On some new advances in self-normalization approaches for inference on time series
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Statistical inference in time series analysis has been an important subject in various fields including climate science, economics, finance and industrial engineering among others. Numerous problems of research interest include statistical inference about unknown quantities, assessing structural stability and forecasting. These problems have been widely studied in the literature, but mainly for independent data, while in many applications involving time series data dependence is not unusual and in fact quite common. In this thesis, we incorporate serial dependence into the analysis by involving self-normalization in time series analysis. We start with the problem of testing whether there are change-points in a given time series. The method we propose does not require the number of change-points to be predefined, and thus is unsupervised. It does not require any tuning parameters and can be applied to a wide class to quantities of interest. The asymptotic distribution of the test statistic is studied and an approximation scheme is proposed to reduce testing procedure complexity. We then consider the problem of construction of confidence intervals, for which the conventional self-normalizer exhibits certain degrees of asymmetry when applied to quantities other than the mean. The method we propose provides a time-symmetric generalization to the conventional self-normalizer and leads to improved finite sample performance for quantities other than the mean.