Chapter 13. Rank, Sign and Permutation Statistics
13.1 Rank Statistics
Simple linear rank statistic (p.173)
N
TN =
cN i aN (RN i )
i=1
where
RN i : rank of Xi among X1 , , XN
aN (1), , aN (N ) : scores
cN 1 , , cN N : constants
Notations
order statistics
Chapter 14. Relative Eciency of Tests
14.1 Asymptotic Power Functions
Terminologies and notations
null vs alternative : H0 : 0
vs H1 : 1
test statistic and critical region : Tn Kn
test : n = n (Tn ), 0 n 1
power function : n () = E n or n () = P (Tn Kn )
Chapter 12. U -Statistics
12.1 One-Sample U -Statistics
X1 , , Xn : r.s. from a population with cdf F
estimand : = EF h(X1 , , Xr )
h : symmetric kernel of order r
U -statistic : Un =
n 1
r
h(X1 , , Xr )
where the summation is with respect to
= (1 , , r
16. The space D[0, )
Denition
Dt = D[0, t]
D = D[0, )
For x, y Dt
x
t
= supst |x(s)|
t = cfw_continuous, strictly function from [0, t] onto itself
= cfw_continuous, strictly function from [0, ) onto itself
t
= supr<st log sr
sr
d (x, y) = inf t cfw_
t
Appendix 4. Metric Space
metric and metric space (S, ) : (, ) 0
(i) (x, y) = (y, x), x, y S
(ii) (x, z) (x, y) + (y, z) x, y, z S
(iii) (x, y) = 0 x = y
open / closed balls
B(x; r) = cfw_y : (y, x) < r, r > 0
B(x; r) = cfw_y : (y, x) r
-neighborhood
A = c
Appendix 3. LAN and Contiguity
3.1 Regular Parametric Model
Regularly parametrized model (p.135, p.94 of (V) (V : Van Der Vaart)
P = cfw_P : , : open in Rk
p = dP /d, P
s()
(i) |
( nite)
p is L2 () continuously dierentiable, i.e., s() L2 () (L2 deriva
ti
Appendix 1. Projections in L2 space
Notations
L2 : space of square integrable functions of X1 , , Xn
[0] : space of constant functions
[A] : space of functions of XA (Xi )iA
[B] = cfw_g = g(X1 , , Xn ) : < g, f >= 0 for all f [B]
[A|B] = [A] [B] (p.42)
[A
Appendix 2. Rank Tests as LMPI Tests
Invariant testing problems
Wish to test
H : 0
vs K : 1 (1 = 0 )
based on X P P = cfw_P : .
(a) (invariance of the model)
P : invariant under G (a group of transformations)
(b) (invariance of the hypotheses)
g (0 ) = 0