On Optimal Tests for Separate
~ypotheses
and Conditional
Probability Integral Transformations
by
C. P. Quesenberry and R. R. Starbuck
Institute of Statistics
Mimeo Series #985
Raleigh - 1975
ON OPTIMAL TESTS FOR SEPARATE HYPOTHESES AND CONDITIONAL
PROBABILITY INTEGRAL TRANSFORMATIONS
C. P. Quesenberry and R. R, Starbuck
North Carolina State University
SUMMARY
Consider the problem of testing the composite null hypothesis that
a random sample Xl' •.. , X is from a parent which is a member of a
n
particular continuous parametric family of distributions against an alternative that it is from a separate family of distributions,
It is shown here that
in many cases a uniformly most powerful similar (UMPS) test exists for
this problem, and, moreover, that this test is equivalent to a uniformly
most powerful invariant (UMPI) test.
It is also seen in the method of proof
used that the UMPS test statistic is a function of the statistics U , . .. ,
l
U _ obtained by the conditional probability integral transformations (CPIT),
n k
and thus that no information is lost by these transformations,
It is also
shown that these optimal tests have power that is a monotone function of the
null hypothesis class of distributions, so that, for example, if one
additional parameter for the distribution is assumed known, then the power
of the test can not decrease.
Two readily established but important properties
of CPIT transformations are given.
It is first shown that the statistics
given by these transformations are independent of the complete sufficient
statistic, and that these statistics have important invariance properties.
Some examples are given for particular
f~ilies.
These include testing the
two-parameter uniform family against the two-parameter exponential family
to illustrate the transformation approach given here for constructing UMPS
tests.
The problem of testing a scale parameter exponential family against
a shape parameter lognormal family is considered fOl' the
~hape
pararLeter
- 2 -
known} and for it unknown.
Some empirical power results have been computed
for the tests proposed here for these two problems, and these results are
compared with those of other writers.
~
key words:
Separate families, similar tests, invariant tests, NeymanPearson Lemma, conditional probability integral transformations, goodness-of-fit.
1.
INTRODUCTION
Let Xl' •.. , X be a sample from a continuous parent distribution. We
n
are interested in testing that this sample is from one family of distributions
against the alternative that it is from another family where these families
are separate as defined by Cox (1961).
Cox (1961, 1962) has given a general
method for deriving tests for these problems.
His tests are based on the
logarithm of the maximum likelihood ratio-MLR tests.
Jackson (1968) and
Atkinson (1970) have both considered the MLR method further and developed
some particular tests.
Invariant tests are also important for the separate families testing
problem.
Lehmann (1959) gave the general theory of uniformly most powerful
invariant (UMPI) tests.
In an earlier paper Lehmann (1950) attributes the
notion of invariance to R. A. Fisher, Hotelling, Pitman and others; and much
of its development to Hunt and Stein in unpublished work.
Hajek and Sidak (1967).
See, also,
Uthoff (1970, 1973) has considered UMPI tests for
several particular problems.
D,yer (1971, 1973, 1974) has considered several
tests for separate hypotheses problems, including MLR and UMPI statistics.
Antle, Dumonceaux, and Haas (1973) compared the power of MLR and UMPI tests
for some location-scale parameter problems and recommended the MLR test
over the UMPI test .(when they
differ), because it is more easily derived
.
- 3 and has relatively good power.
Dumonceaux and Antle (1973) give the MLR
procedure for discriminating between lognormal and Weibull distributions.
Most goodness-of-fit tests can be used as tests of separate hypotheses.
However, since these tests ignore the alternative hypothesis they would be
expected to have less power than tests that utilize knowledge of the alternative family.
There is no assurance that this loss of power will in fact occur,
unless the test with which a goodness-of-fit test is being compared has maximwfi
power among a class of tests to which both tests belong.
Goodness-of-fit
tests can have unexpected power properties, such as those reported by pyer
(1974), and independently by Stephens (1974).
They found empirically that
if the null hypothesis class is contracted by assuming a parameter to be known,
that the power of some tests is decreased.
This cannot happen for the most
powerful tests considered here, and a statement and proof of this is given
in Theorem 4.4. below.
The conditional probability integral transformations (CPIT) were introduced
in O'Reilly and Quesenberry (1973).
form a sample Xl'
... , Xn
These transformations can be used to trans-
from a k-parameter continuous null hypothesis class
to a set U ' ••. , U
of independently and identically distributed uniform
n-k
l
random variables on the unit interval -- i.i.d. U(O,l). Then many goodnessof-fit test statistics can be used to test the uniformity of the U's on the
(n-k) dimensional hypercube, and the test results applied to the original
composite null hypothesis testing problem.
The advantage of this approach is
largely practical, for it allows the use of the same distributional results
(tables, limiting distributions, etc.) to test a large number of different
null hypothesis classes.
exact test for
~
One can, by this approach, obtain immediately an
sample sizes for a large number of composite hypotheses
for which no test is presently available.
In contrast to this, the large
- 4number of goodness-of-fit tests that have been proposed for composite null
hypothesis problems have distributions that change when a different null r({potbezis
class is considered; and, in fact, from this perspective it should be observed
that the tests for separate hypotheses have different distributions when either
the null or the alternative family is changed.
A question arises as to the
power that can be achieved by this CPIT approach.
We derive here a test that
is most powerful among all tests based on the D's for testing separate families,
and show that the test obtained is, in fact, a UMPS test for the problem.
It
is further shown that this UMPS test and a UMPI test are equivalent under conditions commonly met.
The basic approach adopted here of seeking a most powerful test among the
class of similar tests is, we feel, obviously reasonable.
If one is interested
in testing that a sample is from a particular parametric family; such a,s
N(~,cr2), say; against some alternative outside the family, then the test should
treat all members of the family the same,
"equally" normal.
i.!.,
all normal distributions are
Also, all of the test statistics, of which we are aware,
that have been proposed for testing these types of composite null hypotheses
have distributions that are the same for every member of the null hypothesis
class.
The tests therefore are all similar, and only similar tests seem to ever
be used, anyway.
2.
THE MODEL:
TERMINOLOGY, NOTATION AND PRELIMINARY RESULTS
We shall frequently use terminology given in Lehmann (1959); Chapter 6,
invariance, is especially relevant.
Let X denote a Borel set of real numbers,
X
=
a the
Borel subsets of X, and
(Xl' ••• , X ) denote a vector of independently and identically distributed
n
(i.i.d.) random variables, each distributed according to an absolutely continuous
- 5 distribution P on the Borel space (I,u); and, further, suppose that P is
a member of a parametric class of distributions P
o
=
[Pe; 8 e
is assumed to be a k-dimensional Borel set with elements e
o}.
The set
=
(e , ••. , e ).
k
l
It is also assumed that there exists a k-dimensional sufficient statistic
(T , •.. , T ) for P (or 0), defined on the sample space (In,an ) =
k
l
n
n
(X x ••• x I, a x ••• x a). Also, put p = [p ; ~ = P x ••. x P, PeP},
T
=
n
i.~., pn is a class of product measures on (Xn,a ) corresponding to P.
class p
n
is also written as p
n
=
[P~; e eO}.
1he
If (Xl' •.• , X _k ) given T
n
has an absolutely continuous distribution, we say P has absolute continuity
rank n
k, i.~., a.c.r. P is n - k (cf. O'ReillY and Quesenberry (1973)).
Let g:
X ~ X be a one-to-one transformation, and let gn be the
corresponding one-to-one transformation of Xn onto In defined by
gn(x , ••• , x ) = (g(x ), ••• , g(x )). For each gn, suppose there exists
l
n
n
l
a function g: n ~ 0 (or pn ~pn) such that Pge(X e gnA) = Pe(X e A) for
every A e an.
Let G denote a transformation group on X, gn the corresponding
transformation group on Xn, and
G the
corresponding transformation group on
n (that Gn is a transformation group is easily seen, that G is a group follows
from Lehmann (1959), p. 214).
Denote by
as
the sub a-algebra of
a
induced by a statistic S; by
hl,O h2 the composition of a function hI with a function h ,
2
i.~.,
hI oh (·) = h (h (')); by I the indicator function of a set A. With the
A
l 2
2
usual abuse of notation the same symbol, ~.~., g or g-1 will be used to denote
a point function and the corresponding set function.
The following lemma
is a well-known result in probability theory that provides a convenient
starting point for this work.
Lemma 2.1
For g:
n
I ~ I, one-to-one onto, and S a statistic on (Xn,a ),
- 6 n
v
A € a .
e
a.s. P
Proof.
g
n
as
=
(2.1)
From the definition of an induced a-algebra it follows easily that
a
Sog-n
= IA
0
g
-n
,
Then by the Radon-Nikodym Theorem,
and (2.1) follows.
This completes Lemma 2.1.
A transformation group on a space is said to be transitive if its
maximal invariant is constant on the space.
Transitive groups on parameter
spaces will play an important role in this work.
Lemma 2.2.
(a)
If
G is a transformation group of increasing functions on I that induces a
transitive group G on 0; and
(b)
T is a sufficient statistic for 0, and is equivalent to T
every g
€
0
G;
then the distribution function of the conditional distribution of
n
(Xl' ••• , X ) for fixed T is invariant under G , i.~.,
n
gn for
- 7(2.2)
a.s.
~
Proof.
n
Vg e G•
Let
e e n be
fixed and
e'
exists age G such that for the corresponding
J
i
x :: [( Yl' ..., yn ); Y.1 ::;; x.;
1
n.
ge.
also be an element of
= 1,
g, e'
..., n} and x
=
= (xl '
Then there
Let
••• , x n ).
Then
a.s.
e'
a.s. P
e'
a.s. P
By the sUfficiency of T, the sUbscript
e can
by Lemma 2.1 and (b),
by sUfficiency of T,
be omitted on F, leaving
a.s.
~
n
where the exceptional set may depend on gn, and thus F(x , .• " Xn!T) is
l
n
almost invariant under G • However, since pn is a dominated family on a
Euclidean space, and both In and
Lehmann (1959), Theorem
n are Euclidean sets, it follows by
4 and discussion on p. 226, that the exceptional
n
set does not depend on g.
This completes Lemma 2.2.
It may appear that the conditions (a) and (b) are severely restrictive,
but they are satisfied and readily verified for most of the separate
that have been considered in the literature.
~amilies
Important families that are easily
verified include many location-scale families (for which G = [ax + b; a > 0,
_00
< b < oo}), and shape parameter families such as the Pareto (for which
G=
[
a
x ; a>
oJ).
- 8 -
3.
SOME PROPERTIES OF CONDITIONAL PROBABILITY TRANSFORMATIONS
Conditional probability integral transformations are introduced in
O'Reilly and ~uesenberry (1973).
In this section we assume that a.c.r. P
is n - k, i.e., that the conditional distribution of (Xl' ••• , X k) given
- -
n-
T is absolutely continuous.
Put
u n- k(x n- k; T(X l , · · · , x),
n xl' ••• , x n- k -..I..J = P(Xn- k ~ x n-.kiT, xl' •.• , xn- k - 1)
and u(X l ' ••• , xn_~ T)
= (ul ,
••• , u _ ).
n k
In O'Reilly and ~uesenberry (1973) it is
shown that (Ul ; ... , Un_k )' = (u (Xl' T), ... ,
Un_k(X _k ; T, Xl' ... , Xn _k _l })
l
n
are independently and identically distributed U(O,l) random variables. From
this and a result of Basu (1955), the next theorem is immediate.
Theorem 3.1.
If T
=
(T , ••• , T ) is a complete and sufficient statistic for
l
k
0, then (T , .•. , T ) aRd (U , • •. , U k) are independent vectors •
l
k
l
nThis theorem has important applications for constructing inference
procedures that may be alternatives to nonparametric or robust procedures.
The sufficient statistic T contains all the information for making inferences
within the family P (or 0), whereas the statistic U
information about the family P.
= (Ul ,
•.• , U _ ) contains
n k
Thus U may be used to make inferences about
the class P, such as a goodness-of-fit test for the class P, and T to make a
parametric test within P, and the independence exploited to assess overall
error rates.
Inferences based on U are considered in the following sections.
In the next theorem we shall require that the statistic T be dOUbly transitive
(£f.
O'Reilly and~uesenberry (1973)).
- 9 Theorem 3.2.
If the assumptions of Lemma 2.2 are satisfied, and T is doubly
transitive, then u is equivalent to an invariant statistic,
a.s. P
n
Vg
€
l.~.,
G •
By Theorem 2.4 of O'Reilly and Quesenberry (1973),
Proof.
u.
J
= E(F.(X.!T)lx
l,
J J
••• , x. 1)' j = k + 1, ... , n
J-
By Lemma 2.2 above, F.(x.1T) is invariant under G, j
J
J
=k
a. s. P •
+ 1, ••. , n.
The
result follows.
The following lemma is a consequence of the fact that the transforming
functions of (3.1) are (conditional) continuous distribution functions.
Lemma 3.1.
In the conditional space for fixed T
= t,
there is a.s. a one-to-
one correspondence between (u ' ••• , u _ ) and (xl' .•• , x ), l·~., (V , ..• , V _ )
l
n
n k
n k
l
and (Xl' ••• , X ) are equivalent statistics, in this space.
n
4. MOST POWERFUL SIMILAR TESTS FOR SEPARATE FAMILIES
We consider using the sample (Xl' •.• , X ) from a parent probability
n
distribution P on (X,G) to test the null hypothesis
(4.1)
against the composite alternative
(4.2)
and
~
n P l = ¢.
It will sometimes be useful to consider a simple alternative
We assume in the sequel that the a.c.r. of P
o
is n - k.
- 10 Let f(.) and F(.) denote the density and distribution function, respectively,
corresponding to p(.) • We assume that Po and P
and that at every x e
l
are both identifiable classes,
r at least one distribution in each class has a positive
Cox (1961) calls the classes Po and P separate fami~ies of
l
distributions if the density of an arbitrary member of either class cannot
density.
be obtained as the limit of a sequence of densities from the other class.
shall here be concerned entirely with such separate families Po and P
l
We
•
Recall that a test is called similar-a for testing H ~ K if it has
constant probability a of rejection for every distribution in P
precisely, ~ is a similar-a test function for H vs K if Ep (~)
e
e e no
o
=a
More
for every
For (u ' ••• , u _ ) as given by (3.1), let h (u ' ... , u _ ) denote
l
l l
n k
n k
the density of (U ' ••• , U _ ) when Xl' ••• , X are i.i.d. from P of K'.
l
n
l
n k
From the remark preceding Lemma 3.1, it follows that hl(u , ••• , u _ ) is
n k
l
zero a.s. except in the unit hypercube. The next lemma is a direct application of the Neyman-Pearson Lemma.
Lemma
4.1. The most powerful level-a test of
l' if h (u)
l
.(u) =
> c ,
(4.4)
.
{
0, otherwise ,
where c is determined by P[h (U) > c}
l
U(O,l) random variables.
Theorem
H versus K' based on u is
= a,
for U
= (Ul '
••• , Un_k ) i. i. d.
4.1. If T is a boundedly complete sufficient statistic for P o of
(4.1), then the test
~ ~
*
0
u above is a MPS-a test for H versus K' •
- 11 Proof.
By Lehmann (1959), Theorem 2, p. 134,
E [cp(X , .•• , X )}
l
p
n
=a
if PeP
o
,
if and only if
E [cp(X , •.• , X )IT} = a a.s. p
l
P
n
T
•
Thus to find a most powerful test in the class of similar a-tests it
is sufficient to find a most powerful conditional size-a test on the conditional
space of Xl' ••• , X given T, i.~., to find the most powerful Neyman-structure
n
test. But for T = t fixed, (Xl' ••• , X ) and (U , ••• , U _ ) are equivalent
l
n k
n
statistics by Lemma 3.1. Thus, the test cp is a MPS-a test. Theorem 4.1 is
complete.
It will sometimes be the case that cp does not depend on P for P e PI
of (4.2).
Then, of course, cp is a uniformly most powerful similar-a
(UMPS-a) test for H versus K.
Conditions under which such tests exist are
considered in the following theorem.
Theorem 4.2.
If the conditions for both Theorem 3.2 and Theorem 4.1 are
satisfied by P , then a uniformly most powerful invariant level-a (UMPI-a)
o
test exists for testing H versus K, provided the group Ql induced on PI by
G is also transitive.
Moreover, this test is equivalent to the MPS-a test
of Theorem 4.1, which is then UMPS-a.
Proof.
If cp is invarHtntlevel-al, then since G is tiransitive it follows from
Lehmann (1959), Theorem 3, p. 220, that
E (cp)
p
i.~.,
cp i.s a similar-a test.
=a
if PeP
o
Thus, if a test is MPS-a, it will be most powerful
invariant level_a, provided it is invariant.
But by Theorem 4.1, aMPS-a
- 12 -
test can a.s. be written as a function of u ' .•• , u _ only, and is 6 l
U
n k
measurable and invariant a.s. by Theorem 3.2. This completes Theorem 4.2.
If the null and alternative hypothesis classes are interchanged,
if the problem of testing PI vs P
-
0
is considered, then its solution can be
obtained directly from the solution for the problem of testing P
4.3.
vs P
0-
1
4.2. We state this in Theorem 4.3.
given in Theorem
Theorem
l.~.,
If the conditions for Theorem
4.2 are satisfied and if these
conditions are satisfied with Po and PI interchanged, then a UMPS-a' test
for PI ~ Po is given by the test ~'
given in Lemma
Proof.
Po
E
Po
Let ~
= l-W(U(x l ,
••• , x
n
»,
where
Wis
as
4.1 with a' = I-a.
= w(u(x l '
•.• , x » as in Theorem
n
4.1.
Then cp' = l-cp, and if
Po and Pl € PI' i t follows that EPl(cp') =~, say,
(~') = a' = I-a. Or, for ~ fixed a' is a maximum.
€
i~
a minimum for fixed
This completes Theorem
4.3.
The next definition, and, particularly, Theorem
results of pyer
(1974) and Stephens (1974).
4.4 are motivated by
They found empirically that a
number of well-known goodness-of-fit tests have the property that their power
is less when the value of a parameter is assumed known than for the case when
the same parameter is assumed unknown, under the null hypothesis.
definition and Theorem
In the next
4.4 conditions are given which assure that the power
of the UMPS-a test for a smaller null hypothesis family is never less than
that of the UMPS-a test for a larger family.
Two families of distributions on the same space
(X,u) are said to be
conformable if there exists a group G of transformations on X and corresponding groups
Gl
and
G2
on the parameter spaces that are transitive.
Consider two testing problems
- 13 H :
l
r lH
H :
2
r 2H versus
versus K :
l
r lK
'
r 2K
'
and
r lH c r 2H' r lK c r 2K '
distributions, i = 1, 2.
where
o:f
Theorem
Pl
€
4.4.
and
r iH
I:f ~l is UMPS-~ :for
and
r iK
(4.5) and
(4.6)
are con:formable separate :families
~2
is UMPS-~ :for
(4.6), then :for
P
and P e: P2i<:
1K
2
Proo:f.
The class o:f tests that are
similar-~
class o:f tests that are similar-~ :for
on
K :
2
r ik ;
i
= 1,
:for
(4.6) is a subclass o:f the
(4.5). Thus the power
o:f~.
1
is constant
2; and the test with the maximum power in the superclass must
have power not less than the test in the subclass.
This completes Theorem
4.4.
In the next section we will consider some particular examples, and we
conclude this section with some discussion o:f the results obtained.
4.1
Theorem
shows that any MPS test can be written as a :function o:f the CPIT U statistics,
and thus the search :for an MPS test can be made in the space o:f the U's.
In
this power sense, the CPIT··U-trans:formations thus contain all the in:formation
in the sample.
The equivalence o:f UMPS and UMPI tests is o:f considerable theoretical
interest, and in practice it o:f:fers the advantage o:f allowing the derivation
o:f these tests by the approach that is easier :for a particular case.
In
the derivation o:f the UMPS test by the CPIT trans:formation approach, the
- 14 main task is to find the marginal density hl(u , ... , u _ ) of (U , ••. , U _ )
l
l
n k
n k
under the alternative hypothesis. In many problems, but not all, this is a
difficult problem due to the rather complex nature of the u-transformations.
In section 5 we consider some examples using the CPIT transformation approach.
The results of Theorem
4.3
are of some theoretical interest, we feel, in
that it seems quite natural to use the same test statistic to test
as for testing
Cox
r l -vs r 0
r 0vs
Pl' This property is not shared by the MLR tests of
-
(1961), in general. This result also provides another possible method
for deriving the UMPS test, since the transformations and distributions involved
are different and one derivation will sometimes be much easier than the other.
Finally, it should be mentioned that the results of this section may be
very helpful for testing separate families for some cases When the assumptions
required for the optimal tests discussed here do not hold.
The most common
problem where the assumptions do hold is probably that of testing two locationscale parameter families.
However, if the null and alternative families are
not conformable, then some alternative approach
example, by using Theorems
~ay
be attractive.
For
4.1 and 4.2 it is often possible to find a MPS
test against one member of an alternative family, or, even against a subclass
of an alternative family; but the test may dBpend upon further nuisance parameters.
It may then be possible to obtain a nonoptimal but very good test by estimating
the parameters in the forgoing optimal test.
A problem of this type is considered
in section 5 below, where we consider testing a scale parameter exponential
family against a shape parameter lognormal family, and a
obtained
~y
veF~
good test is
estimating the lognormal parameter in the MPS test for testing a
scale parameter exponential family'against'a particular lognormal distribution.
- 15 5.
APPLICATIONS
In this section we consider two examples.
Example 5.1 Unif'orm ~
expoh~ntial
Let P o be the unif'orm distributions with densities
and PI be the exponential family with densities
A exp [-A(X-S)} I(S,co)(x) , A >
°.
Both of these are 10catioR-scale parameter families, and the conditions
f'or Theorem
4.2
are readily verified.
Uthoff (1973) has given the UMPI
test f'or testing Po ~Pl' which rejects for small values of (X-X~)j(X(n)-x(l).
We give the CPIT derivation now.
CPIT transformations for sample observations
Xl' ••• , xn f'or the uniform family are
i
where zl
= x(l)'
Suppose zl
Zi
= xi'
i
= x.J
zn
= x(n)'
and Z
= j+l,
n
= Xk ,
= 2,
••• , n-l
and the other z's are defined as follows.
j <k
••• , k-l; and Zi
Then zi
= x i +l '
= xi_I'
i
= 2,
i = k, ••• , n-l.
••• , j ;
It can be
verified by direct Jacobian methods that these transformations give i.i.d.
U(O,l) random variables.
To find the UMPS test the joint distribution ofu , ••. , u _ must be
2
n l
obtained under the assumption that Xl' ••• , Xnconstitute a random sample from
an exponential (S,A) distribution.
Without loss of' generality, let (S,»
= (0,1)
The constant in the density hI is not needeq and will be omitted in the following
development.
The joint density of zl' .•• , zn is
•
- 16 n
f (zl ' .•. , Zn)
cc
n-l
exp (- I: z.) I (0 CXl) ( zl )I (
CXl) (z ) IT
i=l 2 ,
Zl'
n i=2
The joint density of u '
1
... , u n
g(u , ••• , un)
l
cc
is
n-2
n 2
u - exp [-nul-u (1+ I: u.)}
n
n
.22
2=
n-2
I(O,CXl)(Ul)I(O,CXl)(Un )
i~2 I(O,l)(ui ) •
Integrating out ul and un gives for the joint density of u 2 ' ••• , un_I'
h1(u '
2
.0.,
u n_l
) cc (1
n-l
(
) n-l
+ I: u.)- n-l IT I
()
i=2 2
i=2 . (0,1) u i •
The rhs of (5.3) expressed in terms of zl'
•.• , Z
n
is
or, in terms of the original x's this is
By Theorem
4.1 the hypothesis of uniformity is rejected in favor of the
exponential alternatives if the quantity in (5.5) exceeds a constant, or,
equivalently, if
where P(T
e,u
< c\r 0 ) = a .
Further, by Theorem
4.3
the UMPS-a test for testing PI ~Po' i.~.,
exponential against uniform, is given by rejecting if
.. 17 ..
e,u > c ,
T
where
peTe,u > clP )
l
Eiainpl'e 5.2
Exponent'ial
~
=~ .
lognormal
The problem of deciding whether data is exponentially or lognormally
distributed arises in the study of survival times of microorganisms which have
been exposed to a disinfectant or poison
(£!.
Irwin (1942)).
Cox (1961,
1962) developed an MLR test for this problem and gave the asymptotic distribution of the test statistic.
For this problem P
o
is the family of distributions
with densities
A exp (-AX)I(O,oo)(x), A> 0 ,
and PI is the family with densities,
We shall consider two cases.
First, we consider testing P o of
against a particular member of PI of
(5.9),
For the second case we consider testing Po
(5.8)
i.~., the case for a known.
~Pl
with a unknown.
Since P
o
and PI are not conformable, the test obtained will not be UMPS or UMPI for
this second case.
~
I, a known
Srinivasan (1970) has studied the power of two Kolmogorov-Srnirnov type
goodness-of-fit statistics for this problem for a number of values of a •
See also, Schafer, Finkelstein, and Collins (1972) for corrections to
Srinivasan's results.
... 18 '"
By Theorem 4.2, the MPS and MPI tests are equivalent here, and we
obtain the MPI test by applying a lemma of Raj ek and Sidak (1967, p.' 49),
which says that the MPI test rejects for large values of the statistic
co
So v
n-l
fl
(
vx l '
... ,
co
VXn)dV/S Vn-lfo.(VX ,
l
... ,
o
VX )dv ,
n
(5.10)
where
f
o (Yl' ... , y n )
and
This same formula can also be obtained from Lehmann (1959).
The denominator
of the ratio in (5.10) is
n
(n-l)~( ~ x.)-n .
i=l I
The
numer~tor
n
of the ratio in (5.10) is proportional to
n
co
( It x.I )"'lS v-I exp ['" ~ (In x.I + In v)2/2cl}dV
i=l
0
i=l
~.
n
co
n
= (IT x.)-ls
exp [- r: (t+ln x. )2/2c,.2}dt, letting t = In v,
i=l I
... co
i=l
I
n
n
n
2
2
= (2ncr/n)n( IT x.)"'l exp [_[ ~' In2x. .. ( ~.. In x.) /n]/2cr }
I
. 1
I
i=l I
i=l
l=
Therefore the test reduces to rejecting exponentiality if
- 19 n
n
1
n
(.t x.)n( IT x.)- exp [-[ t
i=l
l
i=l
2
i=l
l
n
In x. - ( t
l
i=l
2
2
In x.) /n]/2a} > c(a)
l
or, equivalently, if
n
n
T (a) == In( t x.-) - ( t In xi)/n
e,L
i=l l
i=l
where P(Te,L(a) >,c(cr)
\fl o )
=Q'
It is readily shown that this test is
equivalent to the RML test of Cox (1961).
We have computed by Monte Carlo methods some significance points for the
statistic T L(a) for n = 10, 20, and for a = .01, .05, .10. For these
e,
points we have fUrther computed the power of the test for seven values of cr
These values are given in Table 5.1.
using 5000 samples.
All values were obtained by simulation
- 20 TABLE 5.1
Critical Values and Power of T L( cr) Test For Discrimimtting
e,
Between Exponential and Lognormal Distributions
H:
(O,A)
Exponential
K:
2
Lognormal (0,0' ) , cr known
01=.05
01::;:.01
<1'=.10
n
0'
c( 0')
Power
ck)
Power
c( 0' )
Power
10
0.4
0.6
0.8
1.0
1.4
2.0
2.4
1.63
2.07
2.24
2.38
2.75
3.07
3.20
.93
.38
.08
.08
.24
.64
.81
1.16
1.91
2.18
2.32
2.63
2.91
3.02
1.00
.80
.36
.21
.40
.78
.90
.84
1.80
2.14
2.30
2.58
2.84
2.93
1.00
.92
.53
.35
.53
.85
.94
20
0.4
0.6
0.8
1.0
1.4
2.0
2.4
1.63
2.53
2.84
3.02
3.36
3.65
3.75
1.00
.93
.43
.23
.48
.91
.98
1.16
2.34
2.78
2.99
3.28
3.54
3.63
1.00
1.00
.76
.45
.68
.96
·99
.83
2.23
2.73
2.96
3.25
3.49
3.58
1.00
1.00
.89
.63
.77
.98
1.00
The second digit after the decimal is in some cases in doubt.
The purpose of this table is to allow comparisons of other tests with
this best test.
The same values of 0' have been used as were used by Srinivasan
(1970), and by Schafer, Finkelstein, and Collins (1972).
5.1 are
~
The powers of Table
least upper bound for any test for these hypotheses.
It should be
pointed out, however, that it is not reasonable to expect a general goodnessof-fit test such as that of Srinivasan to compete well with the MPS test,
because the goodness-of-fit test does not assume knowledge of the alternative,
but is, presumably, effective against a broader range of alternatives.
- 21 ~
II, 0' unknown
The families of distributions with densities given by (5.8) and (5.9)
are not conformable.
The exponential family is a scale parameter family,
g(x) = ax, a > o} induce the transitive group
the transfromations [g:
G=
i.~.,
[g: g(O')
tions [g:
= acr, a > o} on the parameter space. However, the transformaa
g(x) = x , a> o} induce this same transitive group Gfor the
lognormal family.
Therefore the forgoing theory gives no UMPS test, and we
shall construct a (nonoptimal) test by replacing 0'2 in (5.13) by a sample
estimator.
We shall use the estimator
,,2
cr
=[
n
L:
2
n
2
(In x.) -( L: In x.) !n}!(n-l) ,
].
i=l
].
i=I
which has two important properties.
It is invariant under scale transforma-
tions, and it is unbiased for 0'2 under the lognormal alternatives.
Replacing
0'2 in (5.13) by &2, a test for Po of (5.8) against PI of (5.9) is given by
rejecting if
*
T
e,L
n
E
n
In( L: x.) ... ( L: In x.)!n
. 1].
. 1
J.
J.=
J.=
n
2
n
2
-(n-l)ln[ L: (In x.) -( L: In x.) !n}!2n> c ,
i=l
].
i=l
J.
(5.16)
where peT* L > clp } =a .
e,
0
We give in Table 5.2 some significance points for the statistic T*
e,L
for n = 10, 20, and a = .01, .05,.10.
The power of this test has also been
computed for the same values of 0' as were given in Table 5.1, and these
values are also given in Table 5.2.
on 5,000 samples.
doubtful.
The values in Table 5.2 are all based
In some cases the second digit after the decimal is
- 22 TABLE 5.2
*
Critical Values and Power
, of Te, L for Discriminating
Between Exponential and Lognormal Distributions
H:
Exponential
(O,A)
K:
Lognormal (0, (j
2
),
(j
unknown
Reject H if T*
L> c
e,
01=.01
Ct'=.05
Ct'=.10
(j
Power
Power
Power
10
Ct'= .01
c = 2.02
Ct' = .05
c = 1.90
Ct' = .10
c = 1.84
0.4
0.6
0.8
1.0
1.4
2.0
2.4
.99
.34
.08
.04
.17
.55
.75
.99
.69
.26
.13
.29
.67
.83
1.00
.83
.43
.26
.39
.75
.87
20
Ct' : : ;
c =
Ct' =
c =
Ct' =
c =
0.4
0.6
0.8
1.0
1.4
2.0
2.4
1.00
.85
.27
.12
.40
.86
.97
1.00
.98
.63
.35
.55
.91
.98
1.00
.99
.78
.52
.66
.93
.98
n
c
.01
2.17
.05
2.10
.10
2.07
It is suggested that the power results given in Tables 5.1 and 5.2 be
compared with those given by Schafer, ~.~ (1972) for the statistics posed
by Srinivasan and Lilliefors.
Such comparison reveals that the power of the
T
test is largest in all cases, as it must be, and that the power of the
e,L
T*
statistic is generally intermediate between that of T L and the best
e,L
e,
of the other two statistics. It is hoped that this case is representative
of a larger class of situations, and that tests with high power can be obtained
for other separate families problems which are not conformable, by substituting
invariant estimators for nuisance parameters in pointwise optimal tests.
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