Modified sequential change point procedures based on estimating, functions

A large class of sequential change point tests are based on estimating functions
where estimation is computationally efficient as (possibly numeric) optimization
is restricted to an initial estimation. This includes examples as diverse as mean
changes, linear or non-linear autoregressive and binary models. While the stan-
dard cumulative-sum-detector (CUSUM) has recently been considered in this
general setup, we consider several modifications that have faster detection rates
in particular if changes do occur late in the monitoring period. More presicely,
we use three different types of detector statistics based on partial sums of a mon-
itoring function, namely the modified moving-sum-statistic (mMOSUM), Page’s
cumulative-sum-statistic (Page-CUSUM) and the standard moving-sum-statistic
(MOSUM). The statistics only differ in the number of observations included in
the partial sum. The mMOSUM uses a bandwidth parameter which multiplica-
tively scales the lower bound of the moving sum. The MOSUM uses a constant
bandwidth parameter, while Page-CUSUM chooses the maximum over all pos-
sible lower bounds for the partial sums. So far, the first two schemes have only
been studied in a linear model, the MOSUM only for a mean change.
We develop the asymptotics under the null hypothesis and alternatives under
mild regularity conditions for each test statistic, which include the existing the-
ory but also many new examples. In a simulation study we compare all four
types of test procedures in terms of their size, power and run length. Addition-
ally we illustrate their behavior by applications to exchange rate data as well as
the Boston homicide data.