ISRN Probability and Statistics

Volume 2012 (2012), Article ID 926164, 39 pages

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

Received 18 September 2012; Accepted 24 October 2012

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

Copyright © 2012 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.

Abstract.

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 condition ; that is, they may take values outside the real line . 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 a distribution escapes to infinity. The main problem considered in this paper is illustrated in the following in a simplified situation.

Throughout the paper we denote by the indicator of a set . Let be a uniform distribution on the segment . Consider the family of densities , where . The total mass is constant: for any . In Bayesian estimation, improper priors are obtained by letting . In this case, there are two effects at work: the support stretches out indefinitely and the height of the density goes to zero. One might be tempted to think of the limit of , as , as an infinitesimally thin layer smeared over the whole real line. This notion would be wrong because

for any (the set of continuous functions on with bounded support ). Thus, if the limit in distribution of exists, it vanishes on all elements of . By the definition of the support of a distribution [1, Chapter 1, Section 13], the support of does not contain . So instead of spreading the mass over , it is more correct to say that the mass escapes to infinity.

In this paper we provide a rigorous framework for treating more complex situations. To illustrate the arising complexities, let us look at the standard normal variable with the density . Let have the density . Its characteristic function is

The moment generating function of is . None of these expressions is good for characterizing the limit as . Further, let denote an arbitrary continuous and bounded function on . In the expression

the height of the density goes to zero, so the integrand converges to zero everywhere. However, the graph of the density stretches out from the origin. Therefore, the best majorant for the integrand is which is generally not integrable. Thus, the dominated convergence theorem cannot be used to obtain convergence in distribution. While the case of collapsing density is easily handled by the existing theory, the case of stretching out 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 Theorem 2.6 (case of some variables). In their simplest form, those results reveal the main ideas. Let be any summable even function on (it may change sign and not integrate to unity). Let denote the generalized mean of over :

Then for a continuous and bounded function on such that exists, one has

Now suppose that the function depends on two variables and the stretching-out is applied with respect to the first of them. Let be continuous and bounded on . Denote , the result of application of (1.4) to with respect to the first argument,

and let . In cases when is not a density, we call this function a marginal “density.”   Then, (1.5) can be used to prove

if is integrable and spherically symmetric. As we show, the right sides of (1.5) and (1.7) 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; see [3]. Theorem 2.6 is 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

where is an matrix of rank ; a vector and a number are unknown parameters. The matrix is assumed to be a nonnegative function of the parameter , with some . in applications characterizes the degree of autocorrelation. Testing for autocorrelation takes the form versus . The case is of special interest for determining the limiting power of tests.

Assuming that is positive definite for , denote and let be the density of . The density of is then given by [4, Equation (D.2)]

Problem 2. Assuming that

describe the limit in distribution of .

This problem, as an intermediate step in the proof, was considered by Martellosio [4], and his answer is reproduced as follows. Let denote the null space of a matrix and denote . Let be the translation by of the null space of . Reference [4] states that in case (1.10)

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 that (1.11) can be extended to the case . He does not specify the mode of convergence, but, as we argue in Section 2, the convergence in distribution is the right one for the problem of the behavior of the limiting power of tests for autocorrelation. Unfortunately, (1.11) does not hold under the convergence in distribution. We prove this and solve Problem 2 in Theorem 2.7 for the general case .

Let be a critical region for rejecting in favor of . Denote

the probability content of under density (1.9) and define the limiting power as the limit of this probability content: . Specifically, we consider critical regions that arise from the Cliff-Ord test, to be described now. With the regressor matrix from (1.8) denote . Under the spatial autocorrelation assumption, the regression disturbances follow

where is a new disturbance and is some known matrix. The scalar , which is unknown, determines the degree of correlation among the components of . For testing the null against the alternative , Cliff and Ord [5] proposed a test that rejects the null if . Denote

Problem 3. Obtain a closed-form expression for .

Theorem 2.7 is applied in Theorem 2.9 to solve this problem in case (1.10).

Problem 4. Describe the cases when the limiting power disappears, that is, when .

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. Krämer [6] was 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 Krämer 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 of and the critical region . However, both of his main result [4, Theorem 1] and its proof contain errors (see Remark 2.12 in Section 2). Our answer to Problem 4 is given in Theorem 2.11 and corrects [4, Theorem 1] in 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 in [4, 7] have 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 Theorem 2.9. 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 Problem 4, 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 to (1.11).

To this end, we introduce a new convergence concept (which, given its purpose, could be called a retrofit convergence), which may not look intuitively appealing but allows us to prove (1.11) in Theorem 2.13. Under this alternative convergence, there is no analog of (1.12). Therefore, we did not consider Problems 3 and 4 for this convergence.

The previous plan will be implemented under conditions much more restrictive than suggested by Martellosio. All main results that are stated in Sections 2 and 3 contains all proofs.

2. Main Statements

In the multidimensional version of the generalized mean (1.4) instead of averaging over segments , we have to average over balls. The shape of those balls depends on the norm of . Let be an arbitrary norm in . The balls are defined by , where the indication of the space dimension will be important when dealing with more than one space. As an example, one can think of the -norm defined by

In case of the Euclidean norm , we obtain usual balls; in cases and , the balls are cubes. Another useful example is , where is a symmetric, positive definite matrix. We say that a function on is -spherically symmetric if with some function defined on the half-axis . Conditions involving spherical symmetry in the following are similar to Conditions (2.1) and (2.2) from [8].

Let denote an element of the unit sphere and let be the representation of a point in the polar system of coordinates such that and for . Then, the Lebesgue measure of is

where is the volume of the unit ball. The norm gives rise to averages

and to the generalized mean

In the one-dimensional case all balls are segments, and we write simply instead of .

One of the basic properties of generalized means is that they do not depend on the behavior of in any fixed ball :

Other useful properties are (3.16), (3.37), and (3.42).

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

By [9, Theorem  3.6.1], convergence in distribution of random elements is equivalent to the convergence of expected values

[10, Theorem 2.1] asserts that here can be replaced by the set of bounded uniformly continuous functions. is the space of -summable functions on provided with the norm ,  .

For our applications, in the multidimensional version of (1.5), we need to allow to depend on the parameter , as in

Here ,  . ,  , are matrices and ,  , are vectors such that and tend to and , respectively, sufficiently quickly, as stipulated in the next assumption.

Condition 1.

(a) is-spherically symmetric, .(b) , , .

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

where stands for the surface of the unit sphere . The class of densities satisfying (a) includes contaminated normal distribution, multivariate -distribution, multivariate Cauchy distribution, and see; [8]. It does not matter much which norms are used in item (b).

Theorem 2.2. Condition 1 is sufficient for (2.8) to hold on such that exists.

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

The first application of Theorem 2.2 is to the theory of characteristic functions. Let be the distribution function of a (proper, real-valued) random variable . Denote the jump of at point and let be the characteristic function.

Corollary 2.4. If is even, then

where the sum on the right is over all jump points of .

Theorems 3.2.3 and 3.3.4 in [2] are a special case of this corollary with . The proof of Corollary 2.4 is obtained by combining those theorems with our Theorem 2.2.

The second application is to the theory of almost-periodic functions. A complex-valued continuous function on is called almost-periodic if for each there exists such that each interval of length contains at least one number for which . In the space of almost-periodic functions, the formula defines a scalar product ( is a complex conjugate of ), and the numbers are called Fourier coefficients of . It is proved that for each at most; a countable number of these coefficients are nonzero. Denoting them ,  . Bohr's theorem states that ; see [3, page 134]. This fact together with our Theorem 2.2 gives the next corollary.

Corollary 2.5. If is even, then for any almost-periodic function

Further, if , the formula defines a scalar product which is equivalent to .

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 Condition 1(a).

Condition 2. is - spherically symmetric,  .

This assumption allows us to show that when some coordinates of are fixed, as a function of the remaining coordinates satisfies Condition 1(a).

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

is the analog of because . Here, the matrix has two eigenvalues. The limit is a singular matrix because one of these eigenvalues tends to zero as . Generalizing upon this situation and also thinking of applications to invariant tests, in the -dimensional case we consider a symmetric nonnegative matrix of size , where the parameter belongs to the segment . Denote its eigenvalues and let be diagonalized as , where and is an orthogonal matrix. degenerates at the right end of the segment , owing to the following assumption.

Condition 3. The matrix is positive definite for .

The first eigenvalues tend to zero at the same rate, ;   as ; the remaining ones have positive limits: as ,.

The matrices and converge sufficiently quickly as . Namely, with the matrix

one has ,  ,  .

Note that does not exist because of part (b), but part (c) allows us to set by continuity. In line with part (b), we use the partitions

Also, partition conformably with (2.14) ( is and is ). Define a transformation by .   denotes the result of application of the generalized mean operator with respect to , keeping fixed:

and denotes a marginal "density":

Theorem 2.6. Let Conditions 2 and 3 hold and let be such that the limit (2.16) exists for almost all . Then,

Next, we turn to the solution of Problem 2. Denoting and assuming that is positive definite for , we see that condition (1.10) corresponds to the case of our Condition 3. The density (1.9) fits the framework of our Theorem 2.6 because the stretching-out is applied along one variable. Thus, in Theorem 2.7 we apply Theorem 2.6 to (a) characterize the limit in distribution of in case and (b) show that (1.11) does not hold under the weak convergence. By implication, (1.11) is wrong if any convergence stronger than the weak one is considered (e.g., uniform, almost sure, in probability and in ). Even though we use Theorem 2.6, the assumption on the density in part (b) of the next theorem is weaker than that in Theorem 2.6.

Condition 4. is a density on bounded by an integrable, -spherically symmetric function, .

An example of such a density is , where is a nonnegative and nonincreasing function on , which decays at infinity sufficiently quickly for to be integrable. By monotonicity of inequality (3.14) in the following implies that .

Theorem 2.7. Let Condition 3 hold in which .

(a)Denote the projector onto the subspace spanned by the last eigenvectors of and let be defined by

is obtained by replacing with in (2.16). If Condition 2 holds and is such that exists for almost all , then

with defined in (2.17).(b)If Condition 4 holds, then (1.11) cannot be true if the convergence in distribution is understood.

Remark 2.8. Because of the identity , (1.11) correctly captures one feature of the limit distribution: it depends on only through .

Before giving the solution to Problem 3, we need more notation and definitions. Obtaining a closed-form formula for involves a meticulous analysis of based on the representation of given in the following.

Let be the image of . If for a given set (a) the space is represented as an orthogonal sum of two subspaces and and (b) there is a set such that , then we say that is a cylinder with the base and element . For any set , denote . Using , we see that . We say that is cone-like if . In particular, a cone-like set with each of its element contains its opposite . The next representation is a stronger statement than saying that defined in (1.14) is invariant with respect to transformations ,  ,  ,  .

Representation of

The rejection region for the Cliff-Ord test is a cylinder

with a cone-like base , where .

It is convenient to call an aperture the set in representation (2.21). The function is continuous on the unit sphere of and the set is open. By the general property of continuous mappings [11, 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 set (in the relative topology) of the unit sphere of . We need the notations of the interior (defined as the set of points of that belong to with some neighborhood), closure (the set of all limit points of ) and boundary .

Writing (1.12) in the form

we see that Theorem 2.7(a) will be applicable if we manage to extend it from continuous Lipschitz functions to discontinuous functions of type . This is done in Theorem 2.9. In Theorems 2.9 and 2.11, we assume that . Up to the notation, this is the same assumption as (1.10). At least within our method, generalizations of the results in the following to the case are hard to obtain.

Let Condition 3 hold for and let . Then, as and all other eigenvalues have positive limits. Denote the orthonormal eigenvectors of corresponding to the eigenvalues . The partitions (2.14) become , , , . The vector will be called a shift because its role is to shift the line . In the next theorem, we extend (2.20) to with described by (2.21). In the notation of marginal densities, the subscript will indicate the number of integrated-out variables. For example, ,  .

Theorem 2.9 (representation of ). Let Conditions 2 and 3 hold with .

(a)If and

then

where is defined by , and the generalized mean is applied over .(b)If and

then (2.24) is true.

Remark 2.10. (a) As can be seen from the proof, the theorem holds for any test with the critical region satisfying (2.21). (b) Conditions (2.23) and (2.25) are 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 is strict. This implies that in representation (2.21) the set is a nonempty proper subset of .

Theorem 2.11. Let conditions of Theorem 2.9 hold, with (2.23) accompanying and (2.25) accompanying . Besides, we require the function from Condition 2 to be positive on .

(1)If , then .(2)Suppose .

(2.1) If , then .(2.2) If , then examples can be presented such that , or .

(3)If , then .

Remark 2.12. Here, we compare this theorem with [4, Theorem 1].

(1.4) Our conditions on the density and critical region are much more restrictive. Our proof reveals the distinction between the cases and . In particular, in case (2.2), Martellosio excludes the extreme values and , while we provide counter examples showing that they are possible. Note also that we do not impose any conditions on the structure of . What happens to the limiting power in case of that arises in practice needs additional investigation.

(1.5) Martellosio's proof is based on (1.11) which we disprove in Theorem 2.7(b). A series of other propositions from the same paper (see Lemmas D.2, D.3 and E.4, Corollary 1 and Propositions 1, 2 and 5), as well as from [7] (see Lemma 3.2, Theorems 3.3, 3.5 and 4.1, and Proposition 3.6), depend on (1.11) and need a revision. In particular, his claim that his results are true for any invariant critical region and any continuous density that is unimodal at the origin is unwarranted.

(1.7) Even if statement (1.11) were right, the proof of [4, Theorem 1] would be incomplete because it incorrectly uses the -dimensional Lebesgue measure. Its use is inappropriate because for a degenerate density even one point may carry a positive mass. In our proof, we justify the use of -dimensional Lebesgue measure.

Now, we turn to the description of the alternative approach (solution to Problem 5). 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 and let be the maximum of a continuous density . We call an -maximizing set of . The idea of this density-maximizing approach is close to the maximum likelihood principle.

Suppose a decision is taken if the statistic belongs to a set . In case of a favorable decision, is chosen in such a way that the probability is high. The use of probability 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 choose so that is sufficiently close to . For the -maximizing set, we have . Requiring to be close to 1 in the density-maximizing approach is similar to requiring to be close to 1 in the probability-maximizing approach (although normally as ).

As before, we assume that the matrix of size is symmetric for and positive definite for . The idea is to impose conditions ensuring that -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 Conditions 5 through 7 and (2.32).

Condition 5. In the diagonal representation of , the orthogonal matrix satisfies

the first eigenvalues vanish as power functions

with positive constants ; the remaining eigenvalues tend to positive constants

where .

Condition 6. The function in the definition of the set vanishes as a power function

where are positive constants.

Condition 7. The density is -spherically symmetric, , where is continuous and monotonically decreasing on and such that

where .

This assumption implies that , that the inverse function is continuous and monotonically decreasing on , and that

Theorem 2.13. Denote the null subspace of (spanned by the eigenvectors corresponding to its zero eigenvalues). If Conditions 5–7 hold and

then the -maximizing set converges to as .

3. Proofs

The idea of the proof is to approximate with a step function , prove the statement for and then pass to the limit to obtain the statement for . A step function, by definition, is a finite linear combination of indicators of measurable sets. Due to the spherical symmetry of , these sets turn out to be balls. For the method to work, the radii of the balls should be positive and finite. Approximation of by a continuous function in Step 1 is a trick to make sure that takes a finite number of values.

Step 1 ( can be assumed continuous on ). Let and denote the space of measurable functions on such that .   satisfies conditions of [12, Theorem 1]:

(1). (2)Minkowsky inequality: if is a measurable set and is a measurable on function, then .(3)A multiplication operator by a function is bounded in : .(4)Any finite in function is translation-continuous:  .

The first three properties are standard facts of the theory of spaces; the last one follows from the fact that if is a compact subset of , then the weight in the definition of the norm of satisfies on and therefore for such an the norms and are equivalent. Functions from are known to be translation-continuous.

By Burenkov’s theorem there exists a sequence such that . Defining , from (2.9) we have . Hence,

uniformly in . Thus, if we establish

then (2.8) will follow.

Step 2 (approximating with a Step Function). Take an arbitrary . We can assume that is continuous on .

(a)Suppose . We approximate by a step function , which will vanish where is large or small. By summability of , there exist such that

is uniformly continuous on the ring . By Condition 1(a), for any natural we can find and split this ring into smaller rings

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

Put if or ;   for ,  . Combination of (3.3), (3.4), and (3.5) leads to

Consequently, we can fix so that

(b)If , we just put to get (3.7).

Step 3 (replacing rings by balls in the representation of ). Suppose is not identically zero. The sets in (3.4) are concentric rings with finite positive radii of the inner and outer boundaries, and by construction, is a finite linear combination of indicators of such rings. Hence, it can be written as , where are some real numbers and the radii satisfy

Therefore, with some new constants , we can write as a linear combination of indicators of balls:

If , we put formally ,  ,  . Note that ,   and all depend on and that the constants may deviate from the values of significantly.

Step 4 (introducing residuals for generalized means). Define the residual by

Then,

and in both cases

By the Lipschitz condition (2.6) and Condition 1(b),

Here, we have used the fact that on any two norms are equivalent, so

with some . Equations (2.3) and (3.13) imply that

where and do not depend on . From this bound and (3.11), it follows that

where is a new residual satisfying

Step 5 (proving (2.8) for the approximating function). Let . For one term in (3.9) by (2.2), and the first equation in (3.16) we have

Summation of these equations produces

Here, by (3.8), (3.12), and (3.17), ,  ,  , as . From these relations and (3.19), we see that for the given there exists such that

In case , the only difference consists in application of the second equation in (3.16). The conclusion is (3.20) with .

Step 6 (proving (2.8)). Using (3.7) and (3.20), we have

Equation (2.8) follows because is arbitrary.

3.1. Proof of Theorem 2.6

Denote .

Step 1 (replacing by an equivalent limit). Using constancy of on spheres (Condition 2), write as

Denote and replace to get

By Condition 3(b,c),

From (3.23) and (3.24) we deduce

Step 2 (changing the limit variable). Let us have a closer look at the expression in the square brackets in (3.25):

Since , here , and by Condition 3(c),

Considering the independent variable, take as its function. The limit relation translates to , and (3.26) and (3.27) become,

respectively.

Step 3 (proving a preliminary version of (2.18)). Now we check that Theorem 2.2 applies to (3.28) for any fixed . Letting

we see that is -spherically symmetric:

Condition 1(a) is satisfied. As for Condition 1(b), partition conformably with (2.14) and put ,  ,  ,  . Then, , and Condition 1(b) follows from (3.29).

By Theorem 2.2,

where is obtained by applying to with respect to . Functions (3.28) are bounded by an integrable function of :

By the dominated convergence theorem, (3.32) and (3.33) imply that

Now, we combine (3.25), (3.28), (3.32), and (3.34) to get

because

3.2. Proof of Theorem 2.7

3.2.1. Part (a)

Step 1 ( does not depend on the center of balls). Let us make definition (2.4) more precise: ,  . Here, we prove that

For , one has implications . Geometrically, they mean the inclusion . This inclusion implies that

It remains to let to prove (3.37).

Step 2 (proving (2.20)). If , then . By (2.16), we need to look at

Using , we see that

Therefore, replacing , we can continue (3.39) as

Using the definition of and (3.37), we see that this is the same as . Hence, the existence of is equivalent to that of . Application of Theorem 2.6 proves (2.20).

3.2.2. Part (b)

Step 1 (the generalized mean vanishes on -integrable functions). If with some , then by Hölder’s inequality

Step 2 ( vanishes on functions with compact support). By Condition 4,

where and is obtained from (1.9) by replacing with . The function satisfies the Lipschitz condition, , and satisfies Condition 2. Part (a) of this theorem allows us to use (2.20) with in place of and instead of whenever exists for almost all .

Let denote the set of functions with compact support . Suppose . By Step 1, . Equations (2.20) and (3.43) imply that

Suppose (1.11) is true in the weak sense or, more generally, the family converges in distribution, as , to a probabilistic measure supported on some set . Then by (2.7) . This equation and (3.44) imply . Taking here such that and on we have for any . Thus, which is impossible.

3.3. Proof of Theorem 2.9

The proof of Theorem 2.9 will be preceded by a series of lemmas. The main difficulty consists in estimating integrals of the marginal density over cylindrical sets of an infinite Lebesgue measure.

3.3.1. Preliminary Results

Lemma 3.1. For the rejection region defined in (1.14), representation (2.21) holds.

Proof. Since ,   are complementary orthoprojectors, the images and are orthogonal to one another and . Representing as and using ,   we get

where we have denoted . Hence, only the component determines whether belongs to or not, while the component is arbitrary. This proves the equation . Finally, the equation follows by homogeneity from (3.45).

Now we review a few facts about the Moore-Penrose inverse in a form adapted to our needs. Let be a matrix of size , not necessarily square, and denote its rank . Its Moore-Penrose inverse is of size .

Lemma 3.2 (see, [13, pages 24-25]). is unique and has the following properties.

(a) ( is a generalized inverse of ).(b) ( is a generalized inverse of ).(c)The matrix is an orthoprojector, , , and .(d)For a given , a general solution of the equation is given by

where is arbitrary. (Rao writes instead of , but our form is more convenient).

Regarding (3.46), we can remark the following.

(i)The two terms on the right are orthogonal because by (b) and (c)

Here and in the sequel, stands for the scalar product.(ii)Instead of writing the second term in the form ,  , we can equivalently write it as .(iii) implies that . This is because if and , then by (a) .

Recalling that a general solution of the equation is, by definition, the preimage , we state our conclusion in a lemma.

Lemma 3.3. Let . The preimage is a cylinder with the base and element . Further, for implies that , that is, on is one-to-one.

Lemma 3.4. Suppose . Fix a set and a vector . Denote and let and be the accompanying matrices from Lemma 3.2. Then, the set defined by is a cylinder:

Suppose and denote . Then is an orthoprojector with the image

In part (a) replace by , , and , respectively. Then the set has representation (3.48).

Proof. Let us show that implies that

The inclusion is obvious. Conversely, let . Then, . Since is an orthonormal basis, we have . Denoting , it follows that

We have proved (3.50) which implies . Application of Lemma 3.3 proves (3.48).

It is easy to check that is an orthoprojector and its image satisfies (3.49). From (3.49), we conclude that and commute and

see [14, Chapter , Section 3, Lemma 2(a)]. Applying to both sides of (3.50) and using the last equation we get

We have proved an analog of (3.50) for the matrix under consideration:

As previously mentioned, application of Lemma 3.3 proves (3.48).

Lemma 3.5. Let and be the same as in Lemma 3.4(a).

(a)Suppose is of the form where . If (2.23) holds, then for the set from (3.48) one has

(b)If , then ,as , uniformly in .

Proof. Part (a)

Step  1 (proving an auxiliary bound). Denote

We intend to prove that

By Lemma 3.2, (3.50) implies that

By Lemma 3.3, is a one-to-one mapping from to , both of which are of dimension . We use this fact twice.

Firstly, selecting two bases, one in and another in , we can identify with its matrix representation in these bases. This matrix is square of order and rank , so the Jacobian of the transformation ,  , is positive. Therefore,

Secondly, implies that which for implies that . Thus, is a norm on and by (3.14),

As is assumed nonincreasing, is also nonincreasing, and (3.59) leads to

We bound this using polar coordinates:

Here, the quantity

is finite because by condition (2.23), is integrable in the neighborhood of zero and is of order at infinity.

Step  2 (proving the main bound). The matrix from Lemma 3.4(a) is of size and is of size . Equation (3.50) and imply that

Equation (3.48) written point-wise is ,  ,  ,  . Since here are orthogonal, we have

where the right side defines the function .

The transition from to the coordinates is described by an orthogonal matrix. This can be shown as follows. Take in , an orthonormal basis and identify with the set of its coordinates in that basis. Do the same with and . Join the two bases. The equation induces a transformation between the original coordinates of and the ones in the joint basis. Since , the Jacobian of the resulting transformation is unity. Hence, by (3.48), (3.64), (3.65), and (3.57),

Part (b). In the bound (see (3.59) and (3.60)),

we have , so uniformly in . As in (3.66), it follows that

Lemma 3.6. Let , and be the same as in Lemma 3.4(b).

(a)Suppose where . If (2.25) holds, then (3.55) is true with from Lemma 3.4(b).(b)If , then , as , uniformly in .

Proof. The proof essentially follows that of Lemma 3.5(a). From (3.49), and (3.54) we see that instead of (3.58) we have . Instead of (3.64) we have . This leads to the definition of the marginal density used in (2.25). With these changes, (3.62), and (3.66) are true and prove part (a).

With the same changes the proof of Lemma 3.5(b) goes through.

In the course of the proof, we approximate by functions from below and above. Then, we show that the rate of approximation improves as and .

Step 1 (constructing the approximating functions). Denote the distance from a point to a set . For a given , set ,  .

Obviously, . As in [10, page 12 proof of ], we can construct functions that are uniformly continuous on and satisfy

Step 2 (reducing the convergence statement to a special case). Denote

Here, we show that

implies (2.24). From the inequality we have

By Theorem 2.7,

By (3.71), for any , there exists such that

By (3.73), for the same , there is such that

Equations (3.72), (3.74), and (3.75) imply that

Since is arbitrary, this proves (2.24).

Step 3 (further reduction to an indicator function). Let us call a band. Here, we show that (3.71) will follow if we prove that

Note the identity

Since each of the square brackets on the right is nonnegative, (3.77) gives . Now, it suffices to take into account the inequality to see that (3.71) follows.

Step 4 (geometric description of the task). Denote a straight line with a slope and intercept and put

This expression shows that we need to study the set of points common to the straight line and the band . The cylindrical nature of the sets considered implies that the band admits an equivalent definition as a cylinder with the base

and element . From this definition, we see that the condition is equivalent to

We have to sort out points into (a) those for which the set of points satisfying (3.81) is so small that (3.79) is zero and (b) that set is large enough for (3.79) to be positive.

Step 5 (case ). In this case, , and (3.81) reduces to

This condition does not contain . Therefore, if it is not satisfied, the integrand in (3.79) is identically zero and for such . If it is satisfied, the integrand in (3.79) is identically and for such . Hence,

The set of satisfying (3.82) has an infinite Lebesgue measure which precludes us from application of the continuity property of the Lebesgue measure (for an integrable , as ).

As one can see from Figure 1, is not a cone-like set. We divide it into two parts: includes the points inside the unit ball and on the unit sphere and includes the points of outside the unit ball. Put and let us show that

Suppose . Then, and by (3.80), there exists a sequence such that . The sequence is bounded and therefore precompact, while is closed by construction. Hence, we can assume that (passing to a subsequence, if necessary). Then, . This implies  . Let . Since is cone-like, it follows that and . This proves that ,   and (3.84) is true.

From (3.84), we conclude that

This means that the set of solutions of (3.82) is contained in the union of two sets: and where and .

By Lemma 3.4(a), . By Lemma 3.5(a)

By Lemma 3.4(a), . By Lemma 3.5(b)

Equations (3.86) and (3.87) prove that ,  .

Figure 1: consists of two cones bounded by solid lines. is bounded by dashed lines.

Step 6 (case ). Equation (2.5) implies that if the straight line has a bounded intersection with the band , then . Therefore, we are interested in the set of slopes:

Let us prove that

Suppose . Then, there exists a sequence of points which lie on the straight line, , where , and which are -close to : there exists a sequence such that

Equation (2.21) implies that and . Therefore . Because is compact, we can assume that . From (3.90), we have or, dividing both sides by , . This shows that the sequence is bounded and can be assumed convergent to a number . Letting , we get , and (3.89) is true.

Fix and denote . The matrix projects Im  onto the straight line . The projector projects Im   onto Im. Condition (3.81) implies two equations in projections:

The first one can be safely omitted because the shift along a line does not change the property of the line to have an unbounded intersection with . Denoting ,  , and using (3.52), we write the second equation as

The set of points such that (3.81) holds for an unbounded set of values is clearly contained in the set of points that satisfy (3.92). For we use an obvious bound (see (3.79)), and, therefore, . The rest of the proof closely follows that for the case . An inclusion of type (3.85) holds with replaced by in all definitions. The reference to Lemma 3.4(a) is replaced by Lemma 3.4(b). Lemma 3.6 is used instead of Lemma 3.5. The conclusion is that .

We shall use several times the fact that for the density in representation (2.24)will, the conditions and are equivalent. From (2.24), we see that the task is to calculate the function (3.79) with in place of . Because of (2.21), the condition that is equivalent to the condition in projections:

The magnitude of depends on the set

The properties of depend on those of for certain sets , as we will see.

Lemma 3.7. Suppose ( is the projector from Theorem 2.7). Then, for all .

Let . Then, .

Proof. The assumption implies that for , which can be written as . Hence, for any .

If , then and for at least one . The matrix is of size , and of rank at least . From the equation we see that the null space is of dimension at most . The set of satisfying is a partial solution plus an arbitrary element from the null space . Thus the set of all solutions is a hyperplane in of dimension at most and has -dimensional Lebesgue measure zero.

Lemma 3.8. Suppose we are given a set and for almost each , the set is defined by (3.94).

(a)If is bounded for almost all and , then . In particular, if , then .(b)If

and , then . In particular, if , then .(c)Suppose that and for almost all ,

(i.e., either or with some ). Then, .

Proof. If is bounded, then outside a bounded set, and by (2.5),

for almost all . Since outside , we have

Obviously, the integral is null if .

By (3.97), condition (3.95) implies that

Since , we have

In case , the last integral is 1.

Consider, for example, the case of the right half-axis:

Hence, .

Case 1. Let . Then, and . Define . Obviously, and . In the equation

for all large , the term in the square brackets is close to and, because is open in , this term belongs to . Since is cone-like, the left side of (3.102) belongs to for all large . Hence, the set contains the neighborhood of infinity (more precisely, all with where depends on ). By Lemma 3.8(b), where , then .

Case 2. Suppose .

Subcase 2.1. Let or, equivalently, . Condition (3.93) reduces to

By Lemma 3.4(a), the set of solutions of (3.103) is where is the Moore-Penrose inverse of ,, and is the projector from Lemma 3.2. Using Lemma 3.8(b), (3.66), and (3.59) we get

The density is spherically symmetric. It is positive everywhere because

would mean for any , that is, for any , which is impossible by assumption. is one-to-one on , so is a nonempty proper subset of . Therefore, .

Subcase 2.2. Suppose . As in Case  1, is defined. Let us show that it satisfies

Since is a limit point of , there is a sequence such that . Hence, where because . From (2.21), it follows that , and . Letting and using , we get . Since , we have and . Thus, is impossible and (3.106) is true.

In the three examples in the following we assume that . Then, , and by Lemma 3.7(b),

Sub-sub-case (a) (Example of ). Let be symmetric. Assumptions (1.13) imply that whenever , so . If denote eigenvalues of , then the eigenvalues of are . Let the constant in the definition of the critical region be positive. Put and with some integer and real numbers ,   such that ,   define by

Then, for ,

and has eigenvalues

Hence, in the notation of Condition 3,   and are obtained from ,  , by reordering. For example, if , then ,;  ,  . Assuming this ordering, denote the corresponding eigenvectors of by where the minuses and pluses remind us that the corresponding eigenvalues depend on and , respectively.

Let and suppose that

Denote

Because of (3.111), here . Besides, since is orthogonal to , the decomposition

does not contain . Hence, (3.112) becomes

Remembering that the eigenvectors of are just reordered eigenvectors of , we get

for any of form (3.112). From (3.107) and (3.113) we see that here

Hence, normalizing ,  , and assuming that , we get from (3.115)

for all and almost all . This shows that the set (3.94) coincides with for almost all and by Lemma 3.8(b) .

Sub-sub-case (b) Example of . We change (3.111) so as to obtain in (3.115) in place of . Let and suppose that

Instead of (3.113), we get which leads to an analog of (3.114): . Equation(3.115) is replaced by

where a.e. in . The result is that

for all and almost all . Hence, the set (3.94) is empty for almost all and by Lemma 3.8(a), .

Sub-sub-case Example of ]. runs over . If , then , and we can choose the eigenvectors of so that , and then runs over . Let be a union of two cones, as in Figure 2. The line lies on the boundary of the cones. In Figure 2, and . For any nonzero shift , half of the shifted line is contained in because and are not parallel. Here, “half” means “either for all large positive or for all large negative .” By Lemma 3.8(c), .

Figure 2: The base of contains the area between two solid lines and  . The dashed line is .

Case 3. Suppose . Then, as in part 1, and is defined. It satisfies . The term in the square brackets in (3.102) for all large belongs to an open set . Since this set is cone-like, the left side of (3.102) belongs to for all large . The set is bounded for all , and by Lemma 3.8(a), .

Step 1 (the -maximizing set is an ellipsoid). For by orthogonality of and Condition 7, we have

Since is decreasing, this is equivalent to . Denoting ,  , we rewrite the resulting inequality as

In coordinates, this is an ellipsoid with half-axes .

Step 2 (characterizing the rate of decay of ). First we determine the rate with which approaches when . To approximately invert , in the equation , we use representation (2.30): . This is equivalent to . Solving this equation for the involved in the right-hand side produces

Since , we can put in the last equation to get

To obtain , let . Then, by (2.29),

Combining this with (3.124) and (2.29), we have

Step 3 (behavior of half-axes). We intend to show that for the half-axes tend to zero, so that in the limit

whereas for , they tend to , and in the limit, can be any real numbers.

For from (2.28), we see that the condition is equivalent to . Because of (2.31), this condition is the same as which holds by (2.29).

For the remaining half-axes, we get by (2.27), (3.126), and (2.32)

We have proved our statement on convergence of the half-axes. According to (2.26), (3.127) rewrites as ,  . Since the rows of are eigenvectors of , the last equation means that is orthogonal to eigenvectors numbered . Hence, belongs to the null space of , and the set of all such ’s is .

In this proof, the set is obtained as the limit of the ridge of . In general, this ridge can have any shape and the limit will not be .

4. Summary

Theorem 2.2 is the key result in this paper. We tried to obtain it under minimal conditions on . It would be interesting to relax our conditions. In particular, densities that are not spherically symmetric are of interest in applications. Note that when the dependence on is more complex than in , one needs very different techniques and more detailed assumptions about to obtain an analog of Theorem 2.2. Such results can be found in [15]. Another promising direction of research is to consider tests other than the Cliff-Ord one.

Acknowledgments

The author would like to thank F. Martellosio for his remarks that helped him refine Theorem 2.11 and improve the exposition in general.

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