihood estimators (MLEs) can be unstable and highly biased. Uniformly minimum variance unbiased (UMVU) cstiniators offer a viable alternative, and are adaptable to sampling schemes that are typiral of experimental or epidemiologic studies. In this paper, we provide UMVU estimators for the mean and variance under two random effects ANOVA models for logtransformed data. We illustrate substantial mean squared error gains relative to the MLE when estimating the mean under a one-way classification. We illustrate that the results can readily be extended to encompass a useful class of purely random effects models, provided that the study data are balanced.     

A precise estimator for the log-normal mean

The log-normal distribution is frequently encountered in applications. The uniformly minimum variance unbiased (UMVU) estimator for the log-normal mean is given explicitly by a formula found by Finney in 1941. In contrast to this the most commonly used estimator for a log-normal mean is the sample mean. This is possibly due to the complexity of the formula given by Finney. A modified maximum likelihood estimator which approximates the UMVU estimator is derived here. It is sufficiently simple to be implemented in elementary spreadsheet applications. An elementary approximate formula for the root-mean-square error of the suggested estimator and the UMVU estimator is presented. The suggested estimator is compared with the sample mean, the maximum likelihood, and the UMVU estimators by Monte Carlo simulation in terms of root-mean-square error.     

Erratum

For a random variable obeying the inverse Gaussian distribu-tion and its reciprocal, the uniformly minimum variance unbiased (UMVU) estimators of each mode are obtained. The UMVU estimators

of the left and right limits of a certain interval which contains an inverse Gaussian variate with an arbitrary given probability are also proposed.     

Efficient mean estimation in log-normal linear models

Log-normal linear models are widely used in applications, and many times it is of interest to predict the response variable or to estimate the mean of the response variable at the original scale for a new set of covariate values. In this paper we consider the problem of efficient estimation of the conditional mean of the response variable at the original scale for log-normal linear models. Several existing estimators are reviewed first, including the maximum likelihood (ML) estimator, the restricted ML (REML) estimator, the uniformly minimum variance unbiased (UMVU) estimator, and a bias-corrected REML estimator. We then propose two estimators that minimize the asymptotic mean squared error and the asymptotic bias, respectively. A parametric bootstrap procedure is also described to obtain confidence intervals for the proposed estimators. Both the new estimators and the bootstrap procedure are very easy to implement. Comparisons of the estimators using simulation studies suggest that our estimators perform better than the existing ones, and the bootstrap procedure yields confidence intervals with good coverage properties. A real application of estimating the mean sediment discharge is used to illustrate the methodology.     

Variances of UMVU Estimators for Means and Variances After Using a Normalizing Transformation

Suppose that the function f is of recursive type and the random variable X is normally distributed with mean μ and variance α². We set C = f(x). Neyman & Scott (1960) and Hoyle (1968) gave the UMVU estimators for the mean E(C) and for the variance Var(C) from independent and identically distributed random variables X₁,…, X_n(n ≧ 2) having a normal distribution with mean μ and variance σ², respectively. Shimizu & Iwase (1981) gave the variance of the UMVU estimator for E(C). In this paper, the variance of the UMVU estimator for Var(C) is given.     

Efficient Estimation of the PDF and the CDF of the Weibull Extension Model

The Weibull extension model is a useful extension of the Weibull distribution, allowing for bathtub shaped hazard rates among other things. Here, we consider estimation of the PDF and the CDF of the Weibull extension model. The following estimators are considered: uniformly minimum variance unbiased (UMVU) estimator, maximum likelihood (ML) estimator, percentile (PC) estimator, least squares (LS) estimator, and weighted least squares (WLS) estimator. Analytical expressions are derived for the bias and the mean squared error. Simulation studies and real data applications show that the ML estimator performs better than others.     

UNBIASED ESTIMATION OF PARAMETRE FUNCTIONS IN SAMPLING FROM ONE- AND TWO-TRUNCATION PARAMETER FAMILIES

We consider the problem of uniformly minimum variance unbiased (UMVU) estimation of U-estimable functions of three unknown truncation parameters based on two independent random samples: one from a two-truncation parameter family and the other from a one-truncation parameter family. In particular, we obtain the UMVU estimator of the functional Pr{Y > X} and the shortest confidence intervals for some parametric functions.     

The Double Exponential Distribution: Using Calculus to Find a Maximum Likelihood Estimator

Lehmann (1983) discussed several examples of absurd uniform minimum variance unbiased (UMVU) estimators. He argued that these estimators arose because the amount of information available was inadequate for the estimation problem at hand. Here I argue that such absurd UMVU estimators result more from the property of unbiasedness than from inadequate information.     

Estimation of the process incapability index

Greenwich and Jahr-Schaffrath (1995) introduced a new index C _pp a simple transformation of the index C _pm , which provides an uncontaminated separation between information concerning process accuracy and process precision. Under the assumption of normality, we first show that the estimators of C _pp proposed by Greenwich and Jahr-Schaffrath (1995) are UMVU estimators. We also show that for the inaccuracy index, the variance of the unbiased estimator is smaller than the mean squared error (MSE) of the natural (biased) estimator for n > 3. In addition, we obtain the r-th moment and the probability density function of these estimators.     

Separation and Lack of Separation of Subpopulation in the Mixed Distributions

This is a comparative study between the estimates of parameters of mixed distributions in the case of the possibility of separating the units of subpopulation or the absence of that possibility under the progressive type I censored test data. An iterative procedure is developed and tested numerically to obtain new estimators and their variance–covariance matrix. Finally, we will use the exact distribution of the maximum likelihood estimators as well as its asymptotic distribution and the parametric bootstrap method; then, we will discuss the construction of confidence intervals for the mean parameter and their performance is assessed through Monte Carlo simulations.     

On shortest confidence intervals and their relation with uniformly minimum variance unbiased estimators

The aim of this paper is to investigate the possibility of constructing shortest-lenght confidence intervals and give some results and aspects concerning shortest confidence intervals and uniformly minimum variance unbiased (UMVU) estimators.     

Efficient Estimation of the PDF and the CDF of the Exponentiated Gumbel Distribution

The exponentiated Gumbel model has been shown to be useful in climate modeling including global warming problem, flood frequency analysis, offshore modeling, rainfall modeling, and wind speed modeling. Here, we consider estimation of the probability density function (PDF) and the cumulative distribution function (CDF) of the exponentiated Gumbel distribution. The following estimators are considered: uniformly minimum variance unbiased (UMVU) estimator, maximum likelihood (ML) estimator, percentile (PC) estimator, least-square (LS) estimator, and weighted least-square (WLS) estimator. Analytical expressions are derived for the bias and the mean squared error. Simulation studies and real data applications show that the ML estimator performs better than others.     

Estimation for Location-Scale Models with Censored Data

This article considers a class of estimators for the location and scale parameters in the location-scale model based on ‘synthetic data’ when the observations are randomly censored on the right. The asymptotic normality of the estimators is established using counting process and martingale techniques when the censoring distribution is known and unknown, respectively. In the case when the censoring distribution is known, we show that the asymptotic variances of this class of estimators depend on the data transformation and have a lower bound which is not achievable by this class of estimators. However, in the case that the censoring distribution is unknown and estimated by the Kaplan–Meier estimator, this class of estimators has the same asymptotic variance and attains the lower bound for variance for the case of known censoring distribution. This is different from censored regression analysis, where asymptotic variances depend on the data transformation. Our method has three valuable advantages over the method of maximum likelihood estimation. First, our estimators are available in a closed form and do not require an iterative algorithm. Second, simulation studies show that our estimators being moment-based are comparable to maximum likelihood estimators and outperform them when sample size is small and censoring rate is high. Third, our estimators are more robust to model misspecification than maximum likelihood estimators. Therefore, our method can serve as a competitive alternative to the method of maximum likelihood in estimation for location-scale models with censored data. A numerical example is presented to illustrate the proposed method.     

Resampling estimation when observations are m–dependent

For m–dependent, identically distributed random observation, the bootstrap method provides inconsistent estimators of the distribution and variance of the sample mean. This paper proposes an alternative resampling procedure. For estimating the distribution and variance of a function of the sample mean, the proposed resampling estimators are shown to be strongly consistent.     

Randomized quantile regression estimation for heteroskedastic non parametric model

In this paper, we propose robust randomized quantile regression estimators for the mean and (condition) variance functions of the popular heteroskedastic non parametric regression model. Unlike classical approaches which consider quantile as a fixed quantity, our method treats quantile as a uniformly distributed random variable. Our proposed method can be employed to estimate the error distribution, which could significantly improve prediction results. An automatic bandwidth selection scheme will be discussed. Asymptotic properties and relative efficiencies of the proposed estimators are investigated. Our empirical results show that the proposed estimators work well even for random errors with infinite variances. Various numerical simulations and two real data examples are used to demonstrate our methodologies.     

Unbiasedness of two-stage estimation and prediction procedures for mixed linear models

The traditional method for estimating or predicting linear combinations of the fixed effects and realized values of the random effects in mixed linear models is first to estimate the variance components and then to proceed as if the estimated values of the variance components were the true values. This two-stage procedure gives unbiased estimators or predictors of the linear combinations provided the data vector is symmetrically distributed about its expected value and provided the variance component estimators are translation-invariant and are even functions of the data vector. The standard procedures for estimating the variance components yield even, translation-invariant estimators.     

Estimation of the MSE matrix of the stein estimator

Uniformly minimum-variance unbiased (UMVU) estimators of the total risk and the mean-squared-error (MSE) matrix of the Stein estimator for the multivariate normal mean with unknown covariance matrix are proposed. The estimated MSE matrix is helpful in identifying the components which contribute most to the total risk. It also contains information about the performance of the shrinkage estimator with respect to other quadratic loss functions.     

关于捕获再捕获抽样的置信区间

捕获再捕获抽样是一种应用广泛的抽样方法。运用随机模拟,研究捕获再捕获抽样的三种估计量的均值、方差、偏度、峰度、利用近似正态分布构造的置信区间的统计性质。改进了估计量的样本方差计算公式,使得利用近似正态分布构造的置信区间更优。     

On the variance estimate of a wilcoxon-mann whitney statistic for group ordered data

Two consistent estimators for the non-null variance of Wil-coxon-Mann-Whitney’s statistic applied to grouped ordered data, are considered. The first is based on U-statistics and the sec-ond is obtained by the Delta method. Some examples are given to demonstrate the extent of error when using a null variance esti-mate for constructing confidence intervals. It appears that the two consistent estimates are very close, but may both be disting-uishably larger or smaller than the null variance estimate.