On existence of a change in mean of functional data

TL;DR

This paper introduces a new statistical test for detecting changes in the mean function of functional data, improving bias, power, and consistency over existing methods, with applications to temperature data.

Contribution

A novel test statistic for change point detection in functional data using an alternative covariance estimator that is unbiased and more powerful than previous methods.

Findings

Proposed test outperforms existing tests in simulation studies.

Test is asymptotically pivotal under the null hypothesis.

Method performs well on temperature data examples.

Abstract

Functional data often arise as sequential temporal observations over a continuous state-space. A set of functional data with a possible change in its structure may lead to a wrong conclusion if it is not taken in to account. So, sometimes, it is crucial to know about the existence of change point in a given sequence of functional data before doing any further statistical inference. We develop a new methodology to provide a test for detecting a change in the mean function of the corresponding data. To obtain the test statistic we provide an alternative estimator of the covariance kernel. The proposed estimator is asymptotically unbiased under the null hypothesis and, at the same time, has smaller amount of bias than that of the existing estimator. We show here that under the null hypothesis the proposed test statistic is pivotal asymptotically. Moreover, it is shown that under alternative hypothesis the test is consistent for large enough sample size. It is also found that the proposed test is more powerful than the available test procedure in the literature. From the extensive simulation studies we observe that the proposed test outperforms the existing one with a wide margin in power for moderate sample size. The developed methodology performs satisfactorily for the average daily temperature of central England and monthly global average anomaly of temperatures.