Non‐Parametric Selected Ranked Set Sampling

A nonparametric selected ranked set sampling is suggested. The estimator of population mean based on the new approach is compared with that using the simple random sampling (SRS), the ranked set sampling (RSS) and the median ranked set sampling (MRSS) methods. The estimator of population mean using the new approach is found to be more efficient than its counter‐parts for almost all the cases considered.     

A Note on the Estimation of the Population Mean in Stratified Random Sampling Using Two Auxiliary Variables

In this paper, a two‐phase sampling estimator for a stratified population mean using two auxiliary variables x and z is considered when the stratum mean of x is unknown but that of z is known. The suggested estimator under its optimal condition is found to be more efficient than the one using only x.     

An Efficient Regression Type Estimator in Survey Sampling

For the estimation of population mean in simple random sampling, an efficient regression-type estimator is proposed which is more efficient than the conventional regression estimator and hence than mean per unit estimator, ratio and product estimators and many other estimators proposed by various authors. Some numerical examples are included for illustration.     

Variance estimation for systematic designs in spatial surveys

Summary In spatial surveys for estimating the density of objects in a survey region, systematic designs will generally yield lower variance than random designs. However, estimating the systematic variance is well known to be a difficult problem. Existing methods tend to overestimate the variance, so although the variance is genuinely reduced, it is over‐reported, and the gain from the more efficient design is lost. The current approaches to estimating a systematic variance for spatial surveys are to approximate the systematic design by a random design, or approximate it by a stratified design. Previous work has shown that approximation by a random design can perform very poorly, while approximation by a stratified design is an improvement but can still be severely biased in some situations. We develop a new estimator based on modeling the encounter process over space. The new “striplet” estimator has negligible bias and excellent precision in a wide range of simulation scenarios, including strip‐sampling, distance‐sampling, and quadrat‐sampling surveys, and including populations that are highly trended or have strong aggregation of objects. We apply the new estimator to survey data for the spotted hyena (Crocuta crocuta) in the Serengeti National Park, Tanzania, and find that the reported coefficient of variation for estimated density is 20% using approximation by a random design, 17% using approximation by a stratified design, and 11% using the new striplet estimator. This large reduction in reported variance is verified by simulation.     

On an Efficient Regression Type Estimator

The regression type estimator proposed by KAUR (1985) is considered. Another expression for the approximated mean square error (AMSE), to a first degree of approximation, is obtained. This AMSE is also minimized with respect to a parameter α. Three numerical examples are included. These numerical examples show that this estimator is not significantly more efficient than regression estimator and with respect to ratio and sample mean estimators, it does not always exhibit a high efficiency, as was contended by KAUR (1985). Moreover, an upper bound for the relative precision of the proposed estimator with respect to linear regression estimator is derived.     

Case-cohort analysis with accelerated failure time model

Summary .  In a case–cohort design, covariates are assembled only for a subcohort that is randomly selected from the entire cohort and any additional cases outside the subcohort. This design is appealing for large cohort studies of rare disease, especially when the exposures of interest are expensive to ascertain for all the subjects. We propose statistical methods for analyzing the case–cohort data with a semiparametric accelerated failure time model that interprets the covariates effects as to accelerate or decelerate the time to failure. Asymptotic properties of the proposed estimators are developed. The finite sample properties of case–cohort estimator and its relative efficiency to full cohort estimator are assessed via simulation studies. A real example from a study of cardiovascular disease is provided to illustrate the estimating procedure.     

A Combined Product Estimator in Sample Survey

For the estimation of the population mean in stratified random sampling a ‘Combined Product Estimator’ is proposed which is more efficient than the ‘Combined Ratio’ and ‘Separate Ratio’ estimators. Also, the proposed estimator have exact expressions for bias and mean square error. An empirical illustration is given to compare the efficiencies of different estimators.     

Estimation of Ratio Using Rank Set Sampling

Ranked set sampling (RSS) as suggested by McIntyre (1952) and developed by Takahasi and Wakimoto (1968) is used to estimate the ratio. It is proved that by using RSS method the efficiency of the estimator relative to the simple random sampling (SRS) method has increased. Computer simulated results are given. An example using real data is presented to illustrate the computations.     

A Class of Ratio Cum Product-Type Estimator in Double Sampling

A class of ratio cum product-type estimator is proposed in case of double sampling in the present paper. Its bias and variance to the first order of approximation are obtained. For an appropriate weight ‘a’ and a good range of α-values, it is found that the proposed estimator is more efficient than the set of estimator viz., simple mean estimator, usual ratio and product estimators, SRIVASTAVA 's estimator (1967), CHAKARBARTY 's estimator and product-type estimator, which are in fact the particular cases of it. The proposed estimator is as efficient as linear regression estimator in double sampling at optimum value of α.     

Ranked set sampling for replicated sampling designs

In practical ecological sampling studies, a certain design (such as plot sampling or line-intercept sampling) is usually replicated more than once. For each replication, the Horvitz-Thompson estimation of the objective parameter is considered. Finally, an overall estimator is achieved by averaging the single Horvitz-Thompson estimators. Because the design replications are drawn independently and under the same conditions, the overall estimator is simply the sample mean of the Horvitz-Thompson estimators under simple random sampling. This procedure may be wisely improved by using ranked set sampling. Hence, we propose the replicated protocol under ranked set sampling, which gives rise to a more accurate estimation than the replicated protocol under simple random sampling.     

An estimator of number of species from quadrat sampling

We consider the problem of estimating the number of distinct species S in a study area from the recorded presence or absence of species in each of a sample of quadrats. A generalized jackknife estimator of S is derived, along with an estimate of its variance. It is compared with the jackknife estimator for S proposed by Heltshe and Forrester and the empirical Bayes estimator of Mingoti and Meeden. We show that the empirical Bayes estimator has the form of a generalized jackknife estimator under a specific model for species distribution. We compare the new estimators of S to the empirical Bayes estimator via simulation. We characterize circumstances under which each is superior.     

Ranked Set Sampling and the Line Intercept Method: A More Efficient Procedure

Ranked set sampling is a method which may be used to increase the efficiency of the estimator of the mean of a population. Ranked set sampling with size biased probability of selection (i.e., the items are selected with probability proportion to its size) is combined with the line intercept method to increase the efficency of estimating cover, density and total amount of some variable of interest (e.g. biomass). A two-stage sampling plan is suggested with line intercept sampling in the first stage. Simple random sampling and ranked set sampling are compared in the second stage to show that the unbiased estimators of density, cover and total amount of some variable of interest based on ranked set sampling have smaller variances than the usual unbiased estimator based on simple random sampling. Efficiency is increased by reducing the number of items which are measured on a transect or by increasing the number of independent transects utilized in a study area. An application procedure is given for estimation of coverage, density and number of stems of mountain mahogany (Cercocarpus montanus) in a study area east of Laramie, Wyoming.     

Spatially balanced sampling through the pivotal method

A simple method to select a spatially balanced sample using equal or unequal inclusion probabilities is presented. For populations with spatial trends in the variables of interest, the estimation can be much improved by selecting samples that are well spread over the population. The method can be used for any number of dimensions and can hence also select spatially balanced samples in a space spanned by several auxiliary variables. Analysis and examples indicate that the suggested method achieves a high degree of spatial balance and is therefore efficient for populations with trends.     

Optimal sampling and estimation strategies under the linear model

In some cases model-based and model-assisted inferences canlead to very different estimators. These two paradigms are notso different if we search for an optimal strategy rather thanjust an optimal estimator, a strategy being a pair composedof a sampling design and an estimator. We show that, under alinear model, the optimal model-assisted strategy consists ofa balanced sampling design with inclusion probabilities thatare proportional to the standard deviations of the errors ofthe model and the Horvitz–Thompson estimator. If the heteroscedasticityof the model is 'fully explainable’ by the auxiliary variables,then this strategy is also optimal in a model-based sense. Moreover,under balanced sampling and with inclusion probabilities thatare proportional to the standard deviation of the model, thebest linear unbiased estimator and the Horvitz–Thompsonestimator are equal. Finally, it is possible to construct asingle estimator for both the design and model variance. Theinference can thus be valid under the sampling design and underthe model.     

On Double Sampling Ratio-Cum-Product-Type Estimator with Independent Samples

There are two cases in double sampling; case(i) when the second sample is a sub-sample from preliminary large sample, and case(ii) when the second sample is not a sub-sample from the preliminary large sample. Recently SISODIA and DWIVEDI (1981) proposed a ratio cum product-type estimator in double sampling in which they have studied the properties of this estimator under case (i). In this paper, we have made an attempt to study the properties of the same estimator under case (ii). It is found that the estimator is superior than double sampling linear regression estimator, usual ratio estimator, product estimator and among others. The estimator is also compared with simple mean per unit for a given cost of the survey.     

Line Transect Sampling in Small and Large Regions

Melville and Welsh (2001, Biometrics 57, 1130-1137) consider an approach to line transect sampling using a separate calibration study to estimate the detection function g. They present a simulation study contrasting their results with poor results from a traditional estimator, labeled the "Buckland" estimator and referenced to Buckland et al. (1993, Distance Sampling: Estimating Abundance of Biological populations). The poor results from the "Buckland" estimator can be explained by the following observations: (i) the estimator is designated for untruncated distance data, but was applied by Melville and Welsh to truncated distance data; (ii) distance data were not pooled across transects, contrary to standard practice; and (iii) bias of the estimator was evaluated with respect to a fixed rather than a randomized grid of transect lines. We elaborate on the points above and show that the traditional methods perform to expectation when applied correctly. We also emphasize that the estimator labeled the "Buckland" estimator by Melville and Welsh is not an estimator recommended by Buckland et al. for practical survey applications.     

Augmented Inverse Probability Weighted Estimator for Cox Missing Covariate Regression

This article investigates an augmented inverse selection probability weighted estimator for Cox regression parameter estimation when covariate variables are incomplete. This estimator extends the Horvitz and Thompson (1952, Journal of the American Statistical Association 47, 663-685) weighted estimator. This estimator is doubly robust because it is consistent as long as either the selection probability model or the joint distribution of covariates is correctly specified. The augmentation term of the estimating equation depends on the baseline cumulative hazard and on a conditional distribution that can be implemented by using an EM-type algorithm. This method is compared with some previously proposed estimators via simulation studies. The method is applied to a real example.     

Estimation of Prevalence and Relative Risk in Repetitive Surveys

The Theory of double sampling as proposed by Neyman (1938) and subsequently used for successive sampling by Jesson (1942), Yates (1960), Patterson (1950), Eckler (1955), Kuldroff (1963) and Tikkiwal (1960, 1967) has been explored to develop a general estimator which can be used for estimation of parameters such as mean, ratio or double ratio. A simple case of sampling on two occasions has only been considered but the logic can easily be extended for more than two occasions. The results show that the generalised estimator will be very useful for the applied statisticians.     

Estimating the encounter rate variance in distance sampling

Summary .  The dominant source of variance in line transect sampling is usually the encounter rate variance. Systematic survey designs are often used to reduce the true variability among different realizations of the design, but estimating the variance is difficult and estimators typically approximate the variance by treating the design as a simple random sample of lines. We explore the properties of different encounter rate variance estimators under random and systematic designs. We show that a design-based variance estimator improves upon the model-based estimator of Buckland et al. (2001, Introduction to Distance Sampling. Oxford: Oxford University Press, p. 79) when transects are positioned at random. However, if populations exhibit strong spatial trends, both estimators can have substantial positive bias under systematic designs. We show that poststratification is effective in reducing this bias.     

Cox Regression in Nested Case–Control Studies with Auxiliary Covariates

Summary Nested case–control (NCC) design is a popular sampling method in large epidemiological studies for its cost effectiveness to investigate the temporal relationship of diseases with environmental exposures or biological precursors. Thomas' maximum partial likelihood estimator is commonly used to estimate the regression parameters in Cox's model for NCC data. In this article, we consider a situation in which failure/censoring information and some crude covariates are available for the entire cohort in addition to NCC data and propose an improved estimator that is asymptotically more efficient than Thomas' estimator. We adopt a projection approach that, heretofore, has only been employed in situations of random validation sampling and show that it can be well adapted to NCC designs where the sampling scheme is a dynamic process and is not independent for controls. Under certain conditions, consistency and asymptotic normality of the proposed estimator are established and a consistent variance estimator is also developed. Furthermore, a simplified approximate estimator is proposed when the disease is rare. Extensive simulations are conducted to evaluate the finite sample performance of our proposed estimators and to compare the efficiency with Thomas' estimator and other competing estimators. Moreover, sensitivity analyses are conducted to demonstrate the behavior of the proposed estimator when model assumptions are violated, and we find that the biases are reasonably small in realistic situations. We further demonstrate the proposed method with data from studies on Wilms' tumor.