Math 281A Homework 6
Due: Nov 21, in class
1. Let {Xi, Yi}n
i=1be i.i.d. random vectors with Yi∈{0,1}, and
Pα,β(Yi=1∣Xi=x)=1
1+e−α−βx .
The distribution of Xiis non-degenerate, but unknown. Do we have closed form of MLE (ˆα, ˆ
β)? Derive
the asymptotic distribution of (ˆα, ˆ
β).
2. Let {Xi}n
i=1be i.i.d. from Poisson(1/θ).
(a) Calculate the Fisher information Iθin one observation;
(b) Derive the MLE ˆ
θand show its asymptotic distribution.
3. Let {Xi}n
i=1be i.i.d. from N(θ, θ ).
(a) Calculate the Fisher information Iθin one observation;
(b) Derive the MLE ˆ
θand show its asymptotic distribution.
4. (a) Calculate the Kullback-Leibler divergence between two exponential distributions with different
scale parameters, when is it maximal?
(b) Calculate the Kullback-Leibler divergence between two normal distributions with different location
and scale parameters, when is it maximal?