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Showing papers in "Journal of the royal statistical society series b-methodological in 1965"



Journal ArticleDOI
TL;DR: In this article, a linear model with fewer terms if a transformation is applied is described, and it is possible and perhaps useful to find the monotone transformation which minimizes the residual sum of squares after removing the effects explained by the model.
Abstract: Data may sometimes be described by a linear model with fewer terms if a transformation is applied. For example, in a factorial design the need for certain interaction terms may be eliminated by transformation. If attention is restricted to all monotone transformations, it is possible and perhaps useful to find the monotone transformation which minimizes the residual sum of squares after removing the effects explained by the model. Several sets of data from the statistical literature are analysed this way.

464 citations








Journal ArticleDOI
TL;DR: In this article, the authors present an alternative form of design selection for the case of mixtures of three or four factors, which involves the premise that, in the absence of specialist knowledge about the form of the true response function, it is desired to fit a response surface equation of first or second order over the factor space of possible mixtures, and experimental runs are needed to ensure the best surface fit possible.
Abstract: SUMMARY Scheffe, in two recent papers (1958, 1963), has given designs for experimenting with mixtures. The basis of these designs is the choice of symmetrical arrangements of points in the factor space and the fitting of carefully chosen models which have exactly as many coefficients as there are data points. The designs are such that some of the experiments do not contain any of one or more ingredients of the mixture. This may or may not be a disadvantage depending on the problem involved. Here an alternative form of design selection is made for the case of mixtures of three factors. (The method has also been extended to four factors.) This involves the premise that, in the absence of specialist knowledge about the form of the true response function, it is desired to fit a response surface equation of first or second order over the factor space of possible mixtures, and experimental runs are needed which, in a certain sense, ensure the best surface fit possible. The principles used in the choice of appropriate designs will be those originally introduced by Box and Draper (1959).

57 citations






Journal ArticleDOI
TL;DR: In this article, an alternate form of design selection is made for the case of mixtures of four factors, which is similar to the one used in this paper, but with the assumption that there is no specialist knowledge about the true response function.
Abstract: : Scheffe, in two recent papers (1958, 1963), has given designs for experimenting with mixtures. The basis of these designs is the choice of symmetrical arrangements of points in the factor space and the fitting of carefully chosen models which have exactly as many coefficients as there are data points. The designs are such that some of the experiments do not contain any of one or more ingredients of the mixture. This may or may not be a disadvantage depending on the problem involved. Here an alternate form of design selection is made for the case of mixtures of four factors. It extends the method used in the three factor case in an earlier paper, (Draper and Lawrence, 1965). This involves the premise that, in the absence of specialist knowledge about the form of the true response function, it is desired to fit a response surface equation of first or second order over the factor space of possible mixtures, and experimental runs are needed which, in a certain sense, ensure the best surface fit possible. The principles used in the choice of appropriate designs will be those originally introduced by Box and Draper (1959). (Author)


Journal ArticleDOI
B. L. Welch1
TL;DR: In this paper, the authors compared the distributional properties of confidence points with the assumption that a sample S has probability element p(S, 0) dS = exp {L(s, 0)} dS.
Abstract: SUMMARY In a recent paper (Welch and Peers, 1963) formulae were obtained for confidence points depending on the distributional properties of certain integrals of weighted likelihoods. Some comparisons are made here with the SUPPOSE a sample S has probability element p(S, 0) dS = exp {L(S, 0)} dS. We are concerned in the main with situations where L(S, 0) is "in the probability sense" 0(n), where n can become large. For "almost all samples" there is then a single maximumlikelihood estimate T which will differ from 0 by 0(n-A). We write K2(0) = E{(n-iaL/a6)2} = _E(n-E 32L/62), (1) but for convenience we shall often abbreviate K2(T) to K2 and K2(6) simply to K2. Asymptotically ni K'(T- 0) tends to be distributed normally with mean zero and standard deviation unity. The quantity n iK2(6 - T) tends also to have the same distribution. More precisely Pr{n K2(6-T) < x} = 0s+0(n-), (2) where e and cx are related by









Journal ArticleDOI
TL;DR: In this paper, the authors study the statistics N-1 S Xt cos (2tts/N) and Nt t Xt sin (2'rrts/n) as estimators of the sth Fourier coefficients of ft, and of the distributional properties of suitable ratios of these statistics as (sufficient) test statistics for non-constancy of ft.
Abstract: Let Xt (t = 1, 2, ..., N) denote a series of independent normal random variables, with zero means and non-zero finite variances ft (t = 1, 2, ..., N). Then ft and ft 1 admit finite Fourier expansions. The present paper is a study of the statistics N-1 S Xt cos (2tts/N) and Nt t Xt sin (2'rrts/N) as estimators of the sth Fourier coefficients of ft, and of the distributional properties of suitable ratios of these statistics as (sufficient) test statistics for non-constancy of ft, directed at alternatives where the Fourier expansion of ft 1 has a simple form.


Journal ArticleDOI
TL;DR: In this article, it was shown that the number of moment crossings of two symmetrical densities is related to the total number of crossings of the densities, and this generalizes a result of Fisher's recently proved by Finucan (1964) (A note on Kurtosis).
Abstract: : In this paper it is shown how the number of moment crossings of two symmetrical densities is related to the number of crossings of the densities. This generalizes a result of Fisher's recently proved by Finucan (1964) (A note on Kurtosis). (Author)