On relationship between score functions and extremal index

Score functions like the logarithmic derivative of a probability density function and the t-score are applied in many control problems of dynamic systems. However, their estimation by samples of moderate sizes may require nonparametric estimation of the density and its derivative that is difficult. To overcome this problem we attract a new random variable that is equal to inter-cluster times T₁(u) of the underlying process {X_n} normalized by its tail function F(u)̄. A cluster contains consecutive exceedances of the process over a threshold u and inter-cluster issues of the process run under u. We found that the the logarithmic derivative of the F(u)̄T₁(u) may be approximated by extremal index. The latter can be easily estimated by one of the nonparametric estimators. Another aim is to find a relationship between score functions of a marginal variable X_t generating the process and of the normalized T₁(u). The first score function carries the information about distribution, while the other one about dependence structure. We also consider Fisher score that is the gradient, with respect to some parameter, of the logarithm of the likelihood function. The relationships are demonstrated on ARMAX, moving maxima, moving average and AR(1) processes to illustrate this methodology.