An extension of the ranked set sampling theory

This paper is concerned with ranked set sampling theory which is useful to estimate the population mean when the order of a sample of small size can be found without measurements or with rough methods. Consider n sets of elements each set having size m. All elements of each set are ranked but only one is selected and quantified. The average of the quantified elements is adopted as the estimator. In this paper we introduce the notion of selective probability which is a generalization of a notion from Yanagawa and Shirahata (1976). Uniformly optimal unbiased procedures are found for some (n,m). Furthermore, procedures which are unbiased for all distributions and are good for symmetric distributions are studied for (n,m) which do not allow uniformly optimal unbiased procedures.     

A new sampling method for estimating the population mean

A double L ranked set sampling (DLRSS) method is suggested for estimating the population mean. The DLRSS is compared with the simple random sampling (SRS), ranked set sampling (RSS) and L ranked set sampling (LRSS) methods based on the same number of measured units. The conditions for which the suggested estimator performs better than the other estimators are derived. It is found that, the suggested DLRSS estimator is an unbiased of the population mean, and is more efficient than its counterparts using SRS, RSS, and LRSS methods. Real data sets are used for illustration.     

A control in stratified sampling

A new expression of the variance of the units from a stratified finite population wit L strata, is obtained in funcions of the means L–1 strata, and in conseqeuence we obtained formulae which relates the mean of a stratum in a statistical study with the population mean and the ones of the remaining strate. As an application, we obtained a useful checking of the estimation consistency by stratified sampling in any precies survey     

Generalized ratio-type and ratio-exponential-type estimators for population mean under modified Horvitz-Thompson estimator in adaptive cluster sampling

In the present article, we propose the generalized ratio-type and generalized ratio-exponential-type estimators for population mean in adaptive cluster sampling (ACS) under modified Horvitz-Thompson estimator. The proposed estimators utilize the auxiliary information in combination of conventional measures (coefficient of skewness, coefficient of variation, correlation coefficient, covariance, coefficient of kurtosis) and robust measures (tri-mean, Hodges-Lehmann, mid-range) to increase the efficiency of the estimators. Properties of the proposed estimators are discussed using the first order of approximation. The simulation study is conducted to evaluate the performances of the estimators. The results reveal that the proposed estimators are more efficient than competing estimators for population mean in ACS under both modified Hansen-Hurwitz and Horvitz-Thompson estimators.     

On the asymptotic normality of the L1- and L2-errors in histogram density estimation

The L₁ and L₂-errors of the histogram estimate of a density f from a sample X₁,X₂,…,Xn using a cubic partition are shown to be asymptotically normal without any unnecessary conditions imposed on the density f. The asymptotic variances are shown to depend on f only through the corresponding norm of f. From this follows the asymptotic null distribution of a goodness-of-fit test based on the total variation distance, introduced by Györfi and van der Meulen (1991). This note uses the idea of partial inversion for obtaining characteristic functions of conditional distributions, which goes back at least to Bartlett (1938).     

Approximate optimal allocation in repeated sampling from a finite population

Samples of size n are drawn from a finite population on each of two occasions. On the first occasion a variate x is measured, and on the second a variate y. In estimating the population mean of y, the variance of the best linear unbiased combination of means for matched and unmatched samples is itself minimized, with respect to the sampling design on the second occasion, by a certain degree of matching. This optimal allocation depends on the population correlation coefficient, which previous authors have assumed known. We estimate the correlation from an initial matched sample, then an approximately optimal allocation is completed and an estimator formed which, under a bivariate normal superpopulation model, has model expected mean square error equal, apart from an error of order n^-2, to the minimum enjoyed by any linear, unbiased estimator.     

Confidence intervals for the mean of a population containing many zero values under unequal‐probability sampling

In many applications, a finite population contains a large proportion of zero values that make the population distribution severely skewed. An unequal‐probability sampling plan compounds the problem, and as a result the normal approximation to the distribution of various estimators has poor precision. The central‐limit‐theorem‐based confidence intervals for the population mean are hence unsatisfactory. Complex designs also make it hard to pin down useful likelihood functions, hence a direct likelihood approach is not an option. In this paper, we propose a pseudo‐likelihood approach. The proposed pseudo‐log‐likelihood function is an unbiased estimator of the log‐likelihood function when the entire population is sampled. Simulations have been carried out. When the inclusion probabilities are related to the unit values, the pseudo‐likelihood intervals are superior to existing methods in terms of the coverage probability, the balance of non‐coverage rates on the lower and upper sides, and the interval length. An application with a data set from the Canadian Labour Force Survey‐2000 also shows that the pseudo‐likelihood method performs more appropriately than other methods. The Canadian Journal of Statistics 38: 582–597; 2010 © 2010 Statistical Society of Canada     

A method of deriving valid approximate expressions for bias in ratio estimation

The expectation of a ratio of two statistics is usually obtained by the method in which the denominator is expanded into a binomial series around its mean. However, this method can less generally be employed because the expansion of the denominator is not always valid. This paper presents a device in which the expectation of a ratio of two statistics can more generally be obtained. The device is to adopt a positive value as ‘catalyzer’. It can be applied not only to the ratio of estimates based on samples but also to ratio of any two statistics, if the denominator is positive and finite.     

Generalized estimator for the estimation of rare and clustered population variance in adaptive cluster sampling

Adaptive cluster sampling (ACS) is considered to be the most suitable sampling design for the estimation of rare, hidden, clustered and hard-to-reach population units. The main characteristic of this design is that it may select more meaningful samples and provide more efficient estimates for the field investigator as compare to the other conventional sampling designs. In this paper, we proposed a generalized estimator with a single auxiliary variable for the estimation of rare, hidden and highly clustered population variance under ACS design. The expressions of approximate bias and mean square error are derived and the efficiency comparisons have been made with other existing estimators. A numerical study is carried out on a real population of aquatic birds together with an artificial population generated by Poisson cluster process. Related results of numerical study show that the proposed generalized variance estimator is able to provide considerably better results over the competing estimators.     

A matching prior based on the modified profile likelihood in a generalized Weibull stress‐strength model

Bayesian inference of a generalized Weibull stress‐strength model (SSM) with more than one strength component is considered. For this problem, properly assigning priors for the reliabilities is challenging due to the presence of nuisance parameters. Matching priors, which are priors matching the posterior probabilities of certain regions with their frequentist coverage probabilities, are commonly used but difficult to derive in this problem. Instead, we apply an alternative method and derive a matching prior based on a modification of the profile likelihood. Simulation studies show that this proposed prior performs well in terms of frequentist coverage and estimation even when the sample sizes are minimal. The prior is applied to two real datasets. The Canadian Journal of Statistics 41: 83–97; 2013 © 2012 Statistical Society of Canada     

A modified Kolmogorov-Smirnov test for the inverse gaussian density with unknown parameters

The Kolmogorov-Smirnov (KS) test is an empirical distribution function (EDF) based goodness-of-fit test that requires the underlying hypothesized density to be continuous and completely specified. When the parameters are unknown and must be estimated from the data, standard tables of the KS test statistic are not valid. Approximate upper tail percentage points of the KS statistic for the inverse Gaussian (IG) distribution with unknown parameters are tabled in this paper.

A study of the power of the KS test for the IG distribution indicates that the test is able todiscriminate between the IG distribution and distributions such as the uniform and exponentialdistributions that are very different in shape, but is relatively unable to discriminate between the IG distribution and distributions that are similar in shape such as the lognormal and Weibull distributions. In modeling settings the former distinction is typically more important to make than the latter distinction.     

A two-stage unrelated randomized response model for estimating a rare sensitive attribute in probability proportional to size sampling using Poisson distribution

This paper describes the estimating procedures of mean number of entities that possess a rare sensitive attribute using the Mangat (1992) randomized device, when the population consists of some clusters and the population is again stratified with some clusters in each stratum. Unbiased estimation procedures for the mean number of individuals have been discussed and their properties are described when the parameter of a rare unrelated attribute is assumed to be known and unknown. An empirical study is carried out to show the dominance of the proposed estimator over Lee et al. (2013) estimator.     

A representation in beta series for the exact density of some random volume contents

Consider r independent and identically distributed random points in a unit n-ball of which p are in the interior and rp are on the surface. These r points, via their convex hull, generate an r-simplex. This article deals with the exact density of the r-content when the points are uniformly distributed. The exact density of the r-content is obtained for the general values of the parameters r, n and p. A representation of the density is given as a mixture of beta type-1 densities so that one can evaluate various types of probabilities by using incom-plete beta tables.     

A modified likelihood ratio test for the mean direction in the von mises distribution

The likelihood ratio test (LRT) for the mean direction in the von Mises distribution is modified for possessing a common asymptotic distribution both for large sample size and for large concentration parameter. The test statistic of the modified LRT is compared with the F distribution but not with the chi-square distribution usually employed, Good performances of the modified LRT are shown by analytical studies and Monte Carlo simulation studies, A notable advantage of the test is that it takes part in the unified likelihood inference procedures including both the marginal MLE and the marginal LRT for the concentration parameter.