Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-13T12:27:00.330Z Has data issue: false hasContentIssue false
  • English
  • Français

Some classes of conditional inference procedures

Published online by Cambridge University Press:  14 July 2016

E. J. Williams
Get access Rights & Permissions [Opens in a new window]

Abstract

Data classified into several groups usually depend on structural and accessory parameters, the structural parameter θ being a constant for all the data, while the accessory parameters ϕ i vary between groups.

For the structural parameter, inferences that are free of the accessory parameters can be made when the ϕ i admit sufficient statistics or surrogates Si, by conditioning on the Si or by constructing statistics independent of the Si.

When the Si are functions of the structural parameter θ, θ is said to be endomorphic. Conditional likelihood methods are not unequivocal for endomorphic parameters; instead, inference is based on statistics independent of the Si, and so free of the ϕ i, derived from conditional distribution functions.

With endomorphic parameters, since the surrogates and the statistics independent of them are functions of θ, tests of independence of these sets of statistics provide a means of making inferences about θ.

Keywords

STRUCTURAL AND ACCESSORY PARAMETERSCONDITIONAL INFERENCECONDITIONAL DISTRIBUTION FUNCTIONSTATISTICS INDEPENDENT OF A GIVEN STATISTICTESTS OF INDEPENDENCE
Type
Part 5 — Statistical Theory
Information
Journal of Applied Probability , Volume 19 , Issue A , 1982 , pp. 293 - 303
Copyright
Copyright © 1982 Applied Probability Trust 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Andersen, ?. ?. (1970a) Asymptotic properties of conditional maximum-likelihood estimators. J. R. Statist. Soc. B 32, 283–301.Google Scholar
Andersen, ?. B. (1970b) On Fisher's lower bound to asymptotic variances in case of infinitely many incidental parameters. Skand. Aktuarie Tidskr. , 78–85.CrossRefGoogle Scholar
Andersen, ?. B. (1971a) A strictly conditional approach in estimation theory. Skand. Aktuarie Tidskr. , 39–49.CrossRefGoogle Scholar
Andersen, ?. B. (1971b) The asymptotic distribution of conditional likelihood ratio tests. J. Amer. Statist. Assoc. 66, 630–633.Google Scholar
Barnard, G. A. (1963) Some logical aspects of the fiducial argument. J. R. Statist. Soc. B 25, 111–114.Google Scholar
Basu, D. (1955) On statistics independent of a complete sufficient statistic. Sankhya 15, 377–380.Google Scholar
Basu, D. (1958) On statistics independent of sufficient statistics. Sankhya 20, 223–226.Google Scholar
Fisher, R. A. (1953) Dispersion on a sphere. Proc. R. Soc. London A 217, 295–305.Google Scholar
Godambe, V. P. (1980) On sufficiency and ancillarity in the presence of a nuisance parameter. Biometrika 67, 155–162.CrossRefGoogle Scholar
Kalbfleisch, J. D. and Sprott, D. A. (1970) Application of likelihood methods involving large numbers of parameters (with discussion). J. R. Statist. Soc. B 32, 175–208.Google Scholar
Morgan, W. A. (1939) A test for the significance of the difference between the two variances in a sample from a normal bivariate population. Biometrika 31, 13–19.Google Scholar
Pitman, E. J. G. (1939) A note on normal correlation. Biometrika 31, 9–12.Google Scholar
Pitman, E. J. G. and Williams, E. J. (1967) Cauchy-distributed functions of Cauchy variates. Ann. Math. Statist. 38, 916–918.Google Scholar
Williams, E. J. (1962) Exact fiducial limits in nonlinear estimation. J. R. Statist. Soc. B 24, 125–139.Google Scholar
Williams, E. J. (1963) A comparison of the direct and fiducial arguments in the estimation of a parameter. J. R. Statist. Soc. B 25, 95–99.Google Scholar
Williams, E. J. (1969) Cauchy-distributed functions and a characterization of the Cauchy distribution. Ann. Math. Statist. 40, 1083–1085.Google Scholar
Williams, E. J. (1973) Tests of correlation in multivariate analysis. Bull. Int. Statist. Inst. 45, 219–232.Google Scholar
Williams, E. J. (1976) The power of some tests of correlation. In Perspectives in Probability and Statistics , ed. Gani, J., distributed by Academic Press, London, for the Applied Probability Trust, Sheffield, 105–116.Google Scholar