WLLN And Continuous Mapping Theorem - Statistics Assignment Help

Assignment Task

Question I

In the sequence of random variables X1, ...X,,, which only take non-negative values, all components are independent and identically distributed; all Xi's have E(Xi) = afl and var(Xi) = afl2 and finite forth moments. 

(a) find the asymptotic distribution of \Fri (In fC — Ina — In /3). 
(b) Provide one interval estimation for In afl with asymptotic confidence level 1 — c based on the results from (a). 
(c) Construct one consistent estimator for fl and prove its consistency. (Hint: WLLN and continuous mapping theorem.) 
 

Suppose for 3.(d) and 3.(e), we further assume that Xi follows Poisson distribution with parameter A, P (Xi = k) = e-AAk I k!. (Xi now only takes non-negative integer values, with ]EX; = A and var(Xi) = A.)

 
(d) Express a and /3 in terms of A. 
(e) Now we assume n = 1 (only one data point). Show that (-2).xl is one unbiased estimator for e-3A. But this is one silly estimator, why? 

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