Question

beta >0

74. Let X1, X2, ..., Xn be a random sample from the PDF 010105T10 by Disclado Ol betrov , a < x < oo, -o < a < oo, (a) Find the MLE of (a, b). (b) Find the MLE of Pa,p{X1 2 1}. guld brun onheilt f (x; a, B) = 8-1e--(x-a) gmax B>

Answer 1

- 2-a © acrcpo -xa f(x; &if) , je X.X2 , . . . Xn is a random sample dishibulion The likelihood function is from above na L(&, B) : I e t not x t x i=1(1)n. Σχ, και η + na x x < Xa) < X(M) and where xw and xa dendes minimum maximum of Xi's respectively. Maximum

Then keeping to fixed L(&, B) increases with increase in the value of a. Thus, L(0, 3), for fixed ß and X = (X., X.- Xu) a is maximused at the maximum value of x. Now x has the range a (-0, x)) Maximum value of a is Then , MLE f x is xa min xif e Xow Inicion Nous 1(@,m). Le 6 24 1 sn log LCâ, B) = -n luß - 1 Exit na Putting golong Lc&.) 30 MLE H À is ñ . E (Xi = @) (Xi – xo)) n

Now P(x, 71) -C ) Now Invariance properly MLE d a parameter o one-to-one function of the MLE of to) states that it 6 is the and fro is a continuous o, then fo) will be Pa, B (X, > 1) = e Then f( (2,6) is the MLE of - MLE of Pal Pa, (x,z1) is = f((x, (B) Pap(x-> 1) A where 8 2 1 2 ( x - X) и тут