Question

Suppose X1, X2, ..., Xn are independent and identically distributed (iid) with a Uniform -0,0 distri- bution for some unknown e > 0, i.e., the Xi's have pdf Suppose X1, X2,..., Xn are independent and identically distributed (iid f(3) = S 20, if –0 < x < 0; 20 0, otherwise. (a) (4 pts) Briefly explain why or why not this is an exponential family (b) (5 pts) Find one meaningful sufficient statistic for 0. (By "meaningful”, I mean it should reduce the data, e.g., (X1, X2, ..., Xn) is sufficient, but it does not reduce the data, thus it is not meaningful)

Answer 1

Griven, - Xi,... Xr are üd uniform [-o, o ie xi's have pdf f($e) = szaif -OCISO otherwise a for an exponential family, the support of the distribution (ie [x: $(2168oz cannot depend on a. here support of the distribution is LOS « se je 11 so which depends от еСраға ngton] - so the above pdf is not a member of eup onentical family of distribution distribution. Scanned with CamScanner

LAN fcoci)= os sei cooil,..n flacil= - 1 (-os xico) , vi=L(11n - ② 20 هو مطلع 11 ؟ ا في یک خلاء - Lo if otherwise L Indicator function Now flee,,.. no/or= Ñ faci) = { 11-05 zis o) = 29 = 1 16-OSXC1) 1 (x(n) €0) - anon where acco= first order statistic (minimum ofn) rechi- nth order statistic imum ofx) In ean (2) f(x11. un 101= I 16.07, -2ecoD) 1 (xen) so) 2 on 1 1/07-2013) 1 (vecnoso) Scanned with CamScanner g (Tlxfo)

fcxe...and = g (T(x)/6) h(x) - ③ where g(x)/6) = 4(07-2013) ( 8>exin) h(X) = 1 anon By factorization - suficient for the arem (- Xci), Xcnj) is o. CS Scanned with CamScanner