Page 611 - Probability and Statistical Inference

588 13. Sample Size Determination: Two-Stage Procedures

13.3.6 Let X , X , ... be iid random variables with the common pdf f(x; µ,
1
2
σ) = σ exp{ – (x – µ)/σ}I(x > µ) where both the parameters µ(∈ ℜ) and σ(∈
–1
ℜ ) are assumed unknown. Having recorded X , ..., X for n ≥ 2, we estimate
+
1
n
µ and σ by X and T = (n – 1) . Let us denote θ = (µ, σ).
–1
n
n:1
Suppose that the loss incurred in estimating µ by X is given by the function
n:1
L(µ, X ) = a(X - µ) where a and t are fixed and known positive numbers.
t
n:1 n:1
(i) Derive the associated fixed-sample-size risk R (µ, X ) which is
n:1
n
given by E [L(µ, X )];
θ n:1
(ii) The experimenter wants to have the risk R (µ, X ) bounded from
n:1
n
above by ω, a preassigned positive number. Find the expression of
*
n , the optimal fixed-sample-size, had the scale parameter σ been
known.
13.3.7 (Exercise 13.3.6 Continued) For the bounded risk point estimation
problem formulated in Exercise 13.3.6, propose an appropriate two-stage stop-
ping variable N along the lines of (13.3.5) and estimate µ by the smallest order
statistic X . Find the proper choice of “b ” required in the definition of N by
m
N:1
proceeding along (13.3.9) in order to claim that E [L(µ, X )] ≤ ω for all θ, a
θ
N:1
and t. Is it true that b > 1 for all positive numbers t? Obtain the expressions
m
for E [X ] and V [X ].
θ
?
N:1
N:1
13.3.8 Let the random variables X , X , ... be iid having the common
i1
i2
pdf f(x; µ , σ), i = 1, 2, where f(x; µ, σ) = σ exp{-(x - µ)/σ}I(x > µ).
-1
i
Also let the X ’s be independent of the X ’s. We assume that all three
1j
2j
parameters are unknown and (µ , µ , σ) ∈ ℜ × ℜ × ℜ . Having recorded
+
1 2
X , ..., X with n ≥ 2, let us denote =
i1 in
i = 1, 2, and the pooled estimator U = ½
Pn
{U + U } for the scale parameter σ. The customary estimator of µ – µ
1n 2n 1 2
is Let us denote θ = (µ , µ , σ). Suppose that the loss in-
1 2
curred in estimating µ – µ by is given by the function
1 2
where a and t
are fixed and known positive numbers.
(i) Derive the associated fixed-sample-size risk R (µ – µ , –
n 1 2
) which is given by E [L (µ – µ , )];
θ 1 2
(ii) The experimenter wants to have the risk R (µ – µ , )
n 1 2
bounded from above by ω, α preassigned positive number. Find the
expression of n , the optimal fixed-sample-size, had s been known.
*