THl:: POWER OF TEE LIKELIHOOD RATIO TEST
OF WGATION IN
NONLI~
WGRESSION MODELS
by
A. R.
:J!n~ti t'l+te
of Statifltic,s
Ral,igh ,
~~pt~mber ~973
Mimeogr~PA ~eries
No. e89
GA~NT
The' Power of the Likelihood Ratio Test
Of IDeation in Nonlinear
n
by
A. R. Gallant*
ABSTRACT
The
Lik~lihoodRatiQ
a~ain$-qA:
e 1=
eo
Test statistic T for the hypothesisH:
e= eo
iscQnsideredwhen the data are f];enera,ted acco:r;-ding to
the nonlinear model y
= f(x,e)
+ e with variance unk..'1own.
X is obtaineq, such that n.(T-X)
converges in probability to zero; the
distribution function of Xis derived assuming normal errors.
The power of the Likelihood Ratio test is tabulated for selected sample
It
sizes and selected departures from the null hypothesis byusin,g the distribution
tunctionot X to approximate the distributionf'unction ofT.
Monte-Carlo
l?ower estimates for an exponential model are compared to powerpc1ints calculated
us1ngthis approximation to gain a feel for the adequacy of the approximation
. :l?rofessor of Economics and Statistics,
........'* Assistant
North.Carolina·Sta.te University, Raleigh,
1.
INTRODUCTION
This paper considers the hypothesis o£ location:
e == eo
H:
at the
X
t
~
against A:
level o£ signi£icance when the data are responses
generated according to the nonlinear regression model
The unknown parameter
e
is known to be contained in the
which is asubseto£ the p-dimensional reals.
The inputs
X which is a subset o£the k-dimensional reals.
x
t
The errors
et
independent and normally distributed with mean zero
The Likelihood Ratio test and the large sample
statistic are obtained in Section 3.
distribution is tabulated at selected departures
~elected
sample sizes in Section 4.
Monte-Carlo
with the large sample values £or an exponential model in Section 5 in order to
gain a £eel f'or the adequacy o£ the large samPle approximation
Section 6 contains summary and concluding remarks.
The results presented in this paper are £orthe case
known, the large sample distribution of'
but not tabulated, in
2.
[2;3].
NOTATION AND ASSUMPTIONS
notation will be useful
Given the regression model
_1
e
2
where
..
e€
cRP,
0
the observations
(t = 1, 2, ••• , n),
and the hypothesis of location
H:
e = eo against A; 8 j
80
we define:
•• "
y )'
n
(n Xl),
(n X 1),
(n Xl),
~f(x,e) = the
p
Xl vector whose jth element is ~ f(x,e),
<:It; j
F(e) = the n XP matrix whose tth row is
p=
.L
P
F(e)[F'(e)F(e)J-~/(B)
= I-P
V'f(xt,e),.
(n X n) ,
(n X n),
en Xl),
'e
..
g(t;v,X) :: the non-central chi-squared density function with v
degrees freedom and non-centrality
X [4, p- 741,
G(X;v,X) :: r~ g(t;v,X)dt ,
2
n(t;'u' cr ) :: .the normal density function with mean
.;lnd variance·
IJ.
p(i,X) :: the Poisson density function with mean XIn order to obtain asymptotic results, it is necessary to specify the
as n becomes large- A general way of specifying
t
the limiting behavior of nonlinear regression inputs
behavior of the inputs
x
. Malinvaud I S definitions are repeated below for the readers .convenience;
complete discussion and examples are contained in his paper- .
Definition-
Let
a
be the Borel subsets of
sequence of inputs chosen from
a subset A of
X
-
Let IA(x)
The measure 'un on
X
A sequence of measures
weakly to a measure
f'
i,
,
IJ.
on
(x,a)
if for
CXl
X
and· [Xt }t::l be the
be the indicator function of
(x,a) is defined by
f..
~n }.
on
(X, a)
issai.d to
2
Cf,
·~
function
g with domain X
as
•
n
-+ co
The assumptions below are used to obtain the
of the Likelihood Ratio test statistic.
Assumptions.
The parameter space
n
of the p-dimensional and k-dimensional reals, respectively.
function
.
.C)2
f(x,e)
and the partial derivatives
o
The
and
oe.f(x,e)
~
..
0)
oa f(x, e) . are continuous on X X n. The sequence- of inputs tXt It=l
i j
are chosen such that the sequence of measuresVL n J:::;:l converges weakly· to a
o
measure /.L defined over (X,G)
The true value of a, denoted by a,
is
08
n.
contained in an open set which, in turn, is contained in
except on a set of
/.L
measure zero,it is assumed that
If
a = eO.
matrix
is non-singular.
density
As mentioned earlier, the errors
2
n[x;o,a } where
rl
ret}
is non-zero, finite, and unknown.
These assumptions are patterned after those used by Malinvaud
that the Maximum Likelihood (least squares)
it can be shown [2; 3] under these
"..
8(y)
Jii (~(y)
-
8°)1
minimizing
(y -
fCa»
I(y -f(e»
is asymptotically normally
2.&.-1 ..
(J....
over 0
The following theorem is proved in [2; 3] •
Theorem
1. Under the assumptions listed above, the estimator ;2(y) is
consistent for
a2 and is characterized by
....2
J.
cr (y) = e'P e/n +a
n
where
at
n.a
converges in probability to zero.
n
a = eO,
the true parameter value.
is non-singular for all
n
(The matrix
There is. anN
P is evaluated
such that
F/(eo)F(eo)
> N.)
Assumptions which allow 0
to be an unbounded set and do not
the second partial derivatives off(x,e)
re~uire
exist yet are sufficient for the
conclusion of Theorem I are given in [2].
3.
LARGE SAMPLE DISTRIBUTION OF TEE LIKELIHOOD RATIO 'IESTSTATISTIC
The Likelihood of the sample
y
n
L(y;e,c?)= (2ml) -2 exp [ -
is
~0-2(y
-f(a» 'ey - fee))} •
The Maximum Likelihood estimators under Hare
rv
8·= 8
and
0
"12
.
0 (y)
n-l(y _ f( e»)'(y -f( a });
over the entire parameter space they are
minimizing
fee»~
o
.
°
(y - f(e»/(y -
The Likelihood Ratio
over 0
and
is, therefore,
e = e} o <
max{L(y, e, (2 ):
°
0
2
2
O<o<oo}
Thus, the Likeliho9dRatio test has the form:
when
< oo}
=
l
-~J
a(y)
reject the
.
=
~(y)
that
~
T(y) :::
i)xl
2
o(y)
is larger than
pETey) > c
cwhere
Ie
::: e0 ]
.
:::et-
The following lennna is needed to prove the main result of this section.
Lemma 1.
Under the Assumptions listed in Section 2
J.
"'2
I/o (y) ::: n/e'P e + bn
where
n.b
n
Proof.
Let
converges in probability to zeroChoose
and
6 >0
n >N implies
>
whence
e
€
2
0
0
be given.
By Theorem 1, there is an N . • such that
T
0
andlet
a
be as in Theorem 1.
n
n
and
e'pJ.e/ n
in probability to zero_
A
<
such that
P(K.);;::: 1-6 where
,
since "'2
fJ (y)
€
<
,.
converge in probability to
between 0 and 1.
Kn
Thus,
e
€
K
n
e
in
K
n
implies
and n> N imply
1-6 :5 P(K ) .:5
n
Theorem 2.
I
By Taylors theorem, for
o2 . and
P[nll/.~2(y)
Under the Assumptions listed in Section 2tl::le
test statistic may be characterized.by
T(y) ::: X + c
n
where
n.c
n
converges in probability to zero and the distribution-function
of-Xis:
0"
Proof.
By the preceeding Lemma
T(y) = (y - fee »)'(y - fee »/e/pJ.e + b (y - f'(A » '(y - fee »/n
o
0
n'o
°
= (e+6) '(e-t6)/e'pJ.e + b (e+6) '(e+o)/n
n
=X
where/)
+ c
n
is evaluated at
andn.b
n
0
2
and
n
= n.bn(e/e/n + 2 o'ein
The term
e'e/n
by the Strong Law of Large Numbers.
_continuous function of x
00
Nown.c
converges in probability to zero.
probability to
asn ....
eO.
+ o'o/n)
converges in
The term
20 'e/n
we have
by the weak convergence of the measures
Ii-
n
•
Thus,
20 / e/n converges in probability to zero by Chebysheff's inequality.
last term 6 /6/n
converges to a finite constant as shown above.
converges in probability to zero.
Thus,
Set
-
z
--
= fa e
and
----
r = !cr 6
The random variables
-
independent with density
random .variable
n£t;O,l}.
(z + a:y) 'p(z + ay)
is
degrees freedom and non-centrality
rankp
Similarly,
(z+ by )'p.l(z + by)
'.
degrees freedom and non-centrality
independent because
Let
a>
p[X
> a +
.l
pp
= 0;
0.
1]
= p[ (z-J-y) '(Zfy) >
is
·2..l
b
r 'p y /2.
These two
'e
c*
point
can be obtained from a table of
F as follows.
When
p[X> c* ] :: P[e/pe! e/P.l e > c* -1] :: P[F > (n-p)(c*-l)!p] •
Let
Fa, denotethe
with
p
<l?
100 percentage· point of an F random variable
numerator degrees freedom and n-p
denominator
c* is given by
*
C
Note that is
0 <
that
a,<
1
c* ::; 1
:: 1
+ PFa!(n-p ) •
p[X> c* ] :: 1 when. 0=0.
then
It is assumed
* > l throughout the rest of the
c
and hence that
paper.
e·····
4.
PARTIAL TABULATION OF THE POWER FUNCTION
The conclusion of Theorem 2 states that
probability to zero as
n
tends to infinity.
. rapid approach of the difference
that the probability P[X >t]
P[T> t]
n. (T-X)
converges in
This isa relatively
T-X to zero which leads one to expect
would be a good approximation to
even in small samples.
*
The probabilityP[X > c*
] with
c
chosen for
tabulated in Tables 1 through 9 for values of p,
thought to be representative
n,
a,
= .05
is
Al
and· A2
of those occurring most frequently in
applications.
.
'. *
The details of the numerical evaluation of P[X > c)
The density g(t;v'A)
maybe put in the form
g(t;v'A) :: t~::oP(i;A)g(t;V+2i,O)
•
are as
[4, p. 76]
,...
,
fl2
--.-.
0
..
=
-,
---
......
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--
. ......
•
"
..,.
-.. " ,." ,.
"
.
e
Power: p=2, n=30, c* = 1.23860
1.
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A
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1
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2
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4
5
6
8
10
12
-
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~
•
. c* = 1.32893
p=3., n=30,
Power:
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5
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....... _ ~ . _......... ~~R.••.....""' .... ""'!'.,
e
3.
." ....... '~ ..
_h
•• - _ • •
,.~.
* ='. 1.52060
Power:p=5, n=30, c
..'
,."'........,:.,_"'_,..._~~ .. ;,,_....,,"_... -",.'~, ...;,......... ,.->t><"
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l
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e
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ed.
IlL. d
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~.
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e
.
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A2
A1 .
0
·5
1
2
3
4
5
6
8
10
12
.050
.129
.218
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5. Power: p=3, n=60, c*. = 1.14532
A
2
A.l
3
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5
6
8
10
12
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6.
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Power: p=5" n=60, c = 1.21664
0
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1
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3
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p=2, n=l20, c* =1.05331
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0
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1
2
3
4
5
6
8
10
12
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·951
'·982
·994
0.0
: <~'i~:-::~::;;~-;:";:;::~~~;~~.'::,~7;:~':;y.,.' ,?:~.~-.I,f;;.:·;.J,:;1".;..)(~,:.t4~.T~'r,:,,~~~,);'·';;~·:·;"~""k'~:::"~.I:'::~.:·,~y,;,.:·;
•
.
•
j
•
·':~·:~'1o;~r~:~'~1'~'It:;~::;;;.. !~,: ..... , :"'.:t'''~?~.l:'~:,;;~ ;":~i:':":r',~i>~ , ':'.,:" '·~·.~."l"~·r,,\"' .~".~~.t!' ..-":"':~''':;;:
1".1Oll~''''
••
.... ~fa l~'
8.
,
l_~,'" ....~~~,..P""'"f..,."I<': .. ~,......... ~.~;~ .........................1I-·..... ~~ .._...............'...'"'""......,... :ot.~ •.
,.:,~,: ..-~ •. ;~ :;•. ' ~',;; .•." ...•.•:.
~. "-,
..,.;..,''''_<:..:.. '''''''~;.,,, ..,,,.....'''''!"''''W''·.....,...-.;'''~.~l' •• --~,.\ ....-...~...-....:~.,.'..~ ...-...•• ~
- _:.. ,~ .. :',
_
Power: .P=:5, n=l2Q, c* . = 1.06762
X1
. 0.0
1
2
3
'.'...• '.., ... -~• ..;.-;
"~~t
_-,._~-_•• ......----......---~:.-
5
6
8
10
12
.050
.113
.186
•:547
·504
.639
·748
.829
·928
·972
·990
.0001
.050
.113
.186
·347
·504
.• 639
·748
.829
·928
·972
·990
.001
.O~
.113
.187
·348 .
·504
.640
·748
.829
·928
·972
·990
.01
.051
.115
.189
·350
·507
.642
·749
·930
·928
·972
·990
.1
.062
.131
.209
·373
·528
.659
·762
.839
·932
·973
·990
·"'1',
~,"(~~ r"'~T."1~
r-
\'-I~'''~\''~'i''''''
'ttl...
""
< ,\, ...,.
. .,.I',..,...ri:
""~'''',
.
....., '
,..•. ,~.~""
,",'.,....,.•'.,.' ,-;'''''~'.''''-'''''''''
..,'
.~,
'. '.'j".. ·a...·,... ·...,•. ~,.•',.~..... ,....... ,;,..""':.:_'."""~ ••" • ...,;;.;.._,,_...._ _...._ . _ _., ......... ,~~_'
:_"_"
"" :."' " : •. ,.
·e"
9.
A2
p=5, n=120, c* = 1.09925
5
6,
8
10
12
·540
.652
·745
.873
·942
·976
.412
·540
.652
·745
.873
·942
·976
• 277
.412
·540
.653
·745
.874
·942
·976
.152
·2:(9
.414'
·542
.654
·746
.874
-943
·976
.164
.29 4 .
.430
·556
.666
·756
.879
·945
·976
·5
1
2
.050
.096
.150
·m
.412
.0001
.050
.096
.150
·277
.001
.050
.096
.150
.01
.051
.097
.1
.058
.1Cfl
0.0
0
Power:
'~.
!'":
Using' this expression and rearranging terms
X
*'. I *-1] 2 ;
-1] + .2CA2'[C
J 0 G(t [c*.
cx). /
n-p + 2i, 0) get; p+ 2j,
This expression was evaluated on an IBM 370/165 using the IBM Scientif'ic
SUbroutfne. Package [5] subroutines DIG.AM, CDI'R, 'and DQIJ.6.
A listin~ of' the
authors program is available on request.
5.
A MONTE-CARLO COMPARISON
In an ef'f'ort to gain a f'eel f'orthe adeCluacy of the !S.rge sample approximation
P[T >t]
~P[X >~]
in samples of moderate size, Monte-Carlo estimates
for the model
Thirty inputs were chosen from X
three times and the points
= [0,11
.8(.1)1
>
twice•. The paramete:r;ospacewas
8 = (1/2, 1/2).
0
null hypothesis and selected departures from the null hypothesis, five
and the null hypothesis as H:
thousand random. samples were generatedaccoI'ding to the tn~del withrl',
A.
. . ...
*
Thepoint,estimatep ofP[T >c 1
T exceeded c* to five thousand.
11
,..
Var(p)
*
= p[X > c*J • P[XScJ/5000
results are presented in Table 10.
Certain points Should be mentioned about the choice of values of
80
e FI
in the Monte-Carlo study.
1
1
('2' '2)
of the form
tion of "-1 and A2
!.r(cos(~ 1t), sin(€
over
3!».
The ratio
1
(el''2~
is minimi.zed
and, based on a numerical evalua-
maximizedfor
(1 ,
"-!"-l
e
of the form
1
(2
1
., ·.2-)
Three poj.nts were chosen to be of ·the first
form, .and two of the latter form.
Further,tvro sets of points were
Al • This was done to evaluate
power when "-2
changes while
Al is held fixed.
6.
REMARKS
Considering the standard errors of the Monte-Carlo
P[T> c),
*
the estimates support the use of
power in this
in~tance.
p[X> c* ]
Generalizations beyonathis statement
the usual risks of generalizing from Monte-Carlo studies.
Inmost applications,
A2
will be quite small relative t.o
Alas
This being the case, a valueofP[X> c]
*
adequate.
Note that if· "-2= O,then
denotes anon-central 11' with p
i
~
fJ.--;:
.
..
denominator degrees freedom, and non-centrality A [4, p. 77-78].
l
words, the firs·t row of Tables 1 through 9 are
In other
-
10•.. Monte-Carlo Power Estimates
•
Power
.Parameters
~.. ~
e2
81
Non-centralities
1..1
1..
2
Monte-Carlo
A
p[X> c* ]
P
~ -.'-
; .
·5
·5
0
0
.050
.0'32
•5398
·5
·9854
0
.204
.• 2058
.4237
.6849
·9853
.00034
.204
.2114
·5856
·5
4·556
0
·727
·7140
·3473
.8697
4·556
.00537
·728
·7312
.62
·5
8·.958
0
·957
·9530
.00308
12
F-test.
Thus, inmost applications, an adequate indication of .the power of
the Likelihood Ratio test can be obtained from charts of the power of the
F-test such as [1) and [7).
One last point might be.mentioned.
=1
+ P Fa!(n-p )
To reject
is equivalent to rejecting
S(y)
H when
T(y)
H when S(y)
exceeds
exceeds
F
a.
=
This form of the Likelihood Ratio test is ana1agous to theF-test used in .
linear regression and can be compared directly with tab1edF-test critical·
. As stated above, the behavior of· S(y)
A: e .~
eO will be adequately approximated in most applications by anon-central
F with p
nUmerator degrees freedom,
non-centra1itYA1 '
of the
•.
•.,.
under the alternative
n-p
denominator degrees·freedom, and
The size of A wil1give an
2
approximation~
REFERENCES
'.. ··i:
}1
[1]
Fox, M., " Charts of the Power of the F-test, « The Annals
Statistics, 27 (June, 1956), 484-97.
------------
[2]
Gallant, A. R., "Inference for Nonlinear Models,"
Mimeo Series No. 875, Raleigh, 1973.
=;;;....;~~;-.;...;;....;~~;.;;...;..;;.;..;..;.;.
[3]
Gallant, A. R., "Statistical Inference for Nonlinear Regression Models,"
Ph.D. dissertation, Iowa State University, 1971.
[4]
Graybill, F. A., An Introduction to Linear Statistical Models, Vol. I,
New York: McGraw-Hill, 1961.
[5]
IBM Corporation, System 1360 Scientific Subroutine Package, Version III,
. White Plains, New York: IR.\1 Corporation, Technical Publications
Department, 1968•
:. ~4 .
;1
~~.
;~1
::.1
,~
';"
.
~
"". ;
. .
-
. .
[6] Malinvaud, E., "The Consistency of Nonlinear Regressions," The Annals of
Mathematical Statistics, 41 (June, 1970), 956-69.
Pearson, E. S. and Hartley, H. 0., "Charts of the Power Function of the
Ana.lysis of Variance Tests, Derived from the Non-central F-distribution,"
Biometrika, 38(1951),.112-30.