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.

Semiparametric estimation and inference

2019 ◽  
pp. 139-164
Author(s):  
M. D. Edge
Keyword(s):  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Empirical Distribution Function ◽  
Empirical Distribution ◽  
Permutation Test ◽  
Cumulative Distribution ◽  
Permutation Testing ◽  
Unknown Distribution ◽  
Regression Parameters ◽  
Independent Observations

Nonparametric and semiparametric statistical methods assume models whose properties cannot be described by a finite number of parameters. For example, a linear regression model that assumes that the disturbances are independent draws from an unknown distribution is semiparametric—it includes the intercept and slope as regression parameters but has a nonparametric part, the unknown distribution of the disturbances. Nonparametric and semiparametric methods focus on the empirical distribution function, which, assuming that the data are really independent observations from the same distribution, is a consistent estimator of the true cumulative distribution function. In this chapter, with plug-in estimation and the method of moments, functionals or parameters are estimated by treating the empirical distribution function as if it were the true cumulative distribution function. Such estimators are consistent. To understand the variation of point estimates, bootstrapping is used to resample from the empirical distribution function. For hypothesis testing, one can either use a bootstrap-based confidence interval or conduct a permutation test, which can be designed to test null hypotheses of independence or exchangeability. Resampling methods—including bootstrapping and permutation testing—are flexible and easy to implement with a little programming expertise.

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 Minimax estimation of a bivariate cumulative distribution function

Metrika ◽  
2019 ◽  
Vol 83 (5) ◽  
pp. 597-615
Author(s):  
Rafał Połoczański ◽  
Maciej Wilczyński
Keyword(s):  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Empirical Distribution Function ◽  
Empirical Distribution ◽  
Decision Rules ◽  
Cumulative Distribution ◽  
Squared Error Loss ◽  
Error Loss ◽  
Squared Error ◽  
Von Mises

Abstract The problem of estimating a bivariate cumulative distribution function F under the weighted squared error loss and the weighted Cramer–von Mises loss is considered. No restrictions are imposed on the unknown function F. Estimators, which are minimax among procedures being affine transformation of the bivariate empirical distribution function, are found. Then it is proved that these procedures are minimax among all decision rules. The result for the weighted squared error loss is generalized to the case where F is assumed to be a continuous cumulative distribution function. Extensions to higher dimensions are briefly discussed.

scholarly journals A Skewed Model Combining Triangular and Exponential Features: The Two-faced Distribution and its Statistical Properties

2016 ◽  
Vol 35 (4) ◽  
Author(s):  
Maurizio Brizzi
Keyword(s):  
Distribution Function ◽  
Parameter Estimation ◽  
Goodness Of Fit ◽  
Cumulative Distribution Function ◽  
Continuous Distribution ◽  
Cumulative Distribution ◽  
Distribution Model ◽  
Model Combining ◽  
Left And Right ◽  
Expected Values

A new continuous distribution model is introduced, joining triangular and exponential features, respectively on the left and right side of a hinge point. The cumulative distribution function is derived, as well as the first three moments. Expected values and the Pearson index of skewness are tabulated. A possible step-by-step approach to parameter estimation is outlined. An application to Italian geographical data is given, referring to a set of municipalities classified by population, showing a very satisfactory goodness of fit.

DISSONANCE — A MEASURE OF VARIABILITY FOR ORDINAL RANDOM VARIABLES

2001 ◽  
Vol 09 (01) ◽  
pp. 39-53 ◽  
Author(s):  
RONALD R. YAGER
Keyword(s):  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Probability Distributions ◽  
Random Variables ◽  
Cumulative Distribution ◽  
General Properties

We look at the issue of obtaining a variance like measure associated with probability distributions over ordinal sets. We call these dissonance measures. We specify some general properties desired in these dissonance measures. The centrality of the cumulative distribution function in formulating the concept of dissonance is pointed out. We introduce some specific examples of measures of dissonance.

Estimating the cumulative distribution function for the linear combination of gamma random variables

2017 ◽  
Vol 20 (5) ◽  
pp. 939-951
Author(s):  
Amal Almarwani ◽  
Bashair Aljohani ◽  
Rasha Almutairi ◽  
Nada Albalawi ◽  
Alya O. Al Mutairi
Keyword(s):  
Distribution Function ◽  
Linear Combination ◽  
Cumulative Distribution Function ◽  
Random Variables ◽  
Cumulative Distribution
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