Tables of the cumulative non-central chi-square distribution-part 6 (finish)

The cumulative non-central chi-square distribution is tabulated for all combinations of values of α=0(0.1) 1.0 (0.2)3.0(0.5)5.0(1.0)34.0,v=1(1)30(2)50(5)100 and y=0.01 (0.01)0.1(0.1)1.0(0.2)3.0(0.5)10.0(1.0 30.0(2.0)50.0(5.0)165.0. The computations have been correctly rounded to five decimal places. Also, there is a discussion about the error involved in the computations. Furthermore, there is a discussion about possible interpolation in the table using the Lagrange's method     

On the cramer-frechet-rao inequality for translation parameter in the case of finite support

In this paper we investigate the problem of deriving the C-F-R (CRAMER-FRECHET-RAO) bound for the variance of an unbiased estimator of the translation para¬meter for a class of distributions having as support an interval of fixed length. Starting with the general form of the O-F-R, inequality studied earlier by VINCZE (1979) for mixed

densities, we prove some inequalities related to the information quantity occurring in the C-F-R bound. The case when the variance of the unbiased estimator does not depend upon the translation parameter is investigated. The case when the variance depends upon the translation parameter is also briefly discussed. Finally some remarks will be given

concerning the attainability of the variance,bounds given in this paper     

Asymptotic bayesian inference in some nonstandard cases: Bernstein–von Mises Results and regular bayes' estimators

Asymptotically with probability close to one, the convergence in variation (also in distribution) to the multivariate normal, of the aposteriori density function of a parameter agains an apriori density, viz. the BERNSTEIN–VON MISES results are established when observations are not necessarily indenpendent or identically distributed but satisfy weak regularity assumptions on their joint density function. Regular BAYES' estimators are defined with respect to regular loss functions and a positive apriori density and proved consistent, asymptotically efficient and asymptotically normal. Examples and applications to conjugate families of densities, to inference in MARKOV Chains and other nonstandard cases illustrate results     

Stable estimation of location parameter -nonasymptotic approach

Let X₁:, X₂:, …, X_n be iidrv's with cdf F?, F?(x)=F (x-θ), R. Let T be an equivariant median-unbiased estimator of θ. Let πε(F)={G = (1 -ε) F+εH, H any cdf} and let M(G, T) be a median of T if X₁ has cdf G. The oscillation of the bias of T, defined as

Bε(T)=sup (M(G₁ T) :G₁,G₂:∈π_σ:(F)} ,is considered and the estimator with the smallest B$epsi;(T) is explicitly constructed     

On selecting populations better than a control: weibull populations case

Consider k( k ≥ 1) independent Weibull populations and a control population which is also Weibull. The problem of identifying which of these k populations are better than the control using shape parameter as a criterion is considered. We allow the possibility of making at most m(0 ≤ m < k) incorrect identifications of better populations. This allowance results in significant savings in sample size. Procedures based on simple linear unbiased estimators of the reciprocal of the shape parameters of these populations are proposed. These procedures can be used for both complete and Type II-censored samples. A related problem of confidence intervals for the ratio of ordered shape parameters is also considered. Monte Carlo simulations as well as both chi-square and normal approximations to the solutions are obtained.     

Some Further Issues Concerning Likelihood Inference for Left Truncated and Right Censored Lognormal Data

The maximum likelihood estimates (MLEs) of the parameters of a two-parameter lognormal distribution with left truncation and right censoring are developed through the Expectation Maximization (EM) algorithm. For comparative purpose, the MLEs are also obtained by the Newton–Raphson method. The asymptotic variance-covariance matrix of the MLEs is obtained by using the missing information principle, under the EM framework. Then, using asymptotic normality of the MLEs, asymptotic confidence intervals for the parameters are constructed. Asymptotic confidence intervals are also obtained using the estimated variance of the MLEs by the observed information matrix, and by using parametric bootstrap technique. Different confidence intervals are then compared in terms of coverage probabilities, through a Monte Carlo simulation study. A prediction problem concerning the future lifetime of a right censored unit is also considered. A numerical example is given to illustrate all the inferential methods developed here.     

A test of the mean square error criterion for shrinkage estimators

A test for choosing between a shrinkage estimator and the least squares estimator is described and a central-F approximation to the test statistic is considered. An example from the literature was analysed using the test procedure proposed here. The power of the test was studied by means of simulation.     

Missing data estimators in the general linear model: an evaluation of simulated data as an experimental design

Previous simulations have reported second order missing data estimators to be superior to the more straightforward first order procedures such as mean value replacement. These simulations however were based on deterministic comparisonsbetween regression criteria even though simulated sampling is a random procedure. In this paper a simulation structured asan experimental design allows statistical testing of the various missing data estimators for the various regression criteria as well as different regression specifications. Our results indicate that although no missing data estimator is globally best many of the computationally simpler first order methods perform as well as the more expensive higher order estimators, contrary to some previous findings.     

The small sample performance of estimators of the standard errors of structural equation models

In this paper, we compare five asymptotically, under a correctly specified likelihood, equivalent estimators of the standard errors for parameters in structural equation models. The estimators are evaluated under different conditions regarding (i) sample size, varying between N=50 and 3200, (ii) distributional assumption of the latent variables and the disturbance terms, namely normal, and heavy tailed (t), and (iii) the complexity of the model. For the assessment of the five estimators we use overall performance, relative bias, MSE and coverage of confidence intervals. The analysis reveals substantial differences in the performance of the five asymptotically equal estimators. Most diversity was found for t distributed, i.e. heavy tailed, data.     

Optimal sample size allocation for multi-level stress testing with Weibull regression under Type-II censoring

We discuss the optimal allocation problem in a multi-level stress test with Type-II censoring and Weibull (extreme value) regression model. We derive the maximum-likelihood estimators and their asymptotic variance–covariance matrix through the Fisher information. Four optimality criteria are used to discuss the optimal allocation problem. Optimal allocation of units, both exactly for small sample sizes and asymptotically for large sample sizes, for two- and four-stress-level situations are determined numerically. Conclusions and discussions are provided based on the numerical studies.