Blockwise estimation of parameters under abrupt changes in the mean

Abstract

In this thesis we are dealing with the estimation of parameters under shifts in the mean. The results of this work are based on three articles.

The first main chapter of this thesis presents estimation methods for the LRD parameter under shifts in the mean. In the context of long range dependent (LRD) stochastic processes the main task is estimation of the Hurst parameter H, which describes the strength of dependence. When data are contaminated by level shifts ordinary estimators of H, such as the Geweke and Porter-Hudak (GPH) estimator, may fail to distinguish between LRD and structural changes, such as jumps in the mean. As a consequence, the estimator may suffer from positive bias and overestimate the intensity of the LRD. This fact is e.g. a major issue when testing for changes in the mean.

To overcome this problem, we propose to segregate the sample of size N into blocks and then to estimate H on each block separately. Estimates, calculated in different blocks, are then combined and a final estimate of the Hurst parameter is obtained. We investigate several possibilities of segregating the data and assess their performance in a simulation study.

One possibility is segregation into two blocks. The position at which the data are separated into two parts is either estimated using the Wilcoxon change-point test or chosen at any point, yielding estimates, which are combined by averaging. Another possibility is dividing the sequence of observations into many overlapping or non-overlapping blocks and estimating H by averaging estimates from these blocks. In the presence of one or even several jumps this procedure performs well in simulations. When dealing with processes with long memory and short range dependence, such as the fractionally integrated ARMA process (ARFIMA), the proposed estimators do not yield desirable results. Therefore, we follow an ARMA correction procedure and estimate the Hurst parameter in several recursive steps, using the overlapping or the non-overlapping blocks approach.

In the context of LRD we observe that segregation into many blocks improves the ordinary estimators of H considerably under abrupt changes in the mean. We follow this same idea of segregation to estimate the variance of independent or weakly dependent processes under level shifts. The second main chapter of this thesis deals with scale estimation under shifts in the mean.

When dealing with a few level shifts in finite samples we propose usage of the ordinary average of sample variances, obtained from many non-overlapping blocks. Under some conditions on the number of change-points and the number of blocks we prove strong consistency and asymptotic normality for independent data, where full asymptotic efficiency compared to the ordinary sample variance is shown. For weakly correlated processes we prove weak consistency of the blocks estimator. This estimator performs well when the number of level shifts is moderately low. In the presence of many level shifts even better results are obtained by an adaptive trimmed mean of the sample variances from non-overlapping blocks. The fraction of trimmed blockwise estimates is chosen adaptively, where extraordinary high sample variances are removed before calculating the average value. Even though this procedure is developed under the assumption of independence, it performs well also under weak dependence, e.g. when dealing with AR processes.

If the data are additionally contaminated by outliers the proposed estimators fail to estimate the variance properly, since they are not robust. Therefore, we investigate a modified version of the well-known median absolute deviation (MAD) to account for both sources of contamination - level shifts and outliers. The formula of the MAD involves the sample median, which is not a good estimator of location in the presence of level shifts. Our proposal is to calculate the sample median in non-overlapping blocks and to consider absolute differences involving blockwise medians instead of a single median calculated on the whole sample. In this way only some blocks are affected by level shifts and the resulting modified MAD is robust against outliers and level shifts simultaneously. We proved strong consistency and asymptotic normality for independent random variables under some conditions on the number of change-points and the number of blocks. The Bahadur representation of the proposed estimator is shown to be the same as in the case of the ordinary MAD, resulting in the same asymptotic variance. In a simulation study the modified MAD provides very good results. The proposed estimator performs well as compared to other robust methods, which are discussed for comparison, in many simulation scenarios.