AN ESTIMATION PARADOX ARISING IN POPULATION GENETICS THEORY1

We consider a stochastic process describing the evolution of a certain population. A sample is taken from this population in some given generation and is to be used for estimation purposes. It is shown that if one wishes to estimate, from the sample, the actual value of a certain population quantity F in the current generation, estimation using a statistic f is preferred to estimation using a different statistic k, whereas, if one wishes to estimate the mean value of F relative to the stochastic process describing the population evolution, estimation using k is preferred to estimation using f.     

Mixed ranked set sampling design

The main focus of agricultural, ecological and environmental studies is to develop well designed, cost-effective and efficient sampling designs. Ranked set sampling (RSS) is one method that leads to accomplish such objectives by incorporating expert knowledge to its advantage. In this paper, we propose an efficient sampling scheme, named mixed RSS (MxRSS), for estimation of the population mean and median. The MxRSS scheme is a suitable mixture of both simple random sampling (SRS) and RSS schemes. The MxRSS scheme provides an unbiased estimator of the population mean, and its variance is always less than the variance of sample mean based on SRS. For both symmetric and asymmetric populations, the mean and median estimators based on SRS, partial RSS (PRSS) and MxRSS schemes are compared. It turns out that the mean and median estimates under MxRSS scheme are more precise than those based on SRS scheme. Moreover, when estimating the mean of symmetric and some asymmetric populations, the mean estimates under MxRSS scheme are found to be more efficient than the mean estimates with PRSS scheme. An application to real data is also provided to illustrate the implementation of the proposed sampling scheme.     

An odd property of sample median from odd sample sizes

In this paper, we establish some Pitman closeness results concerning the sample median from a symmetric continuous distribution. We show that when an odd sample size is increased by one, the sample median becomes Pitman-closer to the population median, while when an even sample size is increased by one, the sample median need not be Pitman-closer. We establish the former through probabilistic derivations while the latter is through a counterexample. We also discuss the situation when the sample is increased by two observations.     

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.     

Ratio detection for mean change in α mixing observations

A ratio test based on the indicators of the data minus the sample median is proposed to detect the change in the mean of α-mixing stochastic sequences. The asymptotic distribution of the test is derived under the null hypothesis. The consistency of the proposed test is also obtained under the hypothesis that μ changes at some unknown time. We also propose a consistent estimator for the change point on the ratio test. Simulations demonstrate that the test and the estimator behaves well for heavy-tailed sequences. At last, an empirical application demonstrate the validity of the test and the estimator.     

A modified bootstrap estimator for the mean of an asymmetric distribution

A modified bootstrap estimator of the population mean is proposed which is a convex combination of the sample mean and sample median, where the weights are random quantities. The estimator is shown to be strongly consistent and asymptotically normally distributed. The small- and moderate-sample-size behavior of the estimator is investigated and compared with that of the sample mean by means of Monte Carlo studies. It is found that the newly proposed estimator has much smaller mean squared errors and also yields significantly shorter confidence intervals for the population mean.     

Improved estimation of population mean using known median of auxiliary variable

In this article, we consider the problem of estimation of population mean using the known median of auxiliary variable. We proposed an estimator and its efficiency is studied analytically as well as empirically for different conditions. The proposed estimator is found to be more efficient than traditional estimators such as sample mean and linear regression estimator.     

Finite sample correction factors for several simple robust estimators of normal standard deviation

Estimation of the standard deviation of a normal population is an important practical problem that in industrial practice must often be done from small and possibly contaminated data sets. Using multiple estimators is useful, as differences in the estimates may indicate whether the data set is contaminated and the form of the contamination. In this paper, finite sample correction factors have been estimated by simulation for several simple robust estimators of the standard deviation of a normal population. The estimators are the median absolute deviation, interquartile range, shortest half interval (Shorth), and median moving range. Finite sample correction factors have also been estimated for the commonly used non-robust estimators: mean absolute deviation and mean moving range. The simulation has been benchmarked against finite sample correction factors for the sample standard deviation and the sample range.     

Improvement in Estimating the Population Median in Simple Random Sampling and Stratified Random Sampling Using Auxiliary Information

This paper deals with estimation of population median in simple and stratified random samplings by using auxiliary information. Auxiliary information is rarely used in estimating population median, although there have been many studies to estimate population mean using auxiliary information. In this study, we suggest some estimators using auxiliary information such as mode and range of an auxiliary variable and correlation coefficient. We also expand these estimators to stratified random sampling for combined and separate estimators. We obtain mean square error equations for all proposed estimators and find theoretical conditions. These conditions are also supported by using numerical examples.     

Bootstrapping analogs of the one way MANOVA test

Abstract

Analogs of the classical one way MANOVA model have recently been suggested that do not assume that population covariance matrices are equal or that the error vector distribution is known. These tests are based on the sample mean and sample covariance matrix corresponding to each of the p populations. We show how to extend these tests using other measures of location such as the trimmed mean or coordinatewise median. These new bootstrap tests can have some outlier resistance, and can perform better than the tests based on the sample mean if the error vector distribution is heavy tailed.     

Partial groups ranked set sampling and mean estimation

Ranked set sampling (RSS) is an advanced sampling method which is very effective for estimating mean of the population when exact measurement of observation is difficult and/or expensive. Balanced Groups RSS (BGRSS) is one of the modification of RSS where only the lowest, the median and the largest ranked units are taken into account. Although BGRSS is advantageous and useful for some specific cases, it has strict restrictions regarding the set size which could be problematic for sampling plans. In this study, we make an improvement on BGRSS and propose a new design called Partial Groups RSS which offers a more flexible sampling plan providing the independence of the set size and sample size. Partial Groups RSS also has a cost advantage over BGRSS. We construct a Monte Carlo simulation study comparing the performance of the mean estimators of the proposed sampling design and BGRSS according to their sampling costs and mean squared errors for various type of distributions. In addition, we give a biometric data application for investigating the efficiency of Partial Groups RSS in real life applications.     

Some Pitman closeness properties pertinent to symmetric populations

In this paper, we focus on Pitman closeness probabilities when the estimators are symmetrically distributed about the unknown parameter θ. We first consider two symmetric estimators θ?₁ and θ?₂ and obtain necessary and sufficient conditions for θ?₁ to be Pitman closer to the common median θ than θ?₂. We then establish some properties in the context of estimation under the Pitman closeness criterion. We define Pitman closeness probability which measures the frequency with which an individual order statistic is Pitman closer to θ than some symmetric estimator. We show that, for symmetric populations, the sample median is Pitman closer to the population median than any other independent and symmetrically distributed estimator of θ. Finally, we discuss the use of Pitman closeness probabilities in the determination of an optimal ranked set sampling scheme (denoted by RSS) for the estimation of the population median when the underlying distribution is symmetric. We show that the best RSS scheme from symmetric populations in the sense of Pitman closeness is the median and randomized median RSS for the cases of odd and even sample sizes, respectively.     

Inequalities Among Some Measures of Location

Inequalities involving some sample means and order statistics are established. An upper bound of the absolute difference between the sample mean and median is also derived. Interesting inequalities among the sample mean and the median are obtained for cases when all the observations have the same sign. Some other algebraic inequalities are derived by taking expected values of the sample results and then applying them to some continuous distributions. It is also proved that the mean of a non-negative continuous random variable is at least as large as p times 100(1 ? p)th percentile.     

One and two sample confidence intervals for estimating the mean of skewed populations: an empirical comparative study

In this paper we consider and propose some confidence intervals for estimating the mean or difference of means of skewed populations. We extend the median t interval to the two sample problem. Further, we suggest using the bootstrap to find the critical points for use in the calculation of median t intervals. A simulation study has been made to compare the performance of the intervals and a real life example has been considered to illustrate the application of the methods.     

Two zs are Better than One

Given only a random sample of observations, the usual estimator for the population mean is the sample mean. If additional information is provided it might be possible in some situations to obtain a better estimator. The situation considered here is when the variable whose mean is sought is composed of factors that are themselves observable. In the basic case, the variable can be expressed as the product of two, independent, more basic variables, but we also consider the case of more than two, the effect of correlation, and when there are observation costs.     

Generalized preliminary test stochastic restricted estimator in the linear regression model

ABSTRACT

In this paper, we propose three generalized estimators, namely, generalized unrestricted estimator (GURE), generalized stochastic restricted estimator (GSRE), and generalized preliminary test stochastic restricted estimator (GPTSRE). The GURE can be used to represent the ridge estimator, almost unbiased ridge estimator (AURE), Liu estimator, and almost unbiased Liu estimator. When stochastic restrictions are available in addition to the sample information, the GSRE can be used to represent stochastic mixed ridge estimator, stochastic restricted Liu estimator, stochastic restricted almost unbiased ridge estimator, and stochastic restricted almost unbiased Liu estimator. The GPTSRE can be used to represent the preliminary test estimators based on mixed estimator. Using the GPTSRE, the properties of three other preliminary test estimators, namely preliminary test stochastic mixed ridge estimator, preliminary test stochastic restricted almost unbiased Liu estimator, and preliminary test stochastic restricted almost unbiased ridge estimator can also be discussed. The mean square error matrix criterion is used to obtain the superiority conditions to compare the estimators based on GPTSRE with some biased estimators for the two cases for which the stochastic restrictions are correct, and are not correct. Finally, a numerical example and a Monte Carlo simulation study are done to illustrate the theoretical findings of the proposed estimators.     

Transform orders and stochastic monotonicity of statistical functionals

In some inferential statistical methods, such as tests and confidence intervals, it is important to describe the stochastic behavior of statistical functionals, aside from their large sample properties. We study such a behavior in terms of the usual stochastic order. For this purpose, we introduce a generalized family of stochastic orders, which is referred to as transform orders, showing that it provides a flexible framework for deriving stochastic monotonicity results. Given that our general definition makes it possible to obtain some well known ordering relations as particular cases, we can easily apply our method to different families of functionals. These include some prominent inequality measures, such as the generalized entropy, the Gini index, and its generalizations. We also illustrate the applicability of our approach by determining the least favorable distribution, and the behavior of some bootstrap statistics, in some goodness-of-fit testing procedures.     

Visualizing Mean,Median, Mean Deviation,and Standard Deviation of a Set of Numbers

We review the existing visualizations of the mean and the median of a given set of numbers. Then we give an alternative visualization of the mean using the empirical cumulative distribution function of the given numbers. Next, we visualize the mean deviation (MD) and the mean square deviation (MSD) of the given numbers from any arbitrary value, including the variance. In light of these new visualizations, we revisit the well-known optimal properties of the MD from the median and the MSD from the mean. We also give a more elementary explanation of why the denominator of the sample variance of a set of numbers is one less than the sample size.     

On the Mixture Proportional Mean Residual Life Model

In this article, we propose a new mixture model induced by the model of proportional mean residual life. Under some appropriate assumptions, it is shown that the mixing and overall variables in the model admit the positive likelihood ratio dependence structure. To see how the overall variable is affected by the stochastic variation of the mixing variable, we study some stochastic comparisons using these variables. Finally, some useful bounds for tail probability of the overall variable for large values of the mixing variable are derived.     

AFFINE‐EQUIVARIANT SPATIAL MEDIAN AND ITS USE IN THE MULTIVARIATE MULTI‐SAMPLE LOCATION PROBLEM

The classical spatial median is not affine‐equivariant, which often turns out to be an unfavourable property. In this paper, the asymptotic properties of an affine‐equivariant modification of the spatial median are investigated. It is shown that under some weak regularity conditions, the modified spatial median computed by means of the sample norming matrix is asymptotically equivalent to the one computed by means of the population norming matrix, which yields its asymptotic normality. A consistent estimate of the asymptotic covariance matrix of the modified spatial median is also presented. These results are implemented in a scheme, where the sample norm is determined by means of the sample Dümbgen scatter matrix. The results are utilized in the construction of affine‐invariant test statistics for testing the multi‐sample hypothesis of equality of location parameters. The performance of the proposed tests is demonstrated through a simulation study.