This thesis investigates variable stage size multistage hypothesis testing in three different contexts, each building on the previous.

We first consider the problem of sampling a random process in stages until it crosses a predetermined boundary at the end of a stage – first for Brownian motion and later for a sum of i.i.d. random variables. A multistage sampling procedure is derived and its properties are shown to be not only sufficient but also necessary for asymptotic optimality as the distance to the boundary goes to infinity.

Next we consider multistage testing of two simple hypotheses about the unknown parameter of an exponential family. Tests are derived, based on optimal multistage sampling procedures, and are shown to be asymptotically optimal.

Finally we consider multistage testing of two separated composite hypotheses about the unknown parameter of an exponential family. Tests are derived, based on optimal multistage tests of simple hypotheses, and are shown to be asymptotically optimal. Numerical simulations show marked improvement over group sequential sampling in both the simple and composite hypotheses contexts.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, } @phdthesis{10.7907/PY76-DM19, author = {Mei, Yajun}, title = {Asymptotically Optimal Methods for Sequential Change-Point Detection}, school = {California Institute of Technology}, year = {2003}, doi = {10.7907/PY76-DM19}, url = {https://resolver.caltech.edu/CaltechETD:etd-05292003-133431}, abstract = {This thesis studies sequential change-point detection problems in different contexts. Our main results are as follows:

- We present a new formulation of the problem of detecting a change of the parameter value in a one-parameter exponential family. Asymptotically optimal procedures are obtained.
- We propose a new and useful definition of “asymptotically optimal to first-order” procedures in change-point problems when both the pre-change distribution and the post-change distribution involve unknown parameters. In a general setting, we define such procedures and prove that they are asymptotically optimal.
- We develop asymptotic theory for sequential hypothesis testing and change-point problems in decentralized decision systems and prove the asymptotic optimality of our proposed procedures under certain conditions.
- We show that a published proof that the so-called modified Shiryayev-Roberts procedure is exactly optimal is incorrect. We also clarify the issues involved by both mathematical arguments and a simulation study. The correctness of the theorem remains in doubt.

- We develop asymptotic theory for sequential hypothesis testing and change-point problems in decentralized decision systems and prove the asymptotic optimality of our proposed procedures under certain conditions.

- We propose a new and useful definition of “asymptotically optimal to first-order” procedures in change-point problems when both the pre-change distribution and the post-change distribution involve unknown parameters. In a general setting, we define such procedures and prove that they are asymptotically optimal.

- We construct a simple counterexample to a conjecture of Pollak that states that certain procedures based on likelihood ratios are asymptotically optimal in change-point problems even for dependent observations.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Lorden, Gary A.}, } @phdthesis{10.7907/RHBN-ZN06, author = {Pavelich, Janet Mary}, title = {Commuting Equivalence Relations and Scales on Differentiable Functions}, school = {California Institute of Technology}, year = {2002}, doi = {10.7907/RHBN-ZN06}, url = {https://resolver.caltech.edu/CaltechTHESIS:01262012-103823681}, abstract = {This work consists of two independent chapters:

The first is a study of commuting countable Borel equivalence relations, where two equivalence relations R and 5 are said to commute if, as binary relations, they commute with respect to the composition operator , i.e., R ◦ S = S ◦ R. The primary problem considered is, to what extent does the complexity of E = R V S depend on the complexity of R and S , if R and S commute? This is considered both in the case where the underlying space supports no E-invariant probability measure, and the case where it supports at least one such measure. In the first case, the answer is ‘not very much’: any such aperiodic equivalence relation E can be written as R V S, where Rand 5 are smooth aperiodic. In the second case, we frame our study within the context of costs, a system of invariants for countable Borel equivalence relations with invariant probability measures, developed by G. Levitt [12] and D. Gaboriau [5]. One aspect of costs which is not well understood is the extent to which ‘commutativity’ within an equivalence relation (in a more general sense than the definition given above) trivializes its cost. We have shown that, under certain conditions, this is in fact the case. One of the consequences of these investigations is a new, elementary proof of the fact the group SL_2 (Z[^1_2]) is anti-treeable.

The second chapter is motivated by the well known theorem of descriptive set theory that every П^1_1 subset of a Polish (separable, completely metrizable) space admits a П^1_1 scale. We construct a П^1_1 scale on the set of differentiable functions with domain [0,1], which is a П^1_1 subset of the Polish space C([0,1]) . This construction is based on the П^1_1 rank of differentiable functions given by Kechris and Woodin in [4], and, like this rank, is meant to reflect the intrinsic nature of DIFF, and so give a ‘natural’ criterion for determining whether the uniform limit of differentiable functions is itself differentiable. We then attempt to further analyze this ‘scale criterion’ for a sequence of differentiable functions (ƒ_n) by comparing it to the criterion that the sequence (ƒ’_n) converges.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Lorden, Gary A.}, } @phdthesis{10.7907/0DZR-RK42, author = {Asparouhov, Tihomir Zlatev}, title = {Sequential fixed width confidence intervals}, school = {California Institute of Technology}, year = {2000}, doi = {10.7907/0DZR-RK42}, url = {https://resolver.caltech.edu/CaltechTHESIS:01272012-142906804}, abstract = {We consider the problem of constructing confidence intervals of fixed width d and confidence level γ for the success probability p in Bernoulli trials. Algorithms are given for calculating numerical lower bounds on the average expected sample size required and an asymptotic lower bound is obtained as d -> 0. Sequential and two-stage procedures are proposed that attain the asymptotic lower bounds and nearly attain the numerical lower bounds. Asymptotically optimal sequential and two-stage confidence intervals of fixed width and confidence level are proposed for the mean in a general (non-Bernoulli) context.}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Lorden, Gary A.}, }