David H. Young

Bias correction for a generalized log-gamma regression model

A regression model is considered in which the response variable has the generalized log-gamma distribution. Bias approximations for the maximum likelihood estimators of the regression coefficients and scale parameter are presented. The estimator of the scale parameter has a negative bias, which becomes increasingly marked as the number of regressor variates increases. A bias-corrected estimator is proposed that has improved mean squared error properties provided there is at least one regressor variate. Approximations to the percentiles of the unconditional distributions of pivotal random variables used for statistical inference for the parameters in the log-gamma regression model are proposed and evaluated for the normal error and Type I extreme-value error models. The results suggest that bias correction for the estimate of the scale parameter will be important in small samples for all densities in the log-gamma family.

Estimators for the correlation coefficient in a bivariate exponential distribution

The properties of some estimators for the correlation coefficient in a bivariate exponential distribution are considered. The estimators include the maximum likelihood estimator, the method of moments estimator, an estimator based on the sample correlation coefficient and two bias reduction estimators. Results for the means and mean square errors of the estimators are given for sample sizes n =10, 20 as obtained from a Monte Carlo investigation. Finally, some asymptotic efficiency results are presented.

The Power of Approximate Tests for the Regression Coefficients in a Gamma Regression Model

A gamma regression model which has wide application in life-testing and analysis of point processes is considered. Approximate statistical tests for the regression coefficients based on maximum likelihood and weighted least squares fits of the model are considered. The power and significance level properties of the tests are evaluated for a model with a single regressor variate when the underlying gamma distributions have the same shape parameter. The tests give good control over the actual significance levels for all values of the shape parameter. The test using the maximum likelihood estimators has much better power than the test using the weighted least squares estimators for small values of the shape parameter but the power differences are negligible when the shape parameter exceeds five.

Moment properties of estimators for a type 1 extreme value regression model

A regression model is considered in which the response variable has a type 1 extreme-value distribution for smallest values. Bias approximations for the maximum likelihood estimators are pivm and a bias reduction estimator for the scale parameter is proposed. The small sample moment properties of the maximum likelihood estimators are compared with the properties of the ordinary least squares estimators and the best linear unbiased estimators based on order statistics for grouped data.

Bias corrected pearson estimators for the shape parameter in gamma regression

A family of estimators based on the Pearson statistic is considered for estimation of the shape parameter in gamma regression models. Approximations to the biases of the estimators are developed and used to define a bias corrected estimator for the case of a logarithmic link for the means. The new estimator is shown to have markedly better variance and mean square error properties than existing estimators based on the Pearson statistic, in small to moderate size samples.

Improved deviance goodness of fit statistics for a gamma regression model

A gamma regression model with an exponential link function for the means Is considered. Moment properties of the deviance statistics based on maximum likelihood and weighted least squares fits are used to define modified deviance statistics which provide alternative global goodness of fit tests. The null distribution properties of the deviances and modified deviances are compared with those of the approximating chi-square distribution and It is shown that the use of the modified deviances gives much better control over the significance levels of the tests.

Distribution free slippage tests for populations following a lehmann model

Distribution free procedures using the extremes of sample mean rank statistics based on exponential scores are considered for testing whether k continuous populations have identical c.d.f.s against a Lehmann slippage alternative. Tables of critical values of the test statistics are given and various approximations to them are presented and evaluated. Monte Carlo estimates of the power and probability of correct decision are given under a slippage alternative and contrasted with those obtained when ranks are used instead of exponential scores and when a parametric test is used when the underlying distributions are exponential. Finally, some asymptotic efficiency results are developed.

Bias of maximum likelihood estimators of the response standard deviation in regression

The bias of maximum likelihood estimators of the standard deviation of the response in location/scale regression models is considered. Results are obtained for a very wide family of densities for the response variable. These are used to propose point estimators with improved mean square error properties and to demonstrate the importance of bias correction in statistical inference when samples are moderately small.

Estimation efficiency of grouped and censored observation schemes for a generalised regression model

A multiple regression model is considered in which the density of the response variable is a member of a very wide family which includes many well-known distributions. Schemes of observation in which the response observations are grouped or type 1 right censored are examined. Results on the asymptotic variance efficiencies of the maximum likelihood estimators of the regression coefficients and standard deviation of the error distribution are presented for the two schemes.

Testing exponential regression against a gamma alternative

The problem of testing for exponential regression against a gamma alternative is considered, when it is assumed that the link function for the means is correctly specified. Tests based on the deviance, likelihood ratio, Wald and related statistics are exami-ned, together with some modified tests which incorporate bias cor-rection. Simple chi-square approximations to the powers of the tests are proposed and evaluated by simulation for the case of a logarithmic link.