Solution and Analysis of the Fuzzy Volterra Integral Equations viaHomotopy Analysis Method

Homotopy Analysis Method(HAM)is semi-analytic method to solve the linear and nonlinear mathematical models which can be used to obtain the approximate solution.The HAM includes an auxiliary parameter,which is an efficient way to examine and analyze the accuracy of linear and nonlinear problems.The main aim of this work is to explore the approximate solutions of fuzzy Volterra integral equations(both linear and nonlinear)with a separable kernel via HAM.This method provides a reliable way to ensure the convergence of the approximation series.A new general form of HAM is presented and analyzed in the fuzzy domain.A qualitative convergence analysis based on the graphical method of a fuzzy HAM is discussed.The solutions sought by the proposed method show that the HAM is easy to implement and computationally quite attractive.Some solutions of fuzzy second kind Volterra integral equations are solved as numerical examples to show the potential of the method.The results also show that HAM provides an easy way to control and modify the convergence area in order to obtain accurate solutions.

On nonparametric change point estimator based on empirical characteristic functions

We propose a nonparametric change point estimator in the distributions of a sequence of independent observations in terms of the test statistics given by Huˇskov′a and Meintanis(2006) that are based on weighted empirical characteristic functions. The weight function ω(t; a) under consideration includes the two weight functions from Huˇskov′a and Meintanis(2006) plus the weight function used by Matteson and James(2014),where a is a tuning parameter. Under the local alternative hypothesis, we establish the consistency, convergence rate, and asymptotic distribution of this change point estimator which is the maxima of a two-side Brownian motion with a drift. Since the performance of the change point estimator depends on a in use, we thus propose an algorithm for choosing an appropriate value of a, denoted by a_s which is also justified. Our simulation study shows that the change point estimate obtained by using a_s has a satisfactory performance. We also apply our method to a real dataset.

Test for autocorrelation change in discretely observed Ornstein-Uhlenbeck processes driven by Lévy processes

In this paper, we consider the problem of testing for an autocorrelation change in discretely observed Ornstein-Uhlenbeck processes driven by Levy processes. For a test, we propose a class of test statistics constructed by an iterated cumulative sums of squares of the difference between two adjacent observations. It is shown that each of the test statistics weakly converges to the supremum of the square of a Brownian bridge. The test statistics are evaluated by some empirical results.

SOME RESULTS ON ESTIMATION OF THE TAIL INDEX OF A DISTRIBUTION

The author obtains the rate of strong convergence,mean squared error and optimal choice of the“smoothing parameter”(the sample fraction)of a tail index estimator which was proposed by the author from Pickands’estimator,and called modified Pickands’estimator.The similar results about Hill’s estimator are also obtained,which generalize the corresponding results in.Besides,some comparisons between Hill’s estimator and the modified Pickands’estimator are given.