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Loss modeling with mixtures distributions in R package

100%
Sitek G.
Śląski Przegląd Statystyczny
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2017
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issue 15 (21)
183-200
EN
Finite mixtures of probability distributions may be successfully used in the modeling of probability distributions of losses. These distributions are typically heavy tailed and positively skewed. Finding the distribution that fits loss data well is often difficult. The paper shows that the use of mixed models can significantly improve the goodness-of-fit of the loss data. The paper also presents an algorithm to find estimates of parameters of mixture distribution and gives an illustrative example. The analytical approach is probably the most often used in practice and certainly the most frequently adopted in the actuarial literature. It is reduced to finding a suitable analytical expression which fits the observed data well. For parameters estimation we use the maximum likelihood method applying the Newton-Raphson and EM algorithm. Computations of goodness-of-fit can be judged using the Akaike information criterion.
2
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Method of Colour Segmentation in Two Dimensional Images

100%
Korzeniewski J., University of Łódź D. o. S. M.
Acta Universitatis Lodziensis. Folia Oeconomica
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2004
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vol. 175
EN
The paper is divided into two parts. In the first part an overview of some selected methods of segmenting colours in two dimensional images is given. In the second part a new algorithm is proposed. The new algorithm is different from other algorithms due to its stress on accuracy of colour classes, the smallest possible number of colour classes (conditionally on paramameter choice) and due to smaller stress on the small number of eventual segments. The algorithm performance is assessed through applications to the segmentation a couple of colourful images.
3
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Modeling income on the basis of distribution mixture

88%
Sitek G.
Mathematical Economics
|
2014
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issue 10(17)
79-90
EN
Finite mixtures of probability distributions may be successfully used in the modeling of probability distributions of incomes. These distributions are typically heavy tailed and positively skewed. This article deals with the problem of determining the number of components in mixture modeling. This paper considers the likelihood of ratio-based testing of the null hypothesis of homogeneity in mixture models. The number of components is an important parameter in the applications of finite mixture models.
4
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New improved Poisson and negative binomial item count techniques for eliciting truthful answers to sensitive questions

75%
Kowalczyk B., Wieczorkowski R.
Statistics in Transition new series
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2022
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vol. 23
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issue 1
75-88
EN
Item count techniques (ICTs) are indirect survey questioning methods designed to deal with sensitive features. These techniques have gained the support of many applied researchers and undergone further theoretical development. Latterly in the literature, two new item count methods, called Poisson and negative binomial ICTs, have been proposed. However, if the population parameters of the control variable are not provided by the outside source, the methods are not very efficient. Efficiency is an important issue in indirect methods of questioning due to the fact that the protection of respondents’ privacy is usually achieved at the expense of the efficiency of the estimation. In the present paper we propose new improved Poisson and negative binomial ICTs, in which two control variables are used in both groups, although in a different manner. In the paper we analyse best linear unbiased and maximum likelihood estimators of the proportion of the sensitive attribute in the population in the introduced new models. The theoretical findings presented in the paper are supported by a comprehensive simulation study. The improved procedure allowed the increase of the efficiency of the estimation compared to the original Poisson and negative binomial ICTs.
5
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Estimation of regression parameters of two dimensional probability distribution mixtures

75%
Sitek G.
Studia Ekonomiczne
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2016
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vol. 304
30-46
EN
We use two methods of estimation parameters in a mixture regression: maximum likelihood (MLE) and the least squares method for an implicit interdependence. The most popular method for maximum likelihood esti-mation of the parameter vector is the EM algorithm. The least squares method for an implicit interdependence is based solving systems of nonlinear equations. Most frequently used method in the estimation of parameters mixtures regression is the method of maximum likelihood. The article presents the possibility of using a different the least squares method for an implicit interdependence and compare it with the maximum likelihood method. We compare accuracy of two methods of estimation by simulation using bias: root mean square error and bootstrapping standard errors of estimation.
PL
Do estymacji parametrów mieszanek regresji stosujemy dwie metody: metodę największej wiarygodności oraz metodę najmniejszych kwadratów dla zależności niejawnych. Najbardziej popularną metodą polegającą na maksymalizacji funkcji wiarygodności jest algorytm EM. Metoda najmniejszych kwadratów dla zależności niejawnych polega na rozwiązaniu układu równań nieliniowych. Najczęściej stosowaną metodą estymacji parametrów mieszanek regresji jest metoda największej wiarygodności. W artykule pokazano możliwość zastosowania innej metody najmniejszych kwadratów dla zależności niejawnych. Obie metody porównujemy symulacyjnie, używając obciążenia estymatora, pierwiastka błędu średniokwadratowego estymatora oraz bootstrapowe błędy standardowe.
6
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On an Improvement of the Model-Based Clustering Method

75%
Witek E., ent. Department of Statistics T. K. A. U. o. E.
Acta Universitatis Lodziensis. Folia Oeconomica
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2009
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vol. 228
PL
W artykule przedstawiona została modyfikacja metody taksonomii opartej na modelach mieszanych, w przypadku gdy niemożliwym staje się oszacowanie parametrów modelu za pomocą algorytmu EM. Gdy liczba obiektów przypisanych do klasy jest mniejsza niż liczba zmiennych opisujących te obiekty, niemożliwym staje się oszacowanie parametrów modelu. By uniknąć tej sytuacji estymatory największej wiarygodności zastępowane są estymatorami o największym prawdopodobieństwie a posteriori. Wybór modelu o najlepszej parametryzacji i stosownej liczbie klas dokonywany jest wówczas za pomocą zmodyfikowanej statystyki BIC.
EN
An improvement o f the model-based clustering (MBC) method in the case when EM algorithm fails as a result o f singularities is the basic aim o f this paper. Replacement o f the maximum likelihood (MLE) estimator by a maximum a posteriori (MAP) estimator, also found by the EM algorithm is proposed. Models with different number o f components are compared using a modified version o f BIC, where the likelihood is evaluated at the MAP instead o f MLE. A highly dispersed proper conjugate prior is shown to avoid singularities, but when these are not present it gives similar results to the standard method o f MBC.
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Optimal Allocation of the Sample in the Poisson Item Count Technique

45%
Bernardelli M., Kowalczyk B.
Acta Universitatis Lodziensis. Folia Oeconomica
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2018
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vol. 3
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issue 335
35-47
PL
Pośrednie metody ankietowania stanowią podstawowe narzędzie stosowane w przypadku pytań drażliwych. Artykuł nawiązuje do nowej, pośredniej metody zaproponowanej w pracy Tiana i wsp. (2014) i dotyczy optymalnej alokacji próby między grupę badaną i kontrolną. W przypadku gdy alokacji dokonuje się w oparciu o estymatory metodą momentów, rozwiązanie optymalne nie nastręcza trudności i zostało podane w pracy Tiana i wsp. (2014). Jednak to estymacja metodą największej wiarogodności ma lepsze własności, w związku z czym wyznaczenie alokacji optymalnej na jej podstawie jest zadaniem, którego rozwiązanie wydaje się mieć większe znaczenie praktyczne. Zadanie to nie jest trywialne, gdyż w przypadku omawianej metody pośredniej drażliwa zmienna badana ma charakter ukryty i jest zmienną nieobserwowalną. Wzór explicite na wariancję estymatora największej wiarogodności nieznanej frakcji cechy drażliwej nie jest dostępny, a sam estymator wyznaczyć można, używając odpowiednich algorytmów numerycznych. Do określenia optymalnej alokacji próby w oparciu o estymatory NW wykorzystane zostały symulacje Monte Carlo oraz iteracyjny algorytm EM
EN
Indirect methods of questioning are of utmost importance when dealing with sensitive questions. This paper refers to the new indirect method introduced by Tian et al. (2014) and examines the optimal allocation of the sample to control and treatment groups. If determining the optimal allocation is based on the variance formula for the method of moments (difference in means) estimator of the sensitive proportion, the solution is quite straightforward and was given in Tian et al. (2014). However, maximum likelihood (ML) estimation is known from much better properties, therefore determining the optimal allocation based on ML estimators has more practical importance. This problem is nontrivial because in the Poisson item count technique the study sensitive variable is a latent one and is not directly observable. Thus ML estimation is carried out by using the expectation‑maximisation (EM) algorithm and therefore an explicit analytical formula for the variance of the ML estimator of the sensitive proportion is not obtained. To determine the optimal allocation of the sample based on ML estimation, comprehensive Monte Carlo simulations and the EM algorithm have been employed.
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