  Traditional methods for inference in change point detection often rely on a
large number of observed data points and can be inaccurate in non-asymptotic
settings. With the rise of mobile health and digital phenotyping studies, where
patients are monitored through the use of smartphones or other digital devices,
change point detection is needed in non-asymptotic settings where it may be
important to identify behavioral changes that occur just days before an adverse
event such as relapse or suicide. Furthermore, analytical and computationally
efficient means of inference are necessary for the monitoring and online
analysis of large-scale digital phenotyping cohorts. We extend the result for
asymptotic tail probabilities of the likelihood ratio test to the multivariate
change point detection setting, and demonstrate through simulation its
inaccuracy when the number of observed data points is not large. We propose a
non-asymptotic approach for inference on the likelihood ratio test, and compare
the efficiency of this estimated p-value to the popular empirical p-value
obtained through simulation of the null distribution. The accuracy and power of
this approach relative to competing methods is demonstrated through simulation
and through the detection of a change point in the behavior of a patient with
schizophrenia in the week prior to relapse.
