Variance estimation with highly stratified sampling designs with unequal probabilities

It is common practice to design a survey with a large number of strata. However, in this case the usual techniques for variance estimation can be inaccurate. This paper proposes a variance estimator for estimators of totals. The method proposed can be implemented with standard statistical packages without any specific programming, as it involves simple techniques of estimation, such as regression fitting.

Variance estimation for systematic sampling from deliberately ordered populations

The systematic sampling (SYS) design (Madow and Madow, 1944) is widely used by statistical offices due to its simplicity and efficiency (e.g., Iachan, 1982). But it suffers from a serious defect, namely, that it is impossible to unbiasedly estimate the sampling variance (Iachan, 1982) and usual variance estimators (Yates and Grundy, 1953) are inadequate and can overestimate the variance significantly (Särndal et al., 1992). We propose a novel variance estimator which is less biased and that can be implemented with any given population order. We will justify this estimator theoretically and with a Monte Carlo simulation study.

Variance estimation with Chao's sampling scheme

We show that the Hájek (Ann. Math Statist. (1964) 1491) variance estimator can be used to estimate the variance of the Horvitz–Thompson estimator when the Chao sampling scheme (Chao, Biometrika 69 (1982) 653) is implemented. This estimator is simple and can be implemented with any statistical packages. We consider a numerical and an analytic method to show that this estimator can be used. A series of simulations supports our findings.

Polynomial-based noise variance estimation for MIMO-SCBT systems

In this paper, we present a polynomial-based noise variance estimator for multiple-input multiple-output single-carrier block transmission (MIMO-SCBT) systems. It is shown that the optimal pilots for noise variance estimation satisfy the same condition as that for channel estimation. Theoretical analysis indicates that the proposed estimator is statistically more efficient than the conventional sum of squared residuals (SSR) based estimator. Furthermore, we obtain an efficient implementation of the estimator by exploiting its special structure. Numerical results confirm our theoretical analysis.

Genetic heterogeneity of residual variance : estimation of variance components using double hierarchical generalized linear models

Background: The sensitivity to microenvironmental changes varies among animals and may be under genetic control. It is essential to take this element into account when aiming at breeding robust farm animals. Here, linear mixed models with genetic effects in the residual variance part of the model can be used. Such models have previously been fitted using EM and MCMC algorithms. Results: We propose the use of double hierarchical generalized linear models (DHGLM), where the squared residuals are assumed to be gamma distributed and the residual variance is fitted using a generalized linear model. The algorithm iterates between two sets of mixed model equations, one on the level of observations and one on the level of variances. The method was validated using simulations and also by re-analyzing a data set on pig litter size that was previously analyzed using a Bayesian approach. The pig litter size data contained 10,060 records from 4,149 sows. The DHGLM was implemented using the ASReml software and the algorithm converged within three minutes on a Linux server. The estimates were similar to those previously obtained using Bayesian methodology, especially the variance components in the residual variance part of the model. Conclusions: We have shown that variance components in the residual variance part of a linear mixed model can be estimated using a DHGLM approach. The method enables analyses of animal models with large numbers of observations. An important future development of the DHGLM methodology is to include the genetic correlation between the random effects in the mean and residual variance parts of the model as a parameter of the DHGLM.

Variance estimation in ranked set sampling using a concomitant variable

We propose a nonparametric variance estimator when ranked set sampling (RSS) and judgment post stratification (JPS) are applied by measuring a concomitant variable. Our proposed estimator is obtained by conditioning on observed concomitant values and using nonparametric kernel regression.

Estimation simplifiée de la variance dans le cas de l’échantillonnage à deux phases

Dans ce mémoire, nous étudions le problème de l'estimation de la variance pour les estimateurs par double dilatation et de calage pour l'échantillonnage à deux phases. Nous proposons d'utiliser une décomposition de la variance différente de celle habituellement utilisée dans l'échantillonnage à deux phases, ce qui mène à un estimateur de la variance simplifié. Nous étudions les conditions sous lesquelles les estimateurs simplifiés de la variance sont valides. Pour ce faire, nous considérons les cas particuliers suivants : (1) plan de Poisson à la deuxième phase, (2) plan à deux degrés, (3) plan aléatoire simple sans remise aux deux phases, (4) plan aléatoire simple sans remise à la deuxième phase. Nous montrons qu'une condition cruciale pour la validité des estimateurs simplifiés sous les plans (1) et (2) consiste à ce que la fraction de sondage utilisée pour la première phase soit négligeable (ou petite). Nous montrons sous les plans (3) et (4) que, pour certains estimateurs de calage, l'estimateur simplifié de la variance est valide lorsque la fraction de sondage à la première phase est petite en autant que la taille échantillonnale soit suffisamment grande. De plus, nous montrons que les estimateurs simplifiés de la variance peuvent être obtenus de manière alternative en utilisant l'approche renversée (Fay, 1991 et Shao et Steel, 1999). Finalement, nous effectuons des études par simulation dans le but d'appuyer les résultats théoriques.

Estimation de la variance en présence de données imputées pour des plans de sondage à grande entropie

Les travaux portent sur l’estimation de la variance dans le cas d’une non- réponse partielle traitée par une procédure d’imputation. Traiter les valeurs imputées comme si elles avaient été observées peut mener à une sous-estimation substantielle de la variance des estimateurs ponctuels. Les estimateurs de variance usuels reposent sur la disponibilité des probabilités d’inclusion d’ordre deux, qui sont parfois difficiles (voire impossibles) à calculer. Nous proposons d’examiner les propriétés d’estimateurs de variance obtenus au moyen d’approximations des probabilités d’inclusion d’ordre deux. Ces approximations s’expriment comme une fonction des probabilités d’inclusion d’ordre un et sont généralement valides pour des plans à grande entropie. Les résultats d’une étude de simulation, évaluant les propriétés des estimateurs de variance proposés en termes de biais et d’erreur quadratique moyenne, seront présentés.

A simple variance formula for population size estimators by conditioning

This note considers the variance estimation for population size estimators based on capture–recapture experiments. Whereas a diversity of estimators of the population size has been suggested, the question of estimating the associated variances is less frequently addressed. This note points out that the technique of conditioning can be applied here successfully which also allows us to identify sources of variation: the variance due to estimation of the model parameters and the binomial variance due to sampling n units from a population of size N. It is applied to estimators typically used in capture–recapture experiments in continuous time including the estimators of Zelterman and Chao and improves upon previously used variance estimators. In addition, knowledge of the variances associated with the estimators by Zelterman and Chao allows the suggestion of a new estimator as the weighted sum of the two. The decomposition of the variance into the two sources allows also a new understanding of how resampling techniques like the Bootstrap could be used appropriately. Finally, the sample size question for capture–recapture experiments is addressed. Since the variance of population size estimators increases with the sample size, it is suggested to use relative measures such as the observed-to-hidden ratio or the completeness of identification proportion for approaching the question of sample size choice.

A jackknife variance estimator for unequal probability sampling

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.

A simple variance formula for population size estimators by conditioning

This note considers the variance estimation for population size estimators based on capture–recapture experiments. Whereas a diversity of estimators of the population size has been suggested, the question of estimating the associated variances is less frequently addressed. This note points out that the technique of conditioning can be applied here successfully which also allows us to identify sources of variation: the variance due to estimation of the model parameters and the binomial variance due to sampling n units from a population of size N. It is applied to estimators typically used in capture–recapture experiments in continuous time including the estimators of Zelterman and Chao and improves upon previously used variance estimators. In addition, knowledge of the variances associated with the estimators by Zelterman and Chao allows the suggestion of a new estimator as the weighted sum of the two. The decomposition of the variance into the two sources allows also a new understanding of how resampling techniques like the Bootstrap could be used appropriately. Finally, the sample size question for capture–recapture experiments is addressed. Since the variance of population size estimators increases with the sample size, it is suggested to use relative measures such as the observed-to-hidden ratio or the completeness of identification proportion for approaching the question of sample size choice.

Survey estimation and marginal effects in Stata

Stata is a general purpose software package that has become popular among various disciplines such as epidemiology, economics, or social sciences. Users like Stata for its scientific approach, its robustness and reliability, and the ease with which its functionality can be extended by user written programs. In this talk I will first give a brief overview of the functionality of Stata and then discuss two specific features: survey estimation and predictive margins/marginal effects. Most surveys are based on complex samples that contain multiple sampling stages, are stratified or clustered, and feature unequal selection probabilities. Standard estimators can produce misleading results in such samples unless the peculiarities of the sampling plan are taken into account. Stata offers survey statistics for complex samples for a wide variety of estimators and supports several variance estimation procedures such as linearization, jackknife, and balanced repeated replication (see Kreuter and Valliant, 2007, Stata Journal 7: 1-21). In the talk I will illustrate these features using applied examples and I will also show how user written commands can be adapted to support complex samples. Complex can also be the models we fit to our data, making it difficult to interpret them, especially in case of nonlinear or non-additive models (Mood, 2010, European Sociological Review 26: 67-82). Stata provides a number of highly useful commands to make results of such models accessible by computing and displaying predictive margins and marginal effects. In my talk I will discuss these commands provide various examples demonstrating their use.

A decision-support system based on particle swarm optimization for multiperiod hedging in electricity markets

This paper proposes a particle swarm optimization (PSO) approach to support electricity producers for multiperiod optimal contract allocation. The producer risk preference is stated by a utility function (U) expressing the tradeoff between the expectation and variance of the return. Variance estimation and expected return are based on a forecasted scenario interval determined by a price range forecasting model developed by the authors. A certain confidence level is associated to each forecasted scenario interval. The proposed model makes use of contracts with physical (spot and forward) and financial (options) settlement. PSO performance was evaluated by comparing it with a genetic algorithm-based approach. This model can be used by producers in deregulated electricity markets but can easily be adapted to load serving entities and retailers. Moreover, it can easily be adapted to the use of other type of contracts.

A long-term swarm intelligence hedging tool applied to electricity markets

This paper proposes a swarm intelligence long-term hedging tool to support electricity producers in competitive electricity markets. This tool investigates the long-term hedging opportunities available to electric power producers through the use of contracts with physical (spot and forward) and financial (options) settlement. To find the optimal portfolio the producer risk preference is stated by a utility function (U) expressing the trade-off between the expectation and the variance of the return. Variance estimation and the expected return are based on a forecasted scenario interval determined by a long-term price range forecast model, developed by the authors, whose explanation is outside the scope of this paper. The proposed tool makes use of Particle Swarm Optimization (PSO) and its performance has been evaluated by comparing it with a Genetic Algorithm (GA) based approach. To validate the risk management tool a case study, using real price historical data for mainland Spanish market, is presented to demonstrate the effectiveness of the proposed methodology.