Abstract
Abstract. Exponential family statistical distributions, including the well-known normal, binomial, Poisson, and exponential distributions, are overwhelmingly used in data analysis. In the presence of covariates, an exponential family distributional assumption for the response random variables results in a generalized linear model. However, it is rarely ensured that the parameters of the assumed distributions are stable through the entire duration of the data collection process. A failure of stability leads to nonsmoothness and nonlinearity in the physical processes that result in the data. In this paper, we propose testing for stability of parameters of exponential family distributions and generalized linear models. A rejection of the hypothesis of stable parameters leads to change detection. We derive the related likelihood ratio test statistic. We compare the performance of this test statistic to the popular normal distributional assumption dependent cumulative sum (Gaussian CUSUM) statistic in change detection problems. We study Atlantic tropical storms using the techniques developed here, so to understand whether the nature of these tropical storms has remained stable over the last few decades.
Highlights
One important way in which nonlinear structures may be present in data related to many physical and natural phenomena is by structural breaks and changes
The standard framework for applying such change detection techniques requires assuming that the order in which the sampled observations arrive is known, with the question of interest being whether the data generating process has remained stable over time
In order to generalize the scope of statistical change detection tools, in this paper we propose a variant of the sequential industrial monitoring framework, by considering the stability of the data generation process as a problem of detecting the time of the distributional change; in other words, we conduct a hypothesis test, and under the null hypothesis, the data generation process remains stable through the entire sampling time t = 1, . . ., n
Summary
One important way in which nonlinear structures may be present in data related to many physical and natural phenomena is by structural breaks and changes. The standard framework for applying such change detection techniques requires assuming that the order in which the sampled observations arrive is known, with the question of interest being whether the data generating process has remained stable over time. Statistical guarantees are typically expressed in terms of expected run length, i.e., how long it takes on average for a true change to be detected, when there is a control for the expected length of time before false signaling occurs These normality-based sequential monitoring and stability detection techniques originated from industrial process control (Page, 1954), they have far ranging applications at the present time.
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