'tl
,I
}
RATE
O~
CONSISTENCY OF ONE SAMPLE TESTS OF LOCATION
by
v
Jana Jureckova
Department of Biostatistics
University of North Carolina at Chapel Hill
and
Department of Probability and Statistics
Charles University, Prague, Czechoslovakia
Institute of Statistics Mimeo Series No. 1268
February 1980
RATE OF CONSISTENCY OF ONE SAMPLE TESTS OF LOCATION
v
~
Jana Jureckova
Department of Probability and Statistics
Charles University
Prague, Czechoslovakia
~d
Department of Biostatistics
University of North Carolina
Chapel Hill, NC 27514, U.S.A.
ABSTRACT
Let
Xl' ""X n be a sample from a population with continuous
~e
distribution function
j
o <F(x)
F(x -8)
such that
F(x) +F(-x) =1
and
I
<1,
x sRI.
It is shown that the power-function of a monotone
test of H: 8 =8 0 against K: 8 >8 0 cannot tend to 1 as
more than n times faster than the tails of F tend to O.
8 -8 0
Some
-+
standard as well as robust tests are considered with respect to this
rate of convergence.
AMS Subject Classification:
Key Words
62F04, 62GlO
&Phrases: Power-function, Consistent test, Tails of the
distribution, t-test, Sign test, Wilcoxon one-sample test.
".
00
1.
INTRODUCTION
Let
Xl" ",Xn be independent random variables, identically
distributed according to a distribution function
F(x -8),
where
F
is a continuous distribution function satisfying
F(x) +F(-x)
1
=1
o < F (x) <
8
is an unknown parameter.
1
x e:R ;
(1.1)
1
The problem is that of testing the
H: 8 =80 against the alternative K: 8 >8 ,
0
The power-function of a consistent test tends to 1 as
hypothesis
8 -8
~
0
We shall consider the rate of this convergence and show that the rate
at which the tails of the distribution
F tend to
0 form a natural
upper bound on the rate of convergence of the power-function of any
monotone test.
More precisely, we shall show that the probability of
the error of the second kind tends to
0 at most n-times faster than
the tails of the distribution.
Noting this, we are interested in the behavior of some well-known
one-sided tests from this point of view.
It turns up that the
behavior of the standard t-test strongly depends on the underlying
distribution.
While its rate of convergence is near to the upper
bound in the case of normal distribution, this convergence becomes
very slow if the normal distribution is contaminated by a heavy-tailed
distribution.
The rate of convergence of the sign-test is always
slightly below the middle of the range, whatever is the basic
distribution, and similar is the situation of the robust version of
the probability ratio test.
The rate of convergence of the power-
function of the one-sample Wilcoxon test is bounded from below as
well as from above.
00.
-3-
THE UPPER BOUND ON THE RATE OF CONSISTENCY
2.
Xl' ""Xn be independent and identically distributed (i.i.d.)
random variables with continuous distribution function F(x -8) such
Let
that
F satisfies (1.1).
Consider the tests of the hypothesis
H: 8 =8 0 against the alternative
1
", (X) "'n"'" -
Y
n
o
where
and
K: 8 >8
0
which have the form
if Tn(X l -8 0 "",Xn -8 0 ) > c~n)
if T
= C(n)
n
a
if T
< C(n)
n
a
(2.1)
Yn are determined by the condition
E8 l/Jn (~) = a,
o
where
Suppose that the statistic
1
a e: (0, 2)
Tn (Xl' ... ,X)
n
Tn (Xl -t, ... ,Xn -t)
(2.2)
satisfies the conditions
is nonincreasing in
t
(2.3)
and
[X
x(n)
where
(n)
(a)
<8 0 ] => Tn(X l -8 0 "",Xn -8 0 ) < Cn
is the n-th order statistic in the vector
of order statistics corresponding to
'
(2.4)
~
... :::; X
(n)
Xl"",X n '
The following theorem gives an upper bound on the rate of
convergence of the power-functions of tests satisfying (2.1) -(2.4).
Theorem 1.
Under the assumptions made above, it hoZds
(2.5)
lim B(T , 8, a) :::; n
n
8 -8 -+00
0
where
-10gE 8 (1 -l/Jn(~))
B(Tn , 8, a) = -log(l -F(8 -8 )) •
0
Proof:
(2.6)
It follows from the assumption (2.4) that
~
P {T (X -8 ) < C }
a
0
8 n ,...,
~ P (x(n) <8 ) = (1 -F(8 -8 ))n
8
which implies (2.5).
0
0
o
-4-
3.
ONE-SAMPLE t-TEST
The standard one sample t-test has the form
,I. (X) =1
T (X -8 ) > t (n-l)
n ,. ., 0
a'
if
o/n ,...,
(3.1)
where
Tn (X,. ., - 8 ) =
0
IJ1"":T ex:n
(3.2)
,
n
1 n
_
X
-80 ) /Sn
= - L X.,
n n.1= 1 1
S2 =
n
!. L (X. -X) 2
n.1= 1
1
t(n-l) is the upper a-percentile of
a
degrees of freedom.
t
and
(3.3)
n
distribution with
(n -1)
The following theorem shows that, if the sample comes from a
heavy-tailed distribution, the power-function of the t-test tends to
1 only as slowly as the tails of
~
e
Theorem 2.
Let
F tend to
O.
Xl"'" Xn be a sampZe from a popuZation with the
continuous distribution function
F(x -8)
such that
F satisfies
(1.1) and
lim -log(l -F(x)
m·log x
= 1 for some
m >0 •
(3.4)
Then
lim B(T , 8, a) s 1 ,
n
(3.5)
8-8~
o
ho "1d
f., s
or any
J.f'
Proof:
a E: (0 , _1)
2
and for any fixed
n
~3.
It holds
pe{\~eo rn:T
<t n } =
po{~ - X >e _eo} , po{-x" >e -eo} ,
n
and it follows from Theorem 2.2, part (ii) of Jure~kova (1979b) and
from (3.4) that
-5-
lim
B(T, 6, a)
n
6-60~
-en -1)10gF{6 -6 0)
-log(l -FC6 -6 0))
))1
+
-10g[1 -F((2n -1)(6 -6
0
-log(l -F(6 -6 ))
0
J:S;
o
1.
The next theorem shows that, in the case of normal population, the
rate of convergence of the t-test is near to the upper bound given in
Theorem 1.
Theorem 3.
2
N(6, a)
Xl"" , Xn be a sample from the normal distribution
and let Tn be given by (3.2) and (3.3). Then
Let
lim
-
B(T , 6, a)
n
6-60~
holds for any
Proof:
a
E
1
(0, 2)
and n
~ .(1
n
t~n_l)]-2
+
vn-:T
'
(3.6)
~3.
The tails of the normal distribution are exponentially decreasing,
more precisely, it ho1s
lim -log(l 2-F(x))
x~
=1
.
(3.7)
x
2i
Using Markov inequality, we can write for any
E,
0 <E <1,
-6-
~ exp{- _(e_--:80,--)_2 n(l -e:) (1 +
2ci
so that
lim {-:'10g[E (1
8
8-8 0-+00
holds for any
Remark.
E, 0
-l/Jn(~)}] e2ci(8 -e o)-2} ~ n(l
< e:< 1;
Suppose that
distribution function
F(x) =(1
where
-e:) [1 +
o
this implies (3.6).
Xl"",X n
F(x -e}
-A)cI>(~)
is a sample from the population with
such that
+
AG(x),
x ER
1
(3.9)
cI> is the standard normal distribution function,
a fixed number,
a <A
<1,
a
>0,
A is
and G is a heavy-tailed absolutely
continuous distribution function satisfying (3.4).
interprete
;;]-2 . (3.8)
We could then
F as a normal distribution function contaminated by a
heavy-tailed distribution, and it follows from Lemma 2.1 of Jure~kova
(1979b), that
F is again heavy-tailed.
It means that, for a distri-
bution satisfying (3.9), t-test.has a slowly increasing power function
in the sense of (3.5).
4.
SIGN TEST
Let
Xl'" "Xn be LLd. random variables and let
Xl -eo""'Xn -eO'
F is continuous and sYmmetric then, under
denote the number of positive components among
distribution function
Tn(X l -eo'" "Xn -eo)
If
-7-
H: 6 =6 0,
1
bel'
Tn(X l -6 , ""X -6 0)
0
n
has the binomial distribution
n).
denote the largest integer satisfying
C(n)_l
Let
:~o (~)
(;)n
S
1
-a ;
(4.1)
H if T (X -6 ) >C(n).
0
n '"
a
the sign test then rejects
The following theorem shows that the power-function of the sign
test tends to one
of
approXimatively(n-C~n))
times faster than the tails
F tend to zero, whatever is the basic distribution.
Theorem 4.
Let Xl' ... ,Xnbe a sample from a population with distri-
bution function
(1.1).
such that F is continuous and satisfies
Then
n -C
(n)
ex
If
S
B(T , 6, a) S
lim
-
6-60~
holds for any
Proof:
F(x -6)
1
ex t: (0, 2)
n
lim
6-eO~
B(T, 6, a) Sn _cCnY +1
n
ex
and any fixed n.
6 is the true parameter value then
binomial distribution
E6 (1
(4.2)
b(p, n)
Tn(~-60)
with p =F(6 -6 0 ),
C (n)_l
-~n(!)) ~ :~o (~)(F(6
has the
so that
-6 0))i(1 -F(6 _6 0))n-i
n]
c(n)-l
n_C(n)+l
~ cCnY -1 (F(6 -8 )) a
(1 -F(6 -8))
a
0
0
[ ex
which implies
(4.3)
Analogously,
-8-
C(n)
ES (l
-ljJn(~)) S i~O (~J(F(8
-8 0))i(1 -F(S _SO))n-i
(4.4)
holds for
S -SO >K,
where the last inequality follows from the fact
that, for sufficiently large
S,
(4.5)
holds for
0 si SC(n).
a
(4.4) then implies
o
5.
WILCOXON ONE-SAMPLE TEST
Denote
T (~-8 0 )
n
n
=
L sign(X.1
-8 0)R:C1x. -Sol)
. 1
1
1=
1
(5.1)
jX i - 80 1 among IX I -sol~"" IXn -sol.
Consider the Wilcoxon one-sample test with critical values based on the
where
Rr(IX i -Sol)
is the rank of
normal approximation; this test has the form
1 if Tn (X-8
,.., 0) > °nla
ljJn(~)
,
=
if
T
0 if
T
y
n
n
=
°nla
(5.2)
< °nla
where
2
on
=
1
-gn(n +1) (2n +1) ,
l
a
=~
-1
(1 -a)
(5.3)
(5.4)
-9-
~
and
is the standard normal distribution function.
We shall show that the rate of convergence of the Wilcoxon test
is bounded from below as well as from above; this means that the test
is neither too slow nor too fast, whatever is the basic distribution.
Let Xl' ""Xn be a sampZe from a popuZation with
continuous distribution function F(x -8) such that F satisfies
Theorem 5.
(1.1) .
Then" for
~
an (a)
a
~O
and n
.15
lim B(T , 8, a) ~
n
8-80~
>
~
T
- 3(3
a2
-212
j
+1
it hoZds
1im B(Td' 8, a) ~ b
8-80~
n
n
(5.5)
where
(5.6)
and
b
[x]
n
=[0.8n] +1;
(5.7)
denotes the integer part of x.
Proof:· Let
S
denote the number of positive components. among
n
Xl .8 0 "",Xn -8 0 ,
Then
(5.8)
E (1 -lj!n (:s)) ~ PO(Sn ~d)
n
8
where
dn
=
...!-(n
.,12
Actually, if
+
·~3a ).
VS
S
n
>d,
n
then for
o.~0.15
and
n
~nO(o.),
1
l
Tn (X~ - 8 ) ~ ;S2n (Sn +1) - 2 (n -S)(S
n n +n +1)
0
·(S +1) _ n(n +1)
= Sn n
2
r
t
-10'-
~
T (J
a n
which together with (5.2) implies (5.8) and moreover,
E (1 -ljJ (X)) ~ p (x([n-dn]) ~8 <0)
8
8
n"'"
0
and this gives the first inequality in (5.5).
On the other hand, we
have
v
and the rest of the proof follows from Theorem 3.3 of Jureckova (1979b).
6.
A ROBUST TEST
Let
Xl' ""Xn
be i.i.d. random variables, distributed according
to a continuous distribution function
F(x -8),
where
F(x)
satisifes
(1.1) .
F is not fully specified but is supposed to belong to a
If
neighborhood of a given symmetric unimodal distribution (e.g., of the
normal distribution) then we may use the following robust test, less
sensitive to the gross errors; put
n
I
T (X -8 )
0
n '"
. 1
~=
heX. -8 0 )
~
= -------~r ,
2
~
h (X. -X )]
i=l
~
n
where
h:
R1 +Rl
-
(6.1)
.
is a nondecreasing, skew-symmetric and continuous
function such that
hex) = h(k)
where
k > 0 is a given constant.
for x >k
The test then has the following form
-11-
1
= Yn
1/Jn(X)
o
where
T
a
= 4>
-1
if
Tn (X -eO) > Ta
if Tn
= Ta
if T
< T
n
a
AI
(1 -a).
The following theorem shows that in the case of heavy tailed distribution as well as in the case of distribution with exponentially
decreasing tails, the rate of convergence of the power-function of
(6.2) is always slightly below the middle of the range.
Let
Theorem 6.
Xl""'Xn
distribution function
be a sampZe from a popuZation with continuous
such that
F(x -8)
F satisfies (1.1) and
moreover~
it satisfies either (3.4) for some m >0 or
ll'm -log(l -F(x))
x~
box r
Then~
under the above
a'n (a)
.
lim
~
-
e-80~
= 1
p
Jor
some b >0 ,
r >0 .
(6.3)
assumptions~
B(T ,
n
e,
a)
~
lim
B(T , 8, a) ~b'
e-eO~
and n
2
>T ,
a
n
n
(6.4)
where
(6.5)
and
b~
Proof:
i
Let
~
n +1
(6.6)
= -2-
denote the number of i's,
for which
Xi -eO >k,
,
=1, 2, ... , n . Then
Ee(l
-wn(~ll Pe{i~lh(Xi -e01
<
<
"allO 'h(k1}
~ Pe{~ ~ ~ + ~ Ta}·
,
(6.7)
-121
Actually, if Qn > '2(n + In La)'
then
n I L
i~l h(X i -eO) > 2(n +Laln)h(k) - (~- ¥)h(k) =Lalil h(k).
It follows from (6.7) that
i
I
Ee(l
•
where
-ljJn(~)) ~ Pe(X(s) -eO <k)
1
s = [2(n -Lain)];
this implies the first part of (6.4).
On the
other hand,
and this gives the second inequality in (6.4) similarly as in Theorem
o
v
3.2 of Jureckova (1979b).
REFERENCES
i
e
v.
~
Hajek, J. and Sldak, Z.
Prague.
(1967).
Theory of Rank Test.
Academia,
Huber, P.J. (1965). A robust version of the probability ratio test.
Ann. Math. Statist. ~£, 1753-1758.
Jure~kova, J. (1979a). Finite sample comparison of L-estimators of
location. Com. Math. Univ. Carolina 20, 507-518.
""""
Jure~kova, J. (1979b). Estimation of location and criterion of tails.
Rep. SW 66/79, Math. Centre Amsterdam.