DERIVING TESTS OF THE REGRESSION MODEL USING THE DENSITY FUNCTION OF A MAXIMAL INVARIANT

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1 DERIVING TESTS OF THE REGRESSION MODEL USING THE DENSITY FUNCTION OF A MAXIMAL INVARIANT Jahar L. Bhowik and Maxwell L. King Departent of Econoetrics and Business Statistics Monash University Clayton, Victoria 38 Australia Abstract In the context of the linear regression odel in which soe regression coefficients are of interest and others are purely nuisance paraeters, we derive the density function of a axial invariant statistic. This allows the construction of a range of optial test statistics including the locally best invariant test which is equivalent to the wellknown one-sided t-test. The approach is also extended to the general linear regression odel in which the covariance atrix is nonscalar and to non-linear regression odels. Key words: Invariance; linear regression odel; locally best invariant test; non-linear regression odel; nuisance paraeters; t-test. JEL classification: C, C.

2 . Introduction This paper is concerned with the proble of testing the null hypothesis that one regressor coefficient is zero, against the alternative that it is non-negative, in the context of linear and non-linear regression odels. Statistical odels and particularly those used by econoetricians, involve a large nuber of influences. These kinds of odels contain two types of paraeters, those of interest and those not of iediate interest that are known as nuisance paraeters. Their presence causes unexpected coplications in statistical inference. Kalbfleisch and Sprott (97) discussed ethods of eliinating nuisance paraeters fro the likelihood function so that inference can be ade about the paraeters of interest. In practice, any statistical probles including testing of hypothesis, display syetries, which ipose additional restrictions for the choice of proper statistical tests. In the statistical testing literature, the idea of invariance dates back half a century. The unpublished work of Hunt and Stein (946) (see Lehann, 959 and 986) introduced the principle and deonstrated its applicability and eaningfulness in the fraework of hypothesis testing. According to Lehann (95), the notion of invariance was introduced into the statistical literature in the writings of R. A. Fisher, Hotelling (933, 936), Pitan (939), Stein (948) and others, in connection with various special probles. Aong others Lehann (959, 986), King (979, 98, 983a), King and Sith (986), Ara and King (993), and Ara (995) suggested the use of invariance arguents to overcoe the proble of nuisance paraeters. It is a generally accepted principle that if a proble with a unique solution is invariant under a certain group of transforations, then the solution should be invariant under that transforations. A hypothesis testing proble is invariant under a group of

3 transforations acting on the saple space of the observed data vector if for any transforation the probability distribution of transfored data vector belongs to the sae set (null or alternative hypothesis) as the original data vector. The idea behind invariance is that if the hypothesis testing proble under consideration has a particular invariance property, then we should restrict attention to only those tests that share this invariance property. The class of all invariant functions can be obtained as the totality of functions of a axial invariant. A axial invariant is a statistic which takes the sae value for the observed data vectors that are connected by transforations and different values for those data vectors that are not connect by transforations. Consequently any invariant test statistic can be written as a function of the axial invariant. This eans, we can treat the axial invariant as the observed data, find its density and then construct appropriate tests based on this density. The perforance of a statistical test is assessed by its size and power properties. Econoetricians are always interested in optiality of power and for any testing proble they would like to use a uniforly ost powerful (UMP) test. Many testing probles involving the linear regression odel can be reduced, either by conditioning on sufficient statistics or by invariance arguents to testing a siple null hypothesis against one sided alternatives. Ideally we would then like to use a UMP test but unfortunately it is rarely possible to find a UMP test when the alternative hypothesis is coposite andor in the presence of nuisance paraeters. Cox and Hinkley (974, p.) discuss three approaches for constructing tests of siple hypotheses against coposite alternative hypotheses when no UMP test exits. They involve choosing a point at which the power is optiised: 3

4 (i) to pick, soewhat arbitrarily, a typical point in the alternative paraeter space and use it in test construction in order to find a test that at least has optial power for the chosen point; (ii) reoving this arbitrariness by choosing a point to be close to the null hypothesis which leads to the locally best (LB) test, i.e. to axiize the power locally near the null hypothesis; (iii) choosing a test which axiizes soe weighted average of the powers over the alternative paraeter space. Option (i), labelled by King (987b) as the point optial (PO) approach, uses the ost powerful test against a specific alternative solution. Option (ii) is the ost popular for a single paraeter and leads us to a LB or a locally ost powerful test. These tests are constructed by axiizing power locally at the null hypothesis. The LB test is also optial in the sense that its power curve has the steepest slope, at the null hypothesis, of all power curves fro tests with the sae size. Following Neyan and Pearson (936) a nuber of authors, notably Ferguson (967), Efron (975), King (98, 984, 987a), King and Hillier (985), Sengupta (987) and Wu and King (994) aong others, have recoended the use of LB tests. Our interest in this article is to derive the density function of the axial invariant statistic in the context of regression odel and invariant transforation and then construct a LBI test for linear and non-linear regressors, which is called the Locally Best Invariant (LBI) test. Also we show that this LBI test is equivalent to the onesided t-test in the case of testing regression coefficients. 4

5 The plan of this paper is as follows. First of all we derive the density function of the axial invariant in section and in section 3 we construct the LBI test statistic which is shown to also be equivalent to the one-sided t-test. In section 4, we derive the density function of the axial invariant for a non-linear regression function and find the LBI test statistic. Finally, soe concluding rearks are ade in section 5.. Derivation of Density Function Let us consider the linear odel, y = X β + X β + u (.) where y is n, X is an n q nonstochastic atrix, X is an n p nonstochastic atrix, β is a q vector and β is p vector. Here [ X: X] is full colun rank. Considering first the case of p =, we are interested in testing H :β = against H a :β > in the context of the above linear regression odel. It is assued that u~ N(, σ I n here σ is unknown. This proble is invariant under the class of transforations y γ y+ X γ (.) where γ is a positive scalar and γ is a q vector. Let M = I X X ( X ) X. Then My= MXβ + MXβ + Mu = MXβ + Mu. (.3) Multiplying both sides of (.) by P where P is an n atrix such that PP = I, PP = M and = n q,weget Py = PX β + Pu. 5

6 Note that PM y = Py and PM = P. Thus Py ~ N PX (, I β σ ).Let z = Py. Then the joint density function of z is f( z) = ( πσ ) exp{ ( z PX ) ( z PX )} β β. (.4) σ Let r = zz be the usual squared distance of z fro the origin. Now, we change z to the -diensional polar co-ordinates ( r, θ, θ,..., θ ) as follows: z z z = rcosθ j j k j = r( sin θ )cos θ ; = r sinθ k where r, θ k π,for k =,,...,( ) and θ π. for j, (.5) The Jacobian of the transforation is J (, r θ, θ,..., θ ) = ( z, z,..., z) (, r θ, θ,..., θ ) = r k sinθ k (Miller, 964, p.3). To construct the LBI test, we have to find the density function of the axial invariant statistic. So we want to find the distribution of the axial invariant w= z( z z) = z r. Note that z = rw. Now the joint density function of z becoes, after the above change of variables, f (, r θ, θ,..., θ ) = ( πσ ) exp{ ( rw PX β )( rw PX β )} r sinθ k σ k 6

7 =( πσ ) exp{ ( rww wrpx )} β + β X PPX β σ r k k sinθ. (.6) To find the density function of w fro the above joint density function of z,first we have to find the arginal density function of ( θ, θ,..., θ ). The coponents of w are w( θ, θ,..., θ ),, w ( θ, θ,..., θ ) and they are defined by (.5). Therefore the arginal density function of ( θ, θ,..., θ ) can be obtained by integrating out r of (.6), f( θ, θ,..., θ ) = ( πσ ) exp{ ( r ww wrpx β β X PPXβ )} + σ r k sinθ k = ( ) + sin k exp{ ( )} πσ θ k rww wrpx β β X PPXβ r dr σ dr k r r β β = + β ( πσ ) sinθ k exp{ ( )} ww w PX X P PX r dr σ σ σ σ σ z z k = + ( πσ ) sinθ k exp{ ( λ λwpxβ β X PPX β )} λ σ σ dλ; β where ww =, β =, λ = r,and dr = σdλ.ifweset σ σ aw (, β ) = wpx β, (.7) b( w, β ) = ( β X P ww PX β β X P PXβ ) = β X P Mw PXβ (.8) 7

8 and Mw = I ww = I w( w w) w,then bw (, β ) is su of squared errors of the OLS regression of PX β on w and z k f( θ, θ,..., θ ) = ( π) sinθk exp{ ( λ a( w, β )) + b( w, β )} λ dλ; or f ( θ, θ,..., θ ) k = ( π) sinθk exp( bw (, β )) exp{ ( λaw (, β )) } λ dλ k = sinθk exp( bw (, β )) λ ϕ{( λaw (, β )) } dλ = cw (, β ) ( η+ aw (, β )) ϕ( η ) dη, where ϕ{( λ aw (, β )) } = ( π) exp{ ( λaw (, β )) }, (King, 979, chapter 3), cw (, β ) = exp( bw (, β )) sinθk k η= λaw (, β ), λ = η+aw (, β ) and dλ = dη. We can further write f( θ, θ,..., θ ) = c { η + ( ) η a( w, β ) +, (.9) ( )( ) 3 a ( w, ) a ( w, η β β )} ϕ( η ) dη = cw (, β ){ η ϕ( η ) dη+ aw (, β)( ) η ϕ( η ) dη+ a 3 ( w, ) ( )( ) β η ϕ( η ) dη a ( w, β ) ϕ( η ) dη} = cw (, β ){ Γ( ) π + aw (, β )( ) Γ( ) π a 3 ( w, ) ( )( ) ( )... a ( w, )( ) β Γ π + + β π Γ( )}; 4 + 8

9 z where η ϕ( η ) dη= Γ ( ) π, ϕη ( ) η ( π) d = 3 Γ( ) and Γ( ) = π. Therefore the arginal density function of θ, θ,..., θ is, f( θ, θ,..., θ ) = c( w, β ){ Γ( ) π + a( w, β )( ) Γ( ) π 3 + a ( w, β )( )( ) Γ( ) π ( ) a ( w, β )( π) π }; 4 or k f( θ, θ,..., θ ) = exp( b( w, β )) sin θk { Γ( ) π + aw (, β )( ) Γ( ) π + a ( w, β )( )( ) Γ( ) π a ( w, β )( π) π }. (.) ' Now the transforation fro ( θ, θ,..., θ ) to w= z( zz) is straightforward since the coponents of w are defined by (.5). Therefore the density function of w is, f( w) = exp( b( w, β )){ Γ( ) π + a( w, β )( ) Γ( ) π + a ( w, β )( )( ) Γ( ) π a ( w, β )( π) π } 4 or f( w) = π exp( b( w, β )){ Γ( ) + a( w, β )( ) Γ( ) π 3 + a ( w, β )( )( ) Γ( ) π + a ( w, β )( )( )( 3) 6 Γ( ) a π ( w, β ) π } (.) 9

10 where aw (, β ) and bw (, β ) are defined by (.7) and (.8). Here (.) is the density function of the axial invariant w and using this density function we can construct the LBI test statistic. 3.Tests of a Linear Regressor We are interested in testing the hypothesis H :β = against H a :β > in the context of the linear regression odel (.). Let us first consider p =, i.e. β is a scalar. A LBI test of H against H a is that with critical region of the for log f( w) β β = c In the previous section we have derived f( w). α. (3.) 3. Construction of the LBI test statistic The density function of the axial invariant w is given by (.). Taking logs on both sides of (.e get log{ f( w)} =log log π + bw (, β ) + log{ Γ( ) + aw (, β )( ) Γ( ) π + a ( w, )( )( ) ( ) a + β Γ π ( w, β ) π } = + log log π ( β X P ww PXβ β X P PXβ ) + log{ Γ( ) a ( ) Γ( ) π a ( )( ) Γ( ) π... a π } log f( w) = ( X P ww PX X P PX ) + { ( ) + a( w, )( ) ( ) β β Γ β Γ π β a ( w, β )( )( ) Γ( ) π... a ( w, β ) π } { Γ( ) + a( w, β )( ) Γ( ) π + a ( w, β )( )( ) Γ( ) π + β a ( w, β ) π }

11 = ( X P ww PXβ X P PXβ ) + { Γ( ) + a( w, β )( ) Γ( ) π + + a ( w, β )( )( ) Γ( ) π a ( w, β ) π } {( ) Γ( ) π ( wpx ) +( )( ) Γ( ) π( wpx β )( wpx ) π ( )( w PX β ) ( w PX ) }; log f( w) β β = = { Γ( )} ( ) Γ( ) π ( wpx ) = π ( ) Γ( )( wpx ). Γ( ) Therefore, log f( w) β β = Γ( ) π = ( ) Γ( ) ( wpx ). Hence the LBI test rejects H for log f( w) or ( wpx ) d β β = α, Γ( ) π = ( ) Γ( ) ( wpx ) cα (3.) or X P Py X My = d α, (3.3) ( yppy ) ( ymy ) where d α is an appropriate critical value. Thus X My s = is the LBI test statistic. ( ymy )

12 3. Relationship between the t-test and the required LBI test Fro the above discussion we can say that the LBI test rejects H for large values of X My s = (3.4) ( ymy ) for testing H :β = against H a :β > in the linear odel (.) if X is a vector, i.e. p =. The ordinary least squares (OLS) estiator of β in (.) is $ = ( X M X ) X M y, β and the unbiased OLS estiator of the error variance is $ = ( ( y M M X X M X ) X M ) y( ). σ Thus the t test statistic is t = $ {$( X M X ) β σ } ; = ( ) ( ) { ( ( ) X M X X M y y M M X X M X X M ) y} ( X M X ) or t = X M X s X ( ) ( ) { ( M X ) s }; (3.5) where X M X is a positive scalar. So, clearly (3.5) is a onotonic increasing function of the test statistic s. Thus we ay conclude that our LBI test is equivalent to the t-test. 3.3 Whether the test statistic s is UMPI or not We have Py Py w= z( z z) = =. (3.6) ( yppy ) ( ymy ) Replacing (3.6) and (3.4) in (.e get,

13 f( w) exp( s X P PX ){ ( ) s ( = π β β Γ + β ) Γ( ) π s β ( )( ) Γ( ) π + s β ( )( )( 3) Γ( ) π 6 Thus ( sβ ) π }. f( w) exp{ ( s X P PX )}{ ( ) s ( = π β β Γ + β ) Γ( ) π s β ( )( ) Γ( ) π + s β ( )( )( 3) Γ( ) π ( sβ ) π }. (3.7) We know all invariant tests can be written as a function of the axial invariant. To find the best test within the class of invariant tests we have derived the density function of the axial invariant and then we have constructed the LBI test. To prove, our test statistic s is UMPI, according to Lehann (986, section 6.4e have to prove that the function of s, f( w) is a onotonic increasing function of s.sofar, we cannot prove theoretically that our LBI test statistic s is UMPI. We have siulated this function using a nuber of different values of s, β and X P PX and in each case found the function (for these particular valuesas onotonic. This is quite different fro being able to prove that f( w) is onotonic. We do however conjecture that f( w) is onotonic having been unable to disprove this using siulation ethods. We therefore conjecture that the test based on s is UMPI where invariance is with respect to transforations of the for (.). 3.4 Construction of the LBI test when u~ N(, σ Σ ( φ)), where Σ( φ) is known Consider the following odel, y = X β + X β + u (3.8) where X is an n q nonstochastic atrix, X is an n p nonstochastic atrix 3

14 and u~ N[, σ Σ ( φ)] Derivation of Density Function Let M = I X X ( X ) X, using the sae transforation on (3.8) that we used earlier (see section e get, PM u ~ N[, σ ( PΣ ( φ) P )] and Py ~ N{ PX β, σ ( PΣ ( φ) P )}. We want to find the distribution of the axial invariant w= z( z z),where z = Py and zz = r. We get, f( z) = ( πσ ) PΣ( φ) P exp{ ( ) ( ( ) ) z PX β PΣ φ P ( z PXβ)}. (3.9) σ Now, changing z to the -diensional polar co-ordinates ( r, θ, θ,..., θ ) as (.5), where r, θ k π,for k =,,...,( )and θ π, the density of z becoes, f(, r θ, θ,..., θ ) = ( πσ ) PΣ( φ) P exp[ {( rw PX β ) ( P ( φ) P ) ( rw PX β )}] r sinθ Σ k σ k = ( πσ ) PΣ( φ) P exp[ { r σ k k + rw ( PΣ( φ) P ) PX β β X P ( PΣ( φ) P ) PX β }] r sin θ. 4

15 The arginal density function of ( θ, θ,..., θ ) r, can be obtained by integrating out k f( θ, θ,..., θ ) = ( πσ ) PΣ( φ) P sinθk exp[ r { w ( P ( ) P w r β Σ φ ( PΣ( φ) P ) PX + σ σ σ β σ β φ X P ( PΣ( ) P ) PX }] r dr σ z k = ( πσ ) PΣ( φ) P sinθ k exp[ {( r ) σ r ) PX σ β σ ( Σ( ) ) β X P P φ P PX + ( ) σ β }] r σ dr k k ' = ( π) PΣ( φ) P sinθ a( w, φ) exp{ ( (,, )) (,, λ + )} a w φ β b w φ β λ d λ where a ( w, φ) = w ( PΣ ( φ) P ) w, (3.) a w ( P ( φ) P ) PXβ ( w, φβ, ) = Σ, (3.) β X P ( PΣ( φ) P ) PXβ bw (, φβ, ) = [ (3.) β X P ( PΣ( φ) P ) PX β ], λ = r β and β =. σ σ 5

16 k f( θ, θ,..., θ ) = exp( b( w, φ, β ))( π) PΣ( φ) P sinθk a( w, φ) exp{ ( (,, λ )) } a w φ β λ d λ k = exp( bw (, φβ, ))( π) PΣ( φ) P sin θ a ( w, φ) exp( η )( η+ a ( w, φ) a ( w, φ, β ) ) dη a φ where η λ φ β = ( w, ) ( a ( w,, )), λ = ( a( w, φ)) η+ a( w, φ, β ) and dλ = ( a ( w, φ)) dη. Now following a siilar approach to integrating out r to that used earlier (see section e get, k k f( θ, θ,..., θ ) = exp( b( w, φ, β )) PΣ( φ) P sin θk a ( w, φ) π ( )( ) [ Γ( ) + ( ) c( w, φβ, ) Γ( ) + c ( w, φβ, ) Γ( ) c ( w, φβ, ) Γ( )] a ( w, φ) a ( w, φ, β ) where cw (, φβ, ) = = Therefore the density function of axial invariant w is, ) PXβ { }. (3.3) f( w) = π exp( b( w, φ, β )) PΣ( φ) P a ( w, φ)[ Γ( ) + ( )( ) + ( ) c( w, φβ, ) Γ( )+ c ( w, φβ, ) Γ( ) c ( w, φβ, ) Γ ( )]. (3.4) 3.4. Construction of LBI test statistic Taking logs on both sides of (3.4e get, 6

17 log f( w) =log logπ log a( w, φ) + b( w, φ, β ) + log{ PΣ( φ) P } + ( )( ) log{ Γ( ) + ( ) c( w, φβ, ) Γ( ) + c ( w, φβ, ) Γ( ) c ( w, φβ, ) Γ( )}, where a ( w, φ ), bw (,, φβ ) and cw (, φβ, ) are defined by (3.), (3.) and (3.3). log f( w) X P ( PΣ( φ) P ) PXβ = [ X P ( P ( φ) P ) PX β } Σ β ( )( ) + { Γ( ) + ( ) c( w, φβ, ) Γ( ) + c ( w, φβ, ) Γ( ) + F HG... + (,, ) ( )} { ( ) w ( PΣ( φ) P ) PX c w φβ Γ Γ( ) a( w, φ) ( )( ) ) PXβ a( w, φ)... + Γ( )( ) log f( w) β F HG ) PXβ β = F HG ) PX IKJ F w ( PΣ( φ) P ) PX HG IKJ + IKJ F HG } IKJ + a ( w, φ) ) PX = { Γ( )} ( ) Γ ( ) I K J. Hence the LBI test rejects H for, { Γ( )} or equivalently or a F HG ( w, φ) ) PX ( ) Γ ( ) r F HG X P ( PΣ( φ) P d α, c h ) PX IKJ d α IKJ c α 7

18 or X P ( PΣ( φ) P ) Py( y My) { yp ( PΣ( φ) P ) Py} ( ymy ) d α, because w= Py( y M y),or X P ( PΣ( φ) P ) Py { yp ( PΣ( φ) P ) Py} dα. (3.5). Here (see King (98)) X P P P Py = X X X X X ( Σ( φ) ) { Σ( φ) Σ( φ) ( Σ( φ) ) Σ( φ) } y = ( X Σ( φ) ){ Σ( φ) Σ( φ) X ( X Σ( φ) X ) X Σ( φ) } y = X M y where y X X = Σ( φ) y Σ( φ) = Σ( φ) = X X M = I X ( X X ) X. Siilarly yp ( PΣ( φ) P ) Py= y M y. Thus the LBI test (3.) can be written as X M y ( y M y ) d α where d α is an appropriate critical value. 8

19 Therefore our required LBI test statistic is s X M y = ( y M y ). (3.6) In the case of p =, we are interested in testing H :β = against H :β > for the regression equation y = X β + X β + v, (3.7) where v = φ u, v ~ N(, σ I) Σ( ) and φ is known. Using a siilar procedure as the unstared case (3.5), we can show that the test statistic t is a onotonic increasing function of the LBI test s. Thus we ay conclude that our LBI test is equivalent to the one-sided t test. 4.Tests of a Non-linear Regressor Let us consider the following non-linear odel, y = X β + g( X, β ) + u (4.) where X is an n q nonstochastic atrix, X is an n p nonstochastic atrix and g( X, β ) is a non-linear function of β and X, for which g( X, β ) = when β =.Wewishtotest H :β = against H a :β > in the context of the above non-linear regression odel for p =. 4. Derivation of Density Function Let M = I X X ( X ) X. Then My= MXβ + Mg( X, β ) + Mu = Mg( X, β ) + Mu. (4.) Now ultiplying (4.) by P we get, 9

20 Py = Pg( X, β ) + Pu, where P is an n atrix such that PP = I, PP = M and = n q. Then PM y = Py and PM = P. Thus Py ~ N( Pg( X, β), σ I ). (4.3) Let z = Py. We want to find the distribution of w= z( z z).here zz = r. Fro (4.3), f( z) = ( πσ ) exp{ ( z Pg( X, )) ( z Pg( X, ))} β β. (4.4) σ Now, changing z to the -diensional polar co-ordinates ( r, θ, θ,..., θ ) (.5), the density of z becoes, as in f (, r θ, θ,..., θ ) = ( πσ ) exp{ ( rw Pg( X, β )) σ k ( rw Pg( X, ))} r β sinθk = ( πσ ) exp{ ( rww wrpgx (, ) + β σ k g ( X P Pg X r, β) (, β)} sin θk. The density function of ( θ, θ,..., θ ) can be obtained by integrating out r, f( θ, θ,..., θ ) = ( πσ ) exp{ ( r ww w rpg( X, β ) + σ k g ( X, β ) P Pg( X, β )} r sinθ dr. Now using siilar transforations to those we have used earlier for the linear case we get, k

21 z k f( θ, θ,..., θ ) = ( π) sinθk exp{ ( λ a( w, β )) + b( w, β )} λ dλ; where aw (, β ) = wpg ( X, β ), (4.5) bw (, β) = { g ( X, β) PwwPg ( X, β) g ( X, β ) P Pg ( X, β )}, (4.6) g( X g, β ) ( X, β ) =, (4.7) σ λ = r and ww =. σ z k f( θ, θ,..., θ ) = ( π) sinθk exp( b( w, β)) exp{ ( λa( w, β)) } λ dλ = c λ ϕ{( λa( w, β )) } dλ; in which c = exp( b( w, β )) sinθk k and ϕ{( λ aw (, β )) } = ( π) exp{ ( λaw (, β )) }. Alternatively z f( θ, θ,..., θ ) = c ( η+ a( w, β )) ϕ( η ) dη, where η = λ aw (, β ), dλ = dη and λ = η +aw (, β ). Following a siilar procedure as for the linear case (section ), we get k f( θ, θ,..., θ ) = exp( b( w, β)) sin θk { Γ( ) π + aw (, β)( ) Γ( ) π + a ( w, β )( )( ) Γ( ) π a ( w, β)( π) π }. + Therefore the density function of the axial invariant w is,

22 f( w) = π exp( b( w, β)){ Γ( ) + a( w, β)( ) Γ( ) π (4.8) + + a ( w, β )( )( ) Γ( ) π a ( w, β ) π } where aw (, β ) and bw (, β ) are defined by (4.5) and (4.6). Using this density function of the axial invariant statistic, we can construct the LBI test for a nonlinear regressor. 4.. Tests of a Non-Linear Regressor Taking logs of both sides of (4.8e get log{ f( w)} =log log π + bw (, β) + log{ Γ( ) + aw (, β)( ) Γ( ) π + a ( w, )( )( ) ( ) a + β Γ π ( w, β ) π } = + log log π { g ( X, β) P ww Pg ( X, β) g ( X, β) P Pg ( X, β)} + log{ Γ( ) + a( w, β )( ) Γ( ) π + a ( w, β )( )( ) Γ( ) π a w β π... (, ) }. Therefore the LBI test of H :β = against H a :β > rejects H for large values of log f( w) β β = and results in the critical region g ( X, β ) β β = ( ymy ) My cα. (4.9)

23 Thus putting the value of g ( X, ) β β = β in the above function we get the LBI test statistic for testing H :β =,where g ( X, β ) X and β. in (4.7) is a non-linear function of 5. Concluding Rearks In this paper, having derived the density function of the axial invariant statistic, we have constructed an LBI test in the linear regression odel against one-sided alternatives. The principle of invariance is used to eliinate nuisance paraeters in ultiparaeter one-sided testing probles. This allows the construction of the LBI test. The resultant LBI test is found to be equivalent to the one-sided t-test. In the case of a non-linear regression, we have derived the density function of the axial invariant statistic and we have constructed the LBI test statistic for a non-linear regressor. 3

24 References Ara, I. (995), Marginal Likelihood Based Tests of Regression Disturbances, Unpublished Ph.D. thesis, Monash University, Clayton, Melbourne. Ara, I. and King, M.L. (993), Marginal Likelihood Based Tests of Regression Disturbances, Paper presented at the 993 Australasian Meeting of the Econoetric Society, Sydney. Cox, D.R. and Hinkley, D.V. (974), Theoretical Statistics, London: Chapan and Hall. Efron, D. (975), Defining the Curvature of a Statistical Proble (with Application to Second Order Efficiency), The Annals of Statistics, 3, Ferguson, T.S. (967), Matheatical Statistics: A Decision Theoretic Approach, New York: Acadeic Press. Gourieroux, C., Holly, A. and Monfort, A. (98), Likelihood Ratio Test, Wald Test and Kuhn-Tucker Test in Linear Models with Inequality Constraints on the Regression Paraeters, Econoetrica, 5, Hildreth, C. and Houck, J.P. (968), Soe Estiators for a Linear Model with Rando Coefficients, Journal of the Aerican Statistical Association, 63, Hillier, G.H. (986), Joint Tests for Zero Restrictions on Non-Negative Regression Coefficients, Bioetrika, 73, Hotelling, H. (933) Analysis of a Coplex of Statistical Variables into Principal Coponents, Journal of Educational Psychology, 4, and Hotelling, H. (936) Relations Between Two Sets of Variates, Bioetrika, 8, Hunt, G. and Stein, C. (946), Most Stringent Tests of Statistical Hypotheses, unpublished anuscript. Kalbfleisch, J.D. and Sprott, D.A. (97), Application of Likelihood Methods to Models Involving Large Nubers of Paraeters, Journal of the Royal Statistical Society B3, Kariya, T. and Eaton, M.L. (977), Robust Tests for Spherical Syetry, The Annals of Statistics, 5, 6-5. King, M.L. (979), Soe Aspects of Statistical Inference in the Linear Regression Model, Unpublished Ph.D. thesis, University of Canterbury, Christchurch. 4

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26 Sengupta, A. (987), On Tests for Equicorrelation Coefficient of a Standard Syetric Multivariate Noral Distribution, Australian Journal of Statistics, 9, Stein, C. (948), On Sequences of Experients (Abstract), Annals of Matheatical Statistics, 9, 7-8. Wu, P.X. and King, M.L. (994), One-Sided Hypothesis Testing in Econoetrics: A Survey, Pakistan Journal of Statistics,,