摘要:

Supposex1andx2are observations on a bivariate normal distribution with known variance matrix Σ and unknown means vector (μ1, μ2). Suppose an interval estimate off(μ1, μ2), is desired with 1 – α confidence. Let an interval estimate be obtained as [minf(μ1, μ2), maxf(μ1, μ2)] subject to [(x1– μ1) (x2– μ2)] Σ−1[(x1– μ1)(x2– μ2)]′ ≤R2. It is shown that under certain conditions onf(μ1, μ2) and Σ,Rneed be taken only slightly larger thanx1(α), the value of a chi-variate on one degree of freedom exceeded with probabilityα.For example under the given conditions at a nominal .95 confidence one may useR= 2.06 rather than 1.96. Similar results are obtained when variances must be estimated for some special assumptions about Σ.

ISSN:0162-1459    

DOI:10.1080/01621459.1963.10500871

出版商:Taylor & Francis Group    

年代:1963

数据来源: Taylor

摘要:

Probabilistic methods in the evaluation of legal evidence have not yet been employed to any large degree and there is a dearth of potentially useful probabilities that might be estimated from empirical data. The role which mathematical probability can and does play in the court room is illustrated with cases involving combinations of circumstances, forgery, criminal identification, establishment of nonpaternity, and the use of coded messages. ‘Beyond a mathematical doubt’ should be more convincing than ‘beyond a reasonable doubt.’

ISSN:0162-1459    

DOI:10.1080/01621459.1963.10500872

出版商:Taylor & Francis Group    

年代:1963

数据来源: Taylor

摘要:

The likelihood, sufficiency, and conditionality principles proposed by Allan Birnbaum are discussed in the context of two examples that include in their specifications a position relationship between the variable and the parameter. The discussion leads to the suggestion that the likelihood principle is too strong and that the sufficiency and conditionality principles should be used only when the necessary analysis is consistent with the position relationship between variable and parameter.

ISSN:0162-1459    

DOI:10.1080/01621459.1963.10500873

出版商:Taylor & Francis Group    

年代:1963

数据来源: Taylor

摘要:

Previous use of credit appears to be a significant discriminator of subsequent durable goods purchasing behavior and still more important as a discriminator of the financing method used by durable good purchasers, according to an analysis of reinterview data from the 1957 and 1958 Surveys of Consumer Finances. The relation of liquid asset holdings to subsequent purchasing behavior seems to be related to the type of such asset holdings, to previous and to current credit use. These and other relationships with respect to intentions to purchase suggest that surveys more pointedly focussed on the specific questions related to consumer durable goods purchasing behavior could yield better results than those yet attained.

ISSN:0162-1459    

DOI:10.1080/01621459.1963.10500874

出版商:Taylor & Francis Group    

年代:1963

数据来源: Taylor

摘要:

When estimating a location or a scale parameter, an invariant estimate has the desirable property of adjusting itself for changes in the unit of measurement. The mechanism by which it does this is clear. Certain non-invariant estimates also adjust themselves for changes in the unit of measurement. The mechanism by which they do this is illustrated.

ISSN:0162-1459    

DOI:10.1080/01621459.1963.10500875

出版商:Taylor & Francis Group    

年代:1963

数据来源: Taylor

摘要:

Multivariate analysis of variance tests are developed for situations where the underlying covariance structure is uniform (equal variances and covariances) in terms of statistics analogous to Hotelling'sT2andT20. Extensions are made to several populations as well as to blocks of uniform covariance matrices. A special case, which is typical of the test procedures given here, is the problem of testing whether the mean vector of a bivariate normal distribution is equal to some specified vector based onnobservations. The uniform structure assumes that the two unknown variances are equal though the correlation is arbitrary. The testing procedure leads to a statisticUwhich is distributed as the sum of two independentF1,n–1ratios which may be contrasted with theT2statistic proportional toF2,n–2used in the situation where the variances are not assumed equal.

ISSN:0162-1459    

DOI:10.1080/01621459.1963.10500876

出版商:Taylor & Francis Group    

年代:1963

数据来源: Taylor

摘要:

Lehmann and Scheffé have shown how to construct uniformly most powerful unbiased tests of certain hypotheses when the assumed distributions belong to an “exponential family.” The present paper is concerned with particular cases which arise from independent gamma variates with scale parameters θ1and θ2. Conditional distributions are given which are appropriate for testing γ = γ0, where γ =c1/θ1+c2/θ2. As an application, suppose that a series system has two dissimilar components with expected lives θ1and θ2. When component failures are exponentially distributed, so are system failures, the mean being 1/(1/θ1+1/θ2). From independent estimates of θ1and θ2confidence limits can be found for this mean, or for the probability of successful system operation up to any fixed “mission time.” With appropriate restrictions, more general distributions including the Weibull can also be treated.

ISSN:0162-1459    

DOI:10.1080/01621459.1963.10500877

出版商:Taylor & Francis Group    

年代:1963

数据来源: Taylor

摘要:

The Helmert transformation of a set of observations {xj},j= 1, 2, …,Nis applied to give a direct approach to the problem of boundary determination in the case of the χ2-test for location of the normal distribution, i.e. the choice of that set of boundaries which maximizes the power of the test. It is found that no significant increase in power can be achieved by taking a number of classes greater than 20; and even a number as small as 12 is sufficient. Moreover, a power of one-half is achieved for a number of classes much smaller than that required by the Mann-Wald approach. For a preassigned number of classes, symmetrically situated about the origin, it is found that the optimum partition corresponds to equal class-width of about 0.4 standard deviation with pooling of the terminal classes, and this is slightly more powerful than the equal class-probability partition.

ISSN:0162-1459    

DOI:10.1080/01621459.1963.10500878

出版商:Taylor & Francis Group    

年代:1963

数据来源: Taylor

摘要:

A published method for analyzing multiple 2×2 contingency tables arising in retrospective studies of disease is extended in application and form. Extensions of application include comparisons of age-adjusted death rates, life-table analyses, comparisons of two sets of quantal dosage-response data, and miscellaneous laboratory applications as appropriate. Extensions in form involve considering multiple contingency tables with arbitrarily many rows and/or columns, where rows and columns are orderable, and may even be on a continuous scale. The assignment of some score for each row or column is essential to use of the method. With scores assigned, a deviation of the sum of cross products from expectation, and its variance conditioned on all marginal totals, are computed for each table and a chi square is determined corresponding to the grand total of the deviations. For various specific instances and for various scoring procedures, the procedure extends or is equivalent to the asymptotic form of many known non-parametric techniques.

ISSN:0162-1459    

DOI:10.1080/01621459.1963.10500879

出版商:Taylor & Francis Group    

年代:1963

数据来源: Taylor

摘要:

Three non-parametric models are presented for describing the survival experience of patients with chronic disease. Model I assumes that two independent forces of mortality (disease specific and normal) act simultaneously on the total patient population. If after a number of intervals the observed survival experience approximates a normal level, a cure rate can be estimated and the survival pattern described according to Model II: ac-fraction subject only to normal mortality and a (1–c)-fraction subject to an additional mortality risk. If the observed survival appears to be stabilizing at a level somewhat lower than normal, the data fit Model III: a c′-fraction subject to a mortality risk somewhat higher than normal, and a (1–c′)-fraction subject to a high additional mortality risk. When there is no evidence of stabilization of interval survival rates, no cure rate can be estimated and only Model I is applicable. Models I and II can be considered as special cases of Model III.

ISSN:0162-1459    

DOI:10.1080/01621459.1963.10500880

出版商:Taylor & Francis Group    

年代:1963

数据来源: Taylor