2. DISTRIBUTION AND MOMENTS OF t-RATIO UNDER NONNORMALITY

Turning first to the t-ratio τ defined earlier, for purposes of analysis we write w = (w₁,w₂′)′, where w₁ is a scalar, w₂ is a q × 1 vector, and p = 1 + q. Write, conformably,

as the covariance matrix of w, where γ > 0 is an arbitrary scalar, C is a 1 × q vector, and Σ is a q × q positive definite matrix. Also, let

, the rank of B, satisfy 0 < r ≤ q. Furthermore, it is assumed that w has an elliptically symmetric (or simply elliptical) distribution with mean zero and density function of the form

where ϑ is a normalizing constant and h is some arbitrary function. It is conventional to write

. If

, then the density of w reduces to that of spherical symmetry. Special cases of the elliptical class of distributions include the multivariate normal, multivariate Student's t, multivariate Cauchy, and rotated Laplace distributions.

Elliptical distribution has played an important role in distributional theory and other fields of statistics. Fang, Kotz, and Ng (1990) and Fang and Zhang (1990) provide a thorough discussion of the properties of the elliptical class of distributions and related topics. The results given in these books are in fact central to a generation of efforts devoted to various theoretical developments and applications of symmetric nonnormal distributions, and the following list illustrates some of the interesting examples that have been considered. Läuter (1996) and Läuter, Glimm, and Kropf (1996) use the properties of the spherical matrix distribution (Fang and Zhang, 1990) to obtain statistics that are suitable for the analysis of high-dimensional samples of small size and that result in no test size distortion. Li, Fang, and Zhu (1997) propose probability plots for testing spherical symmetry versus elliptical symmetry based on the invariant property⁵

of the spherically symmetric distribution (Fang et al., 1990). Fang, Li, and Liang (1998), on a closely related problem, suggest graphical procedures to detect non-multinormality using the properties of left-spherical matrix distributions (Fang and Zhang, 1990). In addition, a recent potentially important contribution to multinormality testing using low-dimensional projection from Liang, Li, Fang, and Fang (2000) is also based heavily on the results of Fang and Zhang (1990) on the properties of the left-spherical matrix distribution. As we shall see, some of the results presented in this paper also relate well to the contribution of Fang et al. (1990).

The following theorem gives the necessary and sufficient condition for τ to be t distributed.

THEOREM 1. Under the stated assumptions,

if and only if

.

Proof. See the Appendix.

In the case where a is a p × 1 vector with a single nonzero element, B = diag(0,Σ⁻¹), and Σ is of full rank, Theorem 1 simplifies significantly and reduces to the following corollary.

COROLLARY 1. Let a = (1,0′)′ and B = diag(0,Σ⁻¹); then

if and only if C = 0.

Proof. See the Appendix.

So, under the conditions of nonnormality characterized in Corollary 1, the t-ratio τ remains Student's t distributed provided that the random vector forming the t-ratio has a diagonal covariance structure. To consider an application, suppose that w = (w₁,w₂′)′ follows the (spherical) multivariate t distribution, i.e.,

where v > 2 and σ² > 0. Note that cov(w) = (σ²v/(v − 2))I_p and B = diag(0,I_q). Now, by virtue of Corollary 1,

, but w₂′w₂ is neither chi-squared distributed

⁶

nor distributed independently from w₁. So we obtain a counterexample to the situation described by conditions (i)–(iii) in Section 1. Corollary 1 can also be extended to cover the case where the covariance matrix of w is nonspherical. Let

with density function

where

. Formula (2.3) characterizes a multivariate t density with a nonspherical covariance structure provided that

. Straightforwardly, by Corollary 1,

Again, the distribution of τ can remain Student's t with q degrees of freedom despite the violation of the aforementioned conditions (i)–(iii).

Under the normal setup, Magnus (2002) derives the even moments of τ. Note that whatever the distribution of w, if it is elliptically symmetric, the odd moments of τ that exist are zero. The following theorem gives the even moments of τ when w follows the elliptically symmetric distribution.

THEOREM 2. Under the stated assumptions, i.e.,

, and a and B are defined as in Section 1, provided that q ≥ r ≥ 2(s + 1), then E(τ^2s) exists and has the representation

for s = 1,2,3,…, where

are the nonzero eigenvalues of

, and v₁,…,v_r are the eigenvectors of B* such that B*v_i = λ_i v_i,v_i′v_i = 1, and v_i′v_j = 0 for all i ≠ j.

Proof. See the Appendix.

In other words, the moments of τ under the nonnormal elliptical setup are also the same as those under normality. It also follows straightforwardly that

and

These expressions have proved useful for analyzing the sensitivity of τ to the relationship between a′w and w′Bw. Magnus (2002) does so for the special case of w being normally distributed and w′Bw following a chi-square distribution. Magnus's sensitivity analysis is in fact confined to the case where the nonzero eigenvalues of L′BL are all equal, i.e., λ₁ = λ₂ = ··· = λ. In this case w′Bw must be chi-square distributed except for a constant scalar. This is obvious, as

, we have

, where

is an orthogonal matrix such that Q′L′BLQ = λ diag(I_r,0). Magnus (2002) defines

as the dependence parameter in the normal setup. If δ = 0, then a′w and w′Bw are independent; if δ > 0, they are dependent. Magnus (2002) finds that var(τ) decreases linearly from r/(r − 2) to 1 as δ increases from 0 to 1. Hence the more the dependence between a′w and w′Bw, the better the variance of τ is approximated by the standard normal distribution's variance. He also finds that kur(τ) decreases almost linearly from 3(r − 2)/(r − 4) to 3r/(r + 2) and the deviation of the kurtosis from normality decreases as δ increases. So, with dependence the t-ratio τ is better approximated by the standard normal distribution than the Student's t distribution. If w has a (nonnormal) elliptical distribution, then w′Bw may not be chi-square distributed, but Magnus's results are still valid, as τ is distributional invariant for all elliptically symmetric vectors, as is obvious from the proof of Theorem 2. Note that in the elliptical setup, the condition of L′BL having identical nonzero eigenvalues does not imply

for some constant scalar c_s, but it does imply

for some random variable c_r such that c_r ≥ 0 and Pr(c_r = 0) = 0. One such example can be constructed from the (nonspherical) multivariate Student's t density given in (2.3) provided that the nonzero eigenvalues of

are all equal. Thus, the results of Magnus (2002) hold also in the more general context of w having an elliptically symmetric distribution; in the case of L′BL having identical nonzero eigenvalues, the higher δ is, the better the t-ratio is approximated by the standard normal distribution.

⁷

3. APPLICATION TO THE LINEAR MODEL AND THE F-RATIO

In this section we focus our interest on the linear regression model,

where y and ε are n × 1 vectors, X is an n × k fixed matrix of rank k, and β is a k × 1 vector of unknown coefficients. Furthermore, ε ∼ E_n(0,σ²Ω), and σ² is an unknown parameter. When Ω = I, the t-statistic for testing the hypothesis that an element of β, say β_i, equals a particular value β_i^o is given by

where

is the ith element of the ordinary least squares estimator (X′X)⁻¹X′y,

, and [(X′X)⁻¹]_ii is the (i,i)th element of (X′X)⁻¹. Now, under the hypothesis β_i = β_i^o, t_i can be equivalently expressed as

where

is an (n − k) × n matrix such that

. Now, by the properties of the elliptical distributions, (ϑ₁,ϑ₂′)′ ∼ E_n−k+1(0,diag(1,I_n−k)). Clearly, from Corollary 1, t_i ∼ t_n−k. The t-statistic therefore remains valid even without normally distributed disturbances.

Now, suppose that Ω ≠ I but Ω is a known positive definite matrix. A test of the hypothesis

where R is an m × k constant matrix of rank m(< k) and c is an m × 1 vector of constants, is carried out by computing the statistic

where

is the generalized least squares estimator of β. Assuming that H_o is true and the distribution of ε reduces to that of normality, the statistic F is well known to follow a central F distribution with m and n − k degrees of freedom. Ullah and Zinde-Walsh (1985) consider the error terms following the class of spherically symmetric disturbances with density function given by

where g is decreasing and twice differentiable on [0,+∞). It is easy to see that the density given in (3.6) is a special case of the (general) spherically symmetric density. Ullah and Zinde-Walsh (1985) show that F_o, which results from replacing Ω with I in (3.5), is distributed as a central F under H_o whatever g is, provided that it is decreasing and twice differentiable. In fact, an earlier unpublished work of Thomas (1970) has shown that F remains central F distributed under H_o when ε follows the class of elliptical distributions, i.e., ε ∼ E_n(0,σ²Ω). The result of Thomas (1970), however, as discussed in Fang et al. (1990, pp. 54–55), is conditional on the requirement that the distribution's characteristic function is a continuous decreasing function on [0,+∞). In other words, the class of distributions discussed in Thomas (1970) is in fact a subclass of the general class of elliptical distributions with density given in (2.1). Our aim here is to show that whatever the characteristic function, the null distribution of F remains central F under the elliptical distributions of errors.

THEOREM 3. Assume in model (3.1) that ε ∼ E_n(0,σ²Ω). Under H_o : Rβ = c, the distribution of F is F_(m,n−k).

Proof. See the Appendix.

The crucial result for our purpose here is that the distribution of the F-ratio can remain central F even without normally distributed disturbances. Once again, our result relies on the assumption of elliptically symmetric errors. But what if the distribution of ε falls outside that of elliptical symmetry? Zaman (1996, p. 200) argues that the F-test is robust to lack of normality of the errors for a large class of nonnormal densities including all unimodal densities; his argument, however, is incorrect. Let us briefly recall the notation in Zaman's book. He defines the F-ratio as F = ε′Q₁ε/ε′Q₂ε, where Q₁ and Q₂ are projection matrices and ε is a vector of independent and identically distributed (i.i.d.) random errors.⁸

Zaman's arguments are as follows.

1. If ε is replaced by θε, where θ is any random variable, then because θ will cancel out from both the numerator and denominator the F-statistic will remain unchanged;
2. on the basis of the first argument, Zaman concludes that if the errors are i.i.d. with common density f, where f is the density of an arbitrary random variable multiplied by the normal, the F-statistic remains the same as under normality.

One must interpret the arguments of Zaman (1996) cautiously. His first argument is correct provided that one makes the additional qualification that Pr(θ = 0) = 0, but still this does not lead to the conclusion reached in his second argument. Let

for all i's and θ = (θ₁,θ₂,…θ_n)′ be the random vector such that ε = (θ₁ z₁,θ₂ z₂,…θ_n z_n)′, i.e.,

. It is easy to see that ε′Q₁ε/ε′Q₂ε is generally not the same as z′Q₁ z/z′Q₂ z, except for the special case of θ₁ = θ₂ = ··· = θ_n. But the latter situation could not arise at least in the case of continuous θ_i′s. The following counterexample further clarifies our point. Let ε = (ε₁,ε₂)′, where ε₁ and ε₂ are i.i.d. and follow the Cauchy distribution, i.e., f_εi(ε_i) = 1/(π(1 + ε_i²)), i = 1,2. Also, let Q₁ = diag(1,0) and Q₂ = diag(0,1). Now, F = ε₁²/ε₂² has the density

which is not the density of an F_1,1 variate, although f_εi(ε_i) is unimodal.