Determination of the Three-Parameter Weibull Distribution and Confidence Bands From Experimental Fatigue Strength Data

2004 ◽  
Author(s):  
Enayat Mahajerin ◽  
Gary Burgess
Keyword(s):  
Distribution Function ◽  
Fatigue Strength ◽  
Weibull Distribution ◽  
Structural Reliability ◽  
Confidence Bands ◽  
Least Square ◽  
Cumulative Distribution ◽  
Point Estimate ◽  
Cumulative Probability ◽  
Discrete Version

The S-N curve for the material used to make a pressure vessel is approximate because it is drawn from a limited number of test specimens. The resulting curve may be in error due to a variety of factors including surface condition, size, environment conditions, and stress concentrations. As a result, when the fatigue strength like the endurance limit is determined from the S-N curve by observing a definite break in the curve, it will be subject to error. Because of these uncertainties, it is necessary to use appropriate statistical methods to interpret the test results. In this paper, it is assumed that the percentage of failures for a given service life can be approximated by a three-parameter Weibull distribution. The Weibull distribution is flexible and has been shown to be suitable for structural reliability. The distribution is fitted to experimental data using a least square best fit approach applied to a discrete version of the cumulative probability distribution function, F(x). In practice a point-by-point estimate of the cumulative distribution function is used. As a result, it is necessary to establish confidence bands. The true curve of F(x) lies within these bands for a given probability.

scholarly journals Results of the Verification of the Statistical Distribution Model of Microseismicity Emission Characteristics

2016 ◽  
Vol 61 (3) ◽  
pp. 489-496
Author(s):  
Aleksander Cianciara
Keyword(s):  
Distribution Function ◽  
Weibull Distribution ◽  
Goodness Of Fit ◽  
Cumulative Distribution Function ◽  
Statistical Distribution ◽  
Cumulative Distribution ◽  
Distribution Model ◽  
Emission Characteristics ◽  
Goodness Of Fit Test ◽  
Empirical Cumulative Distribution Function

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.

scholarly journals On a Special Weighted Version of the Odd Weibull-Generated Class of Distributions

2021 ◽  
Vol 26 (3) ◽  
pp. 62
Author(s):  
Zichuan Mi ◽  
Saddam Hussain ◽  
Christophe Chesneau
Keyword(s):  
Distribution Function ◽  
Weibull Distribution ◽  
Probability Density ◽  
Hazard Rate ◽  
Cumulative Distribution Function ◽  
Data Fitting ◽  
Environmental Data ◽  
Cumulative Distribution ◽  
Model Parameters ◽  
New Class

In recent advances in distribution theory, the Weibull distribution has often been used to generate new classes of univariate continuous distributions. They find many applications in important disciplines such as medicine, biology, engineering, economics, informatics, and finance; their usefulness is synonymous with success. In this study, a new Weibull-generated-type class is presented, called the weighted odd Weibull generated class. Its definition is based on a cumulative distribution function, which combines a specific weighted odd function with the cumulative distribution function of the Weibull distribution. This weighted function was chosen to make the new class a real alternative in the first-order stochastic sense to two of the most famous existing Weibull generated classes: the Weibull-G and Weibull-H classes. Its mathematical properties are provided, leading to the study of various probabilistic functions and measures of interest. In a consequent part of the study, the focus is on a special three-parameter survival distribution of the new class defined with the standard exponential distribution as a reference. The exploratory analysis reveals a high level of adaptability of the corresponding probability density and hazard rate functions; the curves of the probability density function can be decreasing, reversed N shaped, and unimodal with heterogeneous skewness and tail weight properties, and the curves of the hazard rate function demonstrate increasing, decreasing, almost constant, and bathtub shapes. These qualities are often required for diverse data fitting purposes. In light of the above, the corresponding data fitting methodology has been developed; we estimate the model parameters via the likelihood function maximization method, the efficiency of which is proven by a detailed simulation study. Then, the new model is applied to engineering and environmental data, surpassing several generalizations or extensions of the exponential model, including some derived from established Weibull-generated classes; the Weibull-G and Weibull-H classes are considered. Standard criteria give credit to the proposed model; for the considered data, it is considered the best.

scholarly journals A New Extended Cosine—G Distributions for Lifetime Studies

Mathematics ◽  
2021 ◽  
Vol 9 (21) ◽  
pp. 2758
Author(s):  
Mustapha Muhammad ◽  
Rashad A. R. Bantan ◽  
Lixia Liu ◽  
Christophe Chesneau ◽  
Muhammad H. Tahir ◽  
...  
Keyword(s):  
Weibull Distribution ◽  
Residual Life ◽  
Series Representation ◽  
Absolute Error ◽  
Quantile Function ◽  
Least Square ◽  
Cumulative Distribution ◽  
Reliability Parameter ◽  
Asymptotic Results ◽  
Error Loss

In this article, we introduce a new extended cosine family of distributions. Some important mathematical and statistical properties are studied, including asymptotic results, a quantile function, series representation of the cumulative distribution and probability density functions, moments, moments of residual life, reliability parameter, and order statistics. Three special members of the family are proposed and discussed, namely, the extended cosine Weibull, extended cosine power, and extended cosine generalized half-logistic distributions. Maximum likelihood, least-square, percentile, and Bayes methods are considered for parameter estimation. Simulation studies are used to assess these methods and show their satisfactory performance. The stress–strength reliability underlying the extended cosine Weibull distribution is discussed. In particular, the stress–strength reliability parameter is estimated via a Bayes method using gamma prior under the square error loss, absolute error loss, maximum a posteriori, general entropy loss, and linear exponential loss functions. In the end, three real applications of the findings are provided for illustration; one of them concerns stress–strength data analyzed by the extended cosine Weibull distribution.

scholarly journals A New Approach to Assess Wind Potential

2021 ◽  
Keyword(s):  
Distribution Function ◽  
Maximum Likelihood ◽  
Weibull Distribution ◽  
Wind Power ◽  
Cumulative Distribution Function ◽  
Maximum Likelihood Method ◽  
New Method ◽  
Cumulative Distribution ◽  
Likelihood Method ◽  
Wind Distribution

<p>Weibull Cumulative Distribution Function (C.D.F.) has been employed to assess and compare wind potentials of two wind stations Europlatform and Stavenisse of The Netherland. Weibull distribution has been used for accurate estimation of wind energy potential for a long time. The Weibull distribution with two parameters is suitable for modeling wind data if wind distribution is unimodal. Whereas wind distribution is generally unimodal, random weather changes can make the distribution bimodal. It is always desirable to find a method that accurately represents actual statistical data. Some well-known statistical methods are Method of Moment (MoM), Linear Least Square Method (LLSM), Maximum Likelihood Method (M.L.M.), Modified Maximum Likelihood Method (MMLM), Energy Pattern Factor Method (EPFM), and Empirical Method (E.M.), etc. All these methods employ Probability Density Function (PDF) of Weibull distribution, except LLSM, which uses Cumulative Distribution Function (C.D.F.). In this communication, we are presenting a newly proposed method of evaluating Weibull parameters. Unlike most methods, this new method employs a cumulative distribution function. A MATLAB® GUI-based simulation is developed to estimate Weibull parameters using the C.D.F. approach. It is found that the Mean Square Error (M.S.E.) is the lowest when using the new method. The new method, therefore, estimates wind power density with reasonable accuracy. Wind Power (W.P.) is estimated by considering four different Wind Turbine (W.T.) models for two sites, and maximum W.P. is found using Evance R9000.</p>

scholarly journals Estimasi Fungsi Survival dan Fungsi Hazard Kumulatif Pada Data Survival Pederita Multiple Myeloma Serta Faktor-faktor yang Mempengaruhi Waktu Survivalnya

Intersections ◽  
2019 ◽  
Vol 4 (2) ◽  
pp. 33-43
Author(s):  
Toto Hermawan ◽  
Dwi Nurrohmah ◽  
Ismi Fathul Jannah
Keyword(s):  
Multiple Myeloma ◽  
Distribution Function ◽  
Data Analysis ◽  
Probability Density Function ◽  
Weibull Distribution ◽  
Hazard Function ◽  
Cumulative Distribution Function ◽  
Survival Function ◽  
Random Variable ◽  
Cumulative Distribution

Multiple myeloma is an infectious disease characterized by the accumulation of abnormal plasma cells, a type of white blood cell, in the bone marrow. The main objective of this data analysis is to investigate the effect of Bun, Ca, Pcells and Protein risk factors on the survival time of multiple myeloma patients from diagnosis to death. In the survival data analysis, the observed random variable T is the time needed to achieve success. To explain a random variable, the cumulative distribution function or the probability density function can be used. In survival analysis, the function of the random variable that becomes important is the survival function and the hazard function which can be derived using the cumulative distribution function or the probability density function. In general, it is difficult to determine the survival function or hazard function of a population group with certainty. However, the survival function or hazard function can still be approximated by certain estimation methods. The Kaplan-Meier method can be used to find estimators of the survival function of a population. Meanwhile, to find the estimator of the cumuative hazard function, the Nelson-Aalen method can be used. From the variables studied, it turned out that the one that gave the most significant effect was the Bun variable, namely blood urea nitrogen levels using both the exponential and weibull distribution. However, by using the weibull distribution, the presence of Bence Jones Protein in urine also has a quite real effect

scholarly journals ANALISIS KUMULATIF COVID-19 PROVINSI PAPUA TAHUN 2020 MENGGUNAKAN MODEL DISTRIBUSI JOHNSON SB

2021 ◽  
Vol 21 (1) ◽  
pp. 21-28
Author(s):  
Felix Reba ◽  
Alvian Sroyer
Keyword(s):  
Distribution Function ◽  
Maximum Likelihood ◽  
Cumulative Distribution Function ◽  
Data Distribution ◽  
Secondary Data ◽  
Cumulative Distribution ◽  
Distribution Model ◽  
Cumulative Probability ◽  
Number Of Patients ◽  
Kolmogorov Smirnov

Coronavirus belongs to the coronaviridae family. The coronavirus family groups are alpha (α), beta (β), gamma (γ) and delta (δ) coronavirus. Although research related to covid-19 in several provinces in Indonesia has been conducted by several researchers so far there has been no research related to the Covid-19 model in Papua province. One of the obstacles faced by some researchers is related to the Covid-19 data parameters which are difficult to estimate, so that the model formulated could not describe the outbreak well. Therefore the aim of this study is to conduct a cumulative analysis of the 2020 Papua province Covid-19 using the Johnson SB distribution model. The methods used to perform the analysis are Kolmogorov Smirnov for testing the suitability of the Covid-19 data to the model, Johnson SB to show the data distribution model, Maximum Likelihood to estimate the parameters and the Johnson SB cumulative distribution function to describe the probability of Covid-19 data. 19 Papua Province in 2020. The secondary data on the number of Covid-19 cases in Papua, obtained from the Papua Provincial Health Office is used in this research. The results showed that, the highest increase in the number of patients every day, starting from September 1 2020 to October 31, 2020 for infected cases was on 16-17 September, by 274 patients. Meanwhile, most recovery (308 patients) happened to be on 30-31 October and the highest death (5 people) was on 27-28 September. The highest cumulative probability for cases of infection, recovery and death were (Confirmed <4965) = 0.3, Prob(Cured <6408) = 0.9 and Prob(died <91) = 0.4 respectively.

scholarly journals ON DISCRETE WEIBULL DISTRIBUTION

2014 ◽  
Vol 20 (79) ◽  
pp. 1-9
Author(s):  
ALI HUSEEN ALWAKEEL
Keyword(s):  
Distribution Function ◽  
Weibull Distribution ◽  
Hazard Function ◽  
Random Variable ◽  
Weibull Model ◽  
Cumulative Distribution ◽  
Probability Mass Function ◽  
Failure Rate Function ◽  
Weibull Models ◽  
Discrete Values

Most of the Weibull models studied in the literature were appropriate for modelling a continuous random variable which assumes the variable takes on real values over the interval [0,∞]. One of the new studies in statistics is when the variables take on discrete values. The idea was first introduced by Nakagawa and Osaki, as they introduced discrete Weibull distribution with two shape parameters q and β where      0 < q < 1 and b > 0. Weibull models for modelling discrete random variables assume only non-negative integer values. Such models are useful for modelling for example; the number of cycles to failure when components are subjected to cyclical loading. Discrete Weibull models can be obtained as the discrete counterparts of either the distribution function or the failure rate function of the standard Weibull model. Which lead to different models. This paper discusses the discrete model which is the counterpart of the standard two-parameter Weibull distribution. It covers the determination of the probability mass function, cumulative distribution function, survivor function, hazard function, and the pseudo-hazard function.

scholarly journals Constructing a New Exponentiated Family Distribution with Reliability Estimation

2018 ◽  
Vol 29 (2) ◽  
pp. 155
Author(s):  
Shurooq Ahmed Al-Sultany
Keyword(s):  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Random Variable ◽  
Reliability Estimation ◽  
Least Square ◽  
Cumulative Distribution ◽  
New Family ◽  
Regression Approach ◽  
Two Parameters ◽  
The Given

This paper deals with using one method for transforming two parameters given distribution to another form with three parameters distribution, through using idea of reparameterization with powering the given cumulative distribution function by new parameter, where this work gives a new family through using new parameter which is necessary for generating values of the r.v from given CDF through smoothing the values of the given random variable to obtain new values of r.v using three set of parameters rather than two. The new model Frechet p.d.f is obtained and also its Cumulative distribution function is found then we apply three methods of estimation (which are Maximum likelihood , moments estimator , and the third method is depend on using least square regression approach). Different set of initial values of parameters ( , , ) and different samples size (n=25,50,75,100). The simulation procedure is done using matlab-R2014b, and results are compared using integrated Mean square error.

ON DISCRETE WEIBULL DISTRIBUTION

2014 ◽  
Vol 20 (79) ◽  
pp. 1
Author(s):  
علي عبد الحسين الوكيل
Keyword(s):  
Distribution Function ◽  
Weibull Distribution ◽  
Hazard Function ◽  
Random Variable ◽  
Weibull Model ◽  
Cumulative Distribution ◽  
Probability Mass Function ◽  
Failure Rate Function ◽  
Weibull Models ◽  
Discrete Values

  Most of the Weibull models studied in the literature were appropriate for modelling a continuous random variable which assume the variable takes on real values over the interval [0,∞]. One of the new studies in statistics is when the variables takes on discrete values. The idea was first introduced by Nakagawa and Osaki, as they introduced discrete Weibull distribution with two shape parameters q and β where      0 < q < 1 and b > 0. Weibull models for modelling discrete random variables assume only non-negative integer values. Such models are useful for modelling for example; the number of cycles to failure when components are subjected to cyclical loading. Discrete Weibull models can be obtained as the discrete counter parts of either the distribution function or the failure rate function of the standard Weibull model. Which lead to different models. This paper discusses the discrete model which is the counter part of the standard two-parameter Weibull distribution. It covers the determination of the probability mass function, cumulative distribution function, survivor function, hazard function, and the pseudo-hazard function.

scholarly journals Parameter Estimation Techniques Based on Optimizing Goodness-of-Fit Statistics for Structural Reliability

1993 ◽  
Author(s):  
Alois Starlinger ◽  
Stephen F. Duffy ◽  
Joseph L. Palko
Keyword(s):  
Distribution Function ◽  
Goodness Of Fit ◽  
Cumulative Distribution Function ◽  
Structural Reliability ◽  
Empirical Distribution ◽  
Cumulative Distribution ◽  
Estimation Methods ◽  
Weibull Parameters ◽  
Fit Statistics ◽  
Failure Data

Abstract New methods are presented that utilize the optimization of goodness-of-fit statistics in order to estimate Weibull parameters from failure data. It is assumed that the underlying population is characterized by a three-parameter Weibull distribution. Goodness-of-fit tests are based on the empirical distribution function (EDF). The EDF is a step function, calculated using failure data, and represents an approximation of the cumulative distribution function for the underlying population. Statistics (such as the Kolmogorov-Smirnov statistic and the Anderson-Darling statistic) measure the discrepancy between the EDF and the cumulative distribution function (CDF). These statistics are minimized with respect to the three Weibull parameters. Due to nonlinearities encountered in the minimization process, Powell’s numerical optimization procedure is applied to obtain the optimum value of the EDF. Numerical examples show the applicability of these new estimation methods. The results are compared to the estimates obtained with Cooper’s nonlinear regression algorithm.

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