Question

Delta Theorem's application to MLES Suppose X1, X2, ... are iid Poi(). a. Find the MLE of h() = P(X = 0) b. Find its asymptotic distribution

Answer 1

Let there tee in sample, X., ...xn und Poisson (d). [iid = independent, identical ] By definition, likelihood of d, 4(a)= fx (2) -- fx cew] [: Xy, .., Xn are independent] 6:169) : erobretot en me en sy to vy Kenzo fuL(m): -nit Li And - do team (9) 30 -nt 0 = Q = + Exi: (many) are Camb altres serveis nicht KO [Xix0]. By second derivative check, I ALE = x = 18 sati : Sample Mean Note : 252 (z io realization of ) ay h(x)= P(x=0) = e" o seu By invariance property, if IHLE is MLE of in, then MLE of Ah() is hCânia

:hâule) = e FMLE = et is the MLE of P(x=0) by Central Limiet Theorem: X EN(E(+), V(X1)). on va (< -E(x)) ÔN(o, v(x)) Here, mn(-) @N(o,) [: xi's are iid Poisson (0)] By Delta theorem, In (g(x)-9(+)) & N(o, (g(x)) ) Here, g(x) = ex § (9' (2c)) = (6-22²= 6-22 : g(*)= 6-7 , g(x)=6", (g(x)) = e 21 vale-X_64) N(o, ezt.) et en lecot, terza )
Below is mean and variance of any general Poisson

xso For Prisen(a) : *~P(1) : P(x=2). ed. 20,1,2, .... ED) - ExP(x2) • Izena eta e tetea E((X)3 ) Prst), Ž. ker) et les *V(x)= E(X)-E29 - E(X(x-1)+E(*)- E*(x) = *+d-xsd o