Distributions escaping to infinity and the limiting power of the Cliff-Ord test for autocorrelation

Distributions escaping to infinity and the limiting power of the Cliff-Ord test for autocorrelation

Mynbaev, Kairat

Kazakh-British Technical University

1 January 2011

Online at https://mpra.ub.uni-muenchen.de/44402/

MPRA Paper No. 44402, posted 15 Feb 2013 17:06 UTC

Volume 2012, Article ID 926164,39pages doi:10.5402/2012/926164

Research Article

Distributions Escaping to Infinity and the Limiting Power of the Cliff-Ord Test for Autocorrelation

Kairat T. Mynbaev

International School of Economics, Kazakh-British Technical University, Tole bi 59, Almaty 050000, Kazakhstan

Correspondence should be addressed to Kairat T. Mynbaev,kairat mynbayev@yahoo.com Received 18 September 2012; Accepted 24 October 2012

Academic Editors: J. Hu, A. Hutt, and J. Villarroel

Copyrightq2012 Kairat T. Mynbaev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We consider a family of proper random variables which converges to an improper random variable. The limit in distribution is found and applied to obtain a closed-form expression for the limiting power of the Cliff-Ord test for autocorrelation. The applications include the theory of characteristic functions of proper random variables, the theory of almost periodic functions, and the test for spatial correlation in a linear regression model.

1. Introduction

Improper random variables do not satisfy the conditionPX∈R 1; that is, they may take values outside the real lineR. They are not used much by themselves, but there are situations when they arise as limits of proper random variables. In such cases, we say that adistribution escapes to infinity. The main problem considered in this paper is illustrated in the following in a simplified situation.

Throughout the paper we denote byχ_Athe indicator of a setA. Letg 1/2χ−1,1be a uniform distribution on the segment−1,1. Consider the family of densitiesgλt λgλt, whereλ > 0. The total mass is constant:

Rg_λtdt1 for anyλ > 0. In Bayesian estimation, improper priors are obtained by lettingλ → 0. In this case, there are two effects at work:

the support suppgλ −1/λ,1/λstretches out indefinitely and the height of the density maxg_λ λ/2 goes to zero. One might be tempted to think of the limitGof{g_λ}, asλ → 0, as an infinitesimally thin layer smeared over the whole real line. This notion would be wrong because

R

gλtϕtdt ≤ λ

2

suppϕ

ϕtdt−→0, λ → 0, 1.1

for any ϕ ∈ C0R the set of continuous functions onR with bounded support suppϕ.

Thus, if the limit in distributionGof{gλ}exists, it vanishes on all elements ofC0R. By the definition of the support of a distribution1, Chapter 1, Section 13, the support of Gdoes not containR. So instead of spreading the mass overR, it is more correct to say that the mass escapes to infinity.

In this paper we provide a rigorous framework for treating more complex situa- tions. To illustrate the arising complexities, let us look at the standard normal variable X with the densitygx 2π^−1/2exp−x²/2. LetX_λ have the density g_λx λgλx λ2π^−1/2e^−λx2/2. Its characteristic function is

Ee^itXλ

R

e^it/λxgxdxEe^it/λXe^−t/λ2/2. 1.2

The moment generating function ofX_λisEe^tXλ e^t/λ2/2. None of these expressions is good for characterizing the limit as λ → 0. Further, let ϕ denote an arbitrary continuous and bounded function onR. In the expression

R

g_λtϕtdtλ

R

gλtϕtdt, 1.3

the height of the density goes to zero, so the integrand converges to zero everywhere. How- ever, the graph of the density stretches out from the origin. Therefore, the best majorant for the integrand is|g0ϕt|which is generally not integrable. Thus, the dominated convergence theorem cannot be used to obtain convergence in distribution. While the case ofcollapsing densityλ → ∞is easily handled by the existing theory, the case ofstretching outλ → 0 requires new tools which we develop here.

Problem 1. Describe the limit in distribution when the stretching-out is applied to a density along all or some variables.

This problem is solved in Theorem 2.2 in Section 2 case of all variables and Theorem2.6case of some variables. In their simplest form, those results reveal the main ideas. Letg be any summable even function onRit may change sign and not integrate to unity. LetMϕdenote thegeneralized meanofϕoverR:

Mϕ lim

r→ ∞

1 2r

_r

−r

ϕtdt. 1.4

Then for a continuous and bounded functionϕonRsuch thatMϕexists, one has

R

λgλtϕtdt−→Mϕ

R

gtdt, λ−→0. 1.5

Now suppose that the functiongdepends on two variables and the stretching-out is applied with respect to the first of them. Letϕbe continuous and bounded onR². DenoteM1ϕ, the result of application of1.4toϕwith respect to the first argument,

M1ϕ

t2 lim

r→ ∞

1 2r

r

−r

ϕt1, t2dt1, 1.6

and letg₁t₂

Rgt₁, t₂dt₁. In cases whengis not a density, we call this function amarginal

“density.” Then,1.5can be used to prove

R²

λgλt₁, t₂ϕtdt

R

R

λgλt₁, t₂ϕt₁, t₂dt₁

dt₂

−→

R

M₁ϕ

t₂g₁t₂dt₂, λ−→0,

1.7

ifg is integrable and spherically symmetric. As we show, the right sides of1.5and1.7 determine distributions supported at infinity.

Theorem 2.2 is followed by two applications. One generalizes results from 2 on the link between jumps of a distribution function of a proper random variable and its characteristic function. Another application is to the fundamental theorem of H. Bohr on the Parseval identity for almost-periodic functions; see3. Theorem2.6is applied to the limiting power of the Cliff-Ord test for autocorrelation. To formulate the related problems, we need some notation.

Consider a linear regression model

yXβu, Eu0, varu σ²Σ ρ

, 1.8

whereXis ann×kmatrix of rankk < n; a vectorβ∈R^kand a numberσ²>0 are unknown parameters. The matrixΣρis assumed to be a nonnegative function of the parameterρ ∈ 0, a, with somea >0.ρin applications characterizes the degree of autocorrelation. Testing for autocorrelation takes the formH₀ :ρ0 versusH_a:ρ >0. The caseρ → ais of special interest for determining the limiting power of tests.

Assuming thatΣρis positive definite forρ∈0, a, denoteε Σ^−1/2ρuand letgε be the density ofε. The density ofyis then given by4, EquationD.2

f_ρ y

detΣ^−1/2 ρ

g Σ^−1/2 ρ

y−Xβ

. 1.9

Problem 2. Assuming that

rank Σ⁻¹a

n−1, 1.10

describe the limit in distribution offρ.

This problem, as an intermediate step in the proof, was considered by Martellosio4, and his answer is reproduced as follows. LetNAdenote the null space of a matrixAand denoteAa Σ⁻¹a. LetXβNAabe the translation byXβof the null space ofAa.

Reference4states that in case1.10

asρ−→a, f_ρtends to a degenerate density supported on XβNAa, 1.11 see page 182. Equation 1.11 arises from the confusion between the stretching-out and collapsing. On page 159, rows 10 and 11, he remarks that1.11can be extended to the case rankΣ⁻¹a < n−1. He does not specify the mode of convergence, but, as we argue in Section2, the convergence in distribution is the right one for the problem of the behavior of the limiting power of tests for autocorrelation. Unfortunately,1.11does not hold under the convergence in distribution. We prove this and solve Problem2in Theorem2.7for the general case 0<rankΣ⁻¹a< n.

LetΦbe a critical region for rejectingH₀in favor ofH_a. Denote

β_ρΦ

Φ

f_ρ y

dy, 1.12

the probability content ofΦunder density1.9and define thelimiting powerβΦas the limit of this probability content:βΦ limρ→aβρΦ. Specifically, we consider critical regions that arise from the Cliff-Ord test, to be described now. With the regressor matrix X from1.8 denoteLX XX^′X⁻¹X^′, MX I−LX. Under the spatial autocorrelation assumption, the regression disturbances follow

uρWuε, Eε0, varε σ_ε²I, 1.13

where εis a new disturbance andW is some known n×nmatrix. The scalar ρ, which is unknown, determines the degree of correlation among the components ofu. For testing the nullH₀:ρ0 against the alternativeH_a:ρ >0, Cliffand Ord5proposed a test that rejects the null ifu^′M_XWM_Xu/u^′M_Xu > c. Denote

Φ

u∈Rⁿ: u^′MXWMXu u^′MXu > c

, Φ−

u∈Rⁿ: u^′MXWMXu u^′MXu < c

. 1.14

Problem 3. Obtain a closed-form expression forβΦ.

Theorem2.7is applied in Theorem2.9to solve this problem in case1.10.

Problem 4. Describe the cases whenthe limiting power disappears, that is, whenβΦ 0.

This problem has been the main motivation for this paper. Spatial models in general are peculiar in many respects, and the possibility of the limiting power to disappear is one of those peculiarities that has been attracting researchers’ attention lately. Kr¨amer 6was the first to suggest that the limiting power of tests for spatial autocorrelation may vanish, for some combinations of the regressor matrix and the spatial matrix. Unfortunately, the terms in which Kr¨amer expressed his results do not adequately reflect all the possibilities, and his

proof contains an incorrect argument; see 7, Footnote 5. Martellosio 4 was the first to suggest that the answer is better described in terms of the geometrical relationship between the eigenvectors ofΣ⁻¹a and the critical region Φ. However, both of his main result 4, Theorem 1and its proof contain errorssee Remark2.12in Section2. Our answer to Problem 4is given in Theorem2.11and corrects4, Theorem 1in that where he excludes the extreme values 0 and 1 for the limiting power, we give examples showing that the extreme values are possible. In the light of our result, several statements in4,7have to be reconsidered. The analysis for the Cliff-Ord test is very involved, and it is not feasible to indicate corrections for all Martellosio’s main results that depend on the wrong intermediate statement or his Theorem2.9. The complexity of our analysis necessitates a more detailed notation than that used by Martellosio. In particular, some of his verbal definitions and statements are cast in a more formal way. As a result, our citations are not word for word.

Despite the fact that the convergence in distribution is the right one for Problem4, it would be interesting to know what kind of convergence produces the limit indicated by Martellosio, as is stated next.

Problem 5. Design the mode of convergence that leads to1.11.

To this end, we introduce a new convergence conceptwhich, given its purpose, could be called aretrofit convergence, which may not look intuitively appealing but allows us to prove1.11in Theorem2.13. Under this alternative convergence, there is no analog of1.12. Therefore, we did not consider Problems3and4for this convergence.

The previous plan will be implemented under conditions much more restrictive than suggested by Martellosio. All main results that are stated in Sections2 and3 contains all proofs.

2. Main Statements

In the multidimensional version of the generalized mean 1.4 instead of averaging over segments−r, r, we have to average over balls. The shape of those balls depends on the norm ofRⁿ. Let · _abe an arbitrary norm inRⁿ. The balls are defined byB_a,nx, r {y ∈ Rⁿ : x−y_a ≤ r}, where the indication of the space dimension will be important when dealing with more than one space. As an example, one can think of thelp-norm defined by

x_p

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ _n

i1

|x_i|^p 1/p

, if 1≤p <∞, max1≤i≤n|x_i|, if p∞.

2.1

In case of the Euclidean norm · ₂, we obtain usual balls; in casesp1 andp∞, the balls Bp,nx, rare cubes. Another useful example is x_A x^′Ax^1/2, whereAis a symmetric, positive definite matrix. We say that a functiongonRⁿis · _a-spherically symmetricifgx px_awith some functionpdefined on the half-axis0,∞. Conditions involving spherical symmetry in the following are similar to Conditions2.1and2.2from8.

Letσdenote an element of the unit sphere{σ∈Rⁿ :σ_a1}and letσ, ρσbe the representation of a pointx∈Rⁿin the polar system of coordinates such thatσx/x_aand ρσ x_aforx /0. Then, the Lebesgue measure ofB_a,nx, ris

mesBa,nx, r mesBa,n0, r

σ_a1

rσ 0

ρⁿ⁻¹dρ dσ

rⁿ

σ_a1

rσ/r 0

ρⁿ⁻¹dρ dσ varⁿ,

2.2

wherevamesBa,n0,1is the volume of the unit ball. The norm · _agives rise to averages m

ϕ, B_a,nx, r

1 mesB_a,nx, r

Ba,nx,r

ϕtdt, 2.3

and to the generalized mean

Ma,nϕ lim

r→ ∞m

ϕ, Ba,n0, r

. 2.4

In the one-dimensional case all balls are segments, and we write simplyMϕinstead ofM_a,1ϕ.

One of the basic properties of generalized means is that they do not depend on the behavior ofϕin any fixed ballBa,n0, r:

M_a,nϕ lim

r→ ∞

1 mesBa,n0, r

Ba,n0,r0

ϕtdt

Ba,n0,r\Ba,n0,r0

ϕtdt

lim

r→ ∞

1 mesB_a,n0, r

Ba,n0,r\Ba,n0,r0

ϕtdt.

2.5

Other useful properties are3.16,3.37, and3.42.

CLRⁿ denotes the set of continuous bounded functions on Rⁿ that satisfy the Lipschitz condition

ϕx−ϕ

y≤c1

x−y₂, x, y∈Rⁿ; 2.6

By9, Theorem 3.6.1, convergence in distribution of random elementsX_n−→^d Xis equivalent to the convergence of expected values

EϕXn−→EϕX, ∀ϕ∈CLRⁿ. 2.7

10, Theorem 2.1asserts that hereCLRⁿcan be replaced by the set of bounded uniformly continuous functions.LpRⁿis the space ofp-summable functions onRⁿprovided with the normg_L

p

Rⁿ|gt|^pdt^1/p, 1≤p <∞.

For our applications, in the multidimensional version of1.5, we need to allowϕto depend on the parameterλ, as in

λlim→0λ^d

R^d

ϕ_λtgλtdtM_a,dϕ₀

R^d

gtdt. 2.8

Hereϕ∈CLRⁿ,ϕλt ϕFλtGλ.Fλ,λ≥0, aren×dmatrices andGλ,λ≥0, are n×1 vectors such thatFλandGλtend toF0andG0, respectively, sufficiently quickly, as stipulated in the next assumption.

Condition 1. ag ∈L1R^dis · _a-spherically symmetric,gx px_a. bFλ−F0oλ,Gλ−G0₂o1,λ → 0.

Remark 2.1. Using polar coordinates we see that item a imposes a certain integrability restriction onp:

R^d

gtdt

σ_a1

∞ 0

pρρ^d−1dρ

dσ|S|_a ∞

0

pρρ^d−1dρ <∞, 2.9

where |S|_a stands for the surface of the unit sphere {σ : σ_a 1}. The class of den- sities satisfyingaincludes contaminated normal distribution, multivariate t-distribution, multivariate Cauchy distribution, and see; 8. It does not matter much which norms are used in itemb.

Theorem 2.2. Condition1is sufficient for2.8to hold onϕ∈CLRⁿsuch thatMa,dϕ0exists.

Remark 2.3. The triangle inequalityxy_a≤ x_ay_ais not used in the proof, and a slight generalization of Theorem 2.2in terms of the geometry of balls is possible. The Lipschitz condition 2.6 can be omitted if F, G are constant ϕ can be assumed just bounded and continuous.

The first application of Theorem2.2is to the theory of characteristic functions. LetFbe the distribution function of aproper, real-valuedrandom variableX. Denotejxthe jump ofFat pointxand letϕt Ee^itXbe the characteristic function.

Corollary 2.4. Ifg∈L1Ris even, then

λlim→0

R

e^−itxϕtλgλtdtjx

R

gtdt,

λlim→0

R

ϕt²λgλtdt

k

j²x_k

R

gtdt,

2.10

where the sum on the right is over all jump pointsx_kofF.

Theorems 3.2.3 and 3.3.4 in2are a special case of this corollary withg 1/2χ−1,1. The proof of Corollary2.4is obtained by combining those theorems with our Theorem2.2.

The second application is to the theory of almost-periodic functions. A complex- valued continuous functionfonRis calledalmost-periodicif for eachε >0 there existslε>0 such that each intervala, alεof lengthlεcontains at least one numberτ for which sup_t∈R|ftτ−ft|< ε. In the space of almost-periodic functions, the formulaϕ, ψ Mϕψ defines a scalar productψ is a complex conjugate ofψ, and the numbersαλ ϕ·, e^iλ are calledFourier coefficientsofϕ. It is proved that for eachϕat most; a countable number of these coefficients are nonzero. Denoting themαλ_k,k 1,2, . . . , H. Bohr’s theorem states thatM|ϕ|²

k|αλk|²; see3, page 134. This fact together with our Theorem2.2gives the next corollary.

Corollary 2.5. Ifg∈L₁Ris even, then for any almost-periodic functionϕ

λlim→0

R

ϕt²λgλtdt

k

|αλ_k|²

R

gtdt. 2.11

Further, if

Rgtdt >0, the formulaϕ, ψ_glim_λ→0

Rϕtψtλgλtdtdefines a scalar product which is equivalent toϕ, ψ.

Note that the first part of this corollary applies to characteristic functions of purely discrete distribution functions because from 2, Corollary 2 of Theorem 3.2.3 any such characteristic function is almost periodic. Now, we turn to the multidimensional version of 1.7. For the density, we assume a stronger condition than Condition1a.

Condition 2. g ∈L1Rⁿis · ₂-spherically symmetric,gx px₂.

This assumption allows us to show that when some coordinates ofxare fixed,gxas a function of the remaining coordinates satisfies Condition1a.

Now, we provide the intuition for the next condition. The stretching-out applied in 1.7is described by the transformationsAλtwhere

Aλ λ o

0 1

, λ >0. 2.12

g_λt detAλgAλtis the analog of g_λt λgλt because

R²g_λsds

R²gtdt.

Here, the matrixAλhas two eigenvalues. The limit limλ→0Aλ A0is a singular matrix because one of these eigenvalues tends to zero asλ → 0. Generalizing upon this situation and also thinking of applications to invariant tests, in the n-dimensional case we consider a symmetric nonnegative matrixAρof sizen×n, where the parameter ρbelongs to the segment0, a. Denote its eigenvalues 0≤λ1ρ≤ · · · ≤λnρand letAρbe diagonalized as Aρ PρΛρP^′ρ, whereΛρ diagλ₁ρ, . . . , λ_nρandPρis an orthogonal matrix.

Aρdegenerates at the right end of the segment0, a, owing to the following assumption.

Condition 3. The matrixAρis positive definite for 0≤ρ < a.

The first d eigenvalues tend to zero at the same rate, λ_jρ λ₁ρ1 o1, j 1, . . . , d;λ1ρ → 0 asρ → a; the remaining ones have positive limits:λjρ → λja>0 as ρ → a,jd1, . . . , n.

The matricesPρandΛρconverge sufficiently quickly asρ → a. Namely, with the matrix

Λ ρ

diag

⎡

⎢⎣λ1 ρ

, . . . , λ1 ρ

dtimes

, λd1a, . . . , λna

⎤

⎥⎦ 2.13

one hasΛ⁻¹ρΛ ρ−Ioλ1ρ,Pρ−Paoλ1ρ,ρ → a.

Note that Λ⁻¹a does not exist because of part b, but part c allows us to set Λ⁻¹aΛa Iby continuity. In line with partb, we use the partitions

t t¹

t²

, t¹ t₁, . . . , t_d^′, t² t_d1, . . . , t_n^′, 2.14

Λ ρ

Λ₁

ρ 0 0 Λ₂

ρ

, Λ₁ ρ

isd×d, Λ₂ ρ

isn−d×n−d. 2.15

Also, partitionPa P1a, P2aconformably with2.14 P1aisn×dandP2aisn× n−d. Define a transformationTbyT ϕt ϕP₁at¹P₂aΛ⁻¹₂ at².M_2,dT ϕt² denotes the result of application of the generalized mean operator with respect tot¹, keeping t²fixed:

M2,dT ϕ t² lim

r→ ∞m T ϕ ·, t²

, B2,d0, r

, 2.16

andg₁denotes a marginal ”density”:

g₁ t²

R^d

g t¹, t²

dt¹. 2.17

Theorem 2.6. Let Conditions2and3hold and letϕ ∈CLRⁿbe such that the limit2.16exists for almost allt²∈R^n−d. Then,

ρlim→a

Rⁿ

ϕtdetA ρ

g A

ρ t

dt

R^n−d

M2,dT ϕ t²

g₁ t²

dt². 2.18

Next, we turn to the solution of Problem2. DenotingAρ Σ^−1/2ρ and assuming thatAρis positive definite forρ∈0, a, we see that condition1.10corresponds to the case d1 of our Condition3. The density1.9fits the framework of our Theorem2.6because the stretching-out is applied along one variable. Thus, in Theorem2.7we apply Theorem2.6to acharacterize the limit in distribution offρin case 1≤rankΣ⁻¹a≤n−1 andbshow that1.11does not hold under the weak convergence. By implication,1.11is wrong if any convergence stronger than the weak one is considerede.g., uniform, almost sure, in proba- bility and inLp. Even though we use Theorem2.6, the assumption on the density in partb of the next theorem is weaker than that in Theorem2.6.

Condition 4. gis a density onRⁿbounded by an integrable, · ₂-spherically symmetric func- tion,gx≤px₂.

An example of such a density is gx px_a, where p is a nonnegative and nonincreasing function on 0,∞, which decays at infinity sufficiently quickly for g to be integrable. By monotonicity of p inequality 3.14 in the following implies that gx px_a≤px₂/c₃.

Theorem 2.7. Let Condition3hold in whichAρ Σ^−1/2ρ.

aDenoteQ P₂aP₂^′athe projector onto the subspace spanned by the lastn−deigen- vectors ofAaand letT1be defined by

T₁ϕ

t ϕ P₁at¹P₂aΛ⁻¹₂ at²QXβ

. 2.19

M_2,dT₁ϕis obtained by replacingTwithT₁in2.16. If Condition2holds andϕ∈CLRⁿ is such thatM_2,dT₁ϕt²exists for almost allt²∈R^n−d, then

ρ→lima

Rⁿ

ϕtfρtdt

R^n−d

M2,dT1ϕ t²

g₁ t²

dt², 2.20

withg₁defined in2.17.

bIf Condition 4 holds, then 1.11 cannot be true if the convergence in distribution is understood.

Remark 2.8. Because of the identityXβNAa I−QXβQXβNAa QXβ NAa,1.11correctly captures one feature of the limit distribution: it depends onXβonly throughQXβ.

Before giving the solution to Problem 3, we need more notation and definitions.

Obtaining a closed-form formula forβΦinvolves a meticulous analysis ofΦbased on the representation ofΦgiven in the following.

Let ImX{Xβ:β∈R^k}be the image ofX. If for a given setΦ⊂Rⁿathe spaceRⁿis represented as an orthogonal sumRⁿE1⊕E2of two subspacesE1andE2andbthere is a setB⊂E₁such thatΦ BE₂{be:b∈B, e∈E₂}, then we say thatΦis acylinderwith the baseB andelementE₂. For any setS ⊂ Rⁿ, denote ΓS {γs : s ∈ S, γ ∈ R, γ /0}. Using s1·s, we see thatS ⊂ΓS. We say thatSiscone-likeifS ΓS. In particular, a cone-like set with each of its elementscontains its opposite−s. The next representation is a stronger state- ment than saying thatΦdefined in1.14is invariant with respect to transformationsy → γyXβ,γ∈R,γ /0, β∈R^k.

Representation ofΦ

The rejection regionΦfor the Cliff-Ord test is a cylinder

Φ ΓSImX, 2.21 with a cone-like baseMXΦ ΓS, whereS{s∈ImMX :s₂1, s^′Ws > c}.

It is convenient to call anaperturethe setSin representation2.21. The functions^′Ws is continuous on the unit sphere of ImMXand the setc,∞is open. By the general property of continuous mappings11, Chapter 2,§5, Section 5, Theorem 6, the preimage of an open set under a continuous mapping is open. Thus, the aperture is an open setin the relative topologyof the unit sphere of ImMX. We need the notations of the interior intΦ defined as the set of points ofΦthat belong toΦwith some neighborhood, closure clΦ the set of all limit points ofΦand boundary bdΦ clΦ\Φ.

Writing1.12in the form

β_ρΦ

Rⁿ

χΦ y

f_ρ y

dy, 2.22

we see that Theorem 2.7awill be applicable if we manage to extend it from continuous Lipschitz functions to discontinuous functions of typeχΦ. This is done in Theorem2.9. In Theorems2.9and2.11, we assume thatd1. Up to the notation, this is the same assumption as1.10. At least within our method, generalizations of the results in the following to the cased >1 are hard to obtain.

Let Condition3hold forAρ Σ^−1/2ρand letd1. Then,λ₁ρ → 0 asρ → aand all other eigenvalues have positive limits. Denotef1, ..., fn the orthonormal eigenvectors of Aa Σ^−1/2a corresponding to the eigenvaluesλ1a 0, λ2a, . . . , λna. The partitions 2.14becomet¹ t₁,t² t₂, . . . , t_n^′,P₁a f₁,P₂a f₂, . . . , f_n. The vectorzt² P2aΛ⁻¹₂ at² QXβ will be called ashift because its role is to shift the line P1at¹ P2aΛ⁻¹₂ at²QXβ f1t1zt². In the next theorem, we extend2.20toϕ χΦwith Φdescribed by2.21. In the notation of marginal densities, the subscript will indicate the number of integrated-out variables. For example,g₁t²

Rgt₁, t²dt₁,g_kt

R^kpt² v²₂^1/2dv.

Theorem 2.9representation ofβΦ. Let Conditions2and3hold withd1.

aIff1∈ImXand

₁

0

g_ktdt <∞, 2.23

then

βΦ

Rⁿ⁻¹

MT1χΦ t²

g₁ t²

dt², 2.24

where T₁χΦ is defined by T₁χΦt χΦf₁t₁ zt², and the generalized mean is applied overt₁.

bIff₁∈/ImXand

1 0

g_k1tdt <∞, 2.25

then2.24is true.

Remark 2.10. a As can be seen from the proof, the theorem holds for any test with the critical region satisfying2.21.b Conditions2.23and 2.25are technical assumptions that provide integrability in the neighborhood of the origin of marginal densities that arise in the course of the proof.

To avoid triviality, in the following theorem we assume that the inclusion∅ ⊂Φ⊂Rⁿ is strict. This implies that in representation2.21the setΓSis a nonempty proper subset of ImMX.

Theorem 2.11. Let conditions of Theorem2.9hold, with2.23accompanyingf1∈ImXand2.25 accompanyingf₁∈/ImX. Besides, we require the functionpfrom Condition2to be positive on0,∞.

1Iff1∈Φ, thenβΦ 1.

2Supposef1∈ bdΦ.

2.1Iff₁∈ImX, thenβΦ∈0,1.

2.2Iff₁ ∈/ImX, then examples can be presented such thatβΦ 0,βΦ ∈0,1or βΦ 1.

3Iff₁∈Φ₋, thenβΦ 0.

Remark 2.12. Here, we compare this theorem with4, Theorem 1.

1Our conditions on the density and critical region are much more restrictive. Our proof reveals the distinction between the casesf₁∈ImXandf₁∈/ImX. In particular, in case 2.2, Martellosio excludes the extreme valuesβΦ 0 and βΦ 1, while we provide counter examples showing that they are possible. Note also that we do not impose any con- ditions on the structure ofW. What happens to the limiting power in case ofWthat arises in practice needs additional investigation.

2 Martellosio’s proof is based on1.11 which we disprove in Theorem 2.7b. A series of other propositions from the same papersee Lemmas D.2, D.3 and E.4, Corollary 1 and Propositions 1, 2 and 5, as well as from7 see Lemma 3.2, Theorems 3.3, 3.5 and 4.1, and Proposition 3.6, depend on1.11and need a revision. In particular, his claim that his results are true for any invariant critical region and any continuous densityuthat is unimodal at the origin is unwarranted.

3 Even if statement 1.11 were right, the proof of 4, Theorem 1 would be incomplete because it incorrectly uses then-dimensional Lebesgue measure. Its use is inap- propriate because for a degenerate density even one point may carry a positive mass. In our proof, we justify the use ofn−1-dimensional Lebesgue measure.

Now, we turn to the description of the alternative approachsolution to Problem5. A mode is not a very good characteristic of a distribution when two unimodal densities with the same modes have very different spreads. A set of points where the density is close to its maximum might be a better characteristic in this case, at least for bell-shaped distributions.

Let 0 < ε <1 and letm maxfxbe the maximum of a continuous densityfx. We call Mε {x : fx ≥ 1−εm}anε-maximizing setoff. The idea of this density-maximizing approach is close to the maximum likelihood principle.

Suppose a decisiondis taken if the statisticxbelongs to a setD. In case of a favorable decision,Dis chosen in such a way that the probabilityPx∈Dis high. The use of probabil- ity in this decision rule presumes that the statistic can be calculated repeatedly. However, in

practice, especially in economics, the decision is based on just one value of the statistic in question. In such a case it may be preferable to chooseDso that minx∈Dfxis sufficiently close tommaxfx. For theε-maximizing set, we have min_x∈Mεfx≥1−εm. Requiring εto be close to 1 in the density-maximizing approach is similar to requiringPx∈Dto be close to 1 in the probability-maximizing approachalthough normallyPx ∈ Mε → 0 as ε → 0.

As before, we assume that the matrixAρ Σ^−1/2ρ of sizen×nis symmetric for 0≤ ρ≤aand positive definite for 0≤ ρ < a. The idea is to impose conditions ensuring that ε-maximizing sets are ellipsoids which in the limit give a set of the desired shape. This idea is realized through a delicate balance of the limit behavior of the eigenvalues and density contained in Conditions5through7and2.32.

Condition 5. In the diagonal representation ofAρ, the orthogonal matrixPsatisfies Pa lim

ρ→aP ρ

, 2.26

the firstdeigenvalues vanish as power functions λ_j

ρ c_λ

a−ραλ1o1, j1, . . . , d, asλ−→a, 2.27

with positive constantsc_λ, α_λ; the remaining eigenvalues tend to positive constants λj

ρ

λja1o1, j d1, . . . , n, asλ−→a, 2.28 whereλ_ja>0.

Condition 6. The functionερin the definition of the setM_ερvanishes as a power function ε

ρ c_ε

a−ραε1o1, ρ−→a, 2.29

wherec_ε, α_εare positive constants.

Condition 7. The density g is · ₂-spherically symmetric,gx px₂, wherep is con- tinuous and monotonically decreasing on0,∞and such that

pr m

1−cpr^αp

1or^αε, r−→0, 2.30 wherecp, αp>0.

This assumption implies thatm maxgx p0, that the inverse functionp⁻¹ is continuous and monotonically decreasing on0, m, and that

0p⁻¹m lim

x→mp⁻¹x. 2.31

Theorem 2.13. DenoteNAathe null subspace ofAa(spanned by the eigenvectors correspond- ing to its zero eigenvalues). If Conditions5–7hold and

α_ε

α_p−α_λ <0, 2.32

then theερ-maximizing setM_ερ {x: f_ρx ≥1−ερm}converges toXβNAaas ρ → a.

3. Proofs

The idea of the proof is to approximategwith a step functionh, prove the statement forh and then pass to the limit to obtain the statement forg. Astep function, by definition, is a finite linear combination of indicators of measurable sets. Due to the spherical symmetry ofg, these sets turn out to be balls. For the method to work, the radii of the balls should be positive and finite. Approximation ofpby a continuous function in Step1is a trick to make sure thath takes a finite number of values.

Step 1pcan be assumed continuous on0,∞. LetΩ 0,∞and denoteZΩthe space of measurable functions f on Ω such that f_ZΩ ∞

0 |fρ|ρ^d−1dρ < ∞.ZΩ satisfies conditions of12, Theorem 1:

1C^∞₀ Ω⊂ZΩ⊂L^loc₁ Ω.

2Minkowsky inequality: ifA⊂ R^mis a measurable set andΦx, yis a measurable onΩ×Afunction, then

AΦ·, ydy_ZΩ≤

AΦ·, y_ZΩdy.

3A multiplication operator by a functionϕ∈C₀^∞Ωis bounded inZΩ:fϕ_ZΩ≤ max|ϕ|f_ZΩ.

4Any finite in Ω function f ∈ ZΩ is translation-continuous: limh→0f·h− f·_ZΩ0.

The first three properties are standard facts of the theory ofL_p spaces; the last one follows from the fact that if suppf is a compact subset of Ω, then the weight ρ^d−1 in the definition of the norm ofZΩsatisfiesc1 ≤ρ^d−1 ≤c2on suppfand therefore for such anf the normsf_ZΩandf_L

1are equivalent. Functions fromL₁ are known to be translation- continuous.

By Burenkov’s theorem there exists a sequence {ps} ⊂ C^∞Ω ∩ ZΩ such that ps−p_ZΩ → 0. Defining gsx psx_a, from 2.9 we have gs−g_L

1

|S|_ap_s−p_ZΩ → 0. Hence,

λ^d

R^d

ϕ_λtg_sλtdt−λ^d

R^d

ϕ_λtgλtdt

≤supϕλ^d

R^d

g_sλt−gλtdt

supϕgs−g_L

1 −→0

3.1

uniformly inλ. Thus, if we establish

λlim→0λ^d

R^d

ϕ_λtg_sλtdtM_a,dϕ₀

R^d

g_stdt ∀s, 3.2

then2.8will follow.

Step 2approximatinggwith a Step Function. Take an arbitraryε >0. We can assume that pis continuous on0,∞.

aSupposeg_L

1≥ε. We approximategtby a step functionht, which will vanish wheretis large or small. By summability ofg, there exist 0< N1 N1ε< N2 N2ε<∞such that

t_a<N1

gtdt≤ ε 3,

t_a>N2

gtdt≤ ε

3 3.3

gis uniformly continuous on the ring{N₁≤ t_a≤N₂}. By Condition1a, for any naturalmwe can findδ >0 and split this ring into smaller rings

Am,l{t:N1lδ≤ t_a< N1 l1δ}, l0, . . . , L≡ N2−N1

δ −1, 3.4

in such a way thatgin each ring is close to its value on the inner boundary:

sup

t∈Am,l

gt−p

l m

≤ 1

m. 3.5

Putht 0 ift_a < N1 ort_a > N2;ht pl/m fort ∈ Am,l,l 0, . . . , L.

Combination of3.3,3.4, and3.5leads to h−g

L1

t_a<N1

t_a>N2

gtdt L

l0

Am,l

ht−gtdt

≤ 2ε 3 1

mmesBa,d0, N2

2ε

3 vaN^d₂ε

m .

3.6

Consequently, we can fixmmεso that h−g_L

1 ≤ε. 3.7

bIfg_L

1< ε, we just puth≡0 to get3.7.

Step 3replacing rings by balls in the representation ofh. Supposehis not identically zero.

The sets Am,l in 3.4 are concentric rings with finite positive radii of the inner and outer boundaries, and by construction,his a finite linear combination of indicators of such rings.

Hence, it can be written ashL

l0alχBa,d0,rl−χBa,d0,rl−1, wherealare some real numbers and the radii satisfy

N₁r₀<· · ·< r_LN₂. 3.8

Therefore, with some new constantsbl, we can writehas a linear combination of indicators of balls:

h L

l0

hl, where hlblχBa,d0,rl. 3.9

Ifh ≡0, we put formallyr0 1,b0 0,L 0. Note thatL,bl andrlall depend onε and that the constantsb_lmay deviate from the values ofhsignificantly.

Step 4introducing residuals for generalized means. Define the residualR1rby

R1r

#m

ϕ₀, B_a,d0, r

/M_a,dϕ₀−1, if M_a,dϕ₀/0, m

ϕ0, Ba,d0, r

, if Ma,dϕ00. 3.10

Then,

m

ϕ0, Ba,d0, r

#M_a,dϕ₀R₁r 1, if M_a,dϕ₀/0,

R1r, if Ma,dϕ0 0, 3.11 and in both cases

rlim→ ∞R₁r 0. 3.12

By the Lipschitz condition2.6and Condition1b, ϕFλtGλ−ϕF0tG0

≤c1Fλ−F0tGλ−G0₂

oλt_ao1 oλro1 fort_a≤r.

3.13

Here, we have used the fact that onR^dany two norms are equivalent, so

c2t_a≤ t₂≤c3t_a, 3.14

with somec2, c3>0. Equations2.3and3.13imply that m

ϕFλ·Gλ−ϕ0·, B_a,d0, roλro1 ∀r >0, asλ → 0, 3.15

whereoλando1do not depend onr. From this bound and3.11, it follows that

m

ϕFλ· Gλ, B_a,d0, r

#Ma,dϕ0R1r 1 R2λ, r, if Ma,dϕ0/0,

R₁r R₂λ, r, if M_a,dϕ₀ 0, 3.16 whereR2λ, ris a new residual satisfying

R2λ, r oλro1 ∀r >0, asλ → 0. 3.17 Step 5proving2.8for the approximating function. LetM_a,dϕ₀/0. For one term in3.9 by2.2, and the first equation in3.16we have

λ^d

R^d

ϕFλtGλhlλtdt

λ^dbl

Ba,d0,rl/λ

ϕFλtGλdt

v_ar_l^db_l v_ar_l/λ^d

Ba,d0,rl/λ

ϕFλtGλdt

R^d

hlsds$ Ma,dϕ0

%R1 rl

λ 1&

R2 λ,rl

λ '.

3.18

Summation of these equations produces λ^d

R^d

ϕFλtGλhλtdt

L

l0

R^d

h_lsds$

M_a,dϕ₀% R₁ r_l

λ 1&

R₂ λ,r_l λ

'

Ma,dϕ0

R^d

hsds

L

l0

R^d

h_lsds$

M_a,dϕ₀R₁ r_l λ

R₂ λ,r_l λ

'.

3.19

Here, by3.8,3.12, and3.17,R1rl/λ → 0,R2λ,rl/λ orl o1 o1,l 0, . . . , L, asλ → 0. From these relations and3.19, we see that for the givenεthere existsλε such that

λ^d

R^d

ϕλthλtdt−Ma,dϕ0

R^d

hsds

≤ε for 0< λ≤λε. 3.20

In caseMa,dϕ0 0, the only difference consists in application of the second equation in3.16. The conclusion is3.20withMa,dϕ00.

Step 6proving2.8. Using3.7and3.20, we have

λ^d

R^d

ϕ_λtgλtdt−M_a,dϕ₀

R^d

gtdt

≤ λ^d

R^d

ϕλt(

gλt−hλt) dt

λ^d

R^d

ϕλthλtdt−Ma,dϕ0

R^d

htdt Ma,dϕ0

R^d

(ht−gt) dt

≤ϕ

C

g−h

L1εM_a,dϕ₀g−h

L1

≤ϕ_C1Ma,dϕ0

ε, 0< λ≤λε.

3.21

Equation2.8follows becauseε >0 is arbitrary.

3.1. Proof of Theorem2.6 DenoteJρϕ

RⁿϕtdetAρgAρtdt.

Step 1 replacing limJ_ρϕ by an equivalent limit. Using constancy of g on spheres Condition2, writeJ_ρϕas

Jρ ϕ

Rⁿ

ϕ y

detΛ ρ

g P

ρ Λ

ρ P^′

ρ y

dy

detΛ ρ

Rⁿ

ϕ y

g Λ

ρ P^′

ρ y

dy

detΛ ρ

Rⁿ

ϕ y

g Λ ρ Λ⁻¹

ρ Λ

ρ P^′

ρ y

dy.

3.22

DenoteH⁻¹ρ Λ⁻¹ρΛρP^′ρand replaceH⁻¹ρytto get

Jρ ϕ

detΛ ρ

detH ρ

Rⁿ

ϕ H

ρ t

g Λ ρ

t

dt. 3.23

By Condition3b,c, detΛ

ρ λ^d₁

ρ

detΛ₂a1o1, detH

ρ

det Λ⁻¹ ρ

Λ ρ

detP^′ ρ

1o1.

3.24