13. Consider simple versus simple testing from a Bayesian perspective. Let e
have a Bernoulli distribution with P(B == 1) == p and P(B == 0) == 1 - p.
Given == 0, X will have density p 0 , and given == 1, X will have densit y
e
e
PI·
a) Show that the chance of accepting t he wrong hypothesis in t he
Bayesian model using a test function cp is
R(cp) = E [I{B = O}cp(X) + I{B = 1}(1 - cp(X))].
b) Use smoothing to relate R(cp) to E 0 cp = E [cp(X)
e = OJ and E 1cp =
E[cp(X) B=l J.
1
c) Derive t he test function cp* minimizing R( cp) . Show t h at cp* is a likelihood ratio test , identifying the critical value k.
12.8 Problems
18 Let F be a cumulative distribution function t hat is continuous and strictly
increasing on [O, CX)) with F(O) == 0, and let qa denote t he upper a
quantile for F, so F( qa ) == 1 - a. Suppose we have a single observation X
with
P0(X < x) == F(x/0) ,
x E IR, 0 > 0.
a) Consider testing Ho : 0
< 0o versus H 1 : 0 > 0o. Derive t he significance
level for the test cp == l(c,(X) )· What choice for c will give a specified
level a?
b) Let cpa denote the test with level a in part Ca . Show t hat t he test s
cpa , a E (0 , 1) , are nested in t he sense described in Problem 12.17, and
give a formula to compute the p-value P(X).
2
12.8 Problems
30. Let X and Y be independent with X
N(µy , 1).
Take 11µ11 2 = µ'; + µ~ , and consider testing Ho : µx = µy = 0 versus
H1: llµII > 0. For rotational symmetry, a test based on T == X 2 + Y 2 may
seem natural. The density of T is
rv
N(µ x, 1) and Y
rv
½Io(v'tllµll)exp{-½(t+ 11µ11 2 ) } , t > O·
'
otherwise,
0,
where I 0 is a modified Bessel function given by
ex cos w dw.
3
12.8 Problems
a) Show that Io(x) > Ib(x) and that xI6'(x) + I6(x) == xlo(x).
b) Show that xI6(x)/I0 (x) is increasing in x. Use this to show t hat for
c > 1, I 0 (cx)/ I 0 (x) is an increasing function of x. Hint:
log Jo (ex) - log JO (X)
4
=
c
l
8 log Io (ux) d
a
u
U·
12.8 Problems
c) Show that the densities !11µ11 have monotone likelihood ratios.
d) Derive the uniformly most powerful level a test of Ho versus H 1 based
on T.
e) Derive a level a test of Ho versus H 1 based on X and Y that has power
as high as possible if µ x == µy == l. Is this the same test as t he test in
part (d ?
f) Derive a level a test of Ho : µ x == ex, µy == cy , versus H1 : µ x # Cx
2
2
or µy # cy, based on T == (X - ex) + (Y- cy) .
g) Derive a 1 - a confidence region for (µ x, µy) r dual to the family of tests
in part (fl . What is the shape of your confidence region?
~
5
12.8 Problems
4 7. A random angle X has density
Pe ( x ) ==
exp[0 cos x]
( ) ,
21rio 0
x E [0, 21r) ;
0,
otherwise,
where 0 E IR and I 0 is a modified Bessel function (I0 (0) == 1). Derive t he
uniformly most powerful unbiased test of Ho : 0 == 0 versus H 1 : 0 -# 0
with level a.
6
12.8 P roblems
49. Because a good test of Ho : 0 E n0 versus H 1 : 0 E n 1 should have high
power on n 1 and small power on n0 , a test function ¢ might be chosen
to minimize
/3¢(0)w( 0) dA( 0)
Do
+
(1 - (3cp (0))w(0) dA(0) ,
D1
where A is a measure on n == n0 U n 1 and w > 0 is a weight function.
(With a natural loss structure, Bayes risks would have this form.)
a) Derive a test function ¢ * t hat minimizes this criterion. Assume t hat
Pis a dominated family wit h densities Pe, 0 E
b) Derive the optimal test function¢* explicitly if w is identically one, A is
Lebesgue measure on (0, oo) , Pe is the exponential distribution wit h
failure rate 0, n0 == (0, 1] , and n1 == (1 , oo).
12.8 P r o blems
n.
7