摘要:

General “successive substitutions” iteration equations are developed for obtaining estimates for finite mixtures of distributions from the exponential family. These, in general, correspond to relative maximums of the likelihood function. It is assumed that the number of distributions is known, and that the mixtures are from distributions of the same type, but with different parameter values. The particular equations for the Poisson, binomial, and exponential distributions are given, as well as examples of the results of the procedure for each distribution. From the examples tried, it was observed that the likelihood function increased at each iteration. Graphs of the asymptotic variances of the estimates are given, and two sampling experiments comparing estimates obtained by this scheme with moment estimates are also given.

ISSN:0162-1459    

DOI:10.1080/01621459.1969.10501071

出版商:Taylor & Francis Group    

年代:1969

数据来源: Taylor

摘要:

The concept of “aging,” or progressive shortening of an entity's residual lifetime, is discussed in terms of the entity's survival time distribution. Quantities defined to describe the aging phenomenon include the “specific aging factor,” “hazard rate,” “hazard rate average,” and “mean residual lifetime.” A set of seven criteria for aging is established, based on these quantities, and a chain of implications among the criteria is developed. The hazard rate average and mean residual lifetime are noted as being particularly useful for empirical studies. An application of these two quantities is illustrated for a set of empirical survival time data.

ISSN:0162-1459    

DOI:10.1080/01621459.1969.10501072

出版商:Taylor & Francis Group    

年代:1969

数据来源: Taylor

摘要:

A simple LSD (least significant difference) rule is presented for simultaneously testing the differences betweenntreatments considered in all possible pairs. This rule is a simpler, fully completed, version of the Bayes rule presented for special cases in Duncan [4]. It is based on the same multiple decision theory model except for a modified and extended use of a conjugate chi-square density in the prior. The new rule has the same intuitively appealing dependence on the between-treatmentFratio, varying from a sensitive comparisonwise-α-like rule whenFis large or moderate, to a conservative experimentwise-α-like rule whenFis small. Tables oftfor computing the LSD are presented for three choices of a type-1 to type-2 error-seriousness ratio,k= 50, 100 and 500 (analogous to the usual choices of α = .10, .05 and .01 in testing a single difference), and for full ranges ofFand its degree of freedomqandf.

ISSN:0162-1459    

DOI:10.1080/01621459.1969.10501073

出版商:Taylor & Francis Group    

年代:1969

数据来源: Taylor

摘要:

Results are presented for optimal two-stage sampling, where the size of the second-stage sample can be conditioned on the first-stage sample outcome, in the case where (1) sample observations are drawn from a Normal population or process with known variance but unknown mean, (2) there are two terminal acts whose terminal values are linear functions of the unknown population or process mean, and (3) the cost of a sample of positive size at either stage depends linearly on the sample size. Computer programs for obtaining optimal sampling plans are described, the effects on these optima of changes in cost and statistical parameters are traced, various characteristics of optimal plans are discussed, the sensitivity of two-stage sampling plans to deviations from optimal design is explored, and extensions to a number of closely related problems are given.

ISSN:0162-1459    

DOI:10.1080/01621459.1969.10501074

出版商:Taylor & Francis Group    

年代:1969

数据来源: Taylor

摘要:

A double sample test for a hypothesis concerning the mean μ of a normal random variable with unknown variance σ2is presented in this note. This test is an alternative to the usual single samplettest and is made by taking samples at two stages. After the first sample has been observed the hypothesis is rejected, accepted or a second sample is taken. This double sample test is a modification of a double sample test suggested by D. Owen. However, for the test being suggested here if the second sample is taken the information from the first sample is used in making the final decision.

ISSN:0162-1459    

DOI:10.1080/01621459.1969.10501075

出版商:Taylor & Francis Group    

年代:1969

数据来源: Taylor

摘要:

Weiss [5] and Freeman and Weiss [2] have shown how to construct sampling plans which at least approximately minimize the maximum expected sample size in the case of Bernoulli and Normal populations. In this paper we construct sequential procedures of this type for the testing of an exponential parameter. An invariance property of the stopping region is presented which simplifies the construction of the Bayes regions. Several sampling plans are given and some of their properties are obtained.

ISSN:0162-1459    

DOI:10.1080/01621459.1969.10501076

出版商:Taylor & Francis Group    

年代:1969

数据来源: Taylor

摘要:

Approximate expressions are given for the first four moments of the distribution of sample size required by a sequential probability ratio test. Some possible uses of these moments are also discussed.

ISSN:0162-1459    

DOI:10.1080/01621459.1969.10501077

出版商:Taylor & Francis Group    

年代:1969

数据来源: Taylor

摘要:

The purpose of this paper is the presentation of formulae for obtaining matrix derivatives of the second order to use in making tests for maxima and minima. The theory of such second order derivatives is presented. These formulae require the rearrangment of the parameter elements in vector form and the transformed results feature Kronecker products, which have certain desirable properties. Application is made to several types of problems.

ISSN:0162-1459    

DOI:10.1080/01621459.1969.10501078

出版商:Taylor & Francis Group    

年代:1969

数据来源: Taylor

摘要:

This paper deals with a multivariate extension of Friedman's χ2r-test. A rank permutation distribution and the large sample properties of the criterion are studied. The asymptotic relative efficiency (A.R.E.) for a sequence of translation alternatives is studied and bounds are given for certain special cases. It is shown that, under specified conditions, the A.R.E. of this test with respect to the likelihood ratio test is largest when the block dispersion matrices differ and can be greater than unity when the differences are large.

ISSN:0162-1459    

DOI:10.1080/01621459.1969.10501079

出版商:Taylor & Francis Group    

年代:1969

数据来源: Taylor

摘要:

It is shown that the information contained in the incomplete portion of the sample that is relevant in the estimation of the regression parameters is in the form of a linear restriction. Although the parameters of this restriction are generally unknown they can be estimated in terms of the complete portion of the sample. It is then shown that regression parameter estimates based on first order methods, say, Ĉ, differ from the ordinary least squares estimates based only on the complete portion of the sample, say ĈLS, by a function of the extent to which ĈLSfails to satisfy the above named restriction. The asymptotic variance-co-variance matrix of Ĉ is derived under various conditions and compared to that of ĈLS.

ISSN:0162-1459    

DOI:10.1080/01621459.1969.10501080

出版商:Taylor & Francis Group    

年代:1969

数据来源: Taylor