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On Prediction of Time Series on the Basis of Rank Correlation Coefficients

100%
Wywiał J. L.
Studia Ekonomiczne
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2014
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vol. 189
19-26
EN
In the paper the problem of prediction of a time series is considered. Time series observations can be measured on order scale. On the basis of observed ranks of values of the variables observed in the past periods a forecast of the rank of the observation in the future period is determined. The proposed method results from the derivation of the distribution of the well known Kendall's rank coefficient. The paper was inspired by a lecture of Jean H.P. Paelinck who gave it at the University of Economics in Katowice when he received the title of doctor honoris causa of the University in 1987.
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

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

100%
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.
5
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On Limit Distribution of Horvitz-Thompson Statistic under Poisson Sampling Design

100%
Wywiał J. L.
Studia Ekonomiczne
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2013
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vol. 133
61-70
PL
W pracy na podstawie znanego twierdzenia centralnego Lapunowa jest wyprowadzany rozkład graniczny prawdopodobieństwa znanej statystyki Horvitza-Thompsona (HT). Okazało się, że jeśli określane przez plan losowania Poissona prawdopodobieństwa wylosowania do próby poszczególnych elementów populacji spełniają pewne założenia oraz rozmiar populacji rośnie nieograniczenie, to rozkład standardowej postaci statystyki HT zmierza do rozkładu normalnego standardowego. Taki sam wynik otrzymano przy dodatkowym założeniu narzuconym na prawdopodobieństwa wylosowania elementów populacji do próby, gdy w standardowej postaci statystyki HT jej odchylenie standardowe zastąpimy przez pierwiastek z nieobciążonego estymatora tej wariancji. Rezultaty pracy znajdują zastosowania np. w pewnych typach badań ankietowych, a w szczególności internetowych, wykorzystujących wnioskowanie statystyczne, czyli estymację przedziałową lub testowanie hipotez statystycznych.
6
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Sampling Designs Proportionate to Non-negative Functions of two Quantiles of Auxiliary Variable

100%
Wywiał J. L.
Studia Ekonomiczne
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2013
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vol. 152
175-190
EN
Estimation of the population average in a finite population by means of sampling strategies dependent on sample quantiles of an auxiliary variable are considered. The sampling design proportional to an order statistic of an auxiliary variable was defined by Wywiał (2007, 2008). It was generalized into case of the sampling design proportional to the difference of two order statistics by Wywiał (2009), too. In this paper those results are generalized on the case of a conditional sampling design. Several strategies including the Horvitz-Thompson statistic and regression estimators are considered. Their accuracy is analyzed on the basis of computer simulation experiments.
7
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Generalised spatial autocorrelation coefficients

100%
Wywiał J. L.
Statistics in Transition new series
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2025
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vol. 26
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issue 1
1-8
EN
The article focuses on properties generalised to the multidimensional case of known coefficients of spatial correlation. The main result of the work is the decomposition of the introduced generalised autocorrelation coefficients into the sum of ordinary autocorrelation coefficients, but calculated on the basis of the principal components of the originally observed multidimensional variable. The development is illustrated with an empirical example. The coefficients provide a more detailed description of the spatial relationships of a set of variables defined in a population.
8
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Symulacyjne badanie szybkości zbieżności rozkładu statystyk do rozkładu normalnego

51%
Wywiał J. L., Krzciuk M., Mierzwa M.
Śląski Przegląd Statystyczny
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2013
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issue 11(17)
201-208
EN
The paper discusses studentized sample mean distribution. The sample is from exponential distribution. On the basis of independent replications of the samples empirical distributions studentized mean was calculated. The distance between the empirical distributions and the standard normal distribution was measured by means well known as statistics of Kolmogorov. Under the appropriate sample sizes the degree of the difference between the empirical and theoretical distributions was evaluated. Moreover, the hypothesis on normality of the empirical distributions was tested by means of the Kolmogorov test.
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