Cramér-Rao lower bounds for estimators based on censored data

The Cramér-Rao lower bounds for the variances of unbiased estimators based on censored data are given. Useful techniques of evaluation are then derived for these lower bounds. Examples are given to illustrate these techniques. Small-sample comparisons are made between the resulting lower bounds, the variances of the best linear unbiased estimators, and the variances of unbiased esti-mators which are based on the maximum likelihood estimators.     

Pseudo-bayes estimation of multinomial proportions

In the simultaneous estimation of multinomial proportions, two estimators are developed which incorporate prior means and a prior parameter which reflects the accuracy of the prior means. These estimators possess substantially smaller risk than the standard estimator in a region of the parameter space and are much more robust than the conjugate Bayes estimator with respect to parameter values far from the prior mean. When vague prior information is available, these estimators and confidence regions derived from them appear to be attractive alternatives to the procedures based on the standard estimator.     

Tests for symmetry about an unknown value based on the empirical distribution function

A class of Kolmogorov-Smirnov and Cramér-von Mises type statistics for testing symmetry about an unknown value is described. These statistics are not distribution-free, however, and, indeed, are not readily amenable to calculation. A linear rank statistic analog of the first component of the Cramér-von Mises type statistic is investigated. Asymptotic non-null properties of these procedures in the normal case are studied, and an efficiency comparison of the Cramér-vonMises statistic, the linear rank statistic analog, the modified Wil-coxon statistic, and the likelihood ratio test is reported.     

Comparison of tests for bivariate symmetry versus location and/or scale alternatives

This paper studies the performance of tests which use a null hypothesis of bivariate symmetry and detect the broad class of location and/or scale alternatives . The conditionally distribution-free tests of Sen (1967) and Hollander (1971) and parametric tests related to those of Bell and Haller (1969) are compared in a Monte Carlo study which also includes a new conditionally distribution-free test.     

Maximum Value of Hotelling's T 2 Statistics Based on the Successive Differences Covariance Matrix Estimator

In statistical process control applications, the multivariate T ² control chart based on Hotelling's T ² statistic is useful for detecting the presence of special causes of variation. In particular, use of the T ² statistic based on the successive differences covariance matrix estimator has been shown to be very effective in detecting the presence of a sustained step or ramp shift in the mean vector. However, the exact distribution of this statistic is unknown. In this article, we derive the maximum value of the T ² statistic based on the successive differences covariance matrix estimator. This distributional property is crucial for calculating an approximate upper control limit of a T ² control chart based on successive differences, as described in Williams et al. (2006 Williams , J. D. , Woodall , W. H. , Birch , J. B. , Sullivan , J. H. ( 2006 ). On the distribution of T ² statistics based on successive differences . J. Qual. Technol. 38 : 217 – 229 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]).     

A four-moment approach and other practical solutions to the behrens-f1sher problem

Fixed sample size approximately similar tests for the Behrens-Fisher problem are studied and compared with various other tests suggested in current sttistical methodelogy texts. Several fourmoment approxiamtely similar tests are developed and offered as alternatives. These tests are shown to be good practical solutions which are easily implemented in practice.     

An approximation for analyzing a broad class of implicitly and explicitly defined estimators

An approximation is presented that can be used to gain insight into the characteristics – such as outlier sensitivity, bias, and variability – of a wide class of estimators, including maximum likelihood and least squares. The approximation relies on a convenient form for an arbitrary order Taylor expansion in a multivariate setting. The implicit function theorem can be used to construct the expansion when the estimator is not defined in closed form. We present several finite-sample and asymptotic properties of such Taylor expansions, which are useful in characterizing the difference between the estimator and the expansion.     

Testing for lack of fit in linear multiresponse models based on exact or near replicates

Three procedures for testing the adequacy of a proposed linear multiresponse regression model against unspecified general alternatives are considered. The model has an error structure with a matrix normal distribution which allows the vector of responses for a particular run to have an unknown covariance matrix while the responses for different runs are uncorrelated. Furthermore, each response variable may be modeled by a separate design matrix. Multivariate statistics corresponding to the classical univariate lack of fit and pure error sums of squares are defined and used to determine the multivariate lack of fit tests. A simulation study was performed to compare the power functions of the test procedures in the case of replication. Generalizations of the tests for the case in which there are no independent replicates on all responses are also presented.     

A modification of watson's statistic for goodness-of-fit

A modification of Watson's statistic is suggested for dealing with stochastically ordered alternatives in the goodness-of-fit setting.     

Option pricing for generalized distributions

The Black Scholes formula has been widely used to price financial instruments. The derivation of this formula is based on the assumption of lognormally distributed returns which is often in poor agreement with actual data. An option pricing formula based on the generalized beta of the second kind (GB2) is presented. This formula includes the Black Scholes formula as a special case and accommodates a wide variety of nonlognormally distributed returns. The sensitivity of option values to departures from the skewness and kurtosis associated with the lognormal distribution is investigated.