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Application of Order Statistics of Auxiliary Variable to Estimation the Population Mean

100%
Wywiał J. L.
Statistics in Transition new series
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2012
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vol. 13
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issue 2
279-286
EN
Estimation of the population average in a finite population by means of sampling strategies dependent on an auxiliary variable highly correlated with a variable under study is considered. The sample is drawn with replacement on the basis of the probability distribution of an order statistic of the auxiliary variable. Observations of the variable under study are the values of the concomitant of the order statistic. The mean of the concomitant values is the estimator of a population mean of the variable under study. The expected value and the variance of the estimator are derived. The limit distributions of the considered estimators were considered. Finally, on the basis of simulation analysis, the accuracy of the estimator is considered.
2
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On Conditional Simple Random Sample

100%
Wywiał J. L.
Statistics in Transition new series
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2014
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vol. 15
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issue 4
525–534
EN
Estimation of the population average in a finite and fixed population on the basis of the conditional simple random sampling design dependent on order statistics of the auxiliary variable is studied. The sampling scheme implementing the sampling design is proposed. The inclusion probabilities are derived. The well known Horvitz-Thompson statistic under the conditional simple random sampling designs is considered as the estimator of population mean. Moreover, it was shown that the Horvitz-Thompson estimator under some particular cases of the conditional simple random sampling design is more accurate than the ordinary mean from the simple random sample.
3
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Estimating the population mean using a continuous sampling design dependent on an auxiliary variable

100%
Wywiał J. L.
Statistics in Transition new series
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2020
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vol. 21
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issue 5
1-16
EN
Continuous distribution of variables under study and auxiliary variables are considered. The purpose of the paper is to estimate the mean of the variable under study using a sampling design which is dependent on the observation of a continuous auxiliary variable in the whole population. Auxiliary variable values observed in this population allow to estimate the inclusion density function of the sampling design. The variance of the continuous version of the Horvitz-Thompson estimator under the proposed sampling design is compared with the variance of the mean of a simple random sample. The accuracy of the estimation strategies is analysed by means of simulation experiments.
4
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Simulation analysis of accuracy estimation of population mean on the basis of regression type strategy dependent on order statistic of auxiliary variable

84%
Wywiał J. L., Uniwersytet Ekonomiczny w Katowicach; Wydział Zarządzania; Katedra Statystyki
Acta Universitatis Lodziensis. Folia Oeconomica
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2011
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vol. 255
PL
Problem dotyczy oceny wartości średnie (globalnej) zmiennej w populacji ustalonej I skończonej. Zakład się, że z góry są znane w populacji wartości dodatniej zmiennej pomocniczej. Do estymacji użyto strategia kwantylowej zależnej m.in. od planu losowania proporcjonalnego do nieujemnej funkcji kwantyla z próby zmiennej pomocniczej. Ponadto, brano pod uwagę estymator Horvitza- Thompsona oraz estymator ilorazowy. Porównanie dokładności przeprowadzono na podstawie symulacji komputerowej.
5
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Estimation of Domain Means on The Basis of Strategy Dependent on Depth Function of Auxiliary Variables’ Distribution

84%
Wywiał J. L.
Statistics in Transition new series
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2011
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vol. 12
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issue 1
127-138
EN
The paper deals with the problem of estimation of a domain means in a finite and fixed population. We assume that observations of a multidimensional auxiliary variable are known in the population. The proposed estimation strategy consists of the well known Horvitz-Thompson estimator and the non-simple sampling design dependent on a synthetic auxiliary variable whose observations are equal to the values of a depth function of the auxiliary variable distribution. The well known spherical and Mahalanobis depth functions are considered. A sampling design is proportionate to the maximal order statistic determined on the basis of the synthetic auxiliary variable observations in a simple sample drawn without replacement. A computer simulation analysis leads to the conclusion that the proposed estimation strategy is more accurate for domain means than the well known simple sample means.
6
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Estimation of Mean on The Basis of Conditional Simple Random Sample

84%
Wywiał J. L.
Statistics in Transition new series
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2016
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vol. 17
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issue 3
411-428
EN
Estimation of the population mean in a finite and fixed population on the basis of the conditional simple random sampling design dependent on order statistics (quantiles) of an auxiliary variable is considered. Properties of the well-known Horvitz-Thompson and ratio type estimators as well as the sample mean are taken into account under the conditional simple random sampling designs. The considered examples of empirical analysis lead to the conclusion that under some additional conditions the proposed estimation strategies based on the conditional simple random sample are usually more accurate than the mean from the simple random sample drawn without replacement.
7

On the Evaluation of Sample Size Required for a Good Approximation by the Normal Curve for Some Statistics

84%
Wywiał J. L.
Zeszyty Naukowe Uniwersytetu Ekonomicznego w Krakowie
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2017
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issue 5(965)
17-29
EN
Testing hypotheses or evaluation confidence intervals requires knowledge of some statistics’ distributions. It is convenient if the probability distribution of the statistic converges to normal distribution when the sample size is sufficiently large. This paper examines the problem of how to evaluate sample size in order to determine that a statistic’s distribution does not depart from normal distribution by more than an assumed amount. Two procedures are proposed to evaluate the necessary sample size. The first is based on Berry-Esseen inequality while the second is based on simulation procedure. In order to evaluate the necessary sample size, the distribution of the sample mean is generated by replicating samples of a fixed size. Next, the normal distribution of the evaluated sample means is tested. The size of the generated samples is gradually increased until the hypothesis on the normality of the sample mean distribution is not rejected. This procedure is applied in the cases of statistics other than sample mean.
PL
Podczas testowania hipotez lub wyznaczania przedziałów ufności rozkłady pewnych statystyk zwykle nie są znane. Wygodne jest, gdy rozkłady takich statystyk można przybliżać rozkładem normalnym. Celem pracy jest wyznaczenie takiej liczebności próby, przy której rozkład statystyki jest dostatecznie dobrze aproksymowany rozkładem normalnym. Zaproponowano dwie procedury postępowania. Jedna z nich daje aproksymację liczebności próby na podstawie nierówności Berry-Esseena. Druga metoda polega na generowaniu serii prób o ustalonej liczebności, na podstawie których wyznacza się wartości statystyki. Opierając się na tych wartościach, testuje się normalność rozkładu statystyki. W razie odrzucenia hipotezy o normalności zwiększa się rozmiar generowanych prób. Procedurę tę powtarza się aż do ustalenia liczebności próby, przy której hipoteza o normalności nie jest odrzucona.
8
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Performing quantiles in multiple regression sampling strategy

84%
Wywiał J., Department of Statistics K. U. o. E.
Acta Universitatis Lodziensis. Folia Oeconomica
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2008
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vol. 216
EN
Estimation of the population average in a finite population by means of sampling strategy dependent on the sample quantile of an auxiliary variables is considered. The sampling design is proportionate to the determinant of the matrix dependent on some quantiles of an auxiliary variables. The sampling scheme implementing the sampling design is proposed. The derived inclusion probabilities are applied to estimation the population mean using the well known Horvitz-Thompson estimator. Moreover, the regression estimator is defined as the function of the coefficient dependent on the quantiles of the auxiliary variables. The properties of this estimator under the above defined sampling design are studied. The considerations are supported by empirical examples.
PL
Problem oceny wartości średniej z wykorzystaniem danych o wszystkich wartościach cech pomocniczych jest rozważany. W tym celu znany estymator regresyjny zależny od wielu zmiennych pomocniczych jest wykorzystywany. W odróżnieniu od zwykłego podejścia znanego w metodzie reprezentacyjnej do oceny parametrów regresji są wykorzystywane kwantyle jednej ze zmiennych dodatkowych. Otrzymane na tym polu wyniki są adoptowane do konstrukcji predytorów wartości średniej w nadpopulacji. Wyprowadzono również wariancje różnych odmian proponowanych predykatorów.
9
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Sampling Designs Proportionate to Sum of Two Order Statistics of Auxiliary Variable

84%
Wywiał J. L.
Statistics in Transition new series
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2013
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vol. 14
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issue 2
231-248
EN
In this paper the case of a conditional sampling design proportional to the sum of two order statistics is considered. Several strategies including the Horvitz-Thompson estimator and ratio-type estimators are discussed. The accuracy of these estimators is analyzed on the basis of computer simulation experiments.
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