Application of the group-theoretical method to physical problems

The concept of the theory of continuous groups of transformations has attracted the attention of applied mathematicians and engineers to solve many physical problems in the engineering sciences. Three applications are presented in this paper. The first one is the problem of time-dependent vertical temperature distribution in a stagnant lake. Two cases have been considered for the forms of the water parameters, namely water density and thermal conductivity. The second application is the unsteady free-convective boundary-layer flow on a non-isothermal vertical flat plate. The third application is the study of the dispersion of gaseous pollutants in the presence of a temperature inversion. The results are found in closed form and the effect of parameters are discussed

An Exact Corrected Log-Likelihood Function for Cox's Proportional Hazards Model under Measurement Error and Some Extensions

This paper studies Cox's proportional hazards model under covariate measurement error. Nakamura's ["Biometrika" 77 (1990) 127] methodology of corrected log-likelihood will be applied to the so-called Breslow likelihood, which is, in the absence of measurement error, equivalent to partial likelihood. For a general error model with possibly heteroscedastic and non-normal additive measurement error, corrected estimators of the regression parameter as well as of the baseline hazard rate are obtained. The estimators proposed by Nakamura [Biometrics 48 (1992) 829], Kong "et al." ["Scand. J. Statist." 25 (1998) 573] and Kong & Gu ["Statistica Sinica" 9 (1999) 953] are re-established in the special cases considered there. This sheds new light on these estimators and justifies them as "exact" corrected score estimators. Finally, the method will be extended to some variants of the Cox model. Copyright Board of the Foundation of the Scandinavian Journal of Statistics 2004.