# Hint of HW2: 4. Let X = (X1, ..., Xn) be an i.i.d. sample from the...

###### Question:

Hint of HW2:

4. Let X = (X1, ..., Xn) be an i.i.d. sample from the shifted exponential distri- bution with density fa,1(x) = de-1(z–a) 1(x > a), 0 := (a, 1) E O = R (0,00). Use Neyman-Fisher's theorem to: (a) show that S = X(1) is an SS for the family {fa,1}QER; (b) find an SS for the family {f1,1}x>0; (c) find an SS for the family {fa,x}o=(0,1)€0. (d) In part (a), use the procedure from Rao- Blackwell's theorem and SS S = X(1) to construct an unbiased estimator3 for a. Hints: (d) To apply Rao-Blackwell's theorem, you will need an unbiased estimator for a; see the hints for HW Problem 2 above. Take a "seed" estimator of the form a* = X1 + a. For what value of a will we have a* e Ko (i.e the estimator will be unbiased)? To compute the CE, one may wish to use the result of tutorial Problem 4(a) (rather, of its analog when conditioning on X(1) =s; do that version of Problem 4(a) first). That should do it.
Hint: Recall that measurability of g means that g-'(B) E B(R) for any B E B(R). In fact, for independence of X and Y it suffices that (1) holds for measurable indicators, i.e. functions g of the form g(x) = 1x € C) for some CE B(R) is it clear that such functions are measurable? Just verify the condition from the definition of measurability).

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