SLIDE 1
Applied Asymptotics in R
Examples of the use of higher order likelihood inference Alessandra R. Brazzale
Institute of Biomedical Engineering Italian National Research Council, Padova alessandra.brazzale@isib.cnr.it
The R User Conference 2006 Vienna, 15–17 June 2006
Alessandra R. Brazzale Applied Asymptotics in R
What is this talk about?
an R package bundle
hoa
Higher Order (small sample) Asymptotics
n − → ∞
for likelihood-based parametric inference . . . and where to read more about the subject
Alessandra R. Brazzale Applied Asymptotics in R A toy example Asymptotics Examples Cauchy data Op(n−1/2) Op(n−3/2)
A toy example
i.i.d. sample y1, . . . , yn from the Cauchy distribution f(yi; θ) = 1 π{1 + (yi − θ)2} log likelihood function: ℓ(θ; y) = − n
i=1 log{1 + (yi − θ)2}
maximum likelihood estimator: ˆ θ = argmaxθ ℓ(θ; y)
n = 1
ˆ θ = y F(ˆ θ; θ) = F(y; θ) = π−1 arctan(y − θ)
Alessandra R. Brazzale Applied Asymptotics in R A toy example Asymptotics Examples Cauchy data Op(n−1/2) Op(n−3/2)
Likelihood inference
confidence intervals and p-values are computed using p(θ; ˆ θ) = Pr(ˆ Θ ≤ ˆ θ; θ) exact: p(θ; ˆ θ) = Pr(Y ≤ y; θ) approximate: p(θ; ˆ θ) = Φ(pivot) + Op
- n−1/2
Wald pivot: w(θ) = √ 2(y − θ) likelihood root: r(θ) = sign(ˆ θ − θ)
- 2 log{1 + (y − θ)2}
1/2 score pivot: s(θ) = √ 2(y − θ)/{1 + (y − θ)2}
Alessandra R. Brazzale Applied Asymptotics in R