Chapter 3. Measures of Performance
3.2 Bayes Procedures
Bayesian statistical decision problem (X, X , P, , , A, l) :
prior density (distribution)
Risk function :
R(, ) = E l(, (X)
=
X
l(, (x)dP (x) for : X A
It can be extended to a randomized decision.
D
Chapter 2. Methods of Estimation
2.1 Basic Heuristics
(A) Frequency Substitution
X1 , , Xn r.s. from a population with cdf F
Method-of-moment estimator (MOME) (p.101) :
j
For q = h(m1 , , mr ) with mj = E(X1 ) (population moments),
q M OM E = h(m1 , , mr
Chapter 1. Models, Goals and Performance Criteria
1.1 Models, Parameters and Statistics
Example 1.1.1 (Sampling inspection)
Observe X H(N , N, n) with known N and n
: unknown fraction of defective N = 0, 1, , N
model P = cfw_H(N , N, n) : N = 0, 1, , N
Review C : Matrix Algebra
C.0 References
[1] Linear Algebra, 3rd edition, J. H. Friedberg, A. J. Insel, L. E. Spence,
Prentice Hall (1997)
[2] @< , s3, "@ (2005)
+
A/ f/<
H
$
C.1 Preliminaries
m n (real) matrix
A = (aij ) 1im rectangular array of real
Supplementary Note #2 : Distribution of Quadratic
Forms
1. Noncentral chi-square distribution
Y has the noncentral chi-square distribution with n d.f. and noncentrality
parameter 2 > 0 : Y 2 ( 2 )
n
(i)
d
Y
n
i=1
Zi2 where Zi N(i , 1) (i = 1, , n)
n
2
i
Review B : Topics in Probability and Statistics
B.1 Conditioning by a random vector
Conditional density (pdf ) :
p1|2 (y|z) =
where
p1,2 (y, z)
p2 (z)
for p2 (z) > 0
p1,2 = joint pdf of (Y, Z)
p2 = marginal pdf of Z
Remark : The above denition covers the
Supplementary Note #1 : Exponential Family
Theorem 1 (Completeness of Sucient Statistic) (TSH p.142, 2nd ed.)
dPX (x) = C()expcfw_ T (x)d(x), Rk
If contains a k-dimensional open rectangle,
then T (X) = (T1 (X), , Tk (X) is a C.S.S. for
Proof
May assume (
Review D : Convexity
Properties of a convex function on R1
1 f : convex on (a, b), a < b +
(i) f : continuous on (a, b)
f (x + h) f (x)
f (x + h) f (x)
, f+ (x) lim
(ii) f (x) lim
h0
h0
h
h
h>0
h<0
and f (x) f+ (x) f (y) f+ (y), x, y : x < y, x, y (a, b)
Review A : Basic Probability Theory
A.1 The Basic Model
Probability space : (, A, P )
: sample space
A : -eld over
P : probability measure(distribution) on A
-eld A over : class of subsets of such that
(i) A
(ii) E A E c A
(iii) Ej A (j = 1, 2, )
Ej A
Supplementary Note #3 : Boundary Diminishing
Condition
pdf (x; ) = h(x) expcfw_1 T1 (x) + + k Tk (x) A(), x X , E
with E being the natural parameter space. Assume that E is open and this family is of
rank k. Then the following condition implies the bounda