Uncertainty and sensitivity analysis techniques for use in performance assessment for radioactive waste disposal

The three-parameter generalized extreme-value (GEV) distribution has found wide application for describing annual floods, rainfall, wind speeds, wave heights, snow depths, and other maxima. Previous studies show that small-sample maximum-likelihood estimators (MLE) of parameters are unstable and recommend L moment estimators. More recent research shows that method of moments quantile estimators have for 20.25 , k , 0.30 smaller root-mean-square error than L moments and MLEs. Examination of the behavior of MLEs in small samples demonstrates that absurd values of the GEV-shape parameter k can be generated. Use of a Bayesian prior distribution to restrict k values to a statistically/physically reasonable range in a generalized maximum likelihood (GML) analysis eliminates this problem. In our examples the GML estimator did substantially better than moment and L moment quantile estimators for 20.4 # k # 0.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/1999WR900330

Generalized maximum-likelihood generalized extreme-value quantile estimators for hydrologic data

Estimation of R=P(Y<X) for three-parameter Weibull distribution

There are several measures employed to quantify the degree of skewness of a distribution. These have been based on the expectations or medians of the distributions considered. In 1964, van Zwet showed that all the standardized odd central moments of order 3 or higher maintained the convex or c-ordering of distributions that he introduced. This ordering has been widely accepted as appropriate for ordering two distributions in relation to skewness. More recently, measures based on the medians have been shown to honor the convex ordering. The measure of skewness (μ – M) / [sgrave] where μ, [sgrave], and M are, respectively, the expectation, standard deviation, and mode of the distribution was initially proposed by Karl Pearson. It unfortunately does not maintain the convex ordering. Here we introduce a measure based on the mode of a distribution that maintains the c-ordering. For many classes of right-skewed distributions, it is easily computed as a function of the shape parameter of the family and ...

https://www.jstor.org/stable/info/2684808

Measuring Skewness with Respect to the Mode

We describe and test quantile maximum probability estimator (QMPE), an open-source ANSI Fortran 90 program for response time distribution estimation.1 QMPE enables users to estimate parameters for the ex-Gaussian and Gumbel (1958) distributions, along with three “shifted” distributions (i.e., distributions with a parameter-dependent lower bound): the Lognormal, Wald, and Weibull distributions. Estimation can be performed using either the standard continuous maximum likelihood (CML) method or quantile maximum probability (QMP; Heathcote & Brown, in press). We review the properties of each distribution and the theoretical evidence showing that CML estimates fail for some cases with shifted distributions, whereas QMP estimates do not. In cases in which CML does not fail, a Monte Carlo investigation showed that QMP estimates were usually as good, and in some cases better, than CML estimates. However, the Monte Carlo study also uncovered problems that can occur with both CML and QMP estimates, particularly when samples are small and skew is low, highlighting the difficulties of estimating distributions with parameter-dependent lower bounds.

/pdf/qmpe-estimating-lognormal-wald-and-weibull-rt-distributions-55akaqjpym.pdf

QMPE: estimating Lognormal, Wald, and Weibull RT distributions with a parameter-dependent lower bound.

Previously suggested methods for constructing confidence bands for cumulative distribution functions have been based on the classical Kolmogorov-Smirnov test for an empirical distribution function. This paper gives a method based on maximum likelihood estimation of the parameters. The method is described for a general continuous distribution. Detailed results are given for a location-scale parameter model, which includes the normal and extreme-value distributions as special cases. Results are also given for the related lognormal and Weibull distributions. The formulas derived for these distributions give a band with exact confidence coefftcient. A chi-squared approximation, which avoids the use of special tables, is also described. An example is used to compare the resulting bands with those obtained by previously published methods.

Confidence Bands for Cumulative Distribution Functions of Continuous Random Variables

On suggere que l'utilisation du produit des densites pour la fonction de vraisemblance est une mauvaise application de la methode du maximum de vraisemblance dans des cas non reguliers

Corrected Maximum Likelihood in Non‐Regular Problems

A problem that is often encountered in non-linear regression is instability in the computation of parameter estimates. Such instability may arise from models with too many parameters, resulting in estimating equations that are ill conditioned and consequent problems of identification. This paper shows that in some cases these difficulties can be explained by the presence of an embedded model, a special case of the original model involving fewer parameters but one that is not readily identified from the parameterization of the model

Embedded Models in Non-Linear Regression

SUMMARY Two distinct problems can arise in maximum likelihood estimation in certain threeparameter problems. One problem, associated with the fitting of a threshold parameter, occurs if the fitted distribution has to be very positively skewed. This problem is well known. However, there is a second and unrelated difficulty which occurs quite often in practice and which arises when the distribution is not at all skew. This problem is not so well understood. It is shown in this paper that, within the three-parameter model, there is an embedded twoparameter special case which corresponds to infinite parameter values in the original model. The problem arises when this embedded model is the best fit to the data. The problem is shown to be easily resolved by first carrying out a check to see whether the embedded model should be fitted instead of the three-parameter model. Formal tests for this are discussed. The gamma, inverse Gaussian, log-normal and Weibull distributions are examples where the problem occurs. Numerical examples are provided for illustration.

Embedded Models in Three-parameter Distributions and their Estimation

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Papers

Confidence Bands for Cumulative Distribution Functions of Continuous Random Variables

Corrected Maximum Likelihood in Non‐Regular Problems

Embedded Models in Non-Linear Regression

Embedded Models in Three-parameter Distributions and their Estimation