Question

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 of HW2:

Answer 1

be iid sarrple. X = (X1, X2, عرب 0:-(2, 1) E Ⓡ = R XCOos) 11870) De (a) fa,, ( x ) = e-(2-0) 1 (270) Now, joint pdf f(x) = ir II tau (2 II e (dia) 1 (2270) Åriod) e 1 (2:76,46) 2 and 1 (xeo 70 e g (T(2), d) ń (2) where T(2) = 76) Here, he is independent of a and of (T(2), x) depends on a and on 2 only through (2) Hence, by Neyman Fisher's Theorem taj is ss for the family &1}ER Scanned with CamScanner

f(x) = ne-(4-1) 1 (471) Joint pdf, f(x) = f f , x (26) {"(2194) S n -anlä-1) 1 (2071) 9(T(%) 2 ) where T(26) = ž - 1 Here, h (2) is independent of a and g (T(2, 2) depends on a and on 2 only through T(2) Hence, by Neyman Fisher's Theorem, e-l is ss for family { £ 1,3270 c) fox (2) = xe (2-2) 5(te'Non joint pot fra) i toa (21) 1 169:78) can bu 20v) +(203-w)} 2 (2017) in[ă - Xun) + (Hen-*] 1 (X1970) = g(713), 2, a ). nik) where, T() = ( - X013., ken) cell phone richiamantblecen, (Ž -210), Xen) is 55 for 85 () e