No abstract

A method for estimating the configurational (i.e., non-kinetic) part of the entropy of internal motion in complex molecules is introduced that does not assume any particular parametric form for the underlying probability density function. It is based on the nearest-neighbor (NN) distances of the points of a sample of internal molecular coordinates obtained by a computer simulation of a given molecule. As the method does not make any assumptions about the underlying potential energy function, it accounts fully for any anharmonicity of internal molecular motion. It provides an asymptotically unbiased and consistent estimate of the configurational part of the entropy of the internal degrees of freedom of the molecule. The NN method is illustrated by estimating the configurational entropy of internal rotation of capsaicin and two stereoisomers of tartaric acid, and by providing a much closer upper bound on the configurational entropy of internal rotation of a pentapeptide molecule than that obtained by the standard quasi-harmonic method. As a measure of dependence between any two internal molecular coordinates, a general coefficient of association based on the information-theoretic quantity of mutual information is proposed. Using NN estimates of this measure, statistical clustering procedures can be employed to group the coordinates into clusters of manageable dimensions and characterized by minimal dependence between coordinates belonging to different clusters.

In this paper we discuss some properties of the reversed hazard rate function. This function has been shown to be useful in the analysis of data in the presence of left censored observations. It is also natural in discussing lifetimes with reversed time scale. In fact, ordinary hazard rate functions are most useful for lifetimes, and reverse hazard rates are natural if the time scale is reversed. Mixing up these concepts can often, although not always, lead to anomalies. For example, one result gives that if the reversed hazard rate function is increasing, its interval of support must be (—∞, b) where b is finite. Consequently nonnegative random variables cannot have increasing reversed hazard rates. Because of this result some existing results in the literature on the reversed hazard rate ordering require modification.Reversed hazard rates are also important in the study of systems. Hazard rates have an affinity to series systems; reversed hazard rates seem more appropriate for studying parallel systems. Several results are given that demonstrate this. In studying systems, one problem is to relate derivatives of hazard rate functions and reversed hazard rate functions of systems to similar quantities for components. We give some results that address this. Finally, we carry out comparisons for k-out-of-n systems with respect to the reversed hazard rate ordering.

The methods are applied to real protein data of conformational angles.