This thesis investigates variable stage size multistage hypothesis testing in three different contexts, each building on the previous.
\r\n\r\nWe 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.
\r\n\r\nNext 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.
\r\n\r\nFinally 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.
", "doi": "10.7907/GJAW-6P82", "publication_date": "2004", "thesis_type": "phd", "thesis_year": "2004" }, { "id": "thesis:2231", "collection": "thesis", "collection_id": "2231", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-05292003-133431", "primary_object_url": { "basename": "mei_thesis.pdf", "content": "final", "filesize": 411808, "license": "other", "mime_type": "application/pdf", "url": "/2231/1/mei_thesis.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "Asymptotically Optimal Methods for Sequential Change-Point Detection", "author": [ { "family_name": "Mei", "given_name": "Yajun", "orcid": "0000-0002-1015-990X", "clpid": "Mei-Yajun" } ], "thesis_advisor": [ { "family_name": "Lorden", "given_name": "Gary A.", "clpid": "Lorden-G-A" } ], "thesis_committee": [ { "family_name": "Lorden", "given_name": "Gary A.", "clpid": "Lorden-G-A" }, { "family_name": "Wales", "given_name": "David B.", "clpid": "Wales-D-B" }, { "family_name": "Candes", "given_name": "Emmanuel J.", "clpid": "Candes-E-J" }, { "family_name": "Sherman", "given_name": "Robert P.", "clpid": "Sherman-R-P" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "This thesis studies sequential change-point detection problems in different contexts. Our main results are as follows:
\r\n\r\n- 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.
\r\n\r\n- 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.
\r\n\r\n- 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.
\r\n\r\n- 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.
\r\n\r\n- 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.
", "doi": "10.7907/PY76-DM19", "publication_date": "2003", "thesis_type": "phd", "thesis_year": "2003" }, { "id": "thesis:6783", "collection": "thesis", "collection_id": "6783", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:01262012-103823681", "primary_object_url": { "basename": "Pavelich_j_2001.pdf", "content": "final", "filesize": 25118003, "license": "other", "mime_type": "application/pdf", "url": "/6783/1/Pavelich_j_2001.pdf", "version": "v5.0.0" }, "type": "thesis", "title": "Commuting Equivalence Relations and Scales on Differentiable Functions", "author": [ { "family_name": "Pavelich", "given_name": "Janet Mary", "clpid": "Pavelich-Janet-Mary" } ], "thesis_advisor": [ { "family_name": "Lorden", "given_name": "Gary A.", "clpid": "Lorden-G-A" }, { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" } ], "thesis_committee": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "Clemens", "given_name": "John D.", "clpid": "Clemens-J-D" }, { "family_name": "Luxemburg", "given_name": "W. A. J.", "clpid": "Luxemburg-W-A-J" }, { "family_name": "Ramakrishnan", "given_name": "Dinakar", "orcid": "0000-0002-0159-087X", "clpid": "Ramakrishnan-D" }, { "family_name": "Lorden", "given_name": "Gary A.", "clpid": "Lorden-G-A" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "This work consists of two independent chapters:
\r\n\r\nThe 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 \u25e6 S = S \u25e6 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.
\r\n\r\nThe second chapter is motivated by the well known theorem of descriptive set theory that every \u041f^1_1 subset of a Polish (separable, completely metrizable) space admits a \u041f^1_1 scale. We construct a \u041f^1_1 scale on the set of differentiable functions with domain [0,1], which is a \u041f^1_1 subset of the Polish space C([0,1]) . This construction is based on the \u041f^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 (\u0192_n) by comparing it to the criterion that the sequence (\u0192'_n) converges.
\r\n", "doi": "10.7907/RHBN-ZN06", "publication_date": "2002", "thesis_type": "phd", "thesis_year": "2002" }, { "id": "thesis:6787", "collection": "thesis", "collection_id": "6787", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:01272012-142906804", "type": "thesis", "title": "Sequential fixed width confidence intervals", "author": [ { "family_name": "Asparouhov", "given_name": "Tihomir Zlatev", "clpid": "Asparouhov-T-Z" } ], "thesis_advisor": [ { "family_name": "Lorden", "given_name": "Gary A.", "clpid": "Lorden-G-A" } ], "thesis_committee": [ { "family_name": "Unknown", "given_name": "Unknown" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "We consider the problem of constructing confidence intervals of fixed width d and confidence level \u03b3 for the success probability p in Bernoulli trials. Algorithms are\r\ngiven 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\r\nprocedures are proposed that attain the asymptotic lower bounds and nearly attain the numerical lower bounds. Asymptotically optimal sequential and two-stage\r\nconfidence intervals of fixed width and confidence level are proposed for the mean in a general (non-Bernoulli) context.", "doi": "10.7907/0DZR-RK42", "publication_date": "2000", "thesis_type": "phd", "thesis_year": "2000" }, { "id": "thesis:17611", "collection": "thesis", "collection_id": "17611", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:08112025-214237411", "primary_object_url": { "basename": "Huffman_MD_1980.pdf", "content": "final", "filesize": 12867517, "license": "other", "mime_type": "application/pdf", "url": "/17611/1/Huffman_MD_1980.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "Efficient Approximate Solutions to the Kiefer-Weiss Problem", "author": [ { "family_name": "Huffman", "given_name": "Michael David", "clpid": "Huffman-Michael-David" } ], "thesis_advisor": [ { "family_name": "Lorden", "given_name": "Gary A.", "clpid": "Lorden-G-A" } ], "thesis_committee": [ { "family_name": "Unknown", "given_name": "Unknown" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "The problem is to decide on the basis of repeated independent\r\nobservations whether \u03b80 or \u03b81 is the true value of the parameter e\r\nof a Koopman-Darmois family of densities, where \u03b8 less than \u03b8 less than \u03b8. The\r\nprobability of falsely rejecting e0 is to be at most o:0, and that of\r\nfalsely rejecting \u03b81, at most \u03b11. Procedures are studied from the\r\npoint of view of minimizing the maximum (over \u03b8) expected number of\r\nobservations required when e is the true value of the parameter.
\r\n\r\nTwo types of tests are considered. The first, based on the\r\nwell-known sequential probability ratio test (SPRT), dictates after\r\neach observation whether to stop and ma.ke a decision, or whether to\r\ncontinue sampling. An explicit method is derived for determining a\r\ncombination of one-sided SPRT's, known as a 2-SPRT, which minimizes\r\nthe maximum expected number of observations to within o((n(\u03b10,\u03b11))1/2)\r\nas \u03b10 and \u03b11 go to o, where n(\u03b10,\u03b11) is the minimum of the maximum\r\nexpected sample size, taken over all procedures with error probabilities\r\nat most \u03b10 and \u03b11. The second test uses several stages of observations,\r\ndeciding whether to stop or continue only at the end of each stage.\r\nA procedure designed to \"do what a sequential test would do\", while\r\nusing at most three stages, is defined and shown to minimize the maximum\r\nexpected number of observations to within O((n(\u03b10,\u03b11))1/4(log n(\u03b10,\u03b11))3/2)\r\nas \u03b10 and \u03b11 go to 0.
\r\n\r\nFinally, using backward induction, optimal procedures were\r\ndeveloped on the computer for the case where the mean of an exponential\r\ndensity is tested. Then extensive computer calculations comparing\r\nthe proposed 2-SPRT with these optimal procedures show that the 2-SPRT\r\ncomes within 1% of minimizing the maximum expected sample size over a\r\nbroad range of error probability and parameter values.
", "doi": "10.7907/5nn5-n386", "publication_date": "1980", "thesis_type": "phd", "thesis_year": "1980" } ]