testing a nonlinear regression specification: a nonregular case

Testing a Nonlinear Regression Specification: A Nonregular CaseAuthor(s): A. Ronald GallantSource: Journal of the American Statistical Association, Vol. 72, No. 359 (Sep., 1977), pp. 523-530Published by: American Statistical AssociationStable URL: http://www.jstor.org/stable/2286210 .

Accessed: 14/06/2014 06:21

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

American Statistical Association is collaborating with JSTOR to digitize, preserve and extend access to Journalof the American Statistical Association.

http://www.jstor.org

This content downloaded from 195.34.79.176 on Sat, 14 Jun 2014 06:21:10 AMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=astata

http://www.jstor.org/stable/2286210?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


Testing a Nonlinear Regression Specification: A Nonregular Case

A. RONALD GALLANT*

A statistical test of whether an additive nonlinear term in the response function should be omitted from a nonlinear regression specification is considered. The regularity conditions used to obtain the asymptotic distributions of the customary test statistics are violated when the null hypothesis of omission is true. Moreover, standard iterative algorithms are likely to perform poorly when the data support the null hypothesis. Methods designed to circumvent these mathematical and computational difficulties are described and illustrated.

KEY WORDS: Nonlinear regression; Hypothesis testing; Seg- mented regression.

1. INTRODUCTION

In nonlinear regression analysis it is helpful to be able to choose between two model specifications:

H: Yt = g(xt, A) + et and

A: yt = g(Xt, OP) + Th(Xt, w) + et ,

on the basis of sample data-observed responses yt to k-dimensional inputs xt(t = 1, 2, ..., n). The unknown parameters are 4', r, and w, of dimensions u, 1, and v, respectively. The functional forms, g(x, 4A) and h(x, co), are known. The errors, et, are normally and independently distributed with mean zero and unknown variance a-2. Parametrically, the situation described is equivalent to testing:

H: X = 0 against A:r# 0

regarding 4, co, and a2 as nuisance parameters. It would be natural to employ either the likelihood-

ratio test or the test based on the asymptotic normality of the least-squares estimator (Gallant 1975c). However, both of these tests depend on the unconstrained least- squares estimator 6 = ( a, ,A). When H is true, this dependence causes two difficulties:

1. It is likely that the attempt to fit the full model,

Yt = g(xt, 4') + rh(xt, &o) + et

using a standard first derivative iteritive algorithm (Hartley 1961; Marquardt 1963) will fail or, at best, converge very slowly.

2. The regularity conditions (Gallant 1973; Jennrich 1969; Malinvaud 1970) used to obtain the asymptotic properties of the unconstrained least-squares estimator 0

* A. Ronald Gallant is Associate Professor of Statistics and Economics, Department of Statistics, North Carolina State Uni- versity, Raleigh, NC 27607.

are violated; these asymptotic properties are needed to derive the asymptotic null distribution of the two test statistics (Gallant 1975a, 1975c).

The first difficulty can be illustrated using the model

Yt = 01xlt + 02X2t + 04e 363t + et

where, in the notation above, r = 04, W = 03, I = (01, 02)',

and g(x, It) = - t1x] + {2X2 h(x, w) = ewX3

When the null hypothesis 04 = 0 is true, or nearly true, the sum-of-squares surface is relatively flat with respect to 03. This is the source of computational problems; e.g., put (01, 02, 03) = (O, 1, - 1), a2 = .001, and leave 04 free to vary. A sample of 30 normally distributed errors was generated and added to the response function,

f (x, 0) = 0lXl + 02X2 + 04e63X3

using the inputs xt given in Gallant (1975b, Appendix).' Figure A is a plot of the residual sum of squares against trial values for 03 after fitting for the remaining parameters.

As seen in Figure A, the surface becomes flatter as 04 decreases in absolute value. In all cases, there is a local minimum near zero. In some, it is not a global minimum, e.g., when 04 =-.005. It is not clear what to take as 03 for this case. Machine underflows associated with the computation of exp (03x3t) occur for 03 <- 18.3. Yet any choice of 03 within - 18.3 < 03 < 0 would be rather arbitrary. In practice, one would discard the exponential term in the model, but such a procedure is not a statistical test.

The effect of a decrease in the absolute value of 04 on the performance of a first derivative iterative algorithm, the modified Gauss-Newton method (Hartley 1964), is shown in Table 1. That value of 0 corresponding to the local minimum of the sum-of-squares surface with smallest 1031 was taken as the least-squares estimate 0. From the starting value

(O)0i = &i-.1 (i = 1, 2, 3, 4)

' These errors may be reconstructed from the Appendix of Gallant (1975b) by subtracting f(xt, 0) from yt, with 0 = (0, 1, -1, -.5).

a Journal of the American Statistical Association September 1977, Volume 72, Number 359

Applications Section

523



524 Journal of the American Statistical Association, September 1977

A. Residual Sum of Squares Plotted Against Trial Values for 03 for Various True Values of 04

SSE SSE

-04 9 0i04

3-20 03 83 .03

SSE SSE ^ > ~~~~~~~~- 04 .04

4=3 4= 5 04 43 .00

64= - l i .04 84-- oOI0 .03

-20 0 j 8 20 0

SSE SSE

@4= -.05 -.04 -4.0 04

4"20 03 4'^20 = 03 '-20 0 '-20 0

an attempt was made to recompute the least-squares estimator using the modified Gauss-Newton method and the following rule: Stop when two successive iterations, (i)O and (i+1)O, do not differ in the fifth significant digit (properly rounded) of any component. The performance of the algorithm is seen to deteriorate as 104 1 decreases. Similar behavior is observed in Tables 1 and 2 of Gallant and Fuller (1973).

The mathematical difficulties are caused by the violation of two standard regularity conditions (Gallant

1973; Jennrich 1969; Malinvaud 1970); first, the (almost sure) limit

n

lim (1/n) E [f(.t, 0) - f(xt, 0*)]2 n -*oo t=1

has a unique minimum at the true value 0* of 0 and, second, the (almost sure) limit matrix with typical element

n

lim (1/n) E [(a/(aj)f(i-t, 0*)][(a/1aj)f(xt, 0*)] n ->co t=l

is nonsingular.2 As a result, it does not appear possible to derive the asymptotic properties of c3 when H is true under assumptions of sufficient generality to be useful in applications. Specifically, WA will not converge in probability to a constant so that as n becomes large the response function cannot be approximated by a first-order Taylor's expansion about 0*. Thus, the methods of proof employed in, e.g., Gallant (1975a, 1975c) cannot be used to deduce that, under H, test statistics depending on 0 will eventually behave as their linear regression analogs.

While it may be possible to discover the asymptotic properties of

' using other methods of proof, it would

seem a fruitless task in view of the computational problems noted previously. For this reason, this article does not consider tests for H against A which depend on the computation of 0 but, rather, extends the lack-of-fit test to this problem.

It is useful to consider when the situation of testing H against A using data which support H is likely to arise. It is improbable that one would attempt to fit a nonlinear model which is not supported by the data if one is merely attempting to represent data parametrically without reference to a substantive problem. For example, in the cases considered in Table 1, plots of the observed response Yt against the input x3t fail to reveal any visual impression of exponential growth for values of 104 1 less than .1. Consequently, substantive rather than data analytic considerations will likely have suggested A. As

2The asterisk is used in connection with parametersV*, T*, co*, and O* to emphasize that it is the true, but unknown, value which is meant. The omission of the asterisk does not necessarily denote the contrary.

1. Performance of the Modified Gauss-Newton Method

True value Least-squares estimate Modified Gauss-Newton Tuva/u iterations from a start of 84a 01 02 03 0 42 of Oj-.1

-.5 -.0259 1.02 -1.12 -.505 .00117 4 -.3 -.0260 1.02 -1.20 -.305 .00117 5 -.1 -.0265 1.02 -1.71 -.108 .00118 6 -.05 -.0272 1.02 -3.16 -.0641 .00117 7 -.01 -.0272 1.01 -.0452 .00758 .00120 b

-.005 -.0268 1.01 -.0971 .0106 .00119 b -.001 -.0266 1.01 -.134 .0132 .00119 202 0 -.0266 1.01 -.142 .0139 .00119 69

Parameters other than 04 fixed at 01 = 0, 02 = 1, 03 1, 0-2 =.001. b Algorithm failed to converge after 500 iterations.



Testing a Nonlinear Specification 525

we shall see, it will be helpful if these same substantive considerations also imply probable values for co.

2. AN EXTENSION OF THE LACK-OF-FIT TEST The lack-of-fit method has been discussed by several

authors (Beale 1960; Halperin 1963; Hartley 1964; Turner, Monroe, and Lucas 1961; Williams 1962) in the context of finding exact tests or confidence regions in nonlinear regression analysis. Here, the same idea is employed, but an asymptotic theory is substituted for an exact small-sample theory. The basic idea is straight- forward: If r* = 0, then the least-squares estimator of the parameter 6 in the model

A: yt = g(Xt, 4') + zt'a + et

where the additional regressors zt(w X 1) do not depend on any unknown parameters, is estimating zero provided standard regularity conditions are satisfied. The function g(x, 4) and the regressors Zt are assumed, henceforth, to satisfy the regularity conditions of the Appendix. This assumption is not affected by whether or not 6 = 0 since the additional regressors do not depend on unknown parameters. For the same reason, it is less difficult to compute the least-squares estimators for the parameters of A than it is to compute them for

A: yt = g(xt, 4') + rh(xt, w) + et

from data which support

H: yt = g(xt, 4') + et

The length J1fJ of 6 can move toward zero in the iterations without causing other parameters to become indetermi- nate or the matrix of partial derivatives to become nearly singular.

There are two conventional tests for H against A, the likelihood-ratio test and the test based on the asymptotic normality of the least-squares estimator of 6. Let (4, 6) be the least-squares estimator for the model A, and define:

n

&J = (1/n) E [yt - g(xt, 4) zt 6]- , t=1

= [n/(n - u - w)]6-2,

G(4') = the n by u matrix with typical element

(a/'9j)g(xt iA) , Z = the n by w matrix with tth row Zt' , and

C ={EG(+) Z]'[G(;) *.Z] }-1 Let ; be the least-squares estimator for the model H, and define

n 2 = (1/n) E [yE-y(xt _X,)]2

t=1

In this notation, the likelihood-ratio test rejects when the statistic

T = &/

exceeds the (asymptotic) size a critical point

c=1+wFa/(n-u- w),

where F<, denotes the upper a * 100 percentage point of an F-random variable with w numerator degrees of freedom and n - u - w denominator degrees of freedom. The test based on the asymptotic normality of a rejects when

S= '(C22)1 a/ (ws2)

exceeds the (asymptotic) size a critical point F,, where C22 is obtained by deleting the first u rows and columns of C.

The objective governing the choice of additional regressors is to find those which will maximize the power of the test when A is true. The asymptotic power of the tests using T and S is the same (see the Appendix) and is given by the probability that a doubly noncentral F statistic exceeds F,. The noncentrality parameters of this statistic, using the notational conventions of Graybill (1961, p. 75), are X1 - (r*)2hIQGZ(Z'QGZ)>1Z'QGh/(2a2) for the numerator, and X2 = (r*)2h'QGh/(2o2) - X1 for the denominator, where h = [h(xi, w*), ..., h(xnx, w*)]' and QG-= I - (I*)[G'(I*)G( ,*)ilG'(/*) Thus one should attempt to find those Zt which best approximate h in the sense of maximizing the ratio,

h'QGZ(Z'QGZ)j1Z'QGh/h'QGh

while attempting, simultaneously, to keep the number of columns of Z as small as possible. We consider, next, how this might be done in applications.

In a situation where substantive considerations or previous experimental evidence suggest a single point estimate co for w, the natural choice is Zt = h(xt, W).

If, instead of a point estimate, ranges of plausible values for the components of w are available then a representative selection of values of X

{ i = 1, 2, . . . , K}

whose components fall within these ranges can be chosen -either deterministically or by random sampling from a distribution defined on the plausible values and the vectors h(&i) made the columns of Z. If, following this procedure, the number of columns of Z is unreasonably large, it may be reduced as follows. Decompose the matrix

B-= h (cs 1) * h (cK)

into its principal component vectors, and choose the first few of these to make up Z; equivalently, obtain the singular value decomposition (Bussinger and Golub 1969) B = USV', and choose the first few columns of U to make up Z.

One would be indifferent between a choice of T or o to test H on the basis of the asymptotic theory of the Appendix. Monte Carlo experience does permit a choice if experience acquired in ordinary nonlinear regression analysis carries over to this situation as one would expect;




the Monte Carlo study in Section 4 suggests that it does. To summarize this experience, the power curves of the tests derived from the statistics P and A may be expected to cross when the critical point is adjusted to a percentage point determined by simulation (Blattberg, McGuire, and Weber 1971). Thus power considerations do not permit a choice; however, simulations have revealed that c is a more accurate approximation of the small sample critical point of T than is Fa for S (Gallant 1975c, 1976a); c is a surprisingly accurate approximation even in extremely nonlinear situations (Gallant 1976a). Thus use of the statistic P should yield more accurate probability statements.

3. -EXAMPLES

The first example is a reconsideration of the testing problem described in Gallant and Fuller (1973, sec. 5). The data shown in Figure B are preschool boys' weight/ height ratios plotted against age and were obtained from Eppright et al. (1972); the tabular values are given in the Appendix. The question is whether the data support the choice of a three-segment quadratic-quadratic- linear polynomial response function as opposed to a two- segment quadratic-linear response function. In both cases, the response function is required to be continuous in x (age) and to have a continuous first derivative in x. Formally,

H: Yt = 01 + 02xt + 03T2(04 - Xt) + et

and

A: Yt = 01 + 02xt + 03T2(04 - Xt) + 05T2(06 - Xt) + et

where Tk (Z) = Zk when z > 0

= 0 when z < 0

(see Gallant and Fuller 1973, sec. 2).

B. Preschool Boys' Weight-Height Ratio

POUNDS PER INCH

I.1_

0 0

1.0 0 00 0 0 0 0

0.9 0 oO 0v

0 ~~~0 00

0000~~~ 0.8 00 00

0 0 0 0~~~~~~~

0.7 0 0

0.6 -

0.5

0.3 I O 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72

MONTHS

The correspondence with the notation of Section 1 is:

= (01, 02, 03, 04), r = 05, 0= 66

g (xi s6) = 61 + 462X + 063T2 (s64 X) and

h (x, w) = T2 (w - x)

The parameter w is the join point associated with the quadratic term whose omission is proposed.

Suppose plausible values for w are c =4, = 2 = 8, and C03 = 12. The matrix B of Section 2 has typical row

Bt = [T2(4 - Xt), T2(8 - xt), T2(12 - x,)]

The first principal component vector of B, with elements

Zt = [(2.08) T2(4 - xt) + (14.07)T2(8 - xt) + (39.9) T2(12 -Xt)]1?-4

was chosen as the additional regressor. This choice yields

2= .0005264

2= .0005235

T= 1.006,

P[T > 1.0061H] .485

These data give little support to A.3

C. Specific Retention Volume of Methylene Chloride in Polyethylene Terephthalate

In (cc./gm.) 1.50 -

1.26

1.02 -

0.78 -

0.54

0.30 2.540 2.672 2.804 2936 3.068 3.200

(I/?K) x 103

The second example illustrates how the ideas in Section 2 may be extended in a natural way to a situation studied by Feder (1975). The data shown in Figure C are from

I The model

A: Yt= =V + V2Xg + V3T2(4 - Xt) + 6'Zt + et

does not satisfy the regularity conditions of the Appendix because the second derivative (- 2/C642)T2(V4-x) = 2To(p4 - x) is not continuous in JP4; we are relying on the asymptotic theory in Gallant (1973) and the simulations to justify the use of T in this instance.



Testing a Nonliinear Specification 527

Hsiung (1974) and were obtained to study the specific retention volume of an organic liquid, methylene chloride, in a polymer, polyethylene terephthalate, at selected temperatures; the tabular values are given in the Ap- pendix. The question is whether the data support a single quadratic polynomial model

H: Yt = 01 + 02XI + 03Xt2 + et

or a segmented quadratic-quadratic polynomial model

A: Yt = 01 + 02Xt + 03X 2 + 04T1(06-Xt) + 05T2(06 - Xt) + et

where the latter response function is restricted to be continuous in x but will have a jump discontinuity in the first derivative of height 1041 at r = 06. The answer is obvious from Figure C, but we shall confirm it by means of a statistical test.

Choosing as plausible values for co-06 the points 1 = 2.8, ?'2 -2.85, andC&3 = 2.9, we construct B(n X 6) with typical row

Bt = [T1(2.8 - xt), T1(2.85 - xt), T1(2.9 -x), T2(2.8 - Xt), IT2(2.85 - xt), IT2(2.9 - xt)]

The first two principal component vectors of B, with elements

Zlt = Bt. (.52, .68 .84, .11, .17, .25)'

Z2t = B (-6.4, -.94, 6.6, -2.7, -3.1, -2.8)'

where chosen as additional regressors, yielding = .02381

&2 = .003001

T = 7.935

P[T > 7.935 I H] = .0000895

These data strongly support A, a conclusion which, as was noted, is visually obvious. Observe that the model

A: Yt = 11l + 4I2Xt + It3Xt2 + 6lzlt + 62Z2t + et

is linear in its parameters so that the probability .0000895 is exact, not asymptotic-granted normally distributed errors.

For this problem Feder (1975) has shown that the likelihood-ratio test statistic

T -

(normalized as nT) is not asymptotically distributed as a chi-square random variable when H holds. Moreover, he argues that the distribution of T will depend on the particular arrangement of design points xt. Thus it would appear impossible to obtain a single set of tables for the use of T in applications; it would be necessary to find critical points by Monte Carlo simulation at each instance. The methodology suggested here substitutes the statistic T allowing the use of tables of F. In fact, as was noted, for this particular example the model chosen for A is linear in the parameters; hence exact size, not asymptotic size, is achieved by the test.

2. Power of the Lack-of-Fit Test

Analytic approximation Monte Carlo estimate

X r, X2 Power P(t> ) Std. err. P( > Fa) Std. err.

a. First principal component of B as the additional regressor

0.0 0.0 0.0 .050 .051 .0041 .065 .0069 -0.0001 0.169 0.012 .088 .078 .0048 .148 .0076 -0.0002 0.667 0.049 .208 .198 .0058 .303 .0091 -0.0003 1.523 0.111 .404 .351 .0072 .497 .0091 -0.0004 2.708 0.197 .629 .546 .0086 .689 .0078

b. First two principal components of B as additional regressors

0.0 0.0 0.0 .050 .048 .0049 .118 .0069 -0.0001 0.180 0.001 .076 -0.0002 0.721 0.006 .166 .146 .0059 .295 .0098 -0.0003 1.622 0.013 .330 -0.0004 2.883 0.022 .544

c. All columns of B as additional regressors

0.0 0.0 0.0 .050 -0.0001 0.181 0.0001 .070 -0.0002 0.726 0.0004 .142 -0.0003 1.633 0.001 .280 -0.0004 2.904 0.002 .477

d. The omitted variable as the additional regressor

0.0 0.0 0.0 .050 -0.0001 0.182 0.0 .091 -0.0002 0.726 0.0 .221 -0.0003 1.634 0.0 .430 -0.0004 2.905 0.0 .662




As pointed out by a referee, these examples have exploited the fact that w is univariate and is easy to judge visually being a join point of a segmented polynomial model in both cases. The methods proposed here are applicable when w is multivariate, but the referee's point that application will be more awkward than for univariate X is valid.

4. SAMPLING CHARACTERISTICS OF THE TEST The first example is reexamined in this section to

determine: the power of the test relative to the likelihood- ratio test were w3 known, the variation in power due to variation in the number of principal components of B taken as additional regressors, and the accuracy of the asymptotic approximation to power. The results are reported as Table 2.

The table shows power for the first example at size a = .05 which was computed using the analytic approximation obtained from the asymptotic theory. The parameter 0* was taken from the last entries of Table 2 of Gallant and Puller (1973), save 3*T*, which varies as shown in the table. The standard deviation of the errors was taken as a = .002385. For selected values, Monte Carlo power estimates are compared with the analytic approximations.

These computations indicate that the best choice of additional regressors for the first example is to take as Z the first principal component of B. For practical purposes, the power is as good as if wc* were known. As expected, the asymptotic theory yields a better analytic approximation to the power of the test which uses the statistic T than that which uses S.

APPENDIX A.1. Notation

Y = (Y1, Y2, ..., y)', (n X)

g(#) = [g(xi, t), g(X2, 4I), .* ., g(x, )]', (n X 1) h(w)= [h (xi, w), ..., h(xn, W)]' , (n X 1)

e = (el, e2, ... . en) (n X 1), G(*) = the n by u matrix with typical element (a/3911j)

g(xt, A) where t is the row index

Z = the n by w matrix with rows Zt',

g = W)

h - h(w*) G = G(O*)

PG = G(G'G)-'G', (n X n)

Paz = [G.Z]{[EGZTEG.Z] }-1E{GZ]', (n X n)

QG = I -PG

QGZ = I - PGZ

[G'G G[l- C11 C121

LZ'G ZJZl =_C21 C22..aI = -(C11G' + C22Z') (e + T*h).

A.2. Regularity Conditions

The values of s6 and a are contained in compact sets ' and A, respectively. The sequences x, } and {Z, } have values in the compact sets OC and 5, respectively. The measure Y,n defined on the Borel subsets of SC X a by

1n(A) = the proportion of (x,, z,) in A for t < n

converges weakly to a measure ,u defined on the Borel subsets of 9C X 5; see Gallant (1975a). The functions g(x, 4/), (a/ajPi)g(x, II'), and (2ja6i/abjj)g(x, 4I) are continuous in (x, 4t') on S X T. The function h (x, w*) is continuous on 9C. The true value /A* is contained in an open set which is, in turn, contained in '; A contains an open neighborhood of the zero vector. If g(x, 'I') = g(x, Ab*) except on a set of j, measure zero, it is assumed that 4t' = #*; likewise, g(x, V/) + z'6 = g(x, +P*) ae im- plies + = p6 and 3 = 0. The matrix lim,, [GZ]'EG *Z] is nonsingular. (The limit exists by Lemma 1 of Gallant (1975c)). As n increases, Vn T* tends to a finite limit. The errors {et) are independently and normally distributed each with mean zero and unknown variance U2.

A.3. Asymptotic Theory

(See Gallant (1976b) for proofs.)

Theorem 1: The random variable 42 may be characterized as

42 = (e + r*h)'QGz(e + T*h)//n + a,

where na. converges in probability to zero. The random variable ,2 may be characterized as

2 = (e + r*h)/QG(e + r*h)/n + bn e where nbn converges in probability to zero.

Theorem 2: The statistic P may be characterized as T = X + c., where

X = (e + r*h)'QG(e + r*h)/(e + r*h)'QGz(e + r*h)

and nc converges in probability to zero. The probability P (X > c) is given by the doubly

noncentral F distribution, using the notation of Graybill (1961, p. 75), with numerator degrees of freedom w, and noncentrality parameter

XI = (,r*)2h'(PGz - PG)h/ (2of2)

and denominator degrees freedom n - u - w, and noncentrality parameter

2 = (r*)2h/QGzh/ (2o.2)

Theorem 3: The statistic S may be characterized as Y= + d,, where

Y = [(e + r*h) (PGz - PG) (e + r*h)/w]/ --[(e + r*h)'QGz(e + r*h)/(n- u -w)]

and dn converges in probability to zero. The probabilities P(Y > Fa) and P(X > c) are equal where X is as in Theorem 2.



Testing a Nonlinear Specification 529

Remark: It may be verified algebraically that

Xi = (r*)2h/QGZ(Z QGZ)<1ZIQGh/(2U2)

= (r*)2h'QGh/(2a2)-

A.4. Technical Details of the Simulation

The pseudo-normal random number generator (GGNOF)

(International Mathematical and Statistical Libraries 1975) was employed. The modified Gauss-Newton method (Hartley 1961) was used to compute least- squares estimates using jP* as the starting value of VI and different choices of starting value for 6 depending on the value of T*. All code was written in double pre- cision, except the use of GGNOF, using an IBM 370/165.

The Monte Carlo power estimates are based on 1,000 trials using the control-variate method of variance re- duction (Hammersly and Handscomb 1964, p. 59-60). The control variate employed was the test statistic which takes the value one when V exceeds F,a and is zero other- wise where,

V= (W2'C22-lW2/W)/

[ (e + T*h) QGZ(e + r*h) / (n - u - w)]

The power of this test is given exactly by P(X > c).

A.5. Tabular Values for the Examples

Table 3 presents the data used in Figure B, and Table 4 gives the data used in preparing Figure C.

3. Boys' Weight/Height vs Age

W/H Age W/H Age W/H Age

0.46 0.5 0.88 24.5 0.92 48.5 0.47 1.5 0.81 25.5 0.96 49.5 0.56 2.5 0.83 26.5 0.92 50.5 0.61 3.5 0.82 27.5 0.91 51.5 0.61 4.5 0.82 28.5 0.95 52.5 0.67 5.5 0.86 29.5 0.93 53.5 0.68 6.5 0.82 30.5 0.93 54.5 0.78 7.5 0.85 31.5 0.98 55.5 0.69 8.5 0.88 32.5 0.95 56.5 0.74 9.5 0.86 33.5 0.97 57.5 0.77 10.5 0.91 34.5 0.97 58.5 0.78 11.5 0.87 35.5 0.96 59.5 0.75 12.5 0.87 36.5 0.97 60.5 0.80 13.5 0.87 37.5 0.94 61.5 0.78 14.5 0.85 38.5 0.96 62.5 0.82 15.5 0.90 39.5 1.03 63.5 0.77 16.5 0.87 40.5 0.99 64.5 0.80 17.5 0.91 41.5 1.01 65.5 0.81 18.5 0.90 42.5 0.99 66.5 0.78 19.5 0.93 43.5 0.99 67.5 0.87 20.5 0.89 44.5 0.97 68.5 0.80 21.5 0.89 45.5 1.01 69.5 0.83 22.5 0.92 46.5 0.99 70.5 0.81 23.5 0.89 47.5 1.04 71.5

Source: Eppright et al. (1972).

4. Specific Retention Volume of Methylene Chloride in Polyethylene Terephthalate

Reciprocal x 103 of Natural logarithm of temperature in degrees Kelvin specific volume in cc. per gm.

2.54323 1.16323 2.60960 1.10458 2.67952 0.98832 2.75330 0.87471 2.79173 0.62060 2.82965 0.51175 2.87026 0.35371 2.91120 0.66954 2.94637 0.85555 3.00030 1.07086 3.04228 1.22272 3.09214 1.29113 3.13971 1.38480 3.19081 1.46728

Source: Hsiung (1974).

[Received July 1975. Revised February 1977.]

REFERENCES

Beale, E.M.L. (1960), "Confidence Regions in Nonlinear Esti- mation," Journal of the Royal Statistical Society, Ser. B, 22, No. 1, 41-76.

Blattberg, Robert C., McGuire, Timothy W., Weber, Warren E. (1971), "A Monte-Carlo Investigation of the Power of Alternative Techniques for Testing Hypotheses in Nonlinear Models," Techni- cal Report, Graduate School of Industrial Administration, Carnegie-Mellon University.

Bussinger, Peter A., and Golub, Gene H. (1969), "Singular Value Decomposition of a Complex Matrix," Communications of the ACM, 12, 564-5.

Eppright, E.S., Fox, H.M., Fryer, B.A., Lamkin, G.H., Vivian, V.M., and Fuller, E.S. (1972), "Nutrition of Infants and Preschool Children in the North Central Region of the United States of America," World Review of Nutrition and Dietetics, 14, 269-332.

Feder, Paul I. (1975), "The Log Likelihood Ratio in Segmented Regression," The Annals of Statistics, 3, 84-97.

Gallant, A. Ronald (1973), "Inference for Nonlinear Models," Institute of Statistics Mimeograph Series, No. 875, Raleigh: North Carolina State University.

, and Fuller, W.A. (1973), "Fitting Segmented Polynomial Regression Models Whose Join Points Have to Be Estimated," Journal of the American Statistical Association, 68, 144-7.

(1975a), "The Power of the Likelihood Ratio Test of Location in Nonlinear Regression Models," Journal of the American Sta- tistical Association, 70, 198-203.

(1975b), "Nonlinear Regression," The American Statistician, 29, 73-81.

(1975c), "Testing a Subset of the Parameters of a Nonlinear Regression Model," Journal of the American Statistical Association, 70, 927-32.

(1976a), "Confidence Regions for the Parameters of a Non- linear Regression Model," Institute of Statistics Mimeograph Series, No. 1077, Raleigh: North Carolina State University.

(1976b), "On the Asymptotic Power of the Lack of Fit Test in Nonlinear Regression," Institute of Statistics Mimeograph Series, No. 1083, Raleigh: North Carolina State University.

Graybill, Franklin A. (1961), An Introduction to Linear Statistical Models, Vol. 1, New York: McGraw-Hill Book Co.

Halperin, Max (1963), "Confidence Interval Estimation in Non- linear Regression," Journal of the Royal Statistical Society, Ser. B, 25, No. 2, 330-3.

Hammersly, J.M., and Handscomb, D.C. (1964), Monte-Carlo Methods, New York: John Wiley & Sons.

Hartley, H.O. (1964), "Exact Confidence Regions for the Parameters in Nonlinear Regression Laws," Biometrika, 51, 347-53.




(1961), "The Modified Gauss-Newton Method for the Fitting of Nonlinear Regression Functions by Least Squares," Techno- metrics, 3, 269-80.

Hsiung, P.O. (1974),. "Study of the Interaction of Organic Liquids with Polymers by Means of Gas Chromatography," unpublished Ph.D. dissertation, North Carolina State University.

International Mathematical and Statistical Libraries, Inc. (1975), IMSL Library 1, Fifth Ed., Houston: International Mathematical and Statistical Libraries, Inc.

Jennrich, Richard I. (1969), "Asymptotic Properties of Nonlinear Least-Squares Estimators," The Annals of Mathematical Statistics, 40, 633-43.

Malinvaud, E. (1970), "The Consistency of Nonlinear Regression," The Annals of Mathematical Statistics, 41, 956-69.

Marquardt, Donald W. (1963), "An Algorithm for Least-Squares Estimation of Nonlinear Parameters," Journal of the Society for Industrial and Applied Mathematics, 11, 431-41.

Turner, Malcolm E., Monroe, Robert J., and Lucas, Henry L. (1961), "Generalized Asymptotic Regression and Nonlinear Path Analysis," Biometrics, 17, 120-49.

Williams, E.J. (1962), "Exact Fiducial Limits in Nonlinear Esti- mation," Journal of the Royal Statistical Society, Ser. B, 24, No. 1, 125-39.



testing a nonlinear regression specification: a nonregular case

Documents