A Kolmogorov-Smirnov goodness-of-fit test for the two-parameter Weibull distribution when the parameters are estimated from the data

1983 ◽  
Vol 23 (5) ◽  
pp. 996
2016 ◽  
Vol 45 (3) ◽  
pp. 15-42 ◽  
Author(s):  
Ayse KIZILERSU ◽  
Markus Kreer ◽  
Anthony W. Thomas

The left-truncated Weibull distribution is used in life-time analysis, it has many applications ranging from financial market analysis and insurance claims to the earthquake inter-arrival times.We present a comprehensive analysis of the left-truncated Weibull distribution when the shape, scale or both parameters are unknown and they are determined from the data using the maximum likelihood estimator. We demonstrate that if both the Weibull parameters are unknown then there are sets of sample configurations, with measure greater than zero, for which the maximum likelihood equations do not possess non trivial solutions. The modified critical values of the goodness-of-fit test from the Kolmogorov-Smirnov test statistic when the parameters are unknown are obtained from a quantile analysis. We find that the critical values depend on sample size and truncation level, but  not on the actual Weibull parameters.  Confirming this behavior, we present a complementary analysis using the Brownian bridge approach as an asymptotic limit of the Kolmogorov-Smirnov statistics and find that both approaches are in good agreement. A power testing is performed for our Kolmogorov-Smirnov goodness-of-fit test and  the issues related to the left-truncated data are discussed. We conclude the paper by showing the importance of left-truncated Weibulldistribution hypothesis testing on  the duration times of failed marriages in the US, worldwide terrorist attacks, waiting times between stock market orders, and time intervals of radioactive decay.


FLORESTA ◽  
2011 ◽  
Vol 41 (2) ◽  
Author(s):  
William Thomaz Wendling ◽  
Dartagnan Baggio Emerenciano ◽  
Roberto Tuyoshi Hosokawa

Desenvolve-se uma metodologia traçada por um roteiro em algoritmo factível e amigável para efetivação em planilhas eletrônicas, reconhecidas como uma interface popular para cálculos. Busca-se, assim, apresentar uma ferramenta útil para alunos de graduação e recém-graduados em engenharia florestal, ou engenheiros mais experientes que ainda não dominem a técnica, para ajuste de um modelo de função densidade de probabilidade, com o objetivo de descrever a estrutura da distribuição diamétrica de populações florestais. O modelo adotado é o da função de Weibull, e o método de ajuste é o do percentis, com simulações comparadas por teste de aderência de Kolmogorov-Smirnov. A eficiência do método apresentado é testada por comparação a outro método alternativo.Palavras-chave:  Manejo florestal; florestas - modelos matemáticos; florestas - simulação por computador. AbstractWeibull diameter distribution function adjusts for electronic spreadsheet. This research develops a methodology based on easy and friendly algorithm for spreadsheets, a well known interface for calculus. It aims to present a helpful tool for forestry students, as well as for newly or experienced engineers who haven’t already known adjustment techniques for a density function model of probability, which is useful into diametric distribution structure descriptions of forest population. It has Weibull’s function as main model, percentile as adjustment method, and comparing simulations by Kolmogorov-Smirnov goodness-of-fit test. Efficiency of the presented method was tested by comparison to another method.Keywords: Forest management; forest - mathematical models; forest - computer simulator.


2016 ◽  
Vol 61 (3) ◽  
pp. 489-496
Author(s):  
Aleksander Cianciara

Abstract The paper presents the results of research aimed at verifying the hypothesis that the Weibull distribution is an appropriate statistical distribution model of microseismicity emission characteristics, namely: energy of phenomena and inter-event time. It is understood that the emission under consideration is induced by the natural rock mass fracturing. Because the recorded emission contain noise, therefore, it is subjected to an appropriate filtering. The study has been conducted using the method of statistical verification of null hypothesis that the Weibull distribution fits the empirical cumulative distribution function. As the model describing the cumulative distribution function is given in an analytical form, its verification may be performed using the Kolmogorov-Smirnov goodness-of-fit test. Interpretations by means of probabilistic methods require specifying the correct model describing the statistical distribution of data. Because in these methods measurement data are not used directly, but their statistical distributions, e.g., in the method based on the hazard analysis, or in that that uses maximum value statistics.


Author(s):  
VICENTE SALVADOR E. MONTAÑO ◽  
MICHAEL E. CARTER II

The researchers build an inventory model for retail stores by validating their economicorder quantity through data driven simulation. This paper created an inventoryoptimization model for a personal care retailing business, to avoid stock out and minimize their holding cost and ordering cost. Simulating a thousand different scenarios, the research come up with an optimal inventory model for the two most sellable products in the store. The t-test reveals that product A has a significantly higher demand than product B. The simulation model validates the optimal order quantity of 59 units, with a reorder point of 25 units for product A. However, the simulation model recommends an optimal order quantity of 37 units and a reorder point of 10 units for product B. The Kolmogorov-Smirnov Goodness of Fit Test reveals the normal distribution of the 30 days inventory for Product A but not for Product B. Confirming that stocks out will unlikely happen for product A but will probably occur for product B. The model confirms EOQ findings of product with relatively high demand but low price but a departure for products with low demand but the high price.Keywords: Operations management, retail inventory system, t-test, Monte Carlo Simulation,Kolmogorov-Smirnov Goodness of Fit Test, Davao City, Philippines, Southeast Asia


2007 ◽  
Vol 135 (3) ◽  
pp. 1151-1157 ◽  
Author(s):  
Dag J. Steinskog ◽  
Dag B. Tjøstheim ◽  
Nils G. Kvamstø

Abstract The Kolmogorov–Smirnov goodness-of-fit test is used in many applications for testing normality in climate research. This note shows that the test usually leads to systematic and drastic errors. When the mean and the standard deviation are estimated, it is much too conservative in the sense that its p values are strongly biased upward. One may think that this is a small sample problem, but it is not. There is a correction of the Kolmogorov–Smirnov test by Lilliefors, which is in fact sometimes confused with the original Kolmogorov–Smirnov test. Both the Jarque–Bera and the Shapiro–Wilk tests for normality are good alternatives to the Kolmogorov–Smirnov test. A power comparison of eight different tests has been undertaken, favoring the Jarque–Bera and the Shapiro–Wilk tests. The Jarque–Bera and the Kolmogorov–Smirnov tests are also applied to a monthly mean dataset of geopotential height at 500 hPa. The two tests give very different results and illustrate the danger of using the Kolmogorov–Smirnov test.


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