## Abstract

The paper of T. L. Lai summarizes recent developments in several topics related to sequential analysis. Undoubtedly, the field of sequential statistics looks

very different today than it did only a decade ago. The author’s impressive contribution to the field has much to do with this fact. Naturally, the choice of

topics and results is biased towards those which involve the author’s contributions. I will take the same liberty and remark only on the topic of optimality in the context of sequential change-point detection.

The two main approaches for proving optimality rely either on decision theory, or on the construction of a lower bound and the demonstration that a given

procedure attains the lower bound (or at least approaches it). The first approach

has been successfully applied in the setting of independent observation and simple hypothesis. In more complex situations, only the second approach seems to be working. Indeed, using inequality (3.8) of his paper, Lai was able to demonstrate the asymptotic optimality for detecting changes in the important setting of state-space models.

It should be noted, however, that inequality (3.8) provides only a first-order

asymptotic lower bound. Unfortunately, the resolution of first-order asymptotics

is very limited. Indeed, in this resolution one cannot distinguish between the

performance of most reasonable candidate procedures (like the cusum, Shiryayev Roberts, EWMA, etc.). Only when a higher-order asymptotic expansion is considered can we hope to be able to rank the different procedures.

In these notes we attempt to extend Lai’s results and in the direction of

addressing this drawback. Theorem 1 of the next section formulate an alternative lower bound, motivated by the known optimality properties of the ShiryayevRoberts procedure. This lower bound is demonstrated in Section 3 in the simple setting of a shift of a normal mean. The proof of the theorem is given in an appendix.

very different today than it did only a decade ago. The author’s impressive contribution to the field has much to do with this fact. Naturally, the choice of

topics and results is biased towards those which involve the author’s contributions. I will take the same liberty and remark only on the topic of optimality in the context of sequential change-point detection.

The two main approaches for proving optimality rely either on decision theory, or on the construction of a lower bound and the demonstration that a given

procedure attains the lower bound (or at least approaches it). The first approach

has been successfully applied in the setting of independent observation and simple hypothesis. In more complex situations, only the second approach seems to be working. Indeed, using inequality (3.8) of his paper, Lai was able to demonstrate the asymptotic optimality for detecting changes in the important setting of state-space models.

It should be noted, however, that inequality (3.8) provides only a first-order

asymptotic lower bound. Unfortunately, the resolution of first-order asymptotics

is very limited. Indeed, in this resolution one cannot distinguish between the

performance of most reasonable candidate procedures (like the cusum, Shiryayev Roberts, EWMA, etc.). Only when a higher-order asymptotic expansion is considered can we hope to be able to rank the different procedures.

In these notes we attempt to extend Lai’s results and in the direction of

addressing this drawback. Theorem 1 of the next section formulate an alternative lower bound, motivated by the known optimality properties of the ShiryayevRoberts procedure. This lower bound is demonstrated in Section 3 in the simple setting of a shift of a normal mean. The proof of the theorem is given in an appendix.

Original language | American English |
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Pages (from-to) | 382-388. |

Journal | Statistica Sinica |

Volume | 11 |

Issue number | 2 |

State | Published - 2001 |