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1

On the Bahadur Representation of Quantiles for a Sample from ρ*-Mixing Structure Population

Dudek Dagmara, Kuczmaszewska Anna

Advances in Science and Technology. Research Journal

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2022

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Vol. 16, no 3

316--330

EN

In this paper, we establish the strong consistency and the Bahadur representation of sample quantiles for ρ*-mixing random variables. Additionally, the asymptotic normality and the Berry-Esseen bound of sample quantiles for ρ*-mixing random variables are presented. Additionally, we provide the rate of convergence of sample quantiles to population counterparts. Moreover, numerical simulation is presented to ilustrate and verify obtained results.

2

Proposal of methodology for practical application of nonparametric control charts

Noskievičová Darja, Smajdorová Tereza

Quality Production Improvement - QPI

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2019

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Vol. 1, Iss. 1

464--471

EN

This paper deals with the methodology for practical application of nonparametric control charts. This topic is very important for two reasons: firstly nonparametric control charts are very effective instruments for the realization of the statistical process monitoring phase I due to their robustness against various deviations from the data assumptions that must be met when applying model-based control charts. Secondly nonparametric control charts have very weak SW support and also they are not taught in the frame of training courses not even of the university study programmes. For that reason the practitioners do not know them and do not use them. The paper offers the proposal how to practically apply these control charts which is based on the complex simulation study of various nonparametric control charts performance when various data assumptions have not been met. The study has covered these nonparametric control charts: Shewhart sign control chart, nonparametric EWMA and nonparametric CUSUM control charts, nonparametric progressive mean control chart, control chart based on Mood statistics and robust median absolute deviation control chart. All charts have been studied in condition of not normally distributed data, autocorrelated data and data with nonconstant distribution parameters. The simulations were realized for statistically stable (IC – in control) and also statistically unstable (OC – out of control) processes. For the evaluation of the control charts performance median run length, 0.05-quantile, and 0.95-quantile were used.

3

On adjusting the load bearing capacity of decisive members to reliability classes of statically determinate complex structures

Kowal Z.

Archives of Civil Engineering

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2013

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Vol. 59, nr 1

131--142

EN

The paper provides a solution to the problem of dimensioning decisive bars on the basis of the conditions of meeting the recommended reliability classes [9] of statically determinate structures composed of n members. A theorem was formulated: if a statically determinate structure composed of n decisive members is to attain the reliability greater than, or equal to, the recommended reliability p = 1- q, it is necessary and sufficient that the damage frequency sum qi of decisive members is smaller than the admissible damage frequency q of the structure: Σqi < q. On the basis of this theorem, s coefficients that recommend increase of the load bearing capacity of the decisive bars in a statically determinate structure constructed in order to meet the recommended class [9] of the structure reliability, are estimated and presented in a tabular form.

4

Własności niezawodnościowe wargowych pierścieni uszczelniających

Zdziennicki Z., Maciejczyk A.

Problemy Eksploatacji

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2011

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nr 1

213-220

PL

Artykuł opisuje własności niezawodnościowe wargowych pierścieni uszczelniających -jednego z najbardziej popularnych i najszerzej stosowanych uszczelnień wirujących wałków. W artykule przedstawiono mechanizm uszkodzenia tych uszczelnień, jak i konsekwencje ich uszkodzeń. Opierając się na danych dostępnych w literaturze fachowej, zostały wyznaczone dwie podstawowe funkcyjne charakterystyki niezawodnościowe tych uszczelnień: funkcja prawdopodobieństwa uszkodzeń i funkcja niezawodności. Zmienna losowa opisująca uszkodzenia wargowych pierścieni uszczelniających ma rozkład Gaussa, stąd funkcja prawdopodobieństwa uszkodzeń dla tych pierścieni jest charakterystyczną funkcją gaussowską typu "dzwon". Funkcję niezawodności rozważanych uszczelnień podano w postaci analitycznej, stosując do tego tzw. funkcję błędu. W oparciu o uzyskane funkcyjne charakterystyki niezawodnościowe wyznaczono wartości liczbowe kwantyli rozważanego uszczelnienia - zarówno dla pojedynczego pierścienia, jak i dla jego układów składających się z kilku sztuk. Uzyskane wielkości funkcyjne przedstawiono w postaci wykresów.

EN

The paper describes reliability of lip-type seals, the most popular and most often used seals for rotating shafts. In the paper, a mechanism of seal failure was explained as well as its impact on others elements of mechanical systems. On basis of data from technical literature, these seals have two basic reliability functions, probability density function and reliability function. The random variable, which describes failures of lip seals, has a Gauss distribution. So the probability density function is a "bell" curve. The reliability function of the seals was given in an analytical form with an error function. The resulting functions were presented in their graphical forms as well. On the strength these functions, the probability density function, and the reliability function, the numerical values of the seal quantiles were calculated. It was done for a single lip seal as well as some systems of lip seals. The paper is ended with conclusions.

5

Estymacja parametryczna i nieparametryczna z asymetryczna, funkcja, jądra gamma dla porównawczej oceny górnych kwantyli Q[max,p%] w wybranych przekrojach wodowskazowych dorzecza Wisły i Odry

Jarosińska E.

Czasopismo Techniczne. Środowisko

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2006

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R. 103, z. 10-Ś

121-131

PL

Na bazie danych rocznych przepływów maksymalnych niektórych rzek Polski dokonano porównania górnych kwantyli p% obliczonych tradycyjną metodą parametryczną (rozkład Pearsona III typu) i nieparametryczną metodą jądrową. (z asymetrycznym jądrem gamma K[GAM1]. W połowie przypadków metoda nieparametryczna wykazuje wielomodalny charakter rozkładu. Obliczone nieparametryczne kwantyle p[1%] i p[0,5%] w większości przypadków są wyższe od swoich parametrycznych odpowiedników.

EN

Based on yearly maximum discharge series on some rivers in Poland, a comparison of parametric upper quantiles (Pearson III type) and nonparametric (with the gamma kernel) method of probability distribution estimation was made. In half cases, the nonparametric approach showed multimodality of yearly flow distribution. It was also found that the calculated nonparametric upper 1% and 0,5% quantiles were in most causes higher that their parametric counterparts.

6

Are parsimonious FF models more reliable than true ones? II. Comparative assessment of the performance of simple models versus the parent distributions

Kochanek K., Strupczewski W. G., Singh V. P., Węglarczyk S.

Acta Geophysica Polonica

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2005

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Vol. 53, nr 4

437-457

EN

Applying the methodology described in Strupczewski et al. (2005a; this issue), the performance of various parsimonious models combined with three estima-tion methods versus Flood Parent Distributions is comparatively assessed by simulation experiments. Moments (MOM), L-moments (LMM) and maximum likelihood (MLM) are used as alternative methods. Five four-parameter Specific Wakeby Distributions (SWaD) are employed to serve as Flood Parent Distributions and forty Distribution/Estimation (D/E) procedures are included in respect to the estimation of upper quantiles. The relative bias (RB), relative root mean square error (RRMSE) and reliability of procedures are used for the assessment of the relative performance of alternative procedures. Parsimonious two-parameter models generally perform better for hydrological sample sizes than their three-parameter counterparts with respect to RRMSE. How-ever, the best performing procedures differ for various SWaDs. As far as estimation methods are concerned, MOM usually produces the smallest values of both RB and RRMSE of upper quantiles for all competing methods. The second place in rank is occupied by LMM, whereas, MLM produces usually the highest values. Considerable influence of sampling bias on the value of the total bias has been ascertained. The improper choice of a model fitted to SWaD samples causes that the reliability of some three-parameter parsimonious D/E procedures does not always rise with the sample size. Also odd is that True model does not always give one hundred percent reliability for very large samples, as it should. This means that estimating algorithms still require improvements.

7

Are parsimonious FF models more reliable than true ones? I. Accuracy of Quantiles and Moments Estimation (AQME) - method of assessment

Strupczewskiwgs@igf.edu.pl W. G., Kochanek K., Singh V. P., Węglarczyk S.

Acta Geophysica Polonica

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2005

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Vol. 53, nr 4

419-436

EN

Applying the methodology described in Strupczewski et al. (2005a; this is-sue), the performance of various parsimonious models combined with three estimation methods versus Flood Parent Distributions is comparatively assessed by simulation experiments. Moments (MOM), L-moments (LMM) and maximum likelihood (MLM) are used as alternative methods. Five four-parameter Specific Wakeby Distributions (SWaD) are employed to serve as Flood Parent Distributions and forty Distribution/Estimation (D/E) procedures are included in respect to the estimation of upper quantiles. The relative bias (RB), relative root mean square error (RRMSE) and reliability of procedures are used for the assessment of the relative performance of alternative procedures. Parsimonious two-parameter models generally perform better for hydrological sample sizes than their three-parameter counterparts with respect to RRMSE. How-ever, the best performing procedures differ for various SWaDs. As far as estimation methods are concerned, MOM usually produces the smallest values of both RB and RRMSE of upper quantiles for all competing methods. The second place in rank is occupied by LMM, whereas, MLM produces usually the highest values. Considerable influence of sampling bias on the value of the total bias has been ascertained. The improper choice of a model fitted to SWaD samples causes that the reli-ability of some three-parameter parsimonious D/E procedures does not always rise with the sample size. Also odd is that True model does not always give one hundred percent reliability for very large samples, as it should. This means that estimating algorithms still require improvements.

8

Estymacja parametryczna i nieparametryczna rozkładów prawdopodobieństwa przepływów maksymalnych rocznych wybranych rzek Polski. Jądro symetryczne

Jarosińska E., Węglarczyk S.

Czasopismo Techniczne. Środowisko

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2004

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R. 101, z. 15-Ś

97-106

PL

Korzystając z rzeczywistych danych rocznych przepływów maksymalnych niektórych rzek Polski, dokonano porównania nieparametrycznej (z jądrem Gaussa) i parametrycznej metody estymacji funkcji rozkładu prawdopodobieństwa (rozkład Pearsona, typ III). W większości przypadków metoda nieparametryczna daje dwumodalny obraz rozkładu. Obliczone "nieparametryczne" kwantyle rzędu 1% i 0,5% są na ogół wyższe od swoich "parametrycznych" odpowiedników.

EN

Using yearly maximum discharge series on main rivers in Poland, a comparison was made of nonparametric (with the gaussian kernel) and parametric (Pearson III type) method of probability distribution estimation. In most cases, the nonparametric approach showed bimodality of yearly flow distribution. It was also found that the calculated nonparametric upper 1% and 0,5% ąuantiles were in general higher that their parametric counterparts.

9

Physics of flood frequency analysis. Part I. Linear convective diffusion wave model

Strupczewski W., Węglarczyk S., Singh V.

Acta Geophysica Polonica

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2002

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Vol. 50, nr 3

433-455

EN

It is hypothesized that the impulse response of a linearized convective diffusion wave (CD) model is a probability distribution suitable for flood frequency analysis. This flood frequency model has two parameters, which are derived using the methods of moments and maximum likelihood. Also derived are errors in quantiles for these methods of parameter estimation. The distribution shows an equivalency of the two estimation methods with respect to the mean value - an important property in the case of unknown true distribution function. As the coefficient of variation tends to zero (with the mean fixed), the distribution tends to a normal one, similar to the lognormal and gamma distributions.

10

Model error in flood frequency estimation

Strupczewski W., Węglarczyk S., Singh V.

Acta Geophysica Polonica

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2002

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Vol. 50, nr 2

279-319

EN

Asymptotic bias in large quantiles and moments for three parameter estimation methods, including the maximum likelihood method (MLM), moments method (MOM) and linear moments method (LMM), is derived when a probability distribution function (PDF) is falsely assumed. It is illustrated using an alternative set of PDFs consisting of five two-parameter PDFs that are lower-bounded at zero, i.e., Log-Gumbel (LG), Log-logistic (LL), Log-normal (LN), Linear Diffusion (LD) and Gamma (Ga) distribution functions. The stress is put on applicability of LG and LL in the real conditions, where the hypothetical distribution (H) differs from the true one (T). Therefore, the following cases are considered: H=LG; T=LL, LN, LD and Ga, and H=LL, LN, LD and Ga, T=LG. It is shown that for every pair (H; T) and for every method, the relative bias (RB) of moments and quantiles corresponding to the upper tail is an increasing function of the true value of the coefficient of variation (cv), except that RB of moments for MOM is zero. The value of RB is smallest for MOM and the largest for MLM. The bias of LMM occupies an intermediate position. Since MLM used as the approximation method is irreversible, the asymptotic bias of the MLM-estimate of any statistical characteristic is not asymmetric as is for the MOM and LMM. MLM turns out to be the worst method if the assumed LG or LL distribution is not the true one. It produces a huge bias of upper quantiles, which is at least one order higher than that of the other two methods. However, the reverse case, i.e., acceptance of LN, LD or Ga as a hypothetical distribution while LG or LL as the true one, gives the MLM-bias of reasonable magnitude in upper quantiles. Therefore, one should be highly reluctant in choosing the LG and LL in flood frequency analysis, especially if MLM is to be applied.