Statistical Computing · Quiz 2 · Sep 2024

4. Following the notations from the lectures on weighted importance sampling, let f(z) = exp(—z"/2), for —00 <2 <00 and G be the proposal distribution with density g(z) = Seew-#/), for —c0 <4 <0. Por Z;, Za,...,Zn ‘8 G, what does * ia) N aa 4(Z;) converge to? Enter the answer correct to two decimal places. [4 marks] 5. Consider n iid. samples X,,X5,...,X, from a Rayleigh distribution whose pdf is given by ag f(x | 6) = 7 RP, for >0, 0>0. Compute the MLE of 6. [4 marks] 6. Consider the Poisson regression model: for i = 1,...,n, let x; € R? be the vector of covariates for the i-th observation and 3 € R? be the corresponding vector of regression coefficients. Suppose response y; is a realization of Y; with Y, ~ Poisson (e ), and Y;’s are independent. (i) Find the joint log-likelihood function 1(3). [4 marks] (ii) Find the gradient of [(). [3 marks] (iii) Find the Hessian of [()). [2 marks] (iv) Show that the likelihood function is concave. {2 marks}