Question

Let with Y, Y, ..., Yn be i id random variables the following probability density function, 1 x)/x fyly) = f I y ocyc1 o otherwise a) b) where x>0 is an unknown parameter. Find the maximum likelihood estimator , ã of x. Show this is an unbaised estimator for a. Hint : make use of the fact that in y follows an exponential distribution with mean a. Toe., -lny ~ Exp(x) c) Find the MSE of the manimum likelihood estimator o a Is the maximum likelihood estimator consistent estimator for a ?

Answer 1

Given pdf of random variable vanable yie - 09462947 o otherwise MLE of al - Let ws write function of down the likelihood the given pdf cs Scanned GamScanner

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then FG) = [TCE) [lnych] = 1 ay 3 ] = -1- P1 by L - - 1 set fondy O dy - 1-get, a g - 1 y 2 - 1 tl Te T-(7) a te te Hence pdf of I will be fitle a fill a £ fit het to anned with lehich is a eppotenchial dishni ed with CamScanner bution.

nico how we know that m.gf of exponechal distribution tytt-00)' je My E72 (7-2751 ... how Malt) = M 18 My Mp, (). Met :- M.) (1=254.(-44 (1-4457 at) 2 1 - 2 which is mig f of gamma . dishi bution with parameter namma in) Scanned with CamScanner

it and we know that xn raining Card) then mean of x -and-verte 2 Here fait & leggingamma (hh) - 2) La Hence Elgin 24 Milt in unbiased ATE 2 this proves that for a :() now so uno - ng 4 x 2 In fes Scanned with... CamScanner

now since taLE is unbiased henc Bias (220 So MSE - unit bien a lasta el - MSE 2 = 22 from the nece snary and sufficient condition of comistency know that we ET12 o 1, ash ) & V(T) = 0 the I is said to be comistent for o. Here we get that E D)icima rin naman and v(a) = d2 arn- of volo L | Scanned with CamScanner

ie E (222=2 ? ano . & Nazol nence ă is comistent eirmator food C Scanned with CamScanner