Approximation of epidemic models by diffusion processes
and their statistical inferences
Catherine Larédo
2
1,2
1 UR 341, MaIAGE, INRA, Jouy-en-Josas
UMR 7599, LPMA, Université Paris Diderot
21/11/2015 CIMPA 2015
TALK 3-Part 3, CIMPA 2015
Recap on MLE asymptotics
Catherine Larédo ()
Inference for epidemic models
21/11/2015 CIMPA 2015
1/2
MLE asymptotics
θ0 : True value of the parameter.
Θ: Parameter set included in Rq ; θ0 ∈ Int(Θ) `n (θ): log-likelihood of the
parameter θ given the first n observations
∂g
• Rely on tree basic results
∇θ g (θ) = ∂θ
i
a law of large numbers for the log-likelihood `n (θ),
a central limit theorem for the score function
a law of large numbers for the observed information.
More precisely,
(i) For all θ ∈ Θ, n−1 `n (θ) → J(θ0 , θ) Pθ0 − a.s. uniformly w.r. t. Θ,
θ → J(θ0 , θ) continuous function with a global unique maximum at θ0 .
(ii) n−1/2 ∇θ `n (θ0 ) → N (0, I(θ0 )) in distribution under Pθ0 ,
where I(θ)): Fisher information matrix at θ.
(iii) limn→∞ sup|θ−θ0 |≤δ k − n1 ∇2θ `n (θ) − I(θ0 ) k→ 0 as δ → 0 Pθ0 − a.s.
Catherine Larédo ()
Inference for epidemic models
21/11/2015 CIMPA 2015
2/2