INFERENCE STATISTICS
STA 3102 EXAM f ( x 1 x ) = O Where $ 20 is known and * > 0 is un knowm ( a) The likelihood function can be written as . L ( x X ] = oct. ( T x: ) - e. 2 xi = hex). q ( x, it xi ) where faces = on. if. are, and y ( a, It xi ) = &? ( It * )" dyends on & and on the sample through IT xx. Hence, IT Xi is a sufficient statistic for of by the factorization theo nem / Criteriom (b) let Yr, Yz,. I'm be another sample from the exponential distribution, Then we have ! * -1 L ( 2 1 x ) - e. 2 xi e L(XIY ) is independent of of if IT xi = 11 Y: . This means IT xi is a minimal sufficient statistic for o (C) f ( x 12 ) = 1. x. xe = expelm ( 8 ) + hm ( x ) + ( x- 1) lm * - B . X 4 Note that f(x (x) = f(x). exp for *(x)- 4(0)7Question 3 A distribution belongs to the regular 1-parameter exponential family if among other regularity conditions its pdf or pmf has the form: f (x]B) = g (x) exp (Bt (x) - y ( B) ) , BE QC (0o, 00). Furthermore, for this distribution E[(X)] = y'(B) and Var[t (X)] = w"(B). Refer to Question 2. (a) Show that f(x|0) belongs to the 1-parameter exponential family. Answer: f(x10) = 0(1 - 0)*-1 = (1 -0)*- 1 - 0 = exp { x In(1 - 0) - - In 1 - 0 = g (x) exp It (x) B - y (B)} e where g(x) = 1, B = In(1 -0), t (x) = x and y (B) = - In 1 - 0 = - In {e - 1) . Activ Go toQUESTION 1 A distribution belongs to the regular 1-parameter exponential family if among other regularity conditions its pdf or pmf has the form: f(v10) = g(y) expl0t(y) - y(0)), y ex C (-0o, co) and ( E O c (-00, 00); where g(v) > 0, t(y) is a function of y which does not depend on 0, and y (0) is real valued function of () only. Furthermore, for this distribution, if the first and second derivatives of w (0) exist, then E[t(Y)] = y'(0) and Var[t(Y)] = w"(0). Let X1, X2, ..., X, be the survival times of a random sample of n identical electronic components, and suppose that the survival time of a component has a distribution with probability density function Bax"-le-Bx if0
0 is known and a > 0 is unknown. (a) Use the factorization theorem/criterion to show that X, is a sufficient statistic for a. 1=1 (b) Show that X, is a minimal sufficient statistic for a. i=1 (c) Show that f (x|a) belongs to the regular 1-parameter exponential family by showing that the probability density function can be expressed as: g(x) exp(0t(x) - y (0)} Do not forget to identify: 0; g(x); t(x); and y (0) in the expression
