Question

Let X be a random variable with p.d.f. f(x) = θx^(θ−1) , for 0 < x < 1. Let X1, ..., Xn denote a random sample of size n from this distribution. (a) Find E(X) [2] (b) Find the method of moment estimator of θ [2] (c) Find the maximum likelihood estimator of θ [3] (d) Use the following set of observations to obtain estimates of the method of moment and maximum likelihood estimators of θ. [1 each] 0.0256, 0.3051, 0.0278, 0.8971, 0.0739, 0.3191, 0.7379, 0.3671, 0.9763, 0.0102

Answer 1

O-1 PW)- Ocnel Let X, Y, - . . Xn Xn denote a rundom sample from fal a EG). J x o xo de oja o da N b) Moment estimator o is Oti o x + x = 0 =) out o(1-x) = X х th 22 Źx; n MM el manimum likelihood estimator n ol 210,0) = on ſa 121 log L= n logo n logo + (0-1) { logxi dloge n x =0 =O =) o te logx, dee n =) MLE Eloy ki

d) From the observation (n-10) x = ŕ Exi 1 +3.7401 10 -0.3740 : moment effimater is X 0.3740 NM 1-X I-0.3740 = 0.1899 M LE n - TO O - MLE E logni -7.9068 = 1.2647