On estimating the mean of the selected population with unknown variance

This paper compares four estimators of the mean of the selected population from two normal populations with unknown means and common but unknown variance. The selection procedure is that the population yielding the largest sample mean is selected. The four estimators considered are invariant under both location and scale transformations. The bias and mean square errors of the four estimators are computed and compared. The conclusions are close to those reported by Dahiya ‘1974’, even for small sample sizes     

A new class of unbiased estimators of the variance of the systematic sample mean

We propose a class of estimators of the variance of the systematic sample mean, which is unbiased under the assumption that the population follows a superpopulation model that satisfies some mild conditions. The approach is based on the separate estimation of the portion of the variance due to the systematic component of the model and that due to the stochastic component. In particular, we deal with two estimators belonging to the proposed class that are based on moving averages and local polynomials to estimate the systematic component of the model. The latter estimators are unbiased under the assumption that the population follows a linear trend and the errors are homoscedastic and uncorrelated. Through a simulation study we show that these estimators generally outperform, in terms of bias and mean square error, the usual estimator based on the first differences also when the superpopulation model departs significantly from linearity and the errors are heteroscedastic.     

On estimating the variance of a U-statistic

We compare jackknifing and bootstrapping as methods for estimating the variance of a U-statistic. The use of these estimates in calculating asymptotic confidence intervals is discussed, and the results of a numerical study involving Kendall's tau are reported. For the special case of this statistic, the bootstrap is the estimate of choice.     

On the optimality of the sample variance

The present paper explores the structure of linear exponential families for which the sample variance is a uniformly minimum variance unbiased estimator.     

A quadratic approximation for Jackknife estimators of the variance of sample mean functions

Jackknife estimators of the variance of estimators which are functions of the sample mean are considered. A quadratic approximation of them is proposed and compared with a linear approximation by Monte Carlo experiments carried out by statistical software Minitab.     

Maximum likelihood estimation for the generalized poisson distribution when sample mean is larger than sample variance

The generalized Poisson distribution (GPD), studied by many researchers and containing two parameters θ and λ, has been found to fit very well data sets arising in biological, ecological, social and marketing fields. Consul and Shoukri (1985) have shown that for negative values of λ the GPD gets truncated and the model becomes deficient; however, the truncation error becomes less than 0.0005 if the minimum number of non-zero probability classes ≥ 4 for all values of θ and λ and the GPD model can be safely used in all such cases. The problem of admissible maximum likelihood (ML) estimation when the sample mean is larger than the sample variance is considered in this paper which complements the earlier work of Consul and Shoukri (1984) on the existence of unique ML estimators of θ and λ when the sample mean is smaller than or equal to the sample variance.     

On a new interpretation of the sample variance

It may not be an overstatement that one of the most widely reported measures of variation involves S ², the sample variance, which is also well-known to be alternatively expressed in the form of an U statistic with a symmetric kernel of degree 2 whatever be the population distribution function. We propose a very general new approach to construct unbiased estimators of a population variance by U statistics with symmetric kernels of degree higher than two. Surprisingly, all such estimators ultimately reduce to S ² (Theorem 3.1). While Theorem 3.1 is interesting and novel in its own right, it leads to a newer interpretation of S ² that is much broader than what is known in the statistical literature including economics, actuarial mathematics, and mathematical finance.     

Bias in linear model power and sample size due to estimating variance

Planning a study using the General Linear Univariate Model often involves sample size calculation based on a variance estimated in an earlier study. Noncentrality, power, and sample size inherit the randomness. Additional complexity arises if the estimate has been censored. Left censoring occurs when only significant tests lead to a power calculation, while right censoring occurs when only non-significant tests lead to a power calculation. We provide simple expressions for straightforward computation of the distribution function, moments, and quantiles of the censored variance estimate, estimated noncentrality, power, and sample size. We also provide convenient approximations and evaluate their accuracy. The results allow demonstrating that ignoring right censoring falsely widens confidence intervals for noncentrality and power, while ignoring left censoring falsely narrows the confidence intervals. The new results allow assessing and avoiding the potentially substantial bias that censoring may create.     

On the sample mean of locally stationary long-memory processes

Some asymptotic statistical properties of the sample mean of a class locally stationary long-memory process are studied in this paper. Conditions for consistency are investigated and precise convergence rates of the variance of the sample mean are established for a class of time-varying long-memory parameter functions. A central limit theorem for the sample mean is also established. Furthermore, the calculation of the variance of the sample mean is illustrated through several numerical and simulation experiments.     

On improvement in estimating the population mean in simple random sampling

Kadilar and Cingi [Ratio estimators in simple random sampling, Appl. Math. Comput. 151 (3) (2004), pp. 893–902] introduced some ratio-type estimators of finite population mean under simple random sampling. Recently, Kadilar and Cingi [New ratio estimators using correlation coefficient, Interstat 4 (2006), pp. 1–11] have suggested another form of ratio-type estimators by modifying the estimator developed by Singh and Tailor [Use of known correlation coefficient in estimating the finite population mean, Stat. Transit. 6 (2003), pp. 655–560]. Kadilar and Cingi [Improvement in estimating the population mean in simple random sampling, Appl. Math. Lett. 19 (1) (2006), pp. 75–79] have suggested yet another class of ratio-type estimators by taking a weighted average of the two known classes of estimators referenced above. In this article, we propose an alternative form of ratio-type estimators which are better than the competing ratio, regression, and other ratio-type estimators considered here. The results are also supported by the analysis of three real data sets that were considered by Kadilar and Cingi.     

Improved penalized mean for estimating the mean concentration of contaminants

Chen and Jernigan proposed a non-parametric, conservative method that involved using a penalized mean to estimate the average concentration of contaminants in soils. The method assumes a random sample obtained from a whole site involved in the US Superfund program. However, in some cases, about 10% of known data are collected from the 'hot spots'. In this paper, two procedures are proposed to use the information from hot spots data or an extreme value to estimate the mean concentration of contaminants. These procedures are evaluated using a data set of chromium concentrations from one of the Environmental Protection Agency's toxic waste sites. The simulation results show that these new procedures are cost-eff ective.     

On the ABLUE of the normal mean from a censored sample

The asymptotically best linear unbiased estimate (ABLUE) of the normal mean is discussed. The estimate is based on k selected order statistics chosen from a singly or doubly censored large sample of size n(>k). The coefficients, the asymptotic relative efficiency of the estimate, and the optimum spacing of k real numbers between 0 and 1 which determines the optimum ranks of order statistics, are provided. A comparison between the ABLUE and the iterated maximum likelihood estimate is made.     

A comparison of alternative variance estimation strategies for estimating the slope of a linear regression in sample surveys

The balanced half-sample, jackknife and linearization methods are used to estimate the variance of the slope of a linear regression under a variety of computer generated situations. The basic sampling design is one in which two PSU's are selected from each of a number of strata . The variance estimation techniques are compared with a Monte Carlo experiment. Results show that variance estimates may be highly biased and variable unless sizeable numbers of observations are available from each stratum. The jackknife and linearization estimates appear superior to the balanced half sample method - particularly when the number of strata or the number of available observations from each stratum is small.     

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.     

Estimating the variance of the sample median

The small-sample bias and root mean squared error of several distribution-free estimators of the variance of the sample median are examined. A new estimator is proposed that is easy to compute and tends to have the smallest bias and root mean squared error.     

On estimating the mean of the selected normal population in two-stage adaptive designs

Adaptive design is widely used in clinical trials. In this paper, we consider the problem of estimating the mean of the selected normal population in two-stage adaptive designs. Under the LINEX and L₂ loss functions, admissibility and minimax results are derived for some location invariant estimators of the selected normal mean. The naive sample mean estimator is shown to be inadmissible under the LINEX loss function and to be not minimax under both loss functions.     

On improvement in estimating population mean in stratified random sampling

Gupta and Shabbir 2 Gupta, S. and Shabbir, J. 2008. On improvement in estimating the population mean in simple random sampling. J. Appl. Stat., 35(5): 559–566. [Taylor & Francis Online], [Web of Science ®] [Google Scholar] have suggested an alternative form of ratio-type estimators for estimating the population mean. In this paper, we obtained a corrected version for the mean square error (MSE) of the Gupta–Shabbir estimator, up to first order of approximation, and the optimum case is discussed. We expand this estimator to the stratified random sampling and propose general classes for combined and separate estimators. Also an empirical study is carried out to show the properties of the proposed estimators.