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Jahrbücher f. Nationalökonomie u. Statistik (Lucius & Lucius, Stuttgart 1999) Bd. (Vol.) 218/1+2 Minima, Zero-points, Saddle-points and Singularities of the Variance of the Horvitz- Thompson Estimator in Selection With Unequal Probabilities Without Replacement Minima, Nullpunkte, Sattelpunkte und Singularitäten der Varianz des Horvitz-Thompson Schätzers bei Auswahl mit unterschiedlichen Wahrscheinlichkeiten ohne Zurücklegen By Hans Schneeberger, Holzkirchen JEL C42 Optimization, Horvitz-Thompson estimator. Optimierung, Horvitz-Thompson Schätzer

Jahrb. f. Nationalök. u. Stat. (Lucius & Lucius, Stuttgart 1996) Bd. (Vol.) 215/6 A Method of Minimizing the Variance of the Horvitz-Thompson Estimator in Selection with Unequal Probabilities without Replacement Ein Verfahren zur Minimierung der Varianz des Horvitz-Thompson Schätzers bei Auswahl mit unterschiedlichen Wahrscheinlichkeiten ohne Zurücklegen By Hans Schneeberger, Holzkirchen 1. Introduction There are a number of methods in sampling techniques to reduce the variance of an estimate, for example in comparison with simple random sampling. Of

Abstract

Generalized regression (GREG) estimation uses a model that assumes that the values of the variable of interest are not correlated. An extension of the GREG estimator to the case where the vector of interest has a positive definite covariance structure is presented in this article. This extension can be translated to the calibration estimators. The key to this extension lies in a generalization of the Horvitz-Thompson estimator which, in some sense, also assumes that the values of the variable of interest are not correlated. The Godambe-Joshi lower bound is another result which assumes a model with no correlation. This is also generalized to a vector of interest with a positive definite covariance structure, and it is shown that the generalized calibration estimator asymptotically attains this generalized lower bound. Properties of the new estimators are given, and they are compared with the Horvitz-Thompson estimator and the usual calibration estimator. The new estimators are applied to the Canadian Reverse Record Check survey and to the problem of variance estimation.

asymptotically normally distributed. A simulation study shows that the proposed estimator performs well when compared with competing alternatives. The various methods are illustrated with a real data set. KEYWORDS: Kendall's tau, dependence, Horvitz-Thompson estimator, Kaplan-Meier estimator, martingales Author Notes: Partial funding for this work was provided by the Natural Sciences and Engineering Research Council of Canada and the Fonds québécois de la recherche sur la nature et les technologies. The authors would like to thank Professor Phyllis K. Mansfield for granting

Abstract

To receive federal homeless funds, communities are required to produce statistically reliable, unduplicated counts or estimates of homeless persons in sheltered and unsheltered locations during a one-night period (within the last ten days of January) called a point-in-time (PIT) count. In Los Angeles, a general population telephone survey was implemented to estimate the number of unsheltered homeless adults who are hidden from view during the PIT count. Two estimation approaches were investigated: i) the number of homeless persons identified as living on private property, which employed a conventional household weight for the estimated total (Horvitz-Thompson approach); and ii) the number of homeless persons identified as living on a neighbor’s property, which employed an additional adjustment derived from the size of the neighborhood network to estimate the total (multiplicity-based approach). This article compares the results of these two methods and discusses the implications therein.

Abstract

Model-based and model-assisted methods of survey estimation aim to improve the precision of estimators of the population total or mean relative to methods based on the nonparametric Horvitz-Thompson estimator. These methods often use a linear regression model defined in terms of auxiliary variables whose values are assumed known for all population units. Information on networks represents another form of auxiliary information that might increase the precision of these estimators, particularly if it is reasonable to assume that networked population units have similar values of the survey variable. Linear models that use networks as a source of auxiliary information include autocorrelation, disturbance, and contextual models. In this article we focus on social networks, and investigate how much of the population structure of the network needs to be known for estimation methods based on these models to be useful. In particular, we use simulation to compare the performance of the best linear unbiased predictor under a model that ignores the network with model-based estimators that incorporate network information. Our results show that incorporating network information via a contextual model seems to be the most appropriate approach. We also show that one does not need to know the full population network, but that knowledge of the partial network linking the sampled population units to the non-sampled population units is necessary. Finally, we also provide an estimator for the mean-squared error to make an informed decision about using the contextual information, as well as the results showing that this adaptive strategy leads to higher precision.

. Christianson, M. Colledge, and P. Kott, 153-169. New York: John Wiley. Särndal, C.-E., B. Swensson, and J. Wretman. 1992. Model Assisted Survey Sampling. New York: Springer-Verlag. Tam, S.M. 1984. “On Covariances from Overlapping Samples.” The American Statistician 38: 288-292. DOI: http://dx.doi.org/10.1080/00031305.1984.10483227. Wood, J. 2008. “On the Covariance Between Related Horvitz-Thompson Estimators.” Journal of Official Statistics 24: 53-78. Available at: http://www.jos.nu/Articles/abstract.asp?article¼241053 (accessed September 1, 2014).

, Olaf, Jürgen John, Umverteilungseffekte durch Reformen der Finan- zierung der Gesetzlichen Krankenversicherung Redistributional Effects of Health Care Financing Reforms in Germany 197-214 Winker, Peter, Kai-Tai Fang, Zufall und Quasi-Monte Carlo Ansätze Randomness and Quasi-Monte Carlo Approaches 2 1 5 - 2 3 0 Schneeberger, Hans Minima, Zero-points, Saddle-points and Singularities of the Variance of the Horvitz-Thompson Estimator in Selection With Unequal Probabilities Without Replacement Minima, Nulllpunkte, Sattelpunkte und Singularitäten der Varianz des

the number of types with frequency 1. In this case, the sample coverage reflects the conditional probability of getting a new type if a token is added to the sample 𝑁 . This probability is then multiplied with the simple ML estimate ̂(𝑤𝑖)ML to get the so-called Good-Turing estimated probability of a type ̂(𝑤𝑖)GT = (1 − 𝑚1 𝑁 ) ̂(𝑤𝑖) ML. (12.4) Furthermore, Chao and Shen (2003, p. 431) suggest to use the Horvitz-Thompson estimator to modify the estimated entropy ̂ML. This estimator is based on the ra- tionale that if𝑁 tokenshavebeen sampledwith replacement