We introduce and study the behavior of estimators of changes in the mean value of a sequence of independent random variables in the case of so called epidemic alternatives which is one of the variants of the change point problem. The consistency and the limit distribution of the estimators developed for this situation are shown. Moreover, the classical estimators used for `at most change' are examined for the studied situation.

A number of papers has been published on the estimation problem in location models with abrupt changes (e.g., Cs" orgő and Horváth (1996)). In the present paper we focus on estimators in location models with gradual changes. Estimators of the parameters are proposed and studied. It appears that the limit behavior (both the rate of consistency and limit distribution) of the estimators of the change point in location models with abrupt changes and gradual changes differ substantially.

We investigate estimators of the asymptotic variance ${\sigma }^2$ of a $d$–dimensional stationary point process $\Psi$ which can be observed in convex and compact sampling window $W_n=nW$. Asymptotic variance of $\Psi$ is defined by the asymptotic relation $Var\left(\Psi \left(W_n\right)\right)\sim {\sigma }^2\left|W_n\right|$ (as $n\to \infty$) and its existence is guaranteed whenever the corresponding reduced covariance measure ${\gamma }_{\mathrm{red}}^{\left(2\right)}(·)$ has finite total variation. The three estimators discussed in the paper are the kernel estimator, the estimator based on the second order intesity of the point process and the...