By D. V. Lindley
A learn of these statistical rules that use a likelihood distribution over parameter area. the 1st half describes the axiomatic foundation within the notion of coherence and the consequences of this for sampling thought statistics. the second one half discusses using Bayesian principles in lots of branches of facts.
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Extra resources for Bayesian Statistics, a Review (CBMS-NSF Regional Conference Series in Applied Mathematics)
Sample text
Yn be an iid sample from N(μ, σ 2 ) with σ 2 unknown and we are interested in testing H0 : μ = μ0 versus H1 : μ = μ0 . Under H0 the MLE of σ 2 is 1 (yi − μ0 )2 . σ2 = n i Up to a constant term, max L(θ) H0 max L(θ) ∝ ∝ −n/2 1 n 2 (yi − μ0 ) i −n/2 1 n (yi − y)2 i and W − μ0 )2 2 i (yi − y) i (yi = n log = n log = n log 1 + i (yi − y)2 + n(y − μ0 )2 2 i (yi − y) t2 n−1 , √ where t = n(y − μ0 )/s) and s2 is the sample variance. Now, W is monotone increasing in t2 , so we reject H0 for large values of t2 or |t|.
The table below shows some examples with two representations, one algebraic and one as a model formula, using the notation of Wilkinson and Rogers (1973). In the latter X is a single vector, while A represents a factor with a set of dummy variates, one for each level. Terms in a model formula define vector spaces without explicit definition of the corresponding parameters. For a full discussion see Chapter 3 of McCullagh and Nelder (1989). 2 Aliasing The vector spaces defined by two terms, say P and Q, in a model formula are often linearly independent.
If the likelihood is regular the two intervals will be similar. However, if they are not similar the likelihood-based CI is preferable. One problem with the Wald interval is that it is not invariant with respect to parameter transformation: if (θL , θU ) is the 95% CI for θ, (g(θL ), g(θU )) is not the 95% CI for g(θ), unless g() is linear. This means Wald intervals works well only on one particular scale where the estimate is most normally distributed. In contrast the likelihood-based CI is transformation invariant, so it works similarly in any scale.