A jackknife variance estimator for unequal probability sampling

Summary.  The jackknife method is often used for variance estimation in sample surveys but has only been developed for a limited class of sampling designs. We propose a jackknife variance estimator which is defined for any without-replacement unequal probability sampling design. We demonstrate design consistency of this estimator for a broad class of point estimators. A Monte Carlo study shows how the proposed estimator may improve on existing estimators.     

Reconciliation of asymptotics for unequal probability sampling without replacement

In simple random sampling without replacement (SRSWOR), certain reverse martingale structures render simple asymptotics for the conventional linear statistics. In unequal probability sampling (UPS) WOR, such martingale-based methodology may not be generally adoptable. General asymptotics for UPSWOR sampling schemes, developed by Hartley and Rao (Ann. Math. Statist. 33 (1962) 350), and Hájek (Ann. Math. Statist. 35 (1964) 1491), rest on different sets of regularity assumptions, and they differ in their treatise too. Some anomalies in this context are eliminated here with a reconciliation of both the approaches, and estimation of the asymptotic variance of linear estimators is considered in the same vein. Applications to small area sampling are also stressed.     

A comparison of conditional Poisson sampling versus unequal probability sampling with replacement

The variance of the Horvitz–Thompson estimator for a fixed size Conditional Poisson sampling scheme without replacement and with unequal inclusion probabilities is compared to the variance of the Hansen–Hurwitz estimator for a sampling scheme with replacement. We show, using a theorem by Gabler, that the sampling design without replacement is more efficient than the sampling design with replacement.     

Entropy of unequal probability sampling designs

There exist many designs for unequal probability sampling. In this paper entropy, which is a measure of randomness, is used to compare eight designs. Both old and commonly used designs and more recent designs are included. Several different and general estimates of entropy are presented. In the quest of finding entropy, expressions for the probability function are derived for different designs. One of them is a recent general design called correlated Poisson sampling. Several designs are close to having maximum entropy, which means that the designs are robust. A few designs yield low entropy and should therefore in general be avoided.     

A generalized improvement procedure for variance bounds for minimum variance unbiased estimator in inverse sampling

New bounds are obtained for the variance of the minimum variance unbiased estimator of p i n inverse sampling. A generalized procedure for further improving the bounds is also discussed.     

Chao's list sequential scheme for unequal probability sampling

The problem of unequal probability sampling is reviewed and discussed in the light of the list sequential scheme proposed by Chao. Chao's scheme is described fully, and its statistical properties are compared with systematic piPS sampling by simulation using standard populations.     

A finite population quantile estimation by unequal probability sampling

Abstract

Using a model-assisted approach, this paper studies asymptotically design-unbiased (ADU) estimation of a population “distribution function” and extends to deriving an asymptotic and approximate unbiased estimator for a population quantile from a sample chosen with varying probabilities. The respective asymptotic standard errors and confidence intervals are then worked out. Numerical findings based on an actual data support the theory with efficient results.     

A new ranked set sample estimator of variance

Summary. We develop an unbiased estimator of the variance of a population based on a ranked set sample. We show that this new estimator is better than estimating the variance based on a simple random sample and more efficient than the estimator based on a ranked set sample proposed by Stokes. Also, a test to determine the effectiveness of the judgment ordering process is proposed.     

On inverse sampling without replacement

The inverse sampling (due to Haldane, 1945) was proposed for random sampling with replacement draws when “the numberr of successes in the sample until the process of drawing is stopped” is prefixed. In this paper an extension of that result is proposed when the draws are obtained via random sampling without replacement.     

A confidence interval for the median of a finite population under unequal probability sampling: A model-assisted approach

This paper presents a method for constructing confidence intervals for the median of a finite population under unequal probability sampling. The model-assisted approach makes use of the _L1$L_1$-norm to motivate the estimating function which is then used to develop a unified approach to inference which includes not only confidence intervals but hypothesis tests and point estimates. The approach relies on large sample theory to construct the confidence intervals. In cases when second-order inclusion probabilities are not available or easy to compute, the Hartley–Rao variance approximation is employed. Simulations show that the confidence intervals achieve the appropriate confidence level, whether or not the Hartley–Rao variance is employed.     

Adaptive sampling without replacement of clusters

In a common form of adaptive cluster sampling, an initial sample of units is selected by random sampling without replacement and, whenever the observed value of the unit is sufficiently high, its neighboring units are added to the sample, with the process of adding neighbors repeated if any of the added units are also high valued. In this way, an initial selection of a high-valued unit results in the addition of the entire network of surrounding high-valued units and some low-valued “edge” units where sampling stops. Repeat selections can occur when more than one initially selected unit is in the same network or when an edge unit is shared by more than one added network. Adaptive sampling without replacement of networks avoids some of this repeat selection by sequentially selecting initial sample units only from the part of the population not already in any selected network. The design proposed in this paper carries this step further by selecting initial units only from the population, exclusive of any previously selected networks or edge units.     

A profile likelihood method for normal mixture with unequal variance

It is well known that the normal mixture with unequal variance has unbounded likelihood and thus the corresponding global maximum likelihood estimator (MLE) is undefined. One of the commonly used solutions is to put a constraint on the parameter space so that the likelihood is bounded and then one can run the EM algorithm on this constrained parameter space to find the constrained global MLE. However, choosing the constraint parameter is a difficult issue and in many cases different choices may give different constrained global MLE. In this article, we propose a profile log likelihood method and a graphical way to find the maximum interior mode. Based on our proposed method, we can also see how the constraint parameter, used in the constrained EM algorithm, affects the constrained global MLE. Using two simulation examples and a real data application, we demonstrate the success of our new method in solving the unboundness of the mixture likelihood and locating the maximum interior mode.     

Recurrence relations for the minimum variance unbiased estimator of the probability function of the R distribution

We obtain a new technique to calculate the value of the minimum variance unbiased estimator (MVUE) of the probability function (p.f.) of the R distribution. This technique is based on an investigation of the ratios of r numbers. A recurrence relation for the MVUE of the p.f. of the R distribution is derived. It is interesting that the derived relation does not depend on the r numbers but depends on the ratios of the r numbers. The new method is efficient, convenient and accurate.     

A note on interpenetrating subsamples of unequal sizes drawn with and without replacement

In this article it is shown how one can gain in efficiency in selecting K interpenetrating subsamples of unequal sizes according to Chaudhuri and Adhikary's (1987) scheme rather than selecting them with replacement and considering the estimator based on distinct units as proposed by Bedi(1987).     

A new procedure for variance estimation in simple random sampling using auxiliary information

In this article we have envisaged an efficient generalized class of estimators for finite population variance of the study variable in simple random sampling using information on an auxiliary variable. Asymptotic expressions of the bias and mean square error of the proposed class of estimators have been obtained. Asymptotic optimum estimator in the proposed class of estimators has been identified with its mean square error formula. We have shown that the proposed class of estimators is more efficient than the usual unbiased, difference, Das and Tripathi (Sankhya C 40:139–148, 1978), Isaki (J. Am. Stat. Assoc. 78:117–123, 1983), Singh et al. (Curr. Sci. 57:1331–1334, 1988), Upadhyaya and Singh (Vikram Math. J. 19:14–17, 1999b), Kadilar and Cingi (Appl. Math. Comput. 173:2, 1047–1059, 2006a) and other estimators/classes of estimators. In the support of the theoretically results we have given an empirical study.     

A minimum variance kernel estimator and a discrete frequency polygon estimator for ordinal contingency tables

This paper introduces two estimators, a boundary corrected minimum variance kernel estimator based on a uniform kernel and a discrete frequency polygon estimator, for the cell probabilities of ordinal contingency tables. Simulation results show that the minimum variance boundary kernel estimator has a smaller average sum of squared error than the existing boundary kernel estimators. The discrete frequency polygon estimator is simple and easy to interpret, and it is competitive with the minimum variance boundary kernel estimator. It is proved that both estimators have an optimal rate of convergence in terms of mean sum of squared error, The estimators are also defined for high-dimensional tables.     

Double bootstrap estimation of variance under systematic sampling with probability proportional to size

Variance estimation under systematic sampling with probability proportional to size is known to be a difficult problem. We attempt to tackle this problem by the bootstrap resampling method. It is shown that the usual way to bootstrap fails to give satisfactory variance estimates. As a remedy, we propose a double bootstrap method which is based on certain working models and involves two levels of resampling. Unlike existing methods which deal exclusively with the Horvitz–Thompson estimator, the double bootstrap method can be used to estimate the variance of any statistic. We illustrate this within the context of both mean and median estimation. Empirical results based on five natural populations are encouraging.