Page 610 - Probability and Statistical Inference

13. Sample Size Determination: Two-Stage Procedures 587
587
13. Sample Size Determination: Two-Stage Procedures
. We denote for n ≥ 2
and θ = (µ, σ ). Suppose that the loss incurred in estimating µ by is given
2
by the function where a and t are fixed and known
positive numbers.
(i) Derive the associated fixed-sample-size risk R (µ, ) which is
n
given by E [L(µ, )];
θ
(ii) The experimenter wants to bound the risk R (µ, ) from above
n
by ω, a preassigned positive number. Find an expression of n , the
*
2
optimal fixed-sample-size, if σ were known.
13.3.3 (Exercise 13.3.2 Continued) For the bounded risk point estimation
problem formulated in Exercise 13.3.2, propose an appropriate two-stage stop-
ping variable N along the lines of (13.3.5) and estimate µ by the sample mean
. Find a proper choice of “b ” required in the definition of N by proceeding
m
along (13.3.9) in order to claim that E [L(µ, )] ≤ ω for all θ, a and t. Is it
θ
true that b > 1 for all positive numbers t? Show that is unbiased for µ.
m
2
13.3.4 Let the random variables X , X , ... be iid N(µ , σ ), i = 1, 2, and
i2
i1
i
that the X ’s be independent of the X ’s. We assume that all three parameters
2j
1j
are unknown and (µ , µ , σ ) ∈ ℜ × ℜ × ℜ . Having recorded X , ..., X with
+
2
1 2 i1 in
n ≥ 2, let us denote = (n –
for i = 1, 2 and the pooled sample variance = . The custom-
ary estimator of µ – µ is taken to be – Let us also denote θ = (µ ,
1 2 1
2
µ , σ ). Suppose that the loss incurred in estimating µ – µ by is
2
2
1
given by the function L(µ – µ , ) = a | { } – {µ – µ }| t
1 2 1 2
where a and t are fixed and known positive numbers.
(i) Derive the fixed-sample-size risk R (µ - µ , ) which is
n 1 2
given by E [L(µ – µ , )];
? 1 2
(ii) The experimenter wants to bound R (µ – µ , ) from
n
1
2
above by ω, α preassigned positive number. Find the the expression
*
of n , the optimal fixed-sample-size, had σ been known.
2
13.3.5 (Exercise 13.3.4 Continued) For the bounded risk point esti-
mation problem formulated in Exercise 13.3.4, propose an appropriate
two-stage stopping variable N along the lines of (13.3.5) depending on
and estimate µ – µ by Find a proper choice of “b ”
1 2 m
required in the definition of N by proceeding along (13.3.9) in order to
claim that E [L(µ – µ , )] = θ for all θ, α and t. Is it true that
θ
2
1
b > 1 for all positive numbers t? Show that is unbiased for µ – µ .
m 1 2