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Proofs of the normalization of the function of the thickness classes one-dimensional distributions normal

100%
Wagner W., Parys D., Stępień L., Wiesław Wagner U. o. I. T. a. M. i. R., Lechosław Stępień U. o. Ł., Dariusz Parys U. o. Ł.
Acta Universitatis Lodziensis. Folia Oeconomica
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2009
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vol. 228
PL
Jednowymiarowy dwuparametrowy rozkład normalny należy do podstawowych rozkładów prawdopodobieństwa w statystyce. W ostatnich latach powstało wiele jego uogólnionych wersji, uwzględniających parametry asymetrii i kurtozy. Tworzą one klasę rozkładów normalnych «-parametrowych, odpowiednio z parametrami: m = 1 położenia (przesunięcia), m = 2 - położenia i zmienności (skali), m = 3 - położenia, zmienności i skośności oraz m = 4 - położenia, zmienności, skośności i spłaszczenia. W pracy podajemy 7 wybranych jednowymiarowych rozkładów prawdopodobieństwa z klasy rozkładów normalnych. Dla nich wymieniono funkcje gęstości oraz przedstawiono dowody unormowania pokazując, iż całka po obszarze określoności tych funkcji jest równa jeden.
EN
One-dimensional two parameters the normal distribution assorts basic probability distributions in the statistics, in last years into being many generalized versions, taking into account parameters of the asymmetry and the kurtosis. They there create the class of normal distributions «-parameter, properly with parameters, m = 1 - positions (shifts), m — 2 — positions and variations (scale), m = 3 - positions, variations and skewness and m = 4 - positions, variations, skewnesses and kurtosis. On the job we give 7 chosen one-dimensional probability distributions from the class of normal distributions. For them one mentioned functions of the thickness and one averred normalizations to show, that the integral after area of the determinates of these functions is equal one.
2
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Symulacyjne badanie szybkości zbieżności rozkładu statystyk do rozkładu normalnego

80%
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.
3

Rozkład statystyki t-studenta przy danej wariancji z próby o rozkładzie normalnym

80%
Gardoń A.
Didactics of Mathematics
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2011
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issue 8 (12)
17-30
EN
It is not so easy to lecture on higher mathematics for economy students. Advanced notions must be often presented for people without an appropriate theoretical background, which forces the teacher to simplify. Unfortunately, the praxis shows that the frontier between a simplification and a factual error is often very subtle and it happens this frontier is sometimes crossed. Such a situation occurs just in the case of the problem, which will be described in this paper. It is a known fact that the so called Student’s T statistics from a normal distributed sample is t-Student distributed, without any doubt. But in handbooks for economy students several authors try to use this statistics for exercises with mathematical tables of the t-Student distribution, ordering a calculation of the probability that the sample average will belong to the given interval, in the case when the theoretical variance is unknown but the sample variance has been calculated. Unfortunately, such a situation has nothing to do with the t-Student distribution and this error is systematically copied in successive handbooks.
4
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Sequential Probability Ratio Test for Mean Based on Pseudo-Likelihood Function

80%
Pekasiewicz D., University of Łódź C. o. S. M.
Acta Universitatis Lodziensis. Folia Oeconomica
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2009
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vol. 225
EN
Hypotheses about expected value of random variable can be verified by means of the parametric sequential probability ratio test in case of the known class of this variable's distribution. The problem with verification of such hypotheses occurs when we have no information about random variable distribution. Then, we have to apply non-parametric methods. The author of the paper proposes the application of pseudo-likelihood function instead of likelihood one in the statistic of sequential probability ratio test. Examples of application of the test based on the likelihood function ratio in selected kinds of distributions are presented together with the results of Monte Carlo analysis concerning properties of these tests.
PL
Hipotezy o wartości oczekiwanej zmiennej losowej możemy zweryfikować parametrycznym ilorazowym testem sekwencyjnym, w przypadku znanej klasy rozkładu tej zmiennej. Problem z weryfikacją takich hipotez pojawia się, gdy nie posiadamy informacji o rozkładzie zmiennej losowej i musimy zastosować metody nieparametryczne. W procy proponowane jest wykorzystanie funkcji pseudowiarygodności, zamiast funkcji wiarygodności, w statystyce ilorazowego testu sekwencyjnego. Przykłady zastosowania testu opartego na ilorazie funkcji pseudowiarygodności dla wybranych rodzajów rozkładów s;j zaprezentowane w pracy wraz z wynikami analizy Monte Carlo dotyczącymi własności tych testów.
5
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Sensitivity of the Art Market to Price Volatility

80%
Borowski K.
Finanse i Prawo Finansowe
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2020
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vol. 2
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issue 26
11-36
EN
The purpose of the article: The art market becomes very popular among investors, when there is strong turbulence on the stock market. In times of calm, the art market is used by investors to diversify risk and build more efficient investment portfolios according to the Markovitz’s theory. The aim of this paper is to: (i) present the peculiarity of investment on the art market, represented by art market indexes in comparison to traditional investments in other financial market segments (money market, equity indexes and commodity market), (ii) to verify the hypothesis of normality of the distribution of rates of return of the analyzed art market indices as well as (iii) to analyze calendar effects occurrence on the art market.Methodology: Comparison of rates of return on the stock, bond, commodity and money markets with rates on the art market in four different time intervals. For each of the analyzed periods, an income-risk map was presented, taking into account the spectrum of financial instruments, including six art indexes: Old Masters, 19th Century, Modern art, Post War art, Contemporary art and Global art. The hypothesis of normality of the distribution of rates of return of the art market indices for four analyzed periods was verified with the use of Jarque-Bera test.Results of the research: Comparison of rates of return on the stock market and art market leads to the conclusion that their relationship depends on the period chosen. For two of the analyzed periods, the rates of return on the stock market were higher than on the art market, but for others periods, the opposite. The distribution of quarterly rates of return resulted to be a normal distribution for almost all of analyzed indices and time periods. Calendar effects were observed in the case of four analyzed indexes.
6
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A new test for normality and independence

80%
Ejsmont W.
Didactics of Mathematics
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2017
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issue 14(18)
27-32
EN
In this paper, a new test of normality and independence is proposed. This test is designed through a multivariate empirical characteristic by considering a result form [Ejsmont 2016].
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Sensitivity of the Art Market to Price Volatility

80%
Borowski K.
Finanse i Prawo Finansowe
|
2021
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issue Numer Specjalny
25-50
EN
The purpose of the article: The art market becomes very popular among investors, when there is strong turbulence on the stock market. In times of calm, the art market is used by investors to diversify risk and build more efficient investment portfolios according to the Markovitz’s theory. The aim of this paper is to: (i) present the peculiarity of investment on the art market, represented by art market indexes in comparison to traditional investments in other financial market segments (money market, equity indexes and commodity market), (ii) to verify the hypothesis of normality of the distribution of rates of return of the analyzed art market indices as well as (iii) to analyze calendar effects occurrence on the art market. Methodology: Comparison of rates of return on the stock, bond, commodity and money markets with rates on the art market in four different time intervals. For each of the analyzed periods, an income-risk map was presented, taking into account the spectrum of financial instruments, including six art indexes: Old Masters, 19th Century, Modern art, Post War art, Contemporary art and Global art. The hypothesis of normality of the distribution of rates of return of the art market indices for four analyzed periods was verified with the use of Jarque-Bera test. Results of the research: Comparison of rates of return on the stock market and art market leads to the conclusion that their relationship depends on the period chosen. For two of the analyzed periods, the rates of return on the stock market were higher than on the art market, but for others periods, the opposite. The distribution of quarterly rates of return resulted to be a normal distribution for almost all of analyzed indices and time periods. Calendar effects were observed in the case of four analyzed indexes.
8
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ДИФФЕРЕНЦИАЦИЯ УРОВНЯ ЧЕЛОВЕЧЕСКОГО РАЗВИТИЯ В СТРАНАХ МИРА И ИХ РЕГИОНАХ

70%
Selivanova-Fyodorova N., Komarova V., Vasserman Y., Tutaeva D.
Sociālo Zinātņu Vēstnesis
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2018
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issue 2
73-102
EN
The main research issue within the framework of this article is the following: what happens to human development in countries of the world and their regions which are beyond the average indicators, and why it happens this way but not the other? The authors have tried to answer this question with the help of combining the diachronic and synchronic analyses of the Human Development Index’s growth and state of differentiation in the world countries in the period 1990-2017, as well as the Historical Index’s of Human Development growth and state of differentiation in the world countries in the period 1870ñ2015, and the Sub-national Human Development Index’s growth and state of differentiation in the internal regions of the EU member countries (with the emphasis on Latvia), and the USA (for comparison) in the period 1990-2017. The novelty of the research lies in the analysis of the differentiation of the human development level in countries of the world and their regions with the help of three indices using a common methodological approach. This kind of parallel analysis allows the authors carrying out a more systematic study into peculiarities of a social and economic phenomenon of human development both in the modern world and in the 150-year old historical perspective. The research outcomes showed that: the average human development level is steadily growing, and in the modern world it dramatically exceeds the average HDI which was achieved 150 years ago; the state of differentiation of the human development level in the world is gradually decreasing over the last 150 years; the HDI distribution in the world countries at each time point of the period under study starting from the second half of the 20th century corresponded to the Gauss curve; in the modern world, the level of human development has a normal distribution on sub-national level too (for instance, between internal regions of the EU countries), and metropolitan areas are almost always the leaders of human development; in Latvia, despite the constant increase in the human development level on average in the country and in its regions, the distribution of the HDI itself between Latvia’s internal regions remains normal, which is also typical for internal regions of the United States. The reported study was funded by the Russian Foundation for Basic Research (RFBR) according to the research project 18-011-00548.
9
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Walidować? Weryfikować? Nie ruszać? O niestatystycznych, statystycznych i stochastycznych metodach oceny jakości danych ilościowych opowieść

70%
Gałuszko K., Lewczuk J., Kuźma K. K.
Studia Politologiczne
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2019
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vol. 54
135-174
EN
The article deals with the quality of data obtained in the quantitative research process. The authors decided to raise the subject, because in the Polish political science – in the opposition to sociology – it’s not well described. The text was made mainly on the basis of a literature review devoted to particular parts of the article and is its synthesis. During the work it was discovered that thanks to relatively simple tools, as well as the use of some programs the quality of the data set can be checked in a simple and precise way.
10
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An EWMA control chart for the exponential distribution using repetitive sampling plan

70%
AZAM M., ASLAM M., JUN C.-H.
Operations Research and Decisions
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2017
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vol. 27
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issue 2
5-19
EN
A new EWMA control chart has been proposed under repetitive sampling when a quantitative characteristic follows the exponential distribution. The properties of the proposed chart, including the average run lengths has been is compared with two existing control charts with the help of simulated data. An application of the proposed chart hs been illustrated using a healthcare data set.
11
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ЗАКОНОМЕРНОСТИ ДИФФЕРЕНЦИАЦИИ УСПЕШНОСТИ ВНУТРЕННИХ РЕГИОНОВ СТРАН ЕВРОПЕЙСКОГО СОЮЗА

61%
Komarova V., Selivanova-Fyodorova N., Lonska Y.
Sociālo Zinātņu Vēstnesis
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2020
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issue 1
30-54
EN
The purpose of this article is to study the performance of the internal regions of the EU countries in a broader sense than economic performance. The Human Development Index (HDI) is suitable as a tool for measuring the performance of regions, since its methodology is based on the idea that it is not economic growth in itself, but people and their capabilities that should be determining criteria for assessing the performance of territories. The object of this study is the internal (functional) regions of the EU member states. In turn, the subject of the study is the regularities in the performanceís differentiation of the object of study. To achieve the goal of this article, the authors studied the form and degree of differentiation of the internal regions of the countries of the European Union in terms of their performance measured by the HDI. The study was carried out by checking the correspondence to the Gaussian curve of the distribution of regions according to the probabilities of the average HDI values, as well as by analyzing the degree of differentiation of the internal regions of the European Union countries according to the HDI using the coefficient of variation. The source of empirical information for this study is the database of subnational HDI for the period from 1990 to 2017, created by the Global Data Lab of Radboud University (Netherlands). An analysis of the distribution parameters of the internal regions of the European Union countries (with an emphasis on Latvia and Poland) and the United States (for comparison) on the subnational HDI for the period from 1990 to 2017, as well as an assessment of its [distribution] correspondence to the Gaussian curve, showed that the performanceís differentiation of the internal regions of the EU countries graphically corresponds to the Gaussian curve, i.e. has the form of a normal distribution, while the most successful are almost always the capital regions. In turn, the trajectory of changes in the degree of performance’s differentiation of the internal regions of the countries of the modern EU takes the form of an inverted U-shaped curve, i.e. the differences in the performance of the regions in the territorial space that the EU is now, increased during the last 28 years in the period of collapse of the East European socialist bloc in the early 1990s, and then decreased as the regions adapted to the new conditions of independence and market economy. Thus, the performance’s differentiation of the internal regions of the EU countries over the past three decades was not chaotic, but occurred in accordance with certain regularities empirically proven by the authors for the first time both on an unchanged sample of 278 regions of the EU member states for 2018 and for “cleaned” sample of the internal regions of the EU countries, taking into account the year of their joining the European Union.
12
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Porównanie generatorów liczb pseudolosowych

61%
Sulewski P.
Wiadomości Statystyczne. The Polish Statistician
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2019
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vol. 64
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issue 7
5-31
EN
The sampling in statistical surveys and numerical calculations as well as simulation testing of probabilistic models in virtually all fields of knowledge require a computer endowed with pseudorandom numbers generators. The main goal of the study is to compare the normal random number generators using various criteria. The properties of 12 random number generators for a normal distribution were investigated. Then, the family of generators was extended by two so-called application generators and a new approach for checking the quality of generators was adopted. A ready-made tool prepared in C++ and in Visual Basic for Application (VBA) for conducting self-contained research using generators was presented. All Monte Carlo simulations were carried out in C++, while the calculations were performed in the VBA editor using the Microsoft Excel 2016 spreadsheet. The analysis of the obtained results shows that the generators with best properties are: MP Monty Python, R, Biegun and Ziggurat. The worst generators, are: BM Box-Muller, Wallace, Iloraz and Excel.
PL
Losowanie prób w badaniach statystycznych i w obliczeniach numerycznych, jak również symulacyjne badanie modeli probabilistycznych właściwie we wszystkich dziedzinach wiedzy wymagają wyposażenia komputera w generatory liczb pseudolosowych. Głównym celem pracy jest porównanie generatorów liczb pseudolosowych normalnych na podstawie ich analizy dokonanej za pomocą różnego rodzaju kryteriów. Zbadano właściwości 12 generatorów liczb pseudolosowych o rozkładzie normalnym. Zaproponowano rozszerzenie rodziny generatorów o dwa tzw. generatory aplikacyjne oraz przyjęcie nowego podejścia do sprawdzania jakości generatorów. Przedstawiono narzędzie przygotowane w języku C++ oraz w języku Visual Basic for Application (VBA) do prowadzenia samodzielnych badań z użyciem generatorów. Symulacje Monte Carlo przeprowadzono w języku C++, a obliczenia wykonano w edytorze VBA przy użyciu arkusza kalkulacyjnego Microsoft Excel 2016. Analiza uzyskanych wyników wskazuje, że najlepsze właściwości mają generatory: MP Monty Pythona, R, Biegun oraz Ziggurat. Najmniej użyteczne okazują się generatory: BM Boxa-Mullera, Wallace’a, Iloraz oraz Excel.
13
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Evaluation of Investing Efficiency of Open Pension Funds by Means of the Method of Cluster Analysis

61%
Mikulec A., University of Łódź C. o. S. M.
Acta Universitatis Lodziensis. Folia Oeconomica
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2009
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vol. 225
EN
The following study aims at analyzing the activity of open pension funds so far. To evaluate their efficiency, the author uses profitability indicators of investment portfolio such as Sharpe, Treynor and Jensen Ratio as well as IR (Information Ratio), TE (Tracking Error) and M2 (M2-measure). The analysis was carried out by means of monthly and quarterly data. The next stage includes rating of open pension funds from the point of view of their efficiency and conducted investment policy, analyzing, at the same time, calculated profitability ratios, rates of return and risk measures. In order to do that, the author uses such methods of cluster analysis as Tree Clustering and k-Means Clustering as well as different distance measures and Amalgamation or Linkage Rules.
PL
Celem niniejszego opracowania jest analiza dotychczasowej działalności Otwartych Funduszy Emerytalnych (OFE) z punktu widzenia osiągniętych wyników inwestycyjnych. Do oceny efektywności tych funduszy wykorzystano wskaźniki rentowności portfela inwestycji: Sharpe'a, Treynora i Jensena, a także IR, TE czy M2 (M2-measure). Analizę przeprowadzono na danych miesięcznych i kwartalnych. W kolejnym etapie dokonano klasyfikacji OFE z punktu widzenia ich efektywności i prowadzonej polityki inwestycyjnej, analizując obliczone wskaźniki rentowności, stopy zwrotu i miary ryzyka. W tym celu wykorzystano takie metody analizy skupień, jak: aglomerację, metodę k-średnich oraz różne miary odległości i metod łączenia lub wiązania.
14
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Normal Distribution of Returns of Warsaw Stock Exchange Indexes

51%
Borowski K.
Problemy Zarządzania
|
2018
|
issue 2/2018 (74)
11-45
EN
The paper verified the hypothesis regarding a normal distribution of returns of Warsaw Stock Exchange indexes for the following time intervals: daily, weekly, monthly, quarterly and yearly. The analyzed rates of return were calculated in the following outlines: closing-closing, opening-opening, opening-closing and overnight. The verification of statistical hypotheses was conducted with the use of the following seven statistical tests: Kolmogorov-Smirnov, Lilliefors, Shapiro-Wilk, Chi-squared, Cramer von Mises, Watson, Anderson-Darling. The analyzed indexes were ranked due to the convergence of their return to the normal distribution with the use of the following tests: Jarque-Bera, Shapiro-Wilk and D’Agostino-Pearson.
PL
W artykule zweryfikowano hipotezę o normalności rozkładu stóp zwrotu indeksów publikowanych przez Giełdę Papierów Wartościowych w Warszawie. Badanie przeprowadzono dla stóp zwrotu: dziennych, tygodniowych, miesięcznych, kwartalnych i rocznych oraz uwzględniono stopy zwrotu liczone w ujęciu: zamknięcie–zamknięcie, zamknięcie–otwarcie, otwarcie–otwarcie i otwarcie–zamknięcie (overnight). Weryfikacja hipotez statystycznych została przeprowadzona za pomocą następujących siedmiu testów: Kolmogorova-Smirnova, Lillieforsa, Shapiro-Wilka, Chi-kwadrat, Cramera von Misesa, Watsona, Andersona- Darlinga. Stworzony został również ranking analizowanych indeksów giełdowych z wykorzystaniem trzech testów statystycznych: Jarque-Bera, Shapiro-Wilka and D’Agostino-Pearsona.
15
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ROZKŁAD NORMALNY STÓP ZWROTU Z AKCJI WCHODZĄCYCH W SKŁAD NASTĘPUJĄCYCH INDEKSÓW GIEŁDOWYCH: WIG20, MWIG40 I SWIG80

51%
Borowski K.
Metody Ilościowe w Badaniach Ekonomicznych
|
2017
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vol. 18
|
issue 4
541-560
PL
W artykule zweryfikowana została teza o normalności rozkłdu stóp zwrotu cen akcji komponentów indeksów giełdowych tj. w dnu 17.10.2017 r.) do dnia 31.03.2017 r., dla stóp zwrotu w ujęciu: zamknięcie-zamknięcie, otwarcie-otwarcie, otwarcie-zamknięcie i overnight. Z wykorzystaniem testów Jarque-Bera, Shapiro-Wilka i D’Agostino-Pearsona podjęcto próbę stworzenia rankingu spółek ze względu na zbieżność rozkładu ich stóp zwrotu do rozkładu normalnego.
EN
The article verified the hypothesis regarding normal distribution of returns of shares - components of the following Warsaw Stock Exchange indexes Keywords: normal distribution, return rates, stock indices, ranking of companies
16
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Modelling income distributions based on theoretical distributions derived from normal distributions

51%
Sulewski P., Szymkowiak M.
Wiadomości Statystyczne. The Polish Statistician
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2023
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vol. 68
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issue 6
1-23
PL
W badaniach nad modelowaniem dochodów do aproksymacji ich rozkładów bardzo często wykorzystuje się takie znane rozkłady, jak Daguma, log-normalny czy Zengi. Celem badania omawianego w artykule jest sprawdzenie możliwości posłużenia się innymi typami rozkładów, tj. rozkładami asymetrycznymi wywodzącymi się z rozkładu normalnego (ND), w kontekście modelowania dochodów. Najważniejsze charakterystyki rozpatrywanych rozkładów określono na podstawie danych z badania EU-SILC 2011 dotyczących miesięcznego dochodu brutto na mieszkańca w Polsce. Rozkłady prawdopodobieństwa podzielono na dwie grupy: I – rozkłady powszechnie stosowane do modelowania dochodów (np. rozkład Daguma) i II – rozkłady wywodzące się z ND (np. rozkład SU Johnsona). Oprócz wizualnej oceny przydatności analizowanych rozkładów prawdopodobieństwa zastosowano kryteria liczbowe, takie jak: kryteria informacyjne dla modeli ekonometrycznych (Akaike Information Criterion, Schwarz’s Bayesian Information Criterion oraz Hannan-Quinn Information Criterion), miary zgodności oraz charakterystyki empiryczne i teoretyczne, w tym specjalnie zdefiniowana na potrzeby artykułu autorska miara wykorzystująca kwantyle. Jak wynika z badania, rozkład SU Johnsona (II grupa), może być – tak jak rozkład Daguma (I grupa) – z powodzeniem wykorzystany do modelowania dochodów.
EN
In income modelling studies, such well-known distributions as the Dagum, the lognormal or the Zenga distributions are often used as approximations of the observed distributions. The objective of the research described in the article is to verify the possibility of using other type of distributions, i.e. asymmetric distributions derived from normal distribution (ND) in the context of income modelling. Data from the 2011 EU-SILC survey on the monthly gross income per capita in Poland were used to assess the most important characteristics of the discussed distributions. The probability distributions were divided into two groups: I – distributions commonly used for income modelling (e.g. the Dagum distribution) and II – distributions derived from ND (e.g. the SU Johnson distribution). In addition to the visual evaluation of the usefulness of the analysed probability distributions, various numerical criteria were applied: information criteria for econometric models (such as the Akaike Information Criterion, Schwarz’s Bayesian Information Criterion and the Hannan-Quinn Information Criterion), measures of agreement, as well as empirical and theoretical characteristics, including a measure based on quantiles, specifically defined by the authors for the purposes of this article. The research found that the SU Johnson distribution (Group II), similarly to the Dagum distribution (Group I), can be successfully used for income modelling.
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