Let Yı, ..., Yn be a random sample from an exponential distribution with density $(0,0)(9) = bedla-») y for y > a, - < a < oo and a > 0 where both a and 6 are the unknown parameters. Let Y(1) S... 5 Y(n) be the order statistic. Take 2; = Yi) – Y(i1), for i < Y(i-1), for i = 2, ..., n and 21 = Y(1) - a. Note that the joint probability density function of the order statistic (Y(1), ...,Y(n)) is given by . n2 n! II f(a,0)(yi). i=1 n-1 (a) Show that 21, ..., Zn are independent and determine the distribution of (n-i+1)Zi.
(b) Determine the distribution of (CT-Y8) + (n – r)Y(r) – na). (c) Let U n-i !=1(Y; – Y(1)), show that U and Y(1) are independent. Find the probability density function of 7 (Y(1) – a). (d) Using U, construct a (1 – a)% confidence interval for 6 in term of xảf, that is (1 – a)100% quantile such that P(xảe < xải,q) = 1 – a where xảe is a x? random variable with degree of freedom df. (e) Using both U and Y(1), construct a (1 – a)100% confidence interval for a. (f) Construct a (1 - a)100% confidence region for the two-dimensional parameter (a, 6). ==