Question

Let X1, X2, X3 ∼(iid) Exponential(λ).

(a) Show that T(X1, X2, X3) = X1 + X2 + X3 is a sufficient statistic for λ.

(b) Find the MVUE for λ.

(c) Show that  $X_{(1)}$ is not a sufficient statistic for λ.

(d) Let $\tilde{\lambda }$ = $3X_{(1)}$ and find $E\tilde{\lambda }$ . Give an argument for why $\tilde{\lambda }$ is not the best estimator of λ.

Answer 1

solution : Given that, * Let X1, , X in de Exponential (). (a) Now we have * fx (x, t) = de las There fore, * ~ one parameter exponential family Here, Ten EX is a complete and safficient for do (b) Thus, ar mma xa gamma (:, +). Tagamma(2+) mmo

There fore, 2 as an ure of T. the we can say that * By lehman scheffe E ( IT) is a UMNOE of do Hence, * is a the UMVUE Of to the (0) Now, * conditional distribution of

fxca, (x) = 36 M (1-etx) Co to je vex (A-1)! (3-1)! clearly conditional distribution of xxv / 12° is not independenti. Of T. Hence, yu, for to is not sufficient Comertek ) Centro de*** caso! IX fxcus () 6-1)! (+)" . a 3 de 3tx forco() = 3te3d2] . * E(3.81 ) = 9 3 + x e 31 dx

8 x ²1 esist da. I =9. 13 (37772 d. There fore, of I. ... is the UMVUE * not ) Hence, It is not the heat estimator.