By Yanhong Wu
The most emphasis is at the inference challenge for the swap element and post-change parameters after a metamorphosis has been detected. extra particularly, a result of handy shape and statistical houses, the writer concentrates at the CUSUM strategy. The objective is to supply a few quantitative reviews at the statistical homes of estimators at the switch element and post-change parameters.
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Extra resources for Inference for Change Point and Post Change Means After a CUSUM Test (Lecture Notes in Statistics)
Example text
1, we see that = ∂2 1 ln Pθ (τ− = ∞) = − 2 (1 + o(1)). 2. 4: Locally as θ → 0, and θd1− → ∞ for some √ 5 Eθ [(ZN0 − b0 (θˆ0 )/ d)2 |SN0 > d] = 1 + 4d > 0, 1 + o(1) . 3) A corrected normal pivot can be formed as ∗ ZN = 0 ZN0 − b0 (θˆ0 )/d1/2 , (1 + c0 (θˆ0 )/d)1/2 which gives a normal approximation accurate up to the order o(1/d). A (1 − α)-level corrected confidence interval for θ can be constructed as θˆ0 − b0 (θˆ0 ) 1/2 d1/2 N0 ± (1 + c0 (θˆ0 )/d)1/2 1/2 N0 zα/2 , where zα is the (1 − α)th quantile of standard normal density.
27 of Siegmund (1985) gives Eθ0 (τ− ) = = 1 Eθ Sτ θ0 0 − 1 √ e−θ0 ρ + O(θ0 ). 1 of Chapter 2 give, respectively, Eθ0 (σM ) = 1 + O(1) 2θ02 and Eθ0 [σM Pθ (τ−M < ∞)] = 1 + O(1). 2, we obtain the approximated value as E ν [ν − Ls |Ls ≤ ν < N ] ≈ 1 −θ0 e−θ0 ρ (s − √ eθ(ρ−(θ−θ−0)) { √ 1 − α − 2θ0 2θ(θ − θ0 ) −(θ − θ0 ) (θ−θ0 )θ 1 α e(θ−θ0 )θ }, × 1− e α + 2− θ0 2θ0 −2θ0 (θ − θ0 ) where α = p0 ps denotes the non-coverage probability. At θ = −θ0 , E ν [ν − Ls |Ls ≤ ν < N ] ≈ 1 e−θ0 ρ √ 1 − α − 2θ0 s− e−2θ0 (ρ+2θ0 )/2 √ 1 − 2αe−θ0 −2 2θ0 + 1 αe−θ0 .
We naturally estimate θ by the sample mean after νˆ, that is, θˆ = TN . 3 of Chapter 4, the bias of θˆ is approximately equal to 1 E ν [θˆ − θ|N > ν] ≈ d 2− (θ/θ0 )3 2(θ/θ0 − 1) , which is valid for θ/θ0 within a range. Therefore, we can use the corrected estimator for θ, for example as in Whitehead (1986), by solving 1 θ˜ = θˆ − d 2− ˆ 0 )3 (θ/θ ˆ 0 − 1) 2(θ/θ , ˆ 0 . 5) for ps , under the restriction for the value of θ/θ we can obtain the estimated value of s such that the noncoverage probability is approximately equal to the required significance level α.