SUMMARY We consider maximum likelihood estimation of the parameters of a probability density which is zero for x 2, the information matrix is finite and the classical asymptotic properties continue to hold. For cx = 2 the maximum likelihood estimators are asymptotically efficient and normally distributed, but with a different rate of convergence. For 1 < a < 2, the maximum likelihood estimators exist in general, but are not asymptotically normal, while the question of asymptotic efficiency is still unsolved. For cx < 1, the maximum likelihood estimators may not exist at all, but alternatives are proposed. All these results are already known for the case of a single unknown location parameter 0, but are here extended to the case in which there are additional unknown parameters. The paper concludes with a discussion of the applications in extreme value theory.

https://www.jstor.org/stable/pdfplus/2336336.pdf

Maximum likelihood estimation in a class of nonregular cases

D. B. Owen: Handbook of Statistical Tables. London: Pergamon Press; Reading, Massachusetts: Addison‐Wesley, 1962. Pp. xii+580. 70s.

Handbook of Statistical Tables.

A conditional method of inference is used to derive exact confidence intervals for several life characteristics such as location, scale, quantiles, and reliability when the data are Type II progressively censored. The method is shown to be feasible and practical, although a computer program may be required for its implementation. The method is applied for the purpose of illustration to the extreme-value and the one- and two-parameter exponential models. Prediction limits for the lifelength of future units are also discussed. An example consisting of data from an accelerated test on insulating fluid reported by Nelson is used for illustration and comparison.

Interval Estimation of Parameters of Life From Progressively Censored Data

The best Weibull distribution methods for the assessment of wind energy potential at different altitudes in desired locations are statistically diagnosed in this study. Seven different methods, namely graphical method (GM), method of moments (MOM), standard deviation method (STDM), maximum likelihood method (MLM), power density method (PDM), modified maximum likelihood method (MMLM) and equivalent energy method (EEM) were used to estimate the Weibull parameters and six statistical tools, namely relative percentage of error, root mean square error (RMSE), mean percentage of error, mean absolute percentage of error, chi-square error and analysis of variance were used to precisely rank the methods. The statistical fittings of the measured and calculated wind speed data are assessed for justifying the performance of the methods. The capacity factor and total energy generated by a small model wind turbine is calculated by numerical integration using Trapezoidal sums and Simpson’s rules. The results show that MOM and MLM are the most efficient methods for determining the value of

/pdf/statistical-diagnosis-of-the-best-weibull-methods-for-wind-2mkcz1opu5.pdf

Statistical diagnosis of the best weibull methods for wind power assessment for agricultural applications

SUMMARY This paper discusses the estimation of parameters in hazard rate models with a changepoint. Due to the irregularity of the models, the classical maximum likelihood method and the method of moments cannot be used. A consistent estimator of the change-point is obtained by examining the properties of the density represented as a mixture. The performance of the estimator is checked via simulation.

Estimation in change-point hazard rate models

This study develops maximum likelihood estimators for the three parameter Weibull distribution based on various left, and right, censored data situations. For the case of single censoring from the left and progressive censoring from the right, the developed estimation procedure involves the simultaneous solution of two iterative equations compared to the arduous task of solving three simultaneous iterative equations as outlined by Harter and Moore [6]. For the two parameter Weibull [4], the above case results in an estimation procedure involving only one iterative equation. The asymptotic variance-covariance matrix of the estimators is given. Pivotal functions are provided whose distributions can be used for confidence interval estimation and for tests of hypotheses. An example is included which illustrates the estimation procedure involving left censored samples.

Maximum Likelihood Estimation for the Three Parameter Weibull Distribution Based on Censored Samples

A technique based on the noncentral t distribution for obtaining one-sided tolerance limits for a balanced one-way-anova random-effects model is developed. These limits are such that the probability is γ that at least a proportion p of the population consisting of very many specimens from very many batches exceeds the calculated lower limit. A few selected tables are given, and an example illustrates the use of the derived tolerance factors.

Factors for One-Sided Tolerance Limits for Balanced One-Way-ANOVA Random-Effects Model

A Bayesian reliability estimation technique known as the ``empirical Bayes approach'' is developed which uses previous experience nce to get a Bayesian point estimator. The techniques require no knowledge of the form of the unknown prior distribution and are robust to assumptions about its form. Empirical Bayes techniques are applicable to situations in which prior, independent observations of the random variable X from the random couple (?, X) are available where ? is the observed parameter of interest distributed in accordance with the unknown prior distribution. Performance comparisons of the empirical Bayes and other well established techniques are developed by examples for the binomial, exponential, Normal, and Poisson situations which often occur in reliability problems. In all cases the empirical Bayes estimator performed better than the classical estimator in minimizing the average squared error.

An Empirical Bayes Approach to Reliability

This paper investigates the maximum flaw length distribution which statistically describes the longest flaw occurring in a given material part. The number of flaws in each part is assumed to follow a Poisson distribution and the size of each of these flaws is based upon some initial flaw length distribution. The following two situations are considered: (1) prior to inspection and (2) after inspection-with-part-rejection, where a part is removed from the population if a flaw is found with length equal to or greater than some predetermined threshold value xR , otherwise the part remains in the population; however, all detected flaws of length less than xR are removed. The probability of detection of a flaw during inspection is assumed to be a function of the flaw's length. The probability of a part being accepted is also noted.

A Problem in the Distribution of Maximum Flaw Length After Inspection

A brief overview of the structural reliability problem and its relationship to probabilistic and statistical concepts is given. Selected references on structural reliability are compiled from various sources. References address structural reliability and selected topics associated with implementing the structural reliability approach.

Glen H. Lemon

Papers

Maximum Likelihood Estimation for the Three Parameter Weibull Distribution Based on Censored Samples

Factors for One-Sided Tolerance Limits for Balanced One-Way-ANOVA Random-Effects Model

An Empirical Bayes Approach to Reliability

A Problem in the Distribution of Maximum Flaw Length After Inspection

Literature Survey on Structural Reliability