Advertisement

Large Sample Theory: Constraint-Equation Hypotheses

  • George A. F. Seber
Chapter
Part of the Springer Series in Statistics book series (SSS)

Abstract

Apart from Chap.  8 on nonlinear models we have been considering linear models and hypotheses. We now wish to extend those ideas to non-linear hypotheses based on samples of n independent observations \(x_{1},x_{2},\ldots,x_{n}\) (these may be vectors) from a general probability density function \(f(x,\boldsymbol{\theta })\), where \(\boldsymbol{\theta }= (\theta _{1},\theta _{2},\ldots,\theta _{p})'\) and \(\boldsymbol{\theta }\) is known to belong to W a subset of \(\mathbb{R}^{p}\). We wish to test the null hypothesis H that \(\boldsymbol{\theta }_{T}\), the true value of \(\boldsymbol{\theta }\), belongs to W H , a subset of W, given that n is large. We saw in previous chapters that there are two ways of specifying H; either in the form of “constraint” equations such as \(\mathbf{a}(\boldsymbol{\theta }) = (a_{1}(\boldsymbol{\theta }),a_{2}(\boldsymbol{\theta }),\ldots,a_{q}(\boldsymbol{\theta }))' = \mathbf{0}\), or in the form of “freedom” equations \(\boldsymbol{\theta }=\boldsymbol{\theta } (\boldsymbol{\alpha })\), where \(\boldsymbol{\alpha }= (\alpha _{1},\alpha _{2},\ldots,\alpha _{p-q})'\), or perhaps by a combination of both constraint and freedom equations. Although to any freedom-equation specification there will correspond a constraint-equation specification and vice versa, this relationship is often difficult to derive in practice, and therefore the two forms shall be dealt with separately in this and the next chapter.

References

  1. Aitchison, J. (1962). Large sample restricted parametric tests. Journal of the Royal Statistical Society B, 24, 234–250.zbMATHMathSciNetGoogle Scholar
  2. McManus, D. A. (1991). Who invented local power analysis? Econometric Theory, 7, 265–268.MathSciNetCrossRefGoogle Scholar
  3. Seber, G. A. F. (1963). The linear hypothesis and maximum likelihood theory, Ph.D. thesis, University of ManchesterGoogle Scholar
  4. Silvey, S. D. (1959). The Lagrangian multiplier test. Annals of Mathematical Statistics, 30, 389–407.zbMATHMathSciNetCrossRefGoogle Scholar
  5. Wald, A. (1949). A note on the consistency of the maximum likelihood estimate. Annals of Mathematical Statistics, 20, 595–601.zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • George A. F. Seber
    • 1
  1. 1.Department of StatisticsThe University of AucklandAucklandNew Zealand

Personalised recommendations