Page 361 - Probability and Statistical Inference

6. Sufficiency, Completeness, and Ancillarity 338

independently of . Is ( – ) distrib-
3
uted independently of T? Is {(X – ) – (Y – )} /(X – X ) /T distrib-
2
2
n:n
n:n
n:1
n:n
uted independently of T? Is {(X – ) – (Y – )} /(X – X ) distributed
2
2
n:n
n:n
n:n
n:1
independently of T? {Hint: Can the characteristics of a location-scale family
and (6.5.10) be used here?}
6.6.14 (Exercise 6.6.8 Continued) Let X , ..., X be iid having the common
1
n
Uniform distribution on the interval (–θ, θ) where 0 < θ < ∞ is the unknown
parameter. Let us denote . Is T distributed independently of
2
| X |/X ? Is T distributed independently of (X – X ) /{X X }? {Hint: Can
n:1
n:n
2
n:n
1
n:1
the characteristics of a scale family and (6.5.9) be used here?}
6.6.15 (Exercise 6.6.14 Continued) Let X , ..., X be iid having the com-
1
n
mon Uniform distribution on the interval (–θ, θ) where 0 < θ < ∞ is the
unknown parameter. Let us denote . Is T distributed indepen-
dently of the two-dimensional statistic (U , U ) where U = | X |/X , U =
2
n:1
1
2
1
n:n
2
(X – X ) /{X X }? {Hint: Can the characteristics of a scale family and
n:n
2
n:1
1
(6.5.9) be used here?}
6.6.16 (Exercise 6.6.13 Continued) Let X , ..., X be iid N(µ , σ ), Y , ...,
2
1
1
m
1
Y be iid N(µ , σ ), the X’s be independent of the Y’s where –∞ < µ , µ < ∞,
2
1
2
2
n
0 < σ < ∞ are all unknown parameters. Use Basu’s Theorem to check whether
the following two-dimensional statistics
are distributed independently where .
{Hint: Can the characteristics of a location-scale family and (6.5.10) be used
here?}
6.6.17 Let X , ..., X be iid having a negative exponential distribution with
n
1
the common pdf σ exp{–(x – θ)/σ}I(x > θ) where θ and σ are both unknown
–1
parameters, –∞ < θ < ∞, 0 < σ < ∞. Argue as in the Example 6.6.15 to show
that X and are independent.
n:1
6.6.18 (Exercise 6.3.2 Continued) Let X , ..., X be iid having the common
1
n
pdf σ exp{–(x – µ)/σ}I(x > µ) where –∞ < µ < ∞, 0 < σ < ∞. Show that
–1
(i) X , the smallest order statistic, is complete if µ is unknown
n:1
but σ is known;
-1
(ii) n is complete if σ is unknown but µ is known;
(iii) (X , is complete if both µ, σ are unknown.
n:1