Horst Stenger

Lineare und nichtlineare Schätzfunktionen in der Stichprobentheorie

ZusammenfassungX1, x2, … xN seien die Werte, die den Elementen 1, 2, … N einer Grundgesamtheit durch ein spezielles Merkmal zugeordnet sind. Wir nehmen an, es sei ein Auswahlverfahren p festgelegt, und betrachten die Klasseαp aller Schätzfunktionen für x=Σxi, die bei Verwendung von p unverzerrt sind; wir schreibenαp1 für die Klasse der linearen Schätzfunktionen ausαp undαps für die Klasse der symmetrischen Schätzfunktionen ausαp.Es läßt sich eine lineare symmetrische Schätzfunktion angeben, die effizienter ist als alle anderen Elemente vonαp1∩αps, sofern p gewisse Symmetrieeigenschaften aufweist; das Ziehen einer festen Zahl von Elementen ohne Zurücklegen genügt diesen Symmetriebedingungen, das Ziehen einer festen Zahl mit Zurücklegen dagegen nicht.Wir zeigen weiter, daßαp1∩αpsnicht vollständig ist bezüglichαps, wenn p für das n-malige Ziehen mit Zurücklegen steht, und daßαp1 nicht vollständig ist bezüglichαp, wenn p n-maliges Ziehen ohne Zurücklegen bedeutet.SummaryLet x1, x2, … xN be the values of a variate for the elements 1, 2, … N of a population. Suppose a sampling design p has been chosen and consider the classαp of all estimators for x=Σ xi, which are unbiassed under p, the classαp1 of all linear estimators inαp and the classαps of all symmetric estimators inαp.We give an estimator being more efficient than all estimators inαp1 ∩αps as long as p is symmetric in a certain sense; drawing a fixed number of elements without replacement is symmetric in this sense, drawing a fixed number with replacement is not.We further show thatαp1∩αps is not complete with respect toαps, if p means sampling with replacement, and thatαp1 is not complete with respect toαp, if p means sampling without replacement.RésuméSoient x1, x2, … xN les valeurs d’ une variable en étude pour les éléments 1, 2, … N d, une population. Nous supposons qu’ une méthode p de tirer un échantillon soit fixée et désignons parαp l’ ensemble des estimateurs non biaisés pour x=Σ xi, parαp1 l’ ensemble des estimateurs linéaires et parαps l’ ensemble des estimateurs symétriques qui sont éléments deαp.Nous construisons un estimateur linéaire et symétrique qui est plus efficient que tout autre élément deαp1∩αps, pourvue que p soit symétrique; tirer un échantillon au sort à la manière des boules d’ une urne sans remise est symétrique d’ après notre définition de symétrie, tirer avec remise n’ est pas symétrique.Nous prouvons en plus: L’ ensembleαp1∩αps n’ est pas complet par rapport àαps si p signifie le tirage d’ un échantillon au sort avec remise;αp1 n’ est pas complet par rapport àαp si p signifie le tirage sans remise.РезумеНапример X1, X2, … хN — стоимости, которые при помоши специальной приметы присоединяются Элементам 1, 2, … N основной совокупности. Предполагаем, что был установлен метод выбора р и занимаемся классом аp всех функции оценки для X = Σ x1 которые при применении р не искажаются. Мы пищем αp1 для класса линейных функции оценки из αp и αps для класса симметрических функции оценки из αp.Возможно образовать линейную симметрическую функцию оценки, которая еффективнее всех других Элементов от αp1 ∩ αps, поскольку р имеет известные свойства относительно симметрии выбор уверенного числа Элементов без откладывания удовлетворяет условиям симметрии напротив выбор уверенного числа с откладыванием не удовлетворяет условиям симметрии. Дальнее мы показываем, что αp1 ∩ αps — неполное относительно αрs, если р означает Ы-кратный выбор с откладыванием, и что αp неполное относительно αp, если р означает N-кратный выбор без откладывания.

Asymptotic expansion of the minimax value in survey sampling

SummarySuppose that a real numberyu is associated with each unitu of a populationU and that the functiony:u →yu onU is known to be an element of the parameter space Θ. The statistician has to select a samples ⊂U ofn units and to employyu;u ∈s to estimate the arithmetic mean of allyu,u ∈U.The performance of such a strategy is assessed by its mean square error or, more simply, by the supremum of the mean square error. This supremum cannot be determined exactly for the parameter space of Scott/Smith (1975). We propose, therefore, an asymptotic approximation; this approximation is based on the assumption, that the sample sizen is fixed and that linear estimators have to be used.

Loss functions and admissible estimators in survey sampling

SummaryConsider simple random sampling (without replacement) of a fixed size and lett0 be the sample mean, i.e. the arithmetic mean of all variate values observed in the sample. The class {λt0: λ real} of estimators for the population mean (i.e. The arithmetic mean of all variate values) then surely is of interest.We discuss different types of variates, in particular variates with positive values only. For these variates the usual square error loss gives rise to a strange class of admissible estimators. An other type of loss functions seems far more appropriate. For this (logarithmic) type, λt0 is admissible iff λ=1. We prove that there exists no other type of loss functions with the property that the unbiased estimatort0 is the only admissible element of the class {λt0: λ>0}.ZusammenfassungBei fest vorgegebenem Stichprobenumfang werde uneingeschränkt zufällig ausgewählt (ohne Zurücklegen);t0 bezeichne das Stichprobenmittel, d.h. das arithmetische Mittel aller Stichprobenbeobachtungen des Untersuchungsmerkmals. Von besonderen Interesse ist dann zweifellos die Klasse {λt0: λ reell} von Schätzfunktionen für das Gesamtmittel, d.h. für das arithmetische Mittel aller Ausprägungen des Untersuchungsmerkmals.Wir betrachten verschiedene Typen von Untersuchungsmerkmalen, insbesondere Merkmale, die nur positive Ausprägungen besitzen. Für diese Merkmale führt die Verwendung der üblichen quadratischen verlustfunktion zu einer sehr merkwürdigen Klasse zulässiger Schätzfunktionen. Ein anderer Typ von Verlustfunktionen erscheint weit eher angebracht. Für diesen (logarithmischen) Typ ist λt0 genaudann zulässig, wenn λ=1 gilt. Wir beweisen, daß für keinen anderen Typ von Verlustfunktionent0 das einzige zulässige Element der Klasse {λt0:λ>0} ist.

Survey Sampling: A Linear Game

A linear game consists of two subsets of a vector space with a scalar product. The idea is that players 1 and 2 select, independently, elements of the first and second set, respectively. Then, player 2 has to pay to player 1 the value of the scalar product of the selected elements. We will discuss survey sampling within the framework of linear games with the statistician in the role of player 2. The vector space to be considered is the set of all symmetric matrices of order N x N with a scalar product identical with the usual mean squared error. The subset from which the statisticians selection is to be made is neither convex nor compact. Standard results of the theory of linear games have to be modified appropriately. The existence of minimax strategies will be established. At the same time we hope to improve our understanding of random selection and of the duality between the fixed population approach and model based approaches to the theory of survey sampling.

A minimax property of Lahiri-Midzuno-Sen's sampling scheme

We consider parameter spaces which are generalizations of spaces discussed so far in connection with minimax strategies. We give a lower bound for the minimax value and derive, under weak assumptions, minimax strategies consisting of the expansion estimator and an appropriate design. This design is of the Lahiri-Midzuno-Sen type for an important subclass of parameter spaces.

Improving the RHC-strategy

It is well known that the RHC-strategy is inadmissible. For example, Rao-Blackwellization yields a strategy which has uniformly smaller risk than the RHC-strategy. The complicated form of the Blackwellized RHC-strategy as well as its inadmissibility is a disadvantage of this strategy. We propose a new strategy which has also uniformly smaller risk than the RHC-strategy. Some optimality results are derived.

Anforderungen an eine repräsentative Stichprobe aus der Sicht des Statistikers

Reprasentative Stichproben geniesen heute eine beachtliche Wertschatzung, vor allem auch bei Nicht-Statistikern. Man stellt sich darunter meist Teilmengen vor, die ebenso gegliedert sind wie die Grundgesamtheit, der sie entnommen wurden. In einer reprasentativen Stichprobe von Personen waren beispielsweise alle Auspragungskombinationen von Merkmalen wie Geschlecht, Alter, Ausbildung, Beruf, Haushaltsgrose, Gemeindegrose... vertreten, und zwar jeweils mit Haufigkeiten, die proportional zu den entsprechenden Haufigkeiten der Grundgesamtheit sind. Naturlich liesen sich bei derartiger Strukturgleichheit an der Stichprobe muhelos Aussagen uber die Grundgesamtheit ablesen.

Design effect of randomized systematic sampling

In statistical practice, systematic sampling (SYS) is used in many modifications due to its simple handling. In addition, SYS may provide efficiency gains if it is well adjusted to the structure of the population under study. However, if SYS is based on an inappropriate picture of the population a high decrease of efficiency, i.e. a high increase in variance may result by changing from simple random sampling to SYS. In the context of two-stage designs SYS so far seems often in use for subsampling within the primary units. As an alternative to this practice, we propose to randomize the order of the primary units, then to select systematically a number of primary units and, thereafter, to draw secondary units by simple random sampling without replacement within the primary units selected. This procedure is more efficient than simple random sampling with replacement from the whole population of all secondary units, i.e. the variance of an adequate estimator for a total is never increased by changing from simple random sampling to randomized SYS whatever be the values associated by a characteristic with the secondary units, while there are values for which the variance decreases for the change mentioned. This result should hold generally, even if our proof, so far, is not complete for general sample sizes.

Regression analysis and random sampling

Abstract Consider a linear regression model. We wish to estimate a regression parameter or predict the sum of all N values of an endogeneous variable based on a sample of n (n observations. If the variances of the residual variables are known up to a common factor the sample may be selected purposively. In case of incomplete knowledge random selection is preferable and an optimal random design exists. To demonstrate this we consider a model with only one exogeneous variable, generalizations being obvious. We interpret the situation as a linear game and derive an equilibrium point determining an optimal estimator and an optimal sampling design. Under weak additional conditions we obtain the Hansen–Hurwitz estimator and a design with inclusion probabilities proportionate to the values of the exogeneous variable.

Minimax strategies in survey sampling

The risk of a sampling strategy is a function on the parameter space, which is the set of all vectors composed of possible values of the variable of interest. It seems natural to ask for a minimax strategy, minimizing the maximal risk. So far answers have been provided for completely symmetric parameter spaces. Results available for more general spaces refer to sample size 1 or to large sample sizes allowing for asymptotic approximation. In the present paper we consider arbitrary sample sizes, derive a lower bound for the maximal risk under very weak conditions and obtain minimax strategies for a large class of parameter spaces. Our results do not apply to parameter spaces with strong deviations from symmetry. For such spaces a minimax strategy will prescribe to consider only a small number of samples and takes a non-random and purposive character.