Marginal Likelihood Estimation with the Cross-Entropy Method

We consider an adaptive importance sampling approach to estimating the marginal likelihood, a quantity that is fundamental in Bayesian model comparison and Bayesian model averaging. This approach is motivated by the difficulty of obtaining an accurate estimate through existing algorithms that use Markov chain Monte Carlo (MCMC) draws, where the draws are typically costly to obtain and highly correlated in high-dimensional settings. In contrast, we use the cross-entropy (CE) method, a versatile adaptive Monte Carlo algorithm originally developed for rare-event simulation. The main advantage of the importance sampling approach is that random samples can be obtained from some convenient density with little additional costs. As we are generating independent draws instead of correlated MCMC draws, the increase in simulation effort is much smaller should one wish to reduce the numerical standard error of the estimator. Moreover, the importance density derived via the CE method is grounded in information theory, and therefore, is in a well-defined sense optimal. We demonstrate the utility of the proposed approach by two empirical applications involving women's labor market participation and U.S. macroeconomic time series. In both applications, the proposed CE method compares favorably to existing estimators.     

Variance estimation of DEE and regression estimator when a first-phase cluster sample is restratified

We consider a variance estimation when a stratified single stage cluster sample is selected in the first phase and a stratified simple random element sample is selected in the second phase. We propose explicit formulas of (asymptotically), we propose explicit formulas of (asymptotically) unbiased variance estimators for the double expansion estimator and regression estimator. We perform a small simulation study to investigate the performance of the proposed variance estimators. In our simulation study, the proposed variance estimator showed better or comparable performance to the Jackknife variance estimator. We also extend the results to a two-phase sampling design in which a stratified pps with replacement cluster sample is selected in the first phase.     

A modified unrelated question randomized response device for complex surveys

Singh et al. (Stat Trans 6(4):515–522, 2003) proposed a modified unrelated question procedure and they also demonstrated that the modified procedure is capable of producing a more efficient estimator of the population parameter π _A , namely, the proportion of persons in a community bearing a sensitive character A when π _A  < 0.50. The development of Singh et al. (Stat Trans 6(4):515–522, 2003) is based on simple random samples with replacement and on the assumption that π _B , namely, the proportion of individuals bearing an unrelated innocuous character B is known. Due to these limitations, Singh et al.’s (Stat Trans 6(4):515–522, 2003) procedure cannot be used in practical surveys where usually the sample units are chosen with varying selection probabilities. In this article, following Singh et al. (Stat Trans 6(4):515–522, 2003) we propose an alternative RR procedure assuming that the population units are sampled with unequal selection probabilities and that the value of π _B is unknown. A numerical example comparing the performance of the proposed RR procedure under alternative sampling designs is also reported.     

Synthetic and composite estimation under a superpopulation model

Under a simple superpopulation model for an arbitrary sampling design we derive optimal linear unbiased estimators/predictors of a mean in a domain. They can be viewed as synthetic and composite estimators of small area estimation theory when no auxiliary variable is available. Moreover, we show that the only requirement for optimality of a sampling strategy is to use any sampling plan of fixed sample size together with traditional estimators (as designed for simple random sampling without replacement). Finally, for symmetric sampling plans, simplified formulas (based on the first two moments of sample sizes) for optimal synthetic and composite estimators and their MSE’s are derived. Throughout the paper we consistently use the model-design setup.     

An inequality for random replacement sampling plans

A sample (X₁ …, X_n) is drawn from a population of size N. Karlin (1974) conjectured that for any function ? in a certain class of real-valued functions on the sample space, ? is at least as large for sampling with replacement as for any other random replacement sampling plan. This conjecture is proved under the assumption that ?     

Unbiased Estimation of P(X > Y) for Exponential Populations Using Order Statistics with Application in Ranked Set Sampling

This paper addresses the problem of unbiased estimation of P[X > Y] = θ for two independent exponentially distributed random variables X and Y. We present (unique) unbiased estimator of θ based on a single pair of order statistics obtained from two independent random samples from the two populations. We also indicate how this estimator can be utilized to obtain unbiased estimators of θ when only a few selected order statistics are available from the two random samples as well as when the samples are selected by an alternative procedure known as ranked set sampling. It is proved that for ranked set samples of size two, the proposed estimator is uniformly better than the conventional non-parametric unbiased estimator and further, a modified ranked set sampling procedure provides an unbiased estimator even better than the proposed estimator.     

Probability generating function of GPED<Subscript>2</Subscript>

The Probability generating function of a random variable which has Generalized Polya Eggenberger Distribution of the second kind (GPED ₂) is obtained. The probability density function of the range R, in random sampling from a uniform distribution on (k, l) and exponential distribution with parameter λ is obtained, when the sample size is a random variable from GPED ₂. The results of Bazargan-Lari (2004) follow as special cases.     

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.     

Estimation of the correlation coefficient in probability proportional to size with replacement sampling

g of the population correlation coefficient has been suggested in case of probability proportional to size with replacement sampling. The asymptotic bias, variance and the estimate of the variance of the estimator r_g have been obtained. A comparison of this estimator has been made with the estimator r given by Gupta et al (1993) and usual estimator r₁ for PPSWR sampling. The proposed estimator r_g satisfies the condition −1≤r_g≤1 which the estimator r does not satisfy. Received: September 1, 1999; revised version: May 29, 2001     

Estimation of a sensitive proportion by Warner’s randomized response data through inverse sampling

It is difficult to obtain trustworthy data on stigmatizing matters like habits of tax evasion, drunken driving, etc., through direct queries. To overcome this difficulty, Warner introduced randomized response techniques to estimate the proportion of people bearing such a stigmatizing or sensitive characteristic in a given community. For simplicity in estimation he restricted to SRSWR and since then, several researchers have extended and applied this technique in various ways. All these results involve sampling with a pre-fixed number of draws. In this paper we employ inverse sampling with equal probabilities with replacement and show certain advantages in estimation using randomized response data by Warner’s device gathered through such a simple inverse sampling scheme.     

A design-based randomized response procedure for the estimation of population proportion and sensitivity level

A general design-based approach to randomized response surveys is proposed. The method is tailored for the joint estimation of the proportion of individuals in the population bearing a sensitive attribute and the proportion of individuals in the sensitive group declaring truthfully their status. The proposal is specialized to the case of simple random sampling without replacement, unequal probability sampling without replacement and stratified sampling.     

Estimating sensitive proportions by Warner’s randomized response technique using multiple randomized responses from distinct persons sampled

In 1965 Warner pioneered randomized response techniques to estimate the proportion of people bearing a sensitive characteristic. He restricted applying his randomized response device to gather data on sensitive issues from respondents chosen by simple random sampling with replacement (SRSWR). It has spawned numerous ramifications. We present results for the situation where the distinct persons chosen in an SRSWR are identified but each one independently gives a randomized response by Warner’s device, repeated as many times as he/she is selected. Two new estimators are proposed for the sensitive proportion and compared against relevant competitors.     

Distribution of the range when sample size has a GPED<Subscript>1</Subscript>

The probability density function of the range R, in random sampling from a uniform distribution on (k, l) and exponential distribution with parameter λ is obtained, when the sample size is a random variable having the Generalized Polya Eggenberger Distribution of the first kind (GPED ₁). The results of Raghunandanan and Patil (1972) and Bazargan-lari (1999) follow as special cases. The p.d.f of rangeR is obtained, when the distribution of the sample sizeN belongs to Katz family of distributions, as a special case. An erratum to this article is available at .     

Two-Stage Cluster Sampling with Ranked Set Sampling in the Secondary Sampling Frame

Motivated by a real-life problem, we develop a Two-Stage Cluster Sampling with Ranked Set Sampling (TSCRSS) design in the second stage for which we derive an unbiased estimator of population mean and its variance. An unbiased estimator of the variance of mean estimator is also derived. It is proved that the TSCRSS is more efficient—in the sense of having smaller variance—than the conventional two-stage cluster simple random sampling in which the second-stage sampling is with replacement. Using a simulation study on a real-life population, we show that the TSCRSS is more efficient than the conventional two-stage cluster sampling when simple random sampling without replacement is used in both stages.     

Protection of privacy in efficient application of randomized response techniques

In estimating the proportion of people bearing a sensitive attribute A, say, in a given community, following Warner’s (J Am Stat Assoc 60:63–69, 1965) pioneering work, certain randomized response (RR) techniques are available for application. These are intended to ensure efficient and unbiased estimation protecting a respondent’s privacy when it touches a person’s socially stigmatizing feature like rash driving, tax evasion, induced abortion, testing HIV positive, etc. Lanke (Int Stat Rev 44:197–203, 1976), Leysieffer and Warner (J Am Stat Assoc 71:649–656, 1976), Anderson (Int Stat Rev 44:213–217, 1976, Scand J Stat 4:11–19, 1977) and Nayak (Commun Stat Theor Method 23:3303–3321, 1994) among others have discussed how maintenance of efficiency is in conflict with protection of privacy. In their RR-related activities the sample selection is traditionally by simple random sampling (SRS) with replacement (WR). In this paper, an extension of an essential similarity in case of general unequal probability sample selection even without replacement is reported. Large scale surveys overwhelmingly employ complex designs other than SRSWR. So extension of RR techniques to complex designs is essential and hence this paper principally refers to them. New jeopardy measures to protect revelation of secrecy presented here are needed as modifications of those in the literature covering SRSWR alone. Observing that multiple responses are feasible in addressing such a dichotomous situation especially with Kuk’s (Biometrika 77:436–438, 1990) and Christofides’ (Metrika 57:195–200, 2003) RR devices, an average of the response-specific jeopardizing measures is proposed. This measure which is device dependent, could be regarded as a technical characteristic of the device and it should be made known to the participants before they agree to use the randomization device. The views expressed are the authors’, not of the organizations they work for. Prof Chaudhuri’s research is partially supported by CSIR Grant No. 21(0539)/02/EMR-II.     

Optimization of continuous and discrete scheduled times for a cumulative damage system with age-dependent maintenance

Abstract

This paper presents a preventive replacement problem when a system is operating successive works with random times and suffering stochastic shocks. The works cause random amount additive damage to the system, and the system fails whenever the cumulative damage reaches a failure level threshold. As an external shock occurs, the system experiences one of the two types of shocks with age-dependent maintenance mechanism: type-I (minor) shock is rectified by a minimal repair, or type-II (catastrophic) shock causes the system to fail. To control the deterioration process, preventive replacement is scheduled to replace the system at a continuous age T or at a discrete number N of working cycles, whichever occurs first, and corrective replacement is performed immediately whenever the system fails due to either shock or damage. The optimal preventive replacement schedule that minimizes the expected cost rate is discussed analytically and computed numerically. The proposed model provides a general framework for analyzing maintenance policies and extends several existing results.     

Family of Estimators of Population Mean Using Two Auxiliary Variables in Stratified Random Sampling

A general family of estimators, which use the information of two auxiliary variables in the stratified random sampling, is proposed to estimate the population mean of the variable under study. Under stratified random sampling without replacement scheme, the expressions of bias and mean square error (MSE) up to the first- and second-order approximations are derived. The family of estimators in its optimum case is discussed. Also, an empirical study is carried out to show the properties of the proposed estimators.     

Randomized nomination sampling for finite populations

We propose a randomized minima–maxima nomination (RMMN) sampling design for use in finite populations. We derive the first- and second-order inclusion probabilities for both with and without replacement variations of the design. The inclusion probabilities for the without replacement variation are derived using a non-homogeneous Markov process. The design is simple to implement and results in simple and easy to calculate estimators and variances. It generalizes maxima nomination sampling for use in finite populations and includes some other sampling designs as special cases. We provide some optimality results and show that, in the context of finite population sampling, maxima nomination sampling is not generally the optimum design to follow. We also show, through numerical examples and a case study, that the proposed design can result in significant improvements in efficiency compared to simple random sampling without replacement designs for a wide choice of population types. Finally, we describe a bootstrap method for choosing values of the design parameters.     

Zur Auswirkung von Endlichkeitskorrekturen bei der Analyse einfacher Zufallsstichproben

In statistical inference one usual assumption is, that data relates to a set of independent identically distributed random variables. From the viewpoint of sampling theory this assumption is only satisfied, if we draw a simple random sample with replacement or the population size is infinite. Then it is not necessary to consider a finite population correction when calculating the variance of a given estimator. To examine the effect of simple random sampling without replacement on the above assumption, the exact variances are calculated in the cases of mean value and variance estimation. This may give us information whether finite population correction is neglible or not.     

Biostatistical genetics and genetic epidemiology

It is well-known that when ranked set sampling (RSS) scheme is employed to estimate the mean of a population, it is more efficient than simple random sampling (SRS) with the same sample size. One can use a RSS analog of SRS regression estimator to estimate the population mean of Y using its concomitant variable X when they are linearly related. Unfortunately, the variance of this estimate cannot be evaluated unless the distribution of X is known. We investigate the use of resampling methods to establish confidence intervals for the regression estimation of the population mean. Simulation studies show that the proposed methods perform well in a variety of situations when the assumption of linearity holds, and decently well under mild non-linearity.