CONFIDENCE REGIONS FOR THE PARAMETERS OF A
NONLINEAR REGRESSION MODEL
by
A. RONALD GALLANT
Institute of Statistics
Mimeograph Series Noo 1077
Raleigh - July 1976
CONFIDENCE
REGIONS FOR TIlE PARAMETERS OF A
,
NONLINEAR REGRESSION MODEL
by
,/(
A. RONALD GALIANT
*A. Ronald Gallant is Associate Professor of Statistics and Economics,
Department of Statistics, North Carolina State University, Raleigh,
North Carolina.
Much of this work was done while on leave at the
Graduate School of Business, University of Chicago.
I am indebted
to Professors R. R. Bahadur, Albert Madansky, B. Peter Pashigian,
David L.
topic.
Walla~e,and
Arnold Zellner for helpful discussions of the
All errors are my own.
ABSTRACT
Three methods in common use for finding confidence regions for the
parameters of a nonlinear regression model are studied.
These are the
lack of fit method, the linearization method, and the maximum likelihood method.
The objective is to determine which of the three methods
is the preferred choice in applications.
The conclusion is that the
linearization method has the best structural characteristics but its
nominal confidence level is apt to be inaccurate.
A choice of the
likelihood ratio method is a compromise between structural characteristics
and accuracy.
The lack of fit method would only be chosen if accuracy
was of extreme concern.
-1-
1.
INTRODUCTION
Three methods in common use for finding confidence regions for
the parameters of a nonlinear regression model
(t == 1, 2 , ••• , n)
are studied.
They are the lack of fit method, the linearization method,
and the maximum likelihood method.
The objective of the study is to
determine which of the three methods is the preferred choice in
applications.
For this purpose, it is assumed that the functional form
of the response function
e
t
(t == 1, 2, ••• , n)
. an d un k nown
mean zero
f(x,e)
is known and that the errors
are independently and normally distributed with
.
a2
var~ance
The unknown parameter
e*
is a
p-dimensional vector known to be contained in the convex, compact
parameter space
®.
The asterisk is used to indicate that the true,
but unknown, value of the parameter is meant; its omission means that
e
is to be regarded as a variable for the
differentiation.
"
The inputs
purpo~e.of; ~.&.,
x t (t == 1, 2, ••• , n)
are k-dimensional
vectors whose values are known.
There is a considerable difference in the burden of the notation
between the more general case of finding confidence regions for a subset of the parameters of the model and the restricted case of a joint
confidence region for all the parameters; see,
~.&.,
[5, Sees. 4 and 5J.
Attention will be confined to the latter case for the most part to avoid
obscurity due to a complicated notation.
Little of substance is affected
by this restriction, excepting those cases where the lack of fit method
cannot be applied.
-2Let
y
denote the observed values of
(t· 1, 2, ••• , n)
Yt
arranged as an n-dimensional column vector;
(n x 1) •
~(y,e)
Consider a family of (Borel-measurable) test functions
defined
by
'1(
reject
H:
e =e
accept
H:
e ... e •
~(y,e)
~'(
As is well-known, a confidence procedure is in a one-to-one correspondence with such a family of tests.
observed
which
H:
y
This correspondence is:
put in the confidence region
e* = e
is accepted;
Ry ... {e e 8:
The probability that
~(Y,e)
R
y
e
those
R
Y
in 8
for
for
~.,
... O} •
includes
e
is given by
The nonlinear regression models considered here are presumed to
satisfy the regularily conditions of [4, 6J.
conceptually, range over
8
some values of
satisfaction of these regularily conditions.
Now, were
e
~'(
e*
to ,
will not permit the
For example, those
'1(
e
-:.~
on the boundary of
8
will not; typically, those
be a set of Lebesque measure zero.
e
which do not will
For those conceptual choices of
which do permit the satisfaction of the regularity conditions, the
e
*
-3three families of tests considered will have the property that
* = ole 'Ie ,a 2 1 = I-a
lim p[~(y)e )
t'l-ko
where a
oJ-
is a preassigned number between zero and one.
The notation
2
p('le ',a) indicates probability computed according to the n-dimensional
multivariate normal distribution with mean vector
(n x 1)
2
and variance-covariance matrix a I .
The procedure for finding confidence intervals associated with
(l-~)
such a family of tests will achieve the nominal
x 100%
confidence level in the limit if the true value permits satisfaction
of the regularily conditions.
(l~)
An approximate
x 100% confidence procedure is defined to
be a procedure with this property for all
of @ with Lebesque measure zero.
-Ie
e
in @ excepting a subset
On the other hand, an exact
(l-ce') x 100% confidence procedure is a procedure for which the equality
holds for all
e*
in
@ )
all
a2 > 0
the test to be defined, typically for
and all
n
n
large enough for
larger than
fit methodS, defined later, usually have this property.
p.
Lack of
Obviously, an
exact confidence procedure is included within the definition of an
approximate procedure.
Approximate confidence procedures, adjusting the definition
suitably for context, are in common use in a wide variety of situations.
Nonetheless, Some have argued that the use of such procedures is
invalid; see the discussion and references in [1, p. 305J.
argument is that, since
e*
and
a 2 are unknown, a confidence state-
e*
ment should be simultaneously correct for all
in
® and all
* = ole *,a21
a 2 > 0 ; the asserted probability should bound
from below.
The thrust
PC~(y,e)
If an approximate procedure is to meet this requirement
it must be a uniform approximate procedure.
That is, it must be true
for the family of tests defining it that
lim
inf
r:Hoo
~':
e ,a
*
2
p[~(y,e)
= ole *,a 21 = I-a.
The objection to approximate confidence procedures can, perhaps, be
made more forcefully by noting that if the equation above is not true
then there does not exist any sample size such that
e:
R
y
Ie* ,a 2 )
simultaneously for all
'i(
e
> 1"'Q'-e:
e: ® and
2
a > 0
The response to this objection is the argument that
are fixed i.n any given applicati.on.
e*
and a 2
It is not necessary to be able to
make confidence statements which are simultaneously correct for all
values of the parameters whatever they might be; it is only necessary
that they be correct for the parameters which obtain.
When an
approximate confidence procedure is used in an application there will,
in fact, be a sample size such that
This is all that is required for validity.
One may not accept the counter argument.
In the present context
this would require that lack of fit confidence procedures be employed
-5by default.
A uniform asymptotic theory is not available for the
J.:i.neari.zation and maximum likelihood fami.lies of tests.
Moreover, it
does not appear likely that a uniform asymptotic theory can be developed
which has sufficient generality to be useful :i.n applications.
The
difficulty is that many nonlinear regression models obtained from substantive considerations have singularities over regions of the parameter
space ® natural to the problem; see,
~.&.,
[3, 8,
9J.
Expected length, area, or volume, according as to dimension of e ,
:l.s commonly used to compare confidence procedures and is a tractable
cr:f.terion here.
The criterion will be termed expected volume regardless
of dimension and i.s defined by
Expected volume· Rn JR dm(e)dP(Yle * ,a 2)
y
J
where
m(e)
denotes Lebesque measure on the p-dimensional real
numbers.
Recall that ® is compact
assured.
Given two approximate confidence procedures with the same
nominal
(l~)
x 100%
80
existence of the integral is
confidence level, the one with the,SiMiler
expected volume is judged the better of the two.
As Pratt [19J shows,
by interchanging the order of integration,
Expected volume
=J® Sn
[lN~(y,e)1dP(yle * ,a 2)dm(e)
R
The integrand is the probability of covering
e,
c (9)
cp
p[cp(y,e) ...
:=
Ole
2
'1(
,'1
l ,
and is analgolJs to the operating characteri.stic curve of a test..
The
essential difference .between the coverage probability function c. (9)
cp
and the operating characteristic function lies in the treatment of the
hypothesized value
8
and the true value of the parameter
the coverage function,
e
~'(
is held fixed and
8
e*
For
varies; the converse
is true for the operating characteristic function.
A comparison of two approximate confidence procedures correspond.
ing to the two families of tests
cp(y,e)
by comparing their coverage,functions.
is smaller than
c'l"(8)
for all
corresponding to the family
8
and
'f(y,e)
can be effected
If it can be shown that
c (8)
cp
then the confidence procedure
cp(y,e)
is the better of the two.
to show this, the integrals of the coverage functions over
Failing
® must be
compared.
Some notation which is used throughout the discussion is set
forth here for convenient reference.
Notation:
Given the regression model
(t
:I
1, 2,
the observations
(t ... 1, 2, ""
and the hypothesis
H:
define:
~~
e
:=
e
n) ,
... , n)
,
(n x 1) ,
(n x 1) ,
(n x 1) ,
F(e) = the
n x p matrix with typical element (o!oej)£(xt,e)
where
t
is the row index and
j
the column index,
(n x n) ,
Q(S)
= I-P(e)
C(e)
= [F
1
(n x n) ,
(e)F(S),-1
0(9) = £(e * ) - £(e)
"e = the
p x 1
C
= C(e)
F
= F(e *)
A
,
,
(n
x p) ,
x 1) ,
vector minimizing
;, I (n-p) ,
s 2 = SSE(e)
A
(p
SSE(e)
over ®',
-8P
III
Q
III
pee*) ,
*
Q(e )
p~str1butio~.
~.
with mean
Let
g(t;\),~)
Let
density functi.on with
~
p(i,~)
\)
denote the Poisson density function
denote the non-central chi-squared
degrees freedom and noncentrality parameter
as defined in [10, p. 74).
distribution function.
G(t;",,~)
FI(t;"'1''''2'~)
Let
distribution function with
Let
"'1
be the corresponding
denote the noncentral F-
numerator degrees-freedom,
denominator. degrees-freedom and noncentrality parameter
in [10, p. 77J.
~
as defined
The central F-distribution function with corresponding
F(t;"'1''''2) •
The upper
of the central F-distribution is denoted by
F~("'1''''2)
degrees freedom is denoted by
denote
\)2
throughout.
F (p,n-p)
~
~
percentage
F
~
will
The critical point of the likelihood
ratio test is denoted by
*
~
2.
11:1
1
+ pF
~
I(n-p) •
THREE COMMON METHODS FOR FINDING CONFIDENCE REGIONS
The three methods considered are obtained from families of tests
for the hypothesis of location
H:
regarding
e*
lllII
e
against
A:
a 2 as a nui.saIlce parameter.
e* .p e
The confidence reg:1,ons
associated with these families would be identi,cal i f the response
function
f(x,e)
were 1.:i.near in the parameters.
Thus, the problem
of deciding which to employ :l.n an application ari.ses solely from the
-9-
nonlinearity of the regression model.
If interest centers in some subvector T
e
of the parameter
partioned according to
e
III
(p',T I) I
then the requisite families of tests would be generated from the
hypothesis
H:
regarding
p
~'c
T
III
and a
~
against A:
T
2
* +T
as nuisance parameters.
Of the three methods,
the lack of fit procedure is affected the most substantially in the
shift to the general case.
The details are given in [5, Sees. 4 and
5J.
2.1
Lack of Fit Method
If the hypothesis
H:
e* ,. e
is true then the components of the
n-vector
"e ,. y - f(e)
constitute a random sample from the normal distribution with mean
zero and unknown variance a
2
• In principle, any test of the null
hypothesis
(t
is a test of
H:
III
1, 2 , ••• , n)
e* ,. e
against A:
a r. s. from a
e'l( + e.
In practice, one is
interested in a test which is sensitive to departures from HI
due to
-10changes in location of the form e(e) • 6(8)
is extremely sensitive to,
~.&.,
rather than a test which
non-normality.
A test which is
sensitive to what are, in the present context, irrelevant departures
Ht can have the undesirable property of
from the null hypothesis
generating empty confidence regions with positive
were the assumptions true.
p~obability,
even
It would seem desirable to avoid confidence
procedures with this property if possible.
These are the essential ideas in the articles by Williams [23]
[121. Turner
and Halperin
~!l.
[22J and Hartley
[151 exploit the
same idea but do not necessarily guarantee by the method of construction
that their confidence regions cannot be empty with positive probability.
These citations are not exhaustive; applications of similar ideas can
be traced back further through the references in these articles.
The family of tests recommended by Halperin for finding an exact
(l~)
x 100%
confidence region for 8
*
is based on the F-statistic
R(e) • [[y-f(8)l fP (8)[y-f(e)1/pl/[[y-f(s)l'Q(s)[y-£(e)1/(n-p)} •
It may be computed using a linear regression program by regressing
y-£(e)
on
F(e)
with zero intercept; the numerator is the regression
mean square with
p degrees freedom and the denominator is the error
mean square with
n - p degrees freedom.
R(e) > F (p,n-p)
so that the family of tests defining the confidence
QI
procedure is
~(y,e)
1
R(S) >F
0
R(e)
QI
III
~
F
(III
•
The test rejects when
-11The probability that the procedure covers
e
is
where
.
2
ARl • 8'(e)p(e)8(e)/(2a ) ,
This expression is obtained by :rec'ognizing that
Me)
has the doubly
noncentral F-distribution with noncentrality parameters
Assuming that the least squares estimate '"e
ARl
and
is not a boundary
point of the parameter space a 11 , the test statistic R(e) takes
.,
on the value zero at e - note that V SSE(e) • -2F'(e)[y-f(e)]. Consequently, the lack of fit confidence region will contain '"e and be
nonempty.
The statistic
R(e)
also takes on the value zero at local minima
and maxima of the sum of squares function.
Thus, in an application
it is possible that a lack of fit confidence region consists of a union
of disjoint subsets of
with an example.
e.
Williams [231
illustrates this situation
This feature mayor may not be disconcerting to
statisticians but it is, in my experience, disconcerting to clients.
In many instances, an attribute of the response function
f(x,e)
is that there is a region A of the p-dimensiona1 real numbers with
infinite volume
Consequently,
(m(A). 00)
cR(S)
such that
5'(e)5(e)
~
a
for all
will be bounded away from zero on A and
e eA.
-12-
SA cR<e)dm(e) =
~/.
00
In these cases, expected volume is more a
e
characteristic of the volume of the parameter space
than an
intrinsic property of the confidence procedure.
If the restrictions embodied in the convex compact set
natural to the situation, this is not a problem.
e
are
It is then useful to
know that one confidence procedure has smaller expected volume than
another.
On the other hand, if the bound on
e
assumption that
lIell
implied in the
is compact is merely an artifact of the regularity
conditions rather than intrinsic to the problem, expected volume may
lose meaning as a basis for comparison.
2.2
Linearization Method
The least squares estimator is asymptotically normally distri-
e'Ie
buted with mean vector
s 2"c '2} •
consistently by
8(8)
for testing
= (e-e)
H:
8(e) > F (p,n-p)
r,
and a variance-covariance matrix estimated
This suggests the use of the statistic
C-1....
(e-e) I (ps 2 )
.....
~'(
e =8
A: e
against
~'e
~
e.
The test rejects when
so that the family of tests defining the associated
01
confidence procedure is
8(e) > F
01
8(e)
s: F
Of
The term linearization test stems from the following observation, which
is useful in computations.
F-test for the hypothesis
The test may be computed as the standard
H: 13
=e
against
A: 13 ""
e
i.n the H:near
·13model
z
=XS
+ e obtained by expanding
f(x,e) in a first order
Taylor's series about the least squares estimate
z
=y
-
fee)
+
F(e)e ,
X·
and
itself, and the statistic
s
2
F(e) •
The least squares estimate,
may be computed using either Hartley's
[14J or Marquardt's
.,..- [181 algorithm•
The probability that the procedure covers
by
6
may be approximated
[61
where
~S
• (e *-e)'C -1 (6 *-6)/(2a 2 ) •
The converage function may be computed using charts of the non-central
F-distribution such as in [20, p. 438-4551 •
Confidence regions obtained from the linearization family of tests
are nonempty and contain the least squares estimate.
where
(6:
~S(Y,e)
• OJ
intersects the boundary of
Neglecting cases
e , a
linear1~ation
confidence region is an ellipsoid with center at "e and eigen vectors
of "C as axes.
In snort, a linearization confidence region has the
familiar features of confidence regions customarily employed in linear
regression analysis.
The expected volume of the linearization confidence procedure, computed from the approximation, may be bounded independently of the volume
of the parameter space
e
as follows:
-14-
II
Thus,
th~
2tr(n/2) l[r(p:f~)r«n':'p)12)J}
linearization procedure can not have an expected volume de-
pending primarily on the volume of ® as is possible with the lack of
fit procedure, to the extent the approximation of the coverage
probability by
2.3
CS(e)
is valid.
Likelihood Ratio Method
."
The likelihood ratio test for the hypothesis
A:
e*
~
e
H:
e* • e
against
employs the statistic
and rejects when T(S) > C* •
The family of tests defining the
associated confidence region is
T(e) > c*
The denominator
o
T(e)
SSE(S)
of
Hartley's [141 or Marquardt's [18]
The numerator
SSE(e)
~
*
c
T(e)
•
may be computed using
algorithm, as mentioned previously.
may be computed by using
e
as a starting value
with either of these algorithms and limiting the number of iterationS
to one.
.. 15-
The probability that the likelihoo4 ratio procedure covers
e
may be approximated by [41
where
This function is partially tabulated in [41.
Confidence regions obtained using the likelihood ratio family of
tests are nonempty and contain the least squares estimate.
squares surface
below
G:~
*SSE(e)
"-
SSE(e)
may have local minima.
The sum of
When these minima are
it is possible that the likelihood ratio confidence
region will consist of a union of disjoi.nt regions.
Note~
however, that
the 1:i.kelihood ratio confidence region may include these local minima;
the lack of fit confidence region
local maxima and saddle points.
~
include them as well as include
One' would expect that confidence
regions consisting of diSjoint unions will occur with less frequency
in. applications using the likelihood procedure than using the lack of
fit procedure.
A feature whi.ch the likelihood ratio procedure shares with the
lack of fit procedure is that, for the same reasons, the integral
SA
cT(e)dm(e)
may be infinite for Some response functions.
As noted
before, expected volume then becomes more a characteristic of the
-16-
® than an intrinsic property of the confidence pro-
parameter space
cedure, to the extent the approximation of the coverage probability by
cT(e) is va1idi
/.
3.
EXAMPLES
The difficulty in attempting a general comparison of the three
confidence procedure is that several factors influence the outcome:
the form of the response function
(x : t
t
design points
= 1,
f(x,e) , the configuration of the
2, ••• , n1 , the sample size
2
magnitude of the error variance a .
true parameter
e*
n, the
and the actual location of the
At one extreme, these factors may interact such
that the regression model is so nearly like a model which is linear
in the parameters that there are no essential differences between the
three methods: confidence contours will be ellipsoidal and nearly
,coinciden~,
coverage probabilities computed according to the asymptotic
approximations will be accurate and nearly equal.
At the other
extreme are regression models which are so nonlinear that confidence
contours can be expected to differ markedly among the three methods and
asymptotic approximations will be so inaccurate as to be useless.
Beale [2] has derived measures of nonlinearity which are designed
to reflect the influence of these factors.
Examples have been examined
by Guttman and Meeter [111 to assess the effectiveness of Beale's
measures as a measure of the coincidence of linearization and likelihood
ratio contours.
success1/.
They find that the measures achieve a fair degree of
More so, if attention is restricted to the measures com-
puted from second order partial derivatives in
function
f(x,e) •
e
of the response
Interestingly, Beale obtained the measure N
cp
(~
cp
i.n Guttman and Meeter) in Section 2 of his article by studying the
coincidence of the lack of fit and likelihood rati.o contours in an
effort to approximate
P[~T(y,e *)
= ole * ,a 2 J •
The concern here is with the sampling characteristics of the three
confidence procedures, so an interest in Bealeae measures centers more
in their ability to predict the accuracy of the asymptotic approximations
of the previous section rather than as a measure of coincidence of contours.
This use is more in line with Beale's intention.
However, coin-
cidence of linearization and likelihood ratio contours and accuracy of
asymptotic approximations in finite samples are not orthogonal
dimensions of statistical behavior.
In the Addendum to their article,
Sprott and Kalbfleisch [21J discuss the relationship between coincidence
and accuracy and present an example where a transformation designed to
i.mprove coi.ncidence also improves accuracy.
Thus, it is expected that
Guttman and Meeter's examples, which differ markedly in coincide'nce of
linearization and likelihood ratio contours will display differences
both in contours of the coverage functions and in accuracy of
approximation.
The two models chosen for reexamination are defined in Table 1;
the designations
' \ ' and,,%
areas in Guttman and Meeter.
Too se
models differ with respect to design points and response function;
variance and sample size are the same in both cases.
Model
the more nonlinear; Bea1e 9 s nonlinearity measures evaluated at
using analytic derivatives are: .N
112
and
N
e
,.
.87954 and
e = .00171
N ... .00333
cp
for
and
N ... .00053
e*
for
1\3 •
The natural parameter space for these models is
achieve compactness,
cp
is
~
e1 ,e 2
> O.
To
must be bounded from above and all
-18-
boundary points included.
In view of the symmetry of Model
reasonable to impose the inequality
8
1
~3
it is
~ 92 as wel1~/.
The coverage funct:i.ons for these models are evaluated over the
region
e'*
±5
standard deviations of the least squares estimator from
and shown as Tables 2 and 3.
Contours, obtained by a quadratic
interpolation from these tables, are plotted as Figures A and B.
An
impression of the relative size and location of the tabled regions
with respect to the original scaling of the models may be had by
refer.ence to Figure C.
The shapes of the confidence regions them-
selves, corresponding to a real:i.zation of the errors, are plotted as
Figure 2.6 of Guttman and Meeter [111; they look much the same as the
75% coverage contours of Figures A and B after allowing for the
differences in scaling.
There are three interesting aspects of these computations.
first comes as no surprise, for the nearly linear Hodel 1'\'2
The
the con-
tours of the cover.age functions are nearly coincident while for the
relatively more nonlinear M.ode1 1'\'3
substantial.
the departures are fairly
The second is the near coincidence of the coverage
functions for the lack of fit and likelihood ratio procedures for both
models; they are not distinguIshable within the tolerances used in
the construction of Figures A and B.
The third is the inequalit.y
which holds over the tabled region for both models.
This indicates
that the likelih.ood rati.o procedure i.s preferable to the lack of fit
procedure as it covers false values of
ability in all instances.
e
with equal or smaller prob..
-19-
These examples de19c:dbe relative yields of two substances in a
x and, as such, satisfy the inequality
chemical process at time
o<
f(x,e) < 1 •
It follows that
8'(e)5(e) < 12
over the set A • {e:
and
0
~
a1 ,
0
SAcT<e)dm(e)
~
e2 , el
~
821 ; hence, the integrals
78/
..
are both inf];nit~ A.s me'ntioned
previously, the comparison of the expected volume of the linearization
with the expected volume of the lack of
fit or. maximum liksU.hood procedure is of dubious value when this situation
obtains; the linearization procedure may, in principle, be shown to
be the best by taking the volume of ® suitable large.
Daspite this qualification, the expected volume
over the· region
has intuitive appeal as a summary measure of performance over a
relevant subset of the parameter space.
This measure is the unweighted
average of the probabj.1i.ty of covering false values of
rectangle
® centered at
•
e
over the
Its interpretation is analagou8 to
that of average power of a test with respect to a uniform, informative
prior, and it may be used as a numeric aid to the interpretatio·n of
Figures A and B.
Expected volumes for the two examples an.d three pro-
cedures are given in 'rable,4.
They were computed by numerical inte
gration from the entries of Tables 2,
3~/With
u
respect to the original
·20-
scaHng, not standard deviation scal:tng.
As seen from the table, there is no difference of any importance
in expected volume among the three procedures for MOdel
ing the visual
with MOdel
n3
impression~onv~yed by
Figure A.
~,confirm­
The situation changes
; the ordering from best to worst is linearization,
likelihood ratio, and lack of fit with little difference between the
likelihood ratio and lack of fit procedures.
This ordering is difficult
to discern visually from Figure B without the aid of the expected volume
com.putation.
The last question of interest is the effect of the variation in
degree of non1:i.nearity between
MOdel. 'n2
and
Model '1'\3
on the
accuracy of the asymptotic approximations used to study the statistical
behavior of the confidence procedures.
Ta.ble 5 relates to thi,s
question; the right hand sides of
computed by MOnte-Carlo integration using the control variate method
of variance reduction [13, p. 59-601 is compared wi.th cS<e)
cT(e)
and
in the table.
The parameter values were chosen for inclusion in Table 5 by
random selection from those parameter comb:i:nations in Tables 2 and 3
haVing non-blank entries, subject to the constraint that there be at
least one point satsifying:
0 < c;s<e)s cT<e) s: .25; .25 < c's<e),
QT(e) s: .5; .5 < 08(9), cT(e) s: .75; .75 < cS<e), cT(e) < .95; and
C (9),c (e)
S
T
= .95
•
-21..
The coverage function c;T<e)
of the likelihood ratio p'rocedure
is a remarkably accurate approximation of the actual coverage probability; its accuracy is negligably affected by the difference in
linearity between the
Models
~2
and 11'3'
non~
For any practical pur-
pose, the nominal confidence level of 95% is achieved in both cases;
overall, the approximation
is entirely satisfactory.
The converse is true of the linearization procedure.
nearly linear MOdel
is
~2
For the
' the approximation
only~rginal1y acceptable
overall.
The agreement of the nominal
95% cqh'fidence level with the confi.dence level actually achieved is
c,Vise
enough for practical purposes, but the hypothesis
.'\.
\
H:
*
P[tt>s(y,e ) ..
Ole*,a 2J ... 95
may be rejected at'
the 5% levellQ./.
,;" .
• -
For the more nonlinear Model 1\'3
11/
the approximation :i.s completely unsatisfactory.- •
These findings ate in agreement with the results reported in
There also, the accuracy of the likelihood ratio approximation was
found to be better than that of the linearization approximation.
[61.
D
40
COMPARISON
The argume'nt that approximate confidf'.lnce procedures are invalid
shall be set aside for the moment.
'Were it accepted, the lack of fit
procedure must be employed and there is no basis for discussion.
The linearization procedure has the most desirable structural
characteristics provided there is agreement that a convex confidence
region is preferable to a (possibly) disconnected region and that a
bounded region is preferable to a (possibly) unbounded region.
More'"
over, the linearization region. is the simplest to construct in
applications, especially when the region is an interval.
Arrayed against these desirable structural characteristics is the
serious fault revealed by Monte-Carlo simulat:lon:
there is apt to be
an uncomfortably large discrepancy between the actual and nominal
confidence level of the lineattization procedure.
These same simulations
indicate that the likelihood ratio procedure suffers little, i f any,
from this deficiency and, of course, the lack of fit procedure is completely free of it, being an exact procedure.
One might, then, choose
either the likelihood ratio or lack of fit procedure in preference to
the linearization procedure.
The structural characteristics
of the likelihood ratio procedure
t
are certainly OO1:'e deSirable than those of the lack of fit procedure.
While both may be unbounded and disconnected, the 1:Lkelihood ratio confidence region has the merit of having a boundary which coincides with
a likelihood contour.
In contrast, a lack of fit region contains
neighborhoods of points corresponding to every ripple of the likel:i.hood
surface" every local minimum, every local max:tmum, and every saddle
-23point.
It is hard to find any merit in this.
The only justification
for a choice of the lack of fit procedure is that it is exact.
In summary, the linearization method has the best structural
characteristics but its nominal confidence level is apt to be
inaccurate.
A choice of the likelihood ratio method is a compromise
between structural characteristics and accuracy.
The lack of fit
method would only be chosen if accuracy were of extreme concern.
-24FOOTNOTES
llThiS is true in applications more often than it might appear at first
glance.
..
Boundaries often correspond to degenerate forms of the
response function so that there is always an interior point with
a smaller Sum of squares than any boundary point.
e
is interior to
e
Further,
with probability tending to one as sample size
increases under the regularity conditions.
'f /A
simple example is
31
- MOre formally,
e
A
L:
= e 1e
-e 2x
-1
C1
2
[(1/n)F'<e)F(9)]-1
" - e)
*
,and Vn (e
and
where all
converges almost surely to
almost surely to
to a matrix
f(x,e)
e
'I(
2
s
converges
converges almost surely
converges in distribution to
a p-variate normal with mean vector zero and variance-covariance
.
2 -1
matrLX C1 L:
i / 1n
[7J.
Some instances, such as the examples of the next section, it is
inf P[~T(y,e)
ee A
reliance on the approximation is then avoided.
possible to verify directly that
1 / rn
Table 2.3 of their article, read
~cp = .0011
= 0 Ie* ,C1 2 J >
for 'M
~
= .011
0 ;
•
-25i/Table 3 and Figure B do not reflect the imposition of the bound
e1
82 • To impose it in Table 3, delete all entries in Table .3
with column head:lngs ..5, -4.5, -4, and. .. 3.5. To impose it in
..
~
Figure B, draw a line through the points (-4, -3.3) and (-3.1.,5) •
1/Smal1er bounds can be established on subsets.
bound
For Model
~3 the
a · .034 holds on the set A- (81'e 2 ): 25 s: 81'
.3 s: 92
~
.305} •
§/In this case, it is poss:Lble to verify that
fA
*
Jr n [1 - ~T(y,8)JdP(Yle ,a 2 )dm(e)
R
on the approximation
open set
sup
SeA
E such that for all
lie + 6(8)11 2 ~ e*
~T(Y,e) ~
0T(e).
0
for
:lnf
SeA
lie
*
is infinite without relying
It may be shown that there is an
e
in
E the
+ 8(e)1\2. This implies that
y. fee ) + e , hence, the inner integral is
bounded away from zero on A and the double integral is infinite.
i/Before rounding; the inequality 6 1 ~ 82 was imposed on MOdel ~3 •
jQ/The z.. statietic is
(.9475 - .95)/.001227 • -2.04 •
lllTwo attempts to rem.edy this deff.ciency were em.ployed.
to use c(e) instead of
.e
a in
the formula fot' S(e).
The first was
The second was
to use the cot'rectio'n suggested by Beale [2, Eq. 2.21J.
attem.pts were failures.
Both
In fairness to Beale it must be appended
that his cor.rection i.e for the likelihood ratio method 'not the
linearization method.
10
Examples of Guttman and Meeter
a.
f(X, e)
Model
112
=
e* = (104,.4)
tXt)
= (.25,
.5, J., 105, 2, 4, .25, .5, 1, 105, 2, 4)
n '-.: 12
Model ~3
b.
-X9
1 - (ale
2
-XA
- 82e
l)/(A l - 82 )
f(X, 'a) =.
0*
= (104,.4)
t
= [1,
n'
= 12
(X }
2, 3, 4, 5, 6,1, 2, 3, 4, 5, 6)
81
I
82
2.
C()verage Probabilities for the Model "2'
-(\ - 1.1J)""1
(82-.1,)""2
-s.o
-4.5 .... 0 -3.5 -3.0 -2•• -2.0 -1.5 ·1.0
-~.5
0.0
O.S
1.0
1.5
2.0
a.5
3.0
3.5
.045 .047 .045 .039
.090 .092 .087 .076
.Io~ .166 .155 .135
.272 .271 .251 .217
.408 .401 .370 .319
.556.542 .498.430
.696 .67• • 618 .536
.809 .780 .716 .622
.886 .851 .762 .679
.927 .887 .812 .702
.935 .888 .806 .689
.912 .85a .762 .637
.u50 .781 .678 .550
.744 .664 .557 .434
.594 .513 .412 .30b
.423 .350 .2b8 .190
.260 .206 .150 .101
.136 .102 .071 .0.5
.,059 .0...2 .028 .017
.02' .014 .009 .005
.006 .004 .002
.032
.062
.109
.175
.Z57
.348
.435
.506
.024
.046
.081
.13.
.193
.262
.329
.383
.017
.032
.056
.091
4.0
4.S
5.0
II. LACK OF FIT
5.0
4.S
4.0
3.S
3.0
2.S
... 0
I.S
1.0
0.5
0.0
-0.5
-1.0
-1.5
-2.0
-;>.5
-3.0
-3.5
-4.0
-4.5
-5.0
.003
.006
.013
.OZ4
.042
.066
.O~7 .095
.OSI .125
.O~~ .lS2
.~76 .170
.C80 .175
.078 .16S
.068 .142
.O~• • 1.1
.03b .077
.024 .O.u
.013 .02~
.006 .0'2
.002 .004
.002
.004
.002 .008
.004 .015
.002 .007 .024
.003
.004
.006
.008
.009
.009
.009
.007
.OOS
.00_
.002
.012
.017
.022
.027
.030
.030
.027
.OZ2
.016
.010
.006
.003
.003
.003 .008
.008 .018
.017 .o~,
.033 .070
.059 •• 19
.096 .185
.1• • • 262
.197 .342
.249 •• '3
.290 .464
.313 .488
.315 .400
.292 •••0
.248 .373
.192 .287
.133 .197
.081 .119
.0.2 .Obl
.019 .027
.007 .0.0
.007
.017
.036
.071
.127
.205
.302
.408
.508
.589
.64'
.659
.639
.583
.492
.378
.259
.154
.079
.034
.012
.013
.030
.062
.117
.190
.306
.430
.554
.662
.7.1
.786
.02t
.047
.093
.168
.274
.405
.545
.676
.l80
.849
.883
.79' .882
.765 .845
.695 .767
.5~7 .646
.4~0 ••9~
.306 .331
.180 .191
.090 .093
.037 .038
.013 .012
.030 .039
.0~5 .000
.125 .150
.217 .253
.341 .387
.486 .538
.633 .683
.760 .e04
.854 .888
.911 .935
.934 .950
.926 .935
.883 .~84
.797 .789
.666 .648
.501 .47b
.330 .304
.186 .166
.088 .075
.034 .028
.011 .008
.011
.021
.037
.059
.13~ .087
.183 .118
.230 .149
.268 .172
.~53 .417 .289 .1b~
.~67 .423 .2~O .182
.548 .401 .270 .166
.494 .3~2 .2~O .1~7
.412 .283 .178 .102
.311 .205 .123 .060
.210 .131 .075 .040
.123 .073 .040 .020
.062 .O~5 .018 .009
.026 .014 .007 .003
.009 .005 .002
.003
.007 .004
.013 .007
.022 .012
.035 .020
.052 .02.
.07• • 039
.089 .049
.102 .056
.10d .O~D
.10~ .O~6
.093 .04.
.075 .037
.05• • 026
.O~4 .016
.019 .009
.009 .004
.004 .002
B. LlNEAPllATION
5.0
4.S
4.0
3.S
3.0
2.5
Z.O
1.5
1.0
0.5
0.0
-o.!»>
-1.0
-1.5
-2.0
-2.5
-3.0
-3.5
-4.0
- •• 5
-5.0
.002 .003 .006
.002 .005 .010 .017
.003 .006 .013 .026 .043
.003 .007.016 .032 .osa .093
.002 :gg~ :g~~ :g~~ :::~
.003 .010 .02b .0~8 .IIS
.006 .017 .043 .093 .175
.010 .027 .065 .134 .240
.Ol~ .039 .089 .175 .301
.020 .050 .110 .209 .347
.O~4 .O~8 .124 .228 .370
.025 .061 .127 .230 .366
.025.059.120.214 .337
.022 .0~1 .104 .103 .287
.018 .0&1.081 .1.3 .2~4
.DI~ .O~9 .058 .102 .159
.008 .019 .037 .065 .102
.OO~ .011 .021 .037.058
.002 .005 .011 .019 .029
.002 .~05 .008 .013
.010
.027
.065
.134
.018
.0••
.081
.143
.013
.029
.05$
.102
.008 .005 .002
.019 .011 .005 .002
.037 .021 .011 .OOS
.06$ .0~7 .019 .008
.199 .306 .422 .530 .614 .666 .682 .660 .603 .514 .403
.288 .422 .SS6 .e70 .752 .796 .803 .773 .706 .603 .474
.378 .530 .670 .781 .852 .806 .884 .848 .773 .660 .518
.455 .6'4 .752 .852 .912 .935 .926 .8a4 .893 .682 .529
.508 .666 .796 .886 .935 .~50 .935 .8~o .796.666 .50B
.529 .682 .803 .884 .926 .935 .912 .8~2 .7~2 .614 .455
.518 .~EO .773 .848 .8e4 .886 .652 .781 .670 .530 .378
.47• • 603.706.773.803.796.752 .670 .~56 .422.• 288
.403 .514 .603 .660 .682 .666 .~14 .530 .422 .306" .199
.316 .403 .474 .518.529.508.455.378.268.199.123
.224 .287 .337 .366 .370 .347 .301 .240 .17~ .Il~ .067
.143 .183 .214 .230 .228 .209 •• 75 .134 .093 .058 .032
.081 .104 .120 .127 .124 .110 .089 .065 .043 .026 .013
.041 .051 .059 .061 .O~8 .050 .039 .027 .017 .010 .005
.018 .022 .025 .025 .024 .020 .015 .010 .006 .003 .002
.287
.337
.366
.370
.347
.30)
.240
.175
.115
.Ob7
.035
.016
.806
.002
.183 .104
.214 .I~O
.230 .127
.228 .124
.209 .110
.175 .Oug
.134 .O~
.093.043
.058 .026
.032 .013
.016 .006
.007 .003
.003
:~~~. :f~: :~~= :~;:
.015
.039
.089
.175
::::
.020
.050
.110
.209
.024
.058
.124
.228
.025
.061
.127
.230
.025
.059
.120
.214
:~~: :~~; :~~: :~}~
.022
.051
.104
.183
:::;
:~f: :~~: :::~
:g:1
:g~t
:g::.
.051 .022
.059 .025
.061
.058
.050
.039
.027
.017
.010
.005
.002
.O2~
.024
.020
.OIS
.010
.006
.003
.002.
C. LIKELIHOOD PATIO
5.0
4.'5
4.0
3.5
3.0
Z.5
2.0
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
';'2.0
-2.5
-3.0
. -3.5
-4.0
-4.5
-5.0
·e
.554
.662
.741
.786
.794
.764
.695
.586
.038 .044 .046 .0.4
.080 .089 .091 .086
.150 .164 .165 .IS4
.2~3 .272.270 .250
.387 .408 .40t .369
.537 .~56 .542 .497
.683 .6~6 .673 .618
.004 .809 .779 .716
.888 .aU6 .851 .781
.935 .921 .807 .812
.950 .9.35 .888 .'806
.9~~ .912 .85~ ~762
.884 .U50 .781 .678
.789 .743 .664 .557
.6.7 .~9~ .512 .412
.009 .022 .04& .078 ~116 .153 .179
.002 .0051 .012 .02• • 041 .'060 .0"7 .089
.002 .LOS .011 .018 .026 .033 .037
.002 .00.·.007 .009 .011 .012
.191 .~a~
.093 .087
.037 .034
.012.011
.OT!:» .059 .0""2 .027 .016 .009 .P.05 .002
.002
.003
.004
.0Ob
·.007
.008
.008
.008
::::
.003
.002
.004
.007
.011
.016
.021
.026
.028
.028
.025
.003
.008
.016
.033
.058
.096
.143
.196
.Z4F
.Z88
.311
.312
.289
.2_5
.007
.OJ7
.036
.07t
.127
.205
.18~ .302
.262 .407
.342 .~08
.412 .588
.463 .640
.487 .658
.479 .639
.438 .682
~370 .491
.013
.030
.062
.117
.198
.30b
.43~
.038
.075
.133
.215
.317
.429
.S35
.621
.679
.702
.089
~6~7
.549
.433
.~06
.031
.060
.106
.172
.255
.345
.433
.505
.551
.b66
.547
.494
•• 11
.JII
.209
.023 .016
.0• • • 030
.079 .OS4
.128 .088
.190 .131
.•259 _179
.3Z6 .226
.381 .265
.415 .261
.422 .26&
.400 .2bO
.351 .22a
.282 .177
.20• • 12~
.131 .Ol~
.021 .030
.047 .064
.093 .124
.168 .216
.Z74 .3.0
.405.486
.S4S .632
.676 .760
.780 .854
.84V .911
.883 .934
.882 .926
~845 .883
.767 .797
.646 .666
.003
.002 .006
.00• • 013
.008 .OZ4
.014 .041
.024 .065
~036 .093
.050 .123
.063 .150
.073 .167
.078 .172
.07~ .16Z
.066 .139
.003
.008
.ole
.037
.070
.119
:g~: :~~~ :~~: :l~:~,~;: :~~~ :~:; :~~~ :~~~ :~~: :~~~ :~~~ :~~: :~g: :~:~ :~~~
NOTE: Blanks indicate val1,.les less than .0015.
use 11 = .052957 and °2 = .01.4005
1
.
.010 .006
.01• • 011
.034 .020
.O~6 .033
.084 .049
.t15 .068
.145 .O~5
.1~9 .099
.leU .105
.180 .102
.163 .091
.135 .073
.101 .O~~
.~67 .034
.O~~ .019
.003
.006
.011
.018
.027
.037
.046
.0~3
.056
.O~3
.046
.036
.azs
.015
.Qoa
:gt: :g~g :gg: .00.
.165 .136 .&02 .071 .c.5 .026 .014 .007 .003
.028 .021 .OJ4 .009 .005 .003
.008 .006 .004 .002
To convert frbm standard deviation scaling
e
e
3·
Coverage Probabilities for the Mode1"S"
(8z-·lI)fu'z
(01- 1·~)fu'1
D.D
-5.0 -".5 -4.0 -3.& -3.0 -2.5 -2.0 -1.5 -1.0 -0.5
--
1I.S
I.D
1.5
2.0
2.5
3.0
3.5
4.D
4.5
e.o
.025
.199
.551
.692
.434
.079
.002
.007
.086
.368
.608
.419
.t27
.005
.002
.034
.222
.49_
.479
.170
.011
.013
.124
.316
.449
.202
.019
.OOS
.06b
.272
.399
.221
.028
.002
.034
.188
.342
.227
.037
.00.
.049
.230
.467
.517
.330
.100
.011
.002
.033
.160
.330
.357
'-
A. LACK Of" FIT
.255 .316
.128 .488
.0• • • 59*
.010 .585
.450
.242
.018
.012
5.0
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
-0.5
.DDS
.03D
.124 .002
.330 .015
.~8• • 094
.74• •323
.731 .648
.•543 .853
.2.2 _848
.050 .601
.004 .227
.030
-1.0
-1.5
-2.0
-2.5
.002
.028
.168
.507
.836
.934
.785
.367
.058
.002
-3.0
.D14
.115 .012
~442 .115·.017
.821 .459 .155
.950 .838 .541
.808 .933 .861
.359 .712 .858
.046 .230 .488
.Dll .082
.002
.002
.o~
.248
.652
.845
.663
.206
.012
.006
.086
.396
.129
.721
.339
.038
-3.5
-".0
-4.5
-5.0
8. LINEARIZATION
s.o
4.5
4.0
3.5
.3.0
2.5
2.0
1.5
1.0
0.5
0.0
.022 .024 .012 .002
.024 .055 .C57 .027 .005
.012 .O~7 .117 .116 .055 .011
.209 .100
.357 .330
.330 .517
.160 .467
.002 .033 .230
.002 .049
.G04
.002 .027 .*16 .218
.005 .05~ .209
.011 .100
.020
-0.5
-1.0
-1.5
-2.0
-2.5
-J.o
.020
.160
.467
.673
.002
.033 .002
.230 .049 .004
.599 .301 .064 .005
.599 .800 $706 .361 .077 .005
.301 .706 0887 .779 .402 .08• • 006,
.064 .361 .779 .935 .815 .416 .084 .005
.005 .077 .402 .815 .950
.005 .084 .416 .815
.006 .08• • 402
.00$ .077
.005
'
-3.5
.S15 .402
.935 .779
.779 .887
.361 .706
.064 .301
.00• • 049
.002
-4.0
-4.5
-5.0
.07?
.361
.706
.800
.599
.2ao
.033
.002
.005
.064
.301
.599
.673
.4b7
.160
.020
.002
.020
.100
.Z09
.209 .218
.055 .116
.005 .027
.002
.ola
.055
.116
.117
.OS7
.012
.005
.027 .002
.057 .012
.O~5 .024
.024 .022
.012
.116
.360
•• ~
.196
.018
'.00• • 002
.060 .030
.253 .170
.380 .3J8
.211 .2a~
.02• • 03.
C. LIKELIHOOD RATIO
.2.5 .1I14 .OOS
.122 .485 .030
5.0
•• 5
4.0
3.5
3.0
.0• • • 589 .1211-
.009 .579 .329
.444 .583
.237 .• 743
.075 .736
_912 .540
.240
.049
.004
i!.5
2.0
1.5
1.0
0.5
0.0
-0.5
-1.0
.. 1.5
-2.0
-2.S
.015
.094 .002
.323 .028
.648 .168
.852.507
.848 .836
.607 .934
.226 .785
.014
.115
.442
.821
.950
.012
.115
.469
.838
.030 .367 .808 .933
.058 .359 .712
.002 .046 .230
.017
~3.0
-3.5
-4.0
-4.5
-5.0
•
~OTE:
.
.017
.165
.541
.061
.858
.4as
.082
.002
.002
.036
.247
.651
.844
.662
.206
.012
.006
.OB5 .024
•• 39• • 195
.726 .546
.719 .689
.338 .432
.038 .078
.002
.006
.084
.360
.600
••73
.125
.005
.002
.032
.213
.482
.410
.167
.011
:
Blanks indicate values less than .0015 •. To convert from standard deviation scaling
use 0'1 =.27395 and ,0'2 = .029216
.
e
,
-
4. EXpected Volume of the Lack of Fit, Linearization, and Likelihood Ratio
Confidence Procedures
Model
EXpected Volume
Lack of Fit Procedure
.01857
Linearization Procedure
.01847
Likelihood Ratio
.01851
.07132
.07417
.69235
Parameter Space
Procedure
5. )bnte-Carlo Estimates of' Coverage Probabilities
Linearization
Likelihood Ratio
MOnte-Carlo Estimate
{9l - 1..4)/u1
(92 - .4)/u2
f;S(9)
P[+s(Y, 9)
= 0]
a • Model
'oT(Q)
P[9c:r(Y' Q)
= 0]
Std. Err.
~
-4.5
-3.0
-1..5
1.5
1.0
0.5
-1.5
-0.5
.0215
.3009
.1051
.1521
•0165
.2842
.1262
.1461
.0011
.0021
.0023
.0018
.0111
.2412
.6949
.1621
.0101
.2411
.6952
.1621
.0035
.•0011
.0016
3.0
2.0
-1.5
0.5
-4.0
3.0
1.0
-0.5
.0062
.2813
.6105
.9115
.0052
.2818
.6111
.9110
.0008
.0011
.0022
.0016
.0045
.3111
.6619
.9115
.0052
.3200
.6632
.9108
.0006
.0028
.0015
.0009
0.0
0.0
.9500
.9415
.0012
.9500
.•9493
.0008
b•
e
Std. Err.
Monte-Carlo Estimate
lbleJ.
.0020
~
-2.5
-1.0
2.0
0.5
0.5
0.0
-1.5
-1.0
.0036
.4016
.5981
.1190
•0460
.5418
.5411
.1953
.0009
.0014
.0062
.0056
.0000
.0000
.0000
.2262
.1193
.1123
.2263
.7218
.7108
.0060
.0011
.0041
4.5
0.0
-2.0
-0.5
-3.0
1.0
3.5
1.0
.0055
.4016
.0205
.7790
.1050
.2873
.2355
.6290
.0012
.0054
.0022
.0055
.0264
.4415
.5793
.8359
.0248
.4436
.5808
.8440
.0025
.0032
.0078
.0040
0.0
0.0
.9500
.8655
.0034
.9500
.9498
.0012
e
e
A.
Coverage Probabilities for the Modein ,
2
CS2-·4)/OZ
5.0
3.2
. 1.4
I'.
I
I
I
I
I
I.
I
I
,•,I
-.4
Cs =25'
I
\
\
\
-2,2
\,
" ' ....
I
-4.0
...._--...;.------,
..2.2
,
,
-.4
NOTE': To convert from standard deviation scaling use
•
,
1.4
e
5~O
3.2
0"1
= .052957
and
(12
<S,-1.4)/c;
= ,011l005
•
e
B.
Coverage Probabilities for the Model "3.
(e z-·4}/oz
5.0
3.2
1.4
CR C =.75
' T eR'eT =.25
-.4
-2.2
,
-4.0
NOTE:
•
-2.2
I
-.4
,
1.4
To convert :fro1iI standard deviation scaling use
e e
5:0 (efI.4}/Of
3.'2
a l = .27395
and
a2
= .029216
•
•
c.
an~
Relative Size
Location of the Tabled Regions.
e,
1.0
4
e, =..6 2
TABLE 3
TABLE 2
.5
e
V
I
,
o
.5
"1.0
.
•
1.5
e
,
,
2.0
2.5
.
,
3.0
..
:a. 2
e
-26REFERENCES
[1J
Bahadur, R. R., "Rates of Convergence of E;stimates and Test
Statistics li , The Annals of Mathematical Statistics, 38 (April
1967), 303-23.
[2J
Beale, E. M. L., "Confidence Regions in Non-Lineal' Estimation",
}our~l
of the Royal Statistical Society Series B. 22, No. 1
(1960), 41-76.
[3J
Feder, Paul I., liThe tog Likelihood Ratio in Segmented
Regression", ,Th,e Annals of StatistiC:Ji, 3 (January 1975), 84-97.
[4J
Gallant, A. Ronald, "'The Power of the L:ikelihood Ratio Test of
Location in Nonlinear Regression Models", J.Q.urnal of the
American Statistical Association, 70 (March 1975), 198-203.
[51
Gallant, A. Ronald, "Nonlinear Regression", The American
§.tatistician, 29 (May 1975), 73-81.
[6J
Gallant, A. Ronald, "Testing a Subset of the Parameters of a
Nonlinear Regression Model", Journal of tl!!.
Am~ric.!~atisticll
Associatl£ll, 70 (December 1975), 927-32.
[7J
Gallant, A. Ronald, "Inference for Nonlinear Models lU
Mimeograph
Series No. 875 (revised), Institute of Statistics, North
Carolina State University, 1975.
[8J
Gallant, A. Ronald, "Testing a Nonlinear Regression Specification:
A Nonregular Case, II Mimeograph Series Nc. 1Ol'7, Institute of
Statistics, North Carolina State University, 1915.
[9J
Gallant, A. Ronald and Fuller, Wayne A., "Fitting Segmented
Polynomial Regression Models Whose Join Points Have to be
Estimated",
,Journal of The America'a Statisti£,a1 A.§sociation,
68 (March 1973), 144-41.
-27[10]
Graybill, Franklin A., An Introduction to
!1Q.~lsp
[l1J
LinearJ?tati~tical
Vol. 1, New York: McGraw-Hill Book Company, 1961-
Guttman, Irwin and Meeter, Duane A.,
liOn
Bea1e Gs Measures of
Non-linearity", Technometrics, 7, No.4 (November 1965), 623-37.
[12J
Halperin, Max, "Confidence Interval Estimation in Non-Linear
Regression", J.Q.urna1 of the Royal Statistical Society Series B,
25, No.2 (1963), 330-3.
[13J
Hammersly, J. M. and Handscomb, D. C., Monte
New York:
[141
Hartl~y,
C~~6 ~ethods,
John Wiley and Sons, Inc., 1964.
H. 0., "The Modified Gauss-Newton J'Yf..ethod for the
Fitting of Nonlinear Regression Func.tions by Least Squares",
Technometrics, 3 (May 1961), 269-80.
[15J
Hartley, H. 0., "Exact Confidence Regions for the Parameters in
Non-Linear Regression Laws", Biometrika, 51, Nos. 3 and 4 (1964),
347-53.
[1.6J
I.B.M. Corporation, System 360 Scientific Subroutine Package,
Version III, White Plains, N.Y.:
Technical Publications
Department, 1968.
C17]
International Mathematical and Statistical Libraries, Inc.
IMSL Library 1. Fifth Ed., Houston:
International Mathematical
and Statistical Libraries, Inc., 1975.
[18J
Marquardt, Donald W., "An Algorithm for Least...Squares Estimation
of Nonlinear Pa~ameters", :10u:mal of the Soc:hety.-:f.Q.r Industria.l,
and Applied Mathematics, 11 (June 1963),
[19]
43l~41.
Pratt, John W., "Length of Confi.dence Intervals", J2urnal of
t~
American Statistica!-Association, 56 (September 1961), 549-67.
-28I
[ 20 1 Scheffe, Henry, The Analysis of Variance, New York: John Wiley
and Sons, Inc., 1959.
[21]
Sprott, D. A. and Kalbfleisch, John D., "Examples of Likelihoods
and Comparison with Point Estimates and Large Sample Approximations,"
Journal of the American Statistical AssociatiQg, 64 (June 1969),
468-84.
[22]
Turner, Malcolm E., Monroe, Robert J., and Lucas, Henry L.,
uGeneralized Asymptotic Regression and Non-Linear Path Analysis",
Biometric~,
[23J Williams, E.
17 (March 1961), 120-49.
J.,
"Exact Fiducial Limits in Non-Linear Estimation",
Journal of the Royal Statistical
(1962), 125-39.
So~iety
Series B, 24, No. 1
-29APPENDIX
NUMERICAL METHODS EMPLOYED
The computations were performed on an IBM 370/168, University
of Chicago, and on IBM 370/165, Triangle Universities Computation
Center, using double precision throughout; the exceptions were the use
of two single precision routines, CDTR [161 and GGNOF [17J.
The computations for Tables 2 and 3 were performed using the
series expansion [10, p. 76l
*AT2![c*-11 2 ;n-p+2i,0)g(t;P+2j,P)dt •
x ~ G(t/[c*
-lJ+20
The Poisson weights were computed using
DL~
was evaluated using CDTR [16J and DQL16 [161.
used to compute
cR(e)
[16J and the integral
Similar methods were
cs(e) •
and
Simpson's rule, applied to the entries of Tables 2 and 3 (before
rounding), was used to obtain the integrals shown in Table 4.
necessary adjustment to accomodate the boundary
6
1
~
e2
The
was by means
of a trapazoid rule.
2
The probabilities P[~S(Y,a) • 0la * ,a]
and
of Table 5 were computed by 4000 Monte Carlo trials using the control
variate method of
va~iance
reduction [13, p. 59-60J.
Ihe control
variate was
V(a) • [(F'e + a
*
- a)'C(F'e +
with corresponding test function
e'!~ -
e)!pJ/[e'Qe/(n-p)]
-301
V(e)
> F
01
o
I,et
i
V(e)
~
F
01
•
index the 4000 trials and let
The MOnte-Carlo estimates were
otherwise
-31with estimated standard error
SE
III
otherwise
and
otherwise
with estimated standard error
SE ..
otherwise
The pseudo-random number generator employed was
GGNOF [171.
modified Gauss-Newton algorithm [141 from a start of
e*
The
was used to
compute "e for each Monte-Carlo trial; excepting 14 instances when
the algorithm failed to converge for the computations related to
Model ~3.
the
<6
,e )
1 2
For these 14 cases,
pairs of Table 3 •
.
,
e
was computed by grid search over