Copyright c 2012 iowa state university statistics 511

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Unformatted text preview: known as the likelihood ratio. −2 log Λ is known as the likelihood ratio test statistic. Tests based on −2 log Λ are called likelihood ratio tests. Copyright c 2012 (Iowa State University) Statistics 511 21 / 30 Under the regularity conditions in Shao (2003) mentioned previously, the likelihood ratio test statistic −2 log Λ is approximately distributed as central χ2f −kr under the null hypothesis, where kf k and kr are the dimensions of the parameter space under the full and reduced models, respectively. This approximation can be reasonable if n is “sufficiently large.” Copyright c 2012 (Iowa State University) Statistics 511 22 / 30 Likelihood Ratio Tests and Confidence Regions for a Subvector of the Full Model Parameter Vector θ Suppose θ is k × 1 vector and is partitioned into vectors θ 1 k1 × 1 and θ 2 k2 × 1, where k = k1 + k2 θ1 and θ = . θ2 Consider a test of H0 : θ 1 = θ 10 . Copyright c 2012 (Iowa State University) Statistics 511 23 / 30 ˆ ˆ Suppose θ is the MLE of θ and θ 2 (θ 1 ) maximizes θ1 over θ 2 for any fixed value of θ 1 . θ2 θ 10 is approximately ˆ θ 2 (θ 10 ) χ21 under the null hypothesis by our previous k result when n is “sufficiently large.” Then 2 ˆ (θ ) − Copyright c 2012 (Iowa State University) Statistics 511 24 / 30 Also, Pr 2 ˆ (θ ) − θ1 ˆ 2 (θ 1 ) θ ≤ χ21 ,1−α k ≈1−α which implies Pr θ1 ˆ θ 2 (θ 1 ) Copyright c 2012 (Iowa State University) 1 ˆ ≥ (θ ) − χ21 ,1−α 2k ≈ 1 − α. Statistics 511 25 / 30 Thus, the set of values of θ 1 that, when maximizing over θ 2 , yield a maximized likelihood within 1 χ21 ,1−α of the likelihood maximized over all 2k θ , form a 100(1 − α)% confidence region for θ 1 . Such a confidence region is known as a profile likelihood confidence region because θ1 ˆ 2 (θ 1 ) θ is the profile log likelihood for θ 1 . Copyright c 2012 (Iowa State University) Statistics 511 26 / 30 A Warning The normal and χ2 approximations mentioned in these notes may be crude if sample sizes are not sufficiently large. The regularity conditions mentioned in these notes do not hold if the true parameter falls on the boundary of the parameter space. Thus, as an 2 example, testing H0 : σu = 0 is not covered by the methods presented here. Copyright c 2012 (Iowa State University) Statistics 511 30 / 30...
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This document was uploaded on 03/27/2014.

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