Question: 6. Let Xl' Xm; Yl , . .. , Y be samples from exponential distributions with densities a-Ie-(. there exists a UMP invariant test

6. Let Xl"' " Xm; Yl , . .. , Y" be samples from exponential distributions with densities a-Ie-(.<-€ l/a for X t and T-le-(y-rll/T for y 1}. (i) For testing T/a 5. against T/a > there exists a UMP invariant test with respect to the group G: X: = aX;+

b, J}' = aJ} +

c, a> 0, - 0

b, e < 00. and its rejection region is L: [y - rnin( YI ' ... , Yn) ] c ) > . L:[X; - rnin(xl , · · ·,xm ) ] 339 (ii) This test is also UMP unbiased. (iii) Extend these results to the case that only the r smallest X's and the s smallest Y's are observed. [(ii): See Problem 12 of Chapter 5.]

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