{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/88658"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/88658","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"Multivariate sequential procedures for testing means","abstract":"We consider a multivariate situation with means µ₁,...,µp and covariance matrix Σ. We wish to derive sequential procedures for testing the hypothesis: H₀: (µ̲ - µ̲ₒ)Σ⁻¹(µ̲ - μ̲₀)’= λ₀²( usually zero) against the alternative: μ̲₀ H₁: (µ̲ - µ̲ₒ)Σ⁻¹µ̲ - μ̲₀)’=λ₁² both for the case where Σ is known (the sequential X²-test) and where Σ is unknown and must be estimated from the sample (the sequential T²-test). These sequential procedures should guarantee that the probability of accepting H₁ when H₀ is true is equal to a and the probability of accepting H₀ when H₁ is true is equal to β. For the case where Σ is known, λ₀² = 0 and λ₁² = λ², the test procedure is as follows: for a sample of n observations form the probability ratio: P1n/P0n = e-nλ²0F₁(p/2;nλ²Xn²/4) where p denotes the number of variables, nx[with horizontal bars above and below the x] denotes the vector of the sample means based on n observations, X²n = n(nx[with horizontal bars above and below the x] - μ̲₀) Σ⁻¹(nx[with horizontal bars above and below the x] - μ̲₀)’ and ₀F₁ (c;x) is a type of generalized hypergeometric function. a. If P1n/P0n ≤ β/(1-α), accept H₀; b. If P1n/P0n ≥ (1- β)/α, accept H₁; c. If β/(1-α) < P1n/P0n < (1-β)/α, continue sampling. For the case where Σ is unknown, the procedure is exactly the same except that the probability ratio is now: P1n/P0n = e⁻-nλ²/2 ₁F₁[n/2,p/2;nλ²Tn²/2(n-1+Tn²)] where Tn² = n(nx[with horizontal bars above and below the x] - μ̲₀)Sn⁻¹(nx[with horizontal bars above and below the x] - μ̲₀)' , Sndenotes the sample covariance matrix based on n observations and ₁F₁(a,c;x) is a confluent hypergeometric function. Procedures are also given for the case λ²₀ ≠ 0. Similar procedures are given to test the hypothesis: H₀ = (μ̲₁ - μ̲₂- δ ̲)Σ⁻¹(μ̲₁ - μ̲₂ - δ̲)’ = λ₀² (usually zero) against the alternative: H₁ = (μ̲₁ - μ̲₂- δ ̲)Σ⁻¹(μ̲₁ - μ̲₂ - δ̲)’ = λ₁² It is shown that these sequential procedures all exist in the sense that the risks of accepting H₀ when H₁ is true and of accepting H₁ when H₀ is true are approximately α and β respectively and that these sequential procedures terminate with probability unity. Some of these situation have been generalized to give simultaneous tests and the means and covariance matrix of a sample. No expressions yet exist for the OC or ASN functions although some conjectured values have been determined for the latter and suggest, in comparison with their corresponding fixed-sample tests, substantial reductions in the sample sizes required when either H₀ or H₁ is true. The general problem of tolerances is discussed and then some of these procedures are demonstrated with a numerical example drawn from the field of ballistic missiles. The determination of P1n/P0n is quite laborious for both the sequential X² - and T²-tests since it requires the evaluation of a hypergeometric function each time an observation is made. It would be better for each value of n, given p, α, β and λ² under H₁, to compute the values of X²n or T²n which would correspond to the boundaries of the tests indicated by β/(1-α) and (1-β)/α. Tables to facilitate both the sequential X²- and T²-tests are given for p = 2,3,...,9; λ² = 0.5, l.0, 2.0; α = β = 0.05 for n ranging from the minimum value necessary to reach a decision to 30, 45 and 60 for λ² = 0.5, 1.0, 2.0 respectively. These tables were prepared on the IBM 650 computer using the Newton-Raphson iterative procedure. Finally, a discussion is given for the hypergeometric function ₀F₁ (c;x) and a table given of this function for c = .5(.5)5.0 and x = .1(.1)1(1) 10(10)100(50)1000.","abstract_html":"We consider a multivariate situation with means µ₁,...,µ<sub>p</sub> and covariance matrix Σ. We wish to derive sequential procedures for testing the hypothesis: H₀: (µ̲ - µ̲ₒ)Σ⁻¹(µ̲ - μ̲₀)’= λ₀²( usually zero) against the alternative: μ̲₀ H₁: (µ̲ - µ̲ₒ)Σ⁻¹µ̲ - μ̲₀)’=λ₁² both for the case where Σ is known (the sequential X²-test) and where Σ is unknown and must be estimated from the sample (the sequential T²-test). These sequential procedures should guarantee that the probability of accepting H₁ when H₀ is true is equal to a and the probability of accepting H₀ when H₁ is true is equal to β. For the case where Σ is known, λ₀² = 0 and λ₁² = λ², the test procedure is as follows: for a sample of n observations form the probability ratio: P<sub>1n</sub>/P<sub>0n</sub> = e<sup>-nλ²</sup><sub>0</sub>F₁(p/2;nλ²X<sub>n²</sub>/4) where p denotes the number of variables, <sub>n</sub>x[with horizontal bars above and below the x] denotes the vector of the sample means based on n observations, X²<sub>n</sub> = n(<sub>n</sub>x[with horizontal bars above and below the x] - μ̲₀) Σ⁻¹(<sub>n</sub>x[with horizontal bars above and below the x] - μ̲₀)’ and ₀F₁ (c;x) is a type of generalized hypergeometric function. a. If P<sub>1n</sub>/P<sub>0n</sub> ≤ β/(1-α), accept H₀; b. If P<sub>1n</sub>/P<sub>0n</sub> ≥ (1- β)/α, accept H₁; c. If β/(1-α) < P<sub>1n</sub>/P<sub>0n</sub> < (1-β)/α, continue sampling. For the case where Σ is unknown, the procedure is exactly the same except that the probability ratio is now: P<sub>1n</sub>/P<sub>0n</sub> = e⁻<sup>-nλ²/2</sup> ₁F₁[n/2,p/2;nλ²T<sub>n</sub>²/2(n-1+T<sub>n</sub>²)] where T<sub>n</sub>² = n(<sub>n</sub>x[with horizontal bars above and below the x] - μ̲₀)S<sub>n</n>⁻¹(<sub>n</sub>x[with horizontal bars above and below the x] - μ̲₀)' , S<sub>n</sub>denotes the sample covariance matrix based on n observations and ₁F₁(a,c;x) is a confluent hypergeometric function. Procedures are also given for the case λ²₀ ≠ 0. Similar procedures are given to test the hypothesis: H₀ = (μ̲₁ - μ̲₂- δ ̲)Σ⁻¹(μ̲₁ - μ̲₂ - δ̲)’ = λ₀² (usually zero) against the alternative: H₁ = (μ̲₁ - μ̲₂- δ ̲)Σ⁻¹(μ̲₁ - μ̲₂ - δ̲)’ = λ₁² It is shown that these sequential procedures all exist in the sense that the risks of accepting H₀ when H₁ is true and of accepting H₁ when H₀ is true are approximately α and β respectively and that these sequential procedures terminate with probability unity. Some of these situation have been generalized to give simultaneous tests and the means and covariance matrix of a sample. No expressions yet exist for the OC or ASN functions although some conjectured values have been determined for the latter and suggest, in comparison with their corresponding fixed-sample tests, substantial reductions in the sample sizes required when either H₀ or H₁ is true. The general problem of tolerances is discussed and then some of these procedures are demonstrated with a numerical example drawn from the field of ballistic missiles. The determination of P<sub>1n</sub>/P<sub>0n</sub> is quite laborious for both the sequential X² - and T²-tests since it requires the evaluation of a hypergeometric function each time an observation is made. It would be better for each value of n, given p, α, β and λ² under H₁, to compute the values of X²<sub>n</sub> or T²<sub>n</sub> which would correspond to the boundaries of the tests indicated by β/(1-α) and (1-β)/α. Tables to facilitate both the sequential X²- and T²-tests are given for p = 2,3,...,9; λ² = 0.5, l.0, 2.0; α = β = 0.05 for n ranging from the minimum value necessary to reach a decision to 30, 45 and 60 for λ² = 0.5, 1.0, 2.0 respectively. These tables were prepared on the IBM 650 computer using the Newton-Raphson iterative procedure. Finally, a discussion is given for the hypergeometric function ₀F₁ (c;x) and a table given of this function for c = .5(.5)5.0 and x = .1(.1)1(1) 10(10)100(50)1000.","abstract_has_math":false,"creators":["Jackson, James Edward"],"institution":"Virginia Polytechnic Institute","degree_name":"Doctor of Philosophy","degree_level":"doctoral","degree_discipline":null,"degree_department":"Statistics","school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":1959,"date_issued":"1959","date_published":"1959","updated_at":"2026-07-22T22:20:03Z","subjects":[],"languages":["en_US"],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/10919/88658","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.department","label":"Department","values":["Statistics"]},{"key":"dc:creator","label":"Author","values":["Jackson, James Edward"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2019-03-26T19:53:11Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2019-03-26T19:53:11Z"]},{"key":"dc:date.issued","label":"Date","values":["1959"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Polytechnic Institute"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation"]},{"key":"dc:type.dcmitype","label":"Dc Type Dcmitype","values":["Text"]},{"key":"thesis:degree_level","label":"Degree Level","values":["doctoral"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Virginia Polytechnic Institute"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en_US"]},{"key":"dc:rights","label":"Dc Rights","values":["In Copyright"]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://rightsstatements.org/vocab/InC/1.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/10919/88658"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["We consider a multivariate situation with means µ₁,...,µp and covariance matrix Σ. We wish to derive sequential procedures for testing the hypothesis: H₀: (µ̲ - µ̲ₒ)Σ⁻¹(µ̲ - μ̲₀)’= λ₀²( usually zero) against the alternative: μ̲₀ H₁: (µ̲ - µ̲ₒ)Σ⁻¹µ̲ - μ̲₀)’=λ₁² both for the case where Σ is known (the sequential X²-test) and where Σ is unknown and must be estimated from the sample (the sequential T²-test). These sequential procedures should guarantee that the probability of accepting H₁ when H₀ is true is equal to a and the probability of accepting H₀ when H₁ is true is equal to β. For the case where Σ is known, λ₀² = 0 and λ₁² = λ², the test procedure is as follows: for a sample of n observations form the probability ratio: P1n/P0n = e-nλ²0F₁(p/2;nλ²Xn²/4) where p denotes the number of variables, nx[with horizontal bars above and below the x] denotes the vector of the sample means based on n observations, X²n = n(nx[with horizontal bars above and below the x] - μ̲₀) Σ⁻¹(nx[with horizontal bars above and below the x] - μ̲₀)’ and ₀F₁ (c;x) is a type of generalized hypergeometric function. a. If P1n/P0n ≤ β/(1-α), accept H₀; b. If P1n/P0n ≥ (1- β)/α, accept H₁; c. If β/(1-α) < P1n/P0n < (1-β)/α, continue sampling. For the case where Σ is unknown, the procedure is exactly the same except that the probability ratio is now: P1n/P0n = e⁻-nλ²/2 ₁F₁[n/2,p/2;nλ²Tn²/2(n-1+Tn²)] where Tn² = n(nx[with horizontal bars above and below the x] - μ̲₀)Sn⁻¹(nx[with horizontal bars above and below the x] - μ̲₀)' , Sndenotes the sample covariance matrix based on n observations and ₁F₁(a,c;x) is a confluent hypergeometric function. Procedures are also given for the case λ²₀ ≠ 0. Similar procedures are given to test the hypothesis: H₀ = (μ̲₁ - μ̲₂- δ ̲)Σ⁻¹(μ̲₁ - μ̲₂ - δ̲)’ = λ₀² (usually zero) against the alternative: H₁ = (μ̲₁ - μ̲₂- δ ̲)Σ⁻¹(μ̲₁ - μ̲₂ - δ̲)’ = λ₁² It is shown that these sequential procedures all exist in the sense that the risks of accepting H₀ when H₁ is true and of accepting H₁ when H₀ is true are approximately α and β respectively and that these sequential procedures terminate with probability unity. Some of these situation have been generalized to give simultaneous tests and the means and covariance matrix of a sample. No expressions yet exist for the OC or ASN functions although some conjectured values have been determined for the latter and suggest, in comparison with their corresponding fixed-sample tests, substantial reductions in the sample sizes required when either H₀ or H₁ is true. The general problem of tolerances is discussed and then some of these procedures are demonstrated with a numerical example drawn from the field of ballistic missiles. The determination of P1n/P0n is quite laborious for both the sequential X² - and T²-tests since it requires the evaluation of a hypergeometric function each time an observation is made. It would be better for each value of n, given p, α, β and λ² under H₁, to compute the values of X²n or T²n which would correspond to the boundaries of the tests indicated by β/(1-α) and (1-β)/α. Tables to facilitate both the sequential X²- and T²-tests are given for p = 2,3,...,9; λ² = 0.5, l.0, 2.0; α = β = 0.05 for n ranging from the minimum value necessary to reach a decision to 30, 45 and 60 for λ² = 0.5, 1.0, 2.0 respectively. These tables were prepared on the IBM 650 computer using the Newton-Raphson iterative procedure. Finally, a discussion is given for the hypergeometric function ₀F₁ (c;x) and a table given of this function for c = .5(.5)5.0 and x = .1(.1)1(1) 10(10)100(50)1000."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Doctor of Philosophy"]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Multivariate sequential procedures for testing means"]}]}],"canonical_facts":{"dc:contributor.department":["Statistics"],"dc:creator":["Jackson, James Edward"],"dc:date.accessioned":["2019-03-26T19:53:11Z"],"dc:date.available":["2019-03-26T19:53:11Z"],"dc:date.issued":["1959"],"dc:description.abstract":["We consider a multivariate situation with means µ₁,...,µp and covariance matrix Σ. We wish to derive sequential procedures for testing the hypothesis: H₀: (µ̲ - µ̲ₒ)Σ⁻¹(µ̲ - μ̲₀)’= λ₀²( usually zero) against the alternative: μ̲₀ H₁: (µ̲ - µ̲ₒ)Σ⁻¹µ̲ - μ̲₀)’=λ₁² both for the case where Σ is known (the sequential X²-test) and where Σ is unknown and must be estimated from the sample (the sequential T²-test). These sequential procedures should guarantee that the probability of accepting H₁ when H₀ is true is equal to a and the probability of accepting H₀ when H₁ is true is equal to β. For the case where Σ is known, λ₀² = 0 and λ₁² = λ², the test procedure is as follows: for a sample of n observations form the probability ratio: P1n/P0n = e-nλ²0F₁(p/2;nλ²Xn²/4) where p denotes the number of variables, nx[with horizontal bars above and below the x] denotes the vector of the sample means based on n observations, X²n = n(nx[with horizontal bars above and below the x] - μ̲₀) Σ⁻¹(nx[with horizontal bars above and below the x] - μ̲₀)’ and ₀F₁ (c;x) is a type of generalized hypergeometric function. a. If P1n/P0n ≤ β/(1-α), accept H₀; b. If P1n/P0n ≥ (1- β)/α, accept H₁; c. If β/(1-α) < P1n/P0n < (1-β)/α, continue sampling. For the case where Σ is unknown, the procedure is exactly the same except that the probability ratio is now: P1n/P0n = e⁻-nλ²/2 ₁F₁[n/2,p/2;nλ²Tn²/2(n-1+Tn²)] where Tn² = n(nx[with horizontal bars above and below the x] - μ̲₀)Sn⁻¹(nx[with horizontal bars above and below the x] - μ̲₀)' , Sndenotes the sample covariance matrix based on n observations and ₁F₁(a,c;x) is a confluent hypergeometric function. Procedures are also given for the case λ²₀ ≠ 0. Similar procedures are given to test the hypothesis: H₀ = (μ̲₁ - μ̲₂- δ ̲)Σ⁻¹(μ̲₁ - μ̲₂ - δ̲)’ = λ₀² (usually zero) against the alternative: H₁ = (μ̲₁ - μ̲₂- δ ̲)Σ⁻¹(μ̲₁ - μ̲₂ - δ̲)’ = λ₁² It is shown that these sequential procedures all exist in the sense that the risks of accepting H₀ when H₁ is true and of accepting H₁ when H₀ is true are approximately α and β respectively and that these sequential procedures terminate with probability unity. Some of these situation have been generalized to give simultaneous tests and the means and covariance matrix of a sample. No expressions yet exist for the OC or ASN functions although some conjectured values have been determined for the latter and suggest, in comparison with their corresponding fixed-sample tests, substantial reductions in the sample sizes required when either H₀ or H₁ is true. The general problem of tolerances is discussed and then some of these procedures are demonstrated with a numerical example drawn from the field of ballistic missiles. The determination of P1n/P0n is quite laborious for both the sequential X² - and T²-tests since it requires the evaluation of a hypergeometric function each time an observation is made. It would be better for each value of n, given p, α, β and λ² under H₁, to compute the values of X²n or T²n which would correspond to the boundaries of the tests indicated by β/(1-α) and (1-β)/α. Tables to facilitate both the sequential X²- and T²-tests are given for p = 2,3,...,9; λ² = 0.5, l.0, 2.0; α = β = 0.05 for n ranging from the minimum value necessary to reach a decision to 30, 45 and 60 for λ² = 0.5, 1.0, 2.0 respectively. These tables were prepared on the IBM 650 computer using the Newton-Raphson iterative procedure. Finally, a discussion is given for the hypergeometric function ₀F₁ (c;x) and a table given of this function for c = .5(.5)5.0 and x = .1(.1)1(1) 10(10)100(50)1000."],"dc:description.degree":["Doctor of Philosophy"],"dc:format.mimetype":["application/pdf"],"dc:identifier.uri":["http://hdl.handle.net/10919/88658"],"dc:language.iso":["en_US"],"dc:publisher":["Virginia Polytechnic Institute"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:title":["Multivariate sequential procedures for testing means"],"dc:type":["Dissertation"],"dc:type.dcmitype":["Text"],"thesis:degree_level":["doctoral"],"thesis:degree_name":["Doctor of Philosophy"],"thesis:institution_name":["Virginia Polytechnic Institute"]},"updated_at":"2026-07-22T22:20:03Z"}