Linear inference remains pivotal in statistical practice, despite errors often having excessive tails and thus deficient of moments required in conventional usage. Such errors are modeled here via spherical $\alpha$ –stable measures on ${ℝ}^n$ with stability index $\alpha \text{}\in \text{}\mathrm{<mo\; stretchy="false">(</mo>0,}\mathrm{2<mo\; stretchy="false">]</mo>,}$ arising in turn through multivariate central limit theory devoid of the second moments required for Gaussian limits. This study revisits linear inference under $\alpha$–stable errors, focusing on aspects to be salvaged from the classical theory even without moments. Critical entities include Ordinary Least Squares $\mathrm{<mo\; stretchy="false">(</mo>}O\text{}L\text{}S\mathrm{<mo\; stretchy="false">)</mo>}$ solutions, residuals, and conventional $F$ ratios in inference. Closure properties are seen in that OLS solutions and residual vectors under $\alpha$ –stable errors also have $\alpha$–stable distributions, whereas F ratios remain exact in level and power as for Gaussian errors. Although correlations are undefined for want of second moments, corresponding scale parameters are seen to gauge degrees of association under $\alpha$ –stable symmetry.

AMS subject classification: 62E15, 62H15, 62J20

Keywords:excessive errors, central limit theory, stable laws, linear inference

Models here are $\mathrm{<mo>\{</mo>}Y\text{}\mathrm{<mo>=</mo>}\text{}X\beta +\epsilon \mathrm{<mo>\}</mo>}$with error vector $\epsilon \text{}\in \text{}{ℝ}^n\mathrm{.}$Classical linear inference rests heavily on means, variances, correlations, skewness and kurtosis parameters, these requiring moments to fourth order. To the contrary, distributions having excessive tails, and devoid of moments even of first or second order, arise in a variety of circumstances. These encompass acoustics, image processing, radar tracking, biometrics, portfolio analysis and risk management in finance, and other venues in contemporary practice. Supporting references include^1–6 monographs of note are,^7–9 together with the recent work of Nolan.¹⁰ In these settings the classical foundations necessarily must be reworked.

To place this study in perspective, alternatives to Gaussian laws long have been sought in theory and practice, culminating in the class $\mathrm{<mo>\{</mo>}{\text{S}}_n\mathrm{<mo\; stretchy="false">(</mo>}0\mathrm{,}\Sigma \mathrm{<mo\; stretchy="false">)</mo><mo>\}</mo>}$consisting of elliptically contoured distributions in ${ℝ}^n$centered at 0 with scale parameters $\Sigma .$ These typically are taken to be rich in moments, and to provide alternatives to the use of large-sample approximate Gaussian distributions under conditions for central limit theory. Comprehensive treatises on the theory and applications of these models are.^11–13

In contrast, errors having excessive tails are modeled on occasion via spherically symmetric $\alpha$stable $\mathrm{<mo\; stretchy="false">(</mo>}\text{S}\alpha \text{S}\mathrm{<mo\; stretchy="false">)</mo>}$distributions in ${ℝ}^n$ with index $\alpha \text{}\in \text{}\mathrm{<mo\; stretchy="false">(</mo>0,}2\mathrm{<mo\; stretchy="false">]</mo>.}$ These comprise the limit distributions of standardized vector sums, specifically, Gaussian limits at  Cauchy limits at $\alpha \text{}\mathrm{<mo>=</mo>}\text{}\mathrm{2,}$ and corresponding stable limits otherwise. These distributions are contained in the class $\mathrm{<mo>\{</mo>}{\text{S}}_n\mathrm{<mo\; stretchy="false">(</mo>}0\mathrm{,}{I}_n\mathrm{<mo\; stretchy="false">)</mo><mo>\}</mo>,}$thus sharing its essential geometric features, but instead are deficient in moments usually ascribed to $\mathrm{<mo>\{</mo>}{\text{S}}_n\mathrm{<mo\; stretchy="false">(</mo>}0\mathrm{,}{I}_n\mathrm{<mo\; stretchy="false">)</mo><mo>\}</mo>.}$Despite the venues cited, $\alpha$stable errors have seen limited usage for want of closed expressions for stable density functions, known only in selected cases but topics of continuing research. Nonetheless, findings reported here rest on well defined characteristic functions ($\text{chf}s$), on critical representations for these, and on the inversion of the latter in order to represent the $\alpha$-stable densities themselves. Even here a divide emerges between independent, identically distributed $\alpha$-stable sequences, and dependent $\text{S}\alpha \text{S}$ variables, as reported in Jensen¹⁴ and as summarized here for completeness in an Appendix. In addition, many findings of the present study are genuinely nonparametric, in applying for all or portions of distributions in the range $\alpha \text{}\in \text{}\mathrm{<mo\; stretchy="false">(</mo>0,}\mathrm{2<mo\; stretchy="false">]</mo>,}$ and thus remaining distribution-free within that class. An outline follows.

Notation and technical foundations are provided in the next major section, Preliminaries, to include Notation and accounts of Special Distributions, Central Limit Theory and Essentials of $\text{S}\alpha \text{S}$ Distributions as subsections. The principal sections following these address Linear Models under $\text{S}\alpha \text{S}$ Errors, with a separate subsection on Models Having Cauchy Errors, and Conclusions. Collateral topics are contained for completeness in Appendix A.

Notation

Spaces of note include ${ℝ}^n$as Euclidean $n$space, with ${\mathbb{S}}_n$as the real symmetric $\mathrm{<mo\; stretchy="false">(</mo>}n\text{}\times \text{}n\mathrm{<mo\; stretchy="false">)</mo>}$ matrices and ${\mathbb{S}}_n^{+}$ as their positive definite varieties. Vectors and matrices are set in bold type; the transpose, inverse, trace, and determinant of are $A\mathrm{<mo>\text{'}</mo>,}$ $A^{-1}\mathrm{,}$ $tr\mathrm{<mo\; stretchy="false">(</mo>}A\mathrm{<mo\; stretchy="false">)</mo>,}$ and $\mathrm{<mo>|</mo>}A\mathrm{<mo>|</mo><mo>;</mo>}$ the unit vector in ${ℝ}^n$ is ; and $I_n$is the $\mathrm{<mo\; stretchy="false">(</mo>}n\text{}\times \text{}n\mathrm{<mo\; stretchy="false">)</mo>}$identity.

Moreover, $Diag\mathrm{<mo\; stretchy="false">(</mo>}{A}_1\mathrm{,}\dots \mathrm{,}{A}_k\mathrm{<mo\; stretchy="false">)</mo>}$is a block-diagonal array, and ${\Sigma }^{\frac{1}{2}}$is the spectral square root of

Special distributions

Given $Y\text{}\mathrm{<mo>=</mo>}\text{}\mathrm{<mo\; stretchy="false">[</mo>}{Y}_1\mathrm{,}\dots \mathrm{,}{Y}_n{]}^{\prime }\in \text{}{ℝ}^n\mathrm{,}$its distribution, expected value, and dispersion matrix are designated as $\mathcal{L}\mathrm{<mo\; stretchy="false">(</mo>}Y\mathrm{<mo\; stretchy="false">)</mo>,}$ $E\mathrm{<mo\; stretchy="false">(</mo>}Y\mathrm{<mo\; stretchy="false">)</mo>}\text{}\mathrm{<mo>=</mo>}\text{}\mu \mathrm{,}$ and $V\mathrm{<mo\; stretchy="false">(</mo>}Y\mathrm{<mo\; stretchy="false">)</mo>}\text{}\mathrm{<mo>=</mo>}\text{}\sum \mathrm{,}$with variance $Var\mathrm{<mo\; stretchy="false">(</mo>}Y\mathrm{<mo\; stretchy="false">)</mo>}\text{}\mathrm{<mo>=</mo>}\text{}{\sigma }^2$on ${ℝ}^1\mathrm{.}$Specifically, $\mathcal{L}\mathrm{<mo\; stretchy="false">(</mo>}Y\mathrm{<mo\; stretchy="false">)</mo>}\text{}\mathrm{<mo>=</mo>}\text{}{\text{N}}_n\mathrm{<mo\; stretchy="false">(</mo>}\text{m}\mathrm{,}\sum \mathrm{<mo\; stretchy="false">)</mo>}$is Gaussian on ${ℝ}^n$with parameters $\mathrm{<mo\; stretchy="false">(</mo>}\mu \mathrm{,}\sum \mathrm{<mo\; stretchy="false">)</mo>.}$Distributions on ${ℝ}^1$of note include the ${\chi }^2\mathrm{<mo\; stretchy="false">(</mo>}u\mathrm{<mo>;</mo>}\nu \mathrm{,}\lambda \mathrm{<mo\; stretchy="false">)</mo>}$and related $\chi \mathrm{<mo\; stretchy="false">(</mo>}u\mathrm{<mo>;</mo>}\nu \mathrm{,}\lambda \mathrm{<mo\; stretchy="false">)</mo>}$distributions, together with the Snedecor -Fisher $F\mathrm{<mo\; stretchy="false">(</mo>}u\mathrm{<mo>;</mo>}{\nu }_1\mathrm{,}{\nu }_2\mathrm{,}\lambda \mathrm{<mo\; stretchy="false">)</mo>,}$these having $\mathrm{<mo\; stretchy="false">(</mo>}\nu \mathrm{,}{\nu }_1\mathrm{,}{\nu }_2\mathrm{<mo\; stretchy="false">)</mo>}$as degrees of freedom and $\lambda$a noncentrality parameter. The characteristic function $\mathrm{<mo\; stretchy="false">(</mo>}\text{chf}\mathrm{<mo\; stretchy="false">)</mo>}$for $Y\text{}\in \text{}{ℝ}^n$is the expectation ${\varphi }_Y\text{}\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\text{}\mathrm{<mo>=</mo>}\text{}\text{E}\mathrm{<mo\; stretchy="false">[</mo>}{e}^{\iota t\mathrm{<mo>\text{'}</mo>}Y}\mathrm{<mo\; stretchy="false">]</mo>}$with argument $t\mathrm{<mo>\text{'}</mo>}\text{}\mathrm{<mo>=</mo>}\text{}\mathrm{<mo\; stretchy="false">[</mo>}{t}_1\mathrm{,}\dots \mathrm{,}{t}_n\mathrm{<mo\; stretchy="false">]</mo>}$ and $\iota \text{}\mathrm{<mo>=</mo>}\text{}\sqrt{-1}\mathrm{<mo>;</mo>}$a standard source is Lukacs & Laha.¹⁵ Attention is drawn subsequently to probability density (pdf) and cumulative distribution (cdf) functions. Moreover, the class $\mathrm{<mo>\{</mo>}\mathcal{L}\mathrm{<mo\; stretchy="false">(</mo>}Z\mathrm{<mo\; stretchy="false">)</mo>}\text{}\in \text{}{\text{S}}_n\mathrm{<mo\; stretchy="false">(</mo>}0\mathrm{,}\sum \mathrm{<mo\; stretchy="false">)</mo><mo>\}</mo>}$consists of elliptically contoured distributions in ${ℝ}^n$centered at 0 and having chf’s of type ${\varphi }_Z\text{}\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\text{}\mathrm{<mo>=</mo>}\text{}\psi \mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo>\text{'}</mo>}\text{S}t\mathrm{<mo\; stretchy="false">)</mo>.}$ We adopt the following.

Definition 1 A distribution P on ${ℝ}^n$ is said to be monotone unimodal about $0\text{}\in \text{}{ℝ}^n$if for every $y\text{}\in \text{}{ℝ}^n$and every convex set C symmetric about $0\text{}\in \text{}{ℝ}^n\mathrm{,}$$P\mathrm{<mo\; stretchy="false">[</mo>}C\text{}+\text{}ky\mathrm{<mo\; stretchy="false">]</mo>}$ is no increasing in $k\text{}\in \text{}\mathrm{<mo\; stretchy="false">[</mo>}\mathrm{0,}\infty \mathrm{<mo\; stretchy="false">)</mo>.}$See reference.¹⁶

Central limit theory

For $\text{i}\text{}\text{i}\text{}\text{d}$vectors $\mathrm{<mo>\{</mo>}{Z}_1\mathrm{,}{Z}_2\mathrm{,}{Z}_3\mathrm{,}\dots \mathrm{<mo>\}</mo>}$in ${ℝ}^n\mathrm{,}$let ${\overline{Z}}_{\text{}N}\text{}\mathrm{<mo>=</mo>}\text{}{N}^{-1}\mathrm{<mo\; stretchy="false">[</mo>}{Z}_1+\dots +Z_N\mathrm{<mo\; stretchy="false">]</mo>,}$and consider limit distributions of type $\mathrm{<mo>\{</mo>}{\mathcal{L}}_{\infty }\mathrm{<mo\; stretchy="false">(</mo>}c{\overline{Z}}_{\text{}N}\mathrm{<mo\; stretchy="false">)</mo>}\text{}\mathrm{<mo>=</mo>}\text{}\mathrm{liminf}\mathcal{L}\mathrm{<mo\; stretchy="false">(</mo>}c{\overline{Z}}_{\text{}N}\mathrm{<mo\; stretchy="false">)</mo><mo>\}</mo>}$for suitably chosen C. On specializing from the elliptical class ${\text{S}}_n\mathrm{<mo\; stretchy="false">(</mo>}\text{d}\mathrm{,}\sum \mathrm{<mo\; stretchy="false">)</mo>}$having location-scale parameters $\mathrm{<mo\; stretchy="false">(</mo>}\text{d}\mathrm{,}\text{S}\mathrm{<mo\; stretchy="false">)</mo>,}$we consider $\alpha$stable limit distributions as follow on identifying ${\mathcal{L}}_{\infty }\mathrm{<mo\; stretchy="false">(</mo>}c{\overline{Z}}_{\text{}N}\mathrm{<mo\; stretchy="false">)</mo>}$with

Definition 2 Let $\mathcal{L}\mathrm{<mo\; stretchy="false">(</mo>}Z\mathrm{<mo\; stretchy="false">)</mo>}\text{}\in \text{}{\text{S}}_n^{\alpha }\mathrm{<mo\; stretchy="false">(</mo>}\text{d}\mathrm{,}\sum \mathrm{<mo\; stretchy="false">)</mo>}$designate an elliptical $\alpha$-Stable law on ${ℝ}^n$centered at $\text{d}\text{}\in \text{}{ℝ}^n$with scale parameters $\sum$and stable index $\alpha \text{}\in \text{}\mathrm{<mo\; stretchy="false">(</mo>0,}2\mathrm{<mo\; stretchy="false">]</mo>,}$having the $\text{chf}$ $${\varphi }_Z\text{}\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\text{}\mathrm{<mo>=</mo>}\text{}\exp \mathrm{<mo>\{</mo>}\iota t\mathrm{<mo>\text{'}</mo>}\text{d}\text{}-\text{}\frac{1}{2}{\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo>\text{'}</mo>}\text{S}t\mathrm{<mo\; stretchy="false">)</mo>}}^{\frac{\alpha }{2}}\mathrm{<mo>\}</mo>.}$$ Each marginal distribution of ${\text{S}}_n^{\alpha }\mathrm{<mo\; stretchy="false">(</mo>}\delta 1_{\text{}n}\mathrm{,}{I}_n\mathrm{<mo\; stretchy="false">)</mo>}$ on $${ℝ}^1\mathrm{,}$$ namely ${\text{S}}_1^{\alpha }\mathrm{<mo\; stretchy="false">(</mo>}\delta \mathrm{,1<mo\; stretchy="false">)</mo>,}$has the chf ${\varphi }_{Z_i}\text{}\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\text{}\mathrm{<mo>=</mo>}\text{}\exp \mathrm{<mo>\{</mo>}\iota t\delta -\frac{1}{2}\mathrm{<mo>|</mo>}t{\mathrm{<mo>|</mo>}}^{\alpha }\mathrm{<mo>\}</mo>.}$Let $\text{S}\alpha \text{S}\text{}\mathrm{<mo>=</mo>}\text{}\mathrm{<mo>\{</mo>}{\text{S}}_n^{\alpha }\mathrm{<mo\; stretchy="false">(</mo>}\text{d}\mathrm{,}\text{S}\mathrm{<mo\; stretchy="false">)</mo><mo>;</mo>}\mathrm{<mo\; stretchy="false">(</mo>}\text{d}\mathrm{,}\sum \mathrm{<mo\; stretchy="false">)</mo>}\text{}\in \text{}\mathrm{<mo\; stretchy="false">(</mo>}{ℝ}^n\text{}\otimes \text{}{\sum }_n\mathrm{<mo\; stretchy="false">)</mo><mo>\}</mo>}$designate the class of all such distributions.

Remark 1 $\mathcal{L}\mathrm{<mo\; stretchy="false">(</mo>}Z\mathrm{<mo\; stretchy="false">)</mo>}$is of full rank and has a density in ${ℝ}^n$if and only if $\sum$is of full rank in ${\mathbb{S}}_n^{+}\mathrm{<mo>;</mo>}$otherwise $\mathcal{L}\mathrm{<mo\; stretchy="false">(</mo>}Z\mathrm{<mo\; stretchy="false">)</mo>}$is concentrated in a subspace of ${ℝ}^n$of dimension equal to the rank of

To continue, designate by ${\mathfrak{D}}_{\text{}\alpha }$the domain of attraction of each element $Z_i$ in $\mathrm{<mo>\{</mo>}{Z}_1\mathrm{,}{Z}_2\mathrm{,}{Z}_3\mathrm{,}\dots \mathrm{<mo>\}</mo>}$ in ${ℝ}^n$ having $\mathrm{liminf}\mathcal{L}\mathrm{<mo\; stretchy="false">(</mo>}c{\overline{Z}}_{\text{}N}\mathrm{<mo\; stretchy="false">)</mo>}$in $\text{S}\alpha \text{S}\mathrm{.}$That is, their chfs satisfy $\mathrm{<mo>\{</mo>}\mathrm{liminf}{\varphi }_{c{\overline{Z}}_{\text{}N}}\text{}\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\text{}\mathrm{<mo>=</mo>}\text{}\exp \mathrm{<mo\; stretchy="false">[</mo>}\iota t\mathrm{<mo>\text{'}</mo>}\text{d}\text{}-\text{}\frac{1}{2}{\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo>\text{'}</mo>}\text{S}t\mathrm{<mo\; stretchy="false">)</mo>}}^{\frac{\alpha }{2}}\mathrm{<mo\; stretchy="false">]</mo><mo>\}</mo>}$when scaled suitably. Specifically, the distributions ${\mathfrak{D}}_2$ attracted to Gaussian limits comprise all distributions $\mathcal{L}\mathrm{<mo\; stretchy="false">(</mo>}{Z}_i\mathrm{<mo\; stretchy="false">)</mo>}$in ${ℝ}^n$having second moments. More generally, domains of attraction to distributions in $\text{S}\alpha \text{S}$have been studied in references,^17–20 to include Lindeberg conditions in Barbosa & Dorea,²¹ together with rates of convergence to stable limits in Paulauskas.²²

Remark 2 That ${\Phi }_Z(t)=exp[\iota t\mathrm{<mo>\text{'}</mo>}\text{d}\text{}-\frac{1}{2}{\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo>\text{'}</mo>}\text{S}t\mathrm{<mo\; stretchy="false">)</mo>}}^{\frac{\alpha }{2}}]$ has elliptical contours derives from the spherical chf ${\varphi }_U\text{}\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\text{}\mathrm{<mo>=</mo>}\text{}\exp \mathrm{<mo\; stretchy="false">[</mo>}\iota t\mathrm{<mo>\text{'}</mo>}\text{q}-\frac{1}{2}{\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo>\text{'}</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}}^{\frac{\alpha }{2}}\mathrm{<mo\; stretchy="false">]</mo>}$ through the transformation a

Essentials for $S\text{}\alpha \text{}S$distributions

As noted, closed expressions for $\text{S}\alpha \text{S}$densities are known in selected cases only, to be complemented by results to follow. Here $g_n\mathrm{<mo\; stretchy="false">(</mo>}u\mathrm{<mo>;</mo>}\text{d}\mathrm{,}\sum \mathrm{<mo\; stretchy="false">)</mo>}$is the Gaussian density on ${ℝ}^n$ having parameters $\mathrm{<mo\; stretchy="false">(</mo>}\delta \mathrm{,}\sum \mathrm{<mo\; stretchy="false">)</mo>,}$ and $f_n^{\alpha }\mathrm{<mo\; stretchy="false">(</mo>}\mu \mathrm{<mo>;</mo>}\delta \mathrm{,}\sum \mathrm{<mo\; stretchy="false">)</mo>}$ is the provisional $\text{S}\alpha \text{S}$ density corresponding to ${\varphi }_Z\text{}\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\text{}\mathrm{<mo>=</mo>}\text{}\exp \mathrm{<mo\; stretchy="false">[</mo>}\iota t\mathrm{<mo>\text{'}</mo>}\text{d}-\frac{1}{2}{\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo>\text{'}</mo>}\text{S}t\mathrm{<mo\; stretchy="false">)</mo>}}^{\frac{\alpha }{2}}\mathrm{<mo\; stretchy="false">]</mo>.}$ The following properties are essential.

Theorem 1 Let $\mathcal{L}\mathrm{<mo\; stretchy="false">(</mo>}Z\mathrm{<mo\; stretchy="false">)</mo>}\text{}\in \text{}{\text{S}}_n^{\alpha }\mathrm{<mo\; stretchy="false">(</mo>}\text{d}\mathrm{,}\text{S}\mathrm{<mo\; stretchy="false">)</mo>}$have the chf ${\varphi }_Z\text{}\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\text{}\mathrm{<mo>=</mo>}\text{}\exp \mathrm{<mo\; stretchy="false">[</mo>}\iota t\mathrm{<mo>\text{'}</mo>}\text{d}\text{}-\text{}\frac{1}{2}{\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo>\text{'}</mo>}\text{S}t\mathrm{<mo\; stretchy="false">)</mo>}}^{\frac{\alpha }{2}}\mathrm{<mo\; stretchy="false">]</mo>}$ and density function $f_n^{\alpha }\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo>;</mo>}\delta \mathrm{,}\sum \mathrm{<mo\; stretchy="false">)</mo>}$ if defined. Then the following properties hold.

1. For nonsingular, $\mathcal{L}\mathrm{<mo\; stretchy="false">(</mo>}Z\mathrm{<mo\; stretchy="false">)</mo>}\text{}\in \text{}{\text{S}}_n^{\alpha }\mathrm{<mo\; stretchy="false">(</mo>}\delta \mathrm{,}\sum \mathrm{<mo\; stretchy="false">)</mo>}$ is absolutely continuous in ${ℝ}^n\mathrm{,}$ having a density function $f_n^{\alpha }\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo>;</mo>}\delta \mathrm{,}\sum \mathrm{<mo\; stretchy="false">)</mo><mo>;</mo>}$
2. The Gaussian mixture ${\varphi }_Z\text{}\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\text{}\mathrm{<mo>=</mo>}\text{}{\displaystyle {\int }_0^{\infty }}{e}^{\iota t\mathrm{<mo>\text{'}</mo>}\text{d}-t\mathrm{<mo>\text{'}</mo>}\text{S}t\mathrm{<mo>/</mo>2}s}d\Psi \mathrm{<mo\; stretchy="false">(</mo>}s\mathrm{<mo>;</mo>}\alpha \mathrm{<mo\; stretchy="false">)</mo>}$ holds with $\Psi \mathrm{<mo\; stretchy="false">(</mo>}s\mathrm{<mo>;</mo>}\alpha \mathrm{<mo\; stretchy="false">)</mo>}$as a mixing on ${ℝ}^1\mathrm{<mo>;</mo>}$
3. The Gaussian mixture $f_n^{\alpha }\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo>;</mo>}\delta \mathrm{,}\sum \mathrm{<mo\; stretchy="false">)</mo>}\text{}\mathrm{<mo>=</mo>}\text{}{\displaystyle {\int }_0^{\infty }}{g}_n\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo>;</mo>}\delta \mathrm{,}{s}^{-1}\sum \mathrm{<mo\; stretchy="false">)</mo>}d\Psi \mathrm{<mo\; stretchy="false">(</mo>}s\mathrm{<mo>;</mo>}\alpha \mathrm{<mo\; stretchy="false">)</mo>}$ holds with $\Psi \mathrm{<mo\; stretchy="false">(</mo>}s\mathrm{<mo>;</mo>}\alpha \mathrm{<mo\; stretchy="false">)</mo>}$as a mixing cdf as before;
4. $\mathcal{L}\mathrm{<mo\; stretchy="false">(</mo>}Z\mathrm{<mo\; stretchy="false">)</mo>}\text{}\in \text{}{\text{S}}_n^{\alpha }\mathrm{<mo\; stretchy="false">(</mo>}\delta \mathrm{,}\sum \mathrm{<mo\; stretchy="false">)</mo>}$ is monotone unimodal with mode at $\text{d}\mathrm{,}$ for each $\alpha \text{}\in \text{}\mathrm{<mo\; stretchy="false">(</mo>0,}2)\mathrm{<mo>;</mo>}$
5. Let $T\mathrm{<mo\; stretchy="false">(</mo>}Z\mathrm{<mo\; stretchy="false">)</mo>}\text{}\mathrm{<mo>=</mo>}\text{}U\text{}\in \text{}{ℝ}^k$ be scale-invariant; then for $\mathcal{L}\mathrm{<mo\; stretchy="false">(</mo>}Z\mathrm{<mo\; stretchy="false">)</mo>}\text{}\in \text{}{\text{S}}_n^{\alpha }\mathrm{<mo\; stretchy="false">(</mo>}\delta \mathrm{,}\sum \mathrm{<mo\; stretchy="false">)</mo>,}$ the distribution $\mathcal{L}\mathrm{<mo\; stretchy="false">(</mo>}U\mathrm{<mo\; stretchy="false">)</mo>}$is identical to its normal-theory form under $\mathcal{L}\mathrm{<mo\; stretchy="false">(</mo>}Z\mathrm{<mo\; stretchy="false">)</mo>}\text{}\mathrm{<mo>=</mo>}\text{}{\text{N}}_n\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo>;</mo>}\delta \mathrm{,}\sum \mathrm{<mo\; stretchy="false">)</mo>.}$

Proof: Conclusion (i) is Theorem 6.5.4 of Press.²³ Conclusion (ii) invokes a result of Hartman et al.²⁴ namely, the process $\mathrm{<mo>\{</mo>}{Z}_t\mathrm{<mo>;</mo>}t\text{}\mathrm{<mo>=</mo>}\text{}\mathrm{1,2,}\dots \mathrm{<mo>\}</mo>}$is spherically invariant if and only if, for each n and$Z\text{}\mathrm{<mo>=</mo>}\text{}\mathrm{<mo\; stretchy="false">[</mo>}{Z}_1\mathrm{,}\dots \mathrm{,}{Z}_n\mathrm{<mo\; stretchy="false">]</mo>,}$the chf ${\varphi }_Z\text{}\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}$ is a scale mixture of spherical Gaussian s on ${ℝ}^n\mathrm{,}$ to give conclusion (ii) on transforming from spherical to elliptical symmetry. To continue, $f_Z\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo>}\text{}\mathrm{<mo>=</mo>}\text{}{\mathrm{<mo\; stretchy="false">(</mo>2}\pi \mathrm{<mo\; stretchy="false">)</mo>}}^{-n}{\displaystyle {\int }_{{ℝ}^n}}{e}^{-it\mathrm{<mo>\text{'}</mo>}z}{\varphi }_Z\text{}\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\Lambda \mathrm{<mo\; stretchy="false">(</mo>}dt\mathrm{<mo\; stretchy="false">)</mo>}$is the standard inversion formula from s to densities in ${ℝ}^n$ with $\Lambda \mathrm{<mo\; stretchy="false">(</mo>}\cdot \mathrm{<mo\; stretchy="false">)</mo>}$ as Lebesgue measure, so that from conclusion (ii) we recover

$f_n^{\alpha }\mathrm{<mo\; stretchy="false">(</mo>}\mu \mathrm{<mo>;</mo>}\delta \mathrm{,}{I}_n\mathrm{<mo\; stretchy="false">)</mo>}\text{}\mathrm{<mo>=</mo>}\text{}\frac{1}{{\mathrm{<mo\; stretchy="false">(</mo>2}\pi \mathrm{<mo\; stretchy="false">)</mo>}}^n}{\displaystyle {\int }_{{ℝ}^k}}{e}^{-it\mathrm{<mo>\text{'}</mo>}x}{\displaystyle {\int }_0^{\infty }}{e}^{it\mathrm{<mo>\text{'}</mo>}\delta -t\mathrm{<mo>\text{'}</mo>}t\mathrm{<mo>/</mo>2}s}d\Psi \mathrm{<mo\; stretchy="false">(</mo>}s\mathrm{<mo>;</mo>}\alpha \mathrm{<mo\; stretchy="false">)</mo>}\Lambda \mathrm{<mo\; stretchy="false">(</mo>}dt\mathrm{<mo\; stretchy="false">)</mo>.}$(1)

Reversing the order of integration inverts the Gaussian chf to give conclusion (iii). Conclusion (iv) follows as in Wolfe²⁵ conjunction with conclusion (iii). Finally observe from conclusion (iii), with ${\displaystyle {\int }_0^{\infty }}{g}_n\mathrm{<mo\; stretchy="false">(</mo>}Z\mathrm{<mo>;</mo>}\delta \mathrm{,}\sum \mathrm{<mo>/</mo>}s\mathrm{<mo\; stretchy="false">)</mo>}d\Psi \mathrm{<mo\; stretchy="false">(</mo>}s\mathrm{<mo>;</mo>}\alpha \mathrm{<mo\; stretchy="false">)</mo>,}$ that the change of variables $Z\text{}\to \text{}U\text{}\mathrm{<mo>=</mo>}\text{}T\mathrm{<mo\; stretchy="false">(</mo>}Z\mathrm{<mo\; stretchy="false">)</mo>}$ behind the integral is independent of $\Psi \mathrm{<mo\; stretchy="false">(</mo>}s\mathrm{<mo>;</mo>}\alpha \mathrm{<mo\; stretchy="false">)</mo>}$ since $T\mathrm{<mo\; stretchy="false">(</mo>}Z\mathrm{<mo\; stretchy="false">)</mo>}$ is scale-invariant independently of s to give conclusion (v).

It remains to reconsider degrees of association in $\text{S}\alpha \text{S}$ distributions, as distinct from the classical second-moment correlation parameters $\mathrm{<mo>\{</mo>}{\rho }_{ij}\text{}\mathrm{<mo>=</mo>}{\sigma }_{ij}\mathrm{<mo>/</mo><mo\; stretchy="false">(</mo>}{\sigma }_{ii}{\sigma }_{jj}{\mathrm{<mo\; stretchy="false">)</mo>}}^{\frac{1}{2}}\mathrm{<mo>\}</mo>.}$ For $\mathcal{L}\mathrm{<mo\; stretchy="false">(</mo>}Z\mathrm{<mo\; stretchy="false">)</mo>}\text{}\in \text{}{\text{S}}_k^{\alpha }\mathrm{<mo\; stretchy="false">(</mo>}\delta \mathrm{,}\sum \mathrm{<mo\; stretchy="false">)</mo>}$ with $\alpha \text{}\mathrm{<mo><</mo>}\text{}\mathrm{2,}$ the elements of s serve instead as scale parameters, since $U\text{}\mathrm{<mo>=</mo>}\text{}{\sum }^{-\frac{1}{2}}Z$ and $U\mathrm{<mo>\text{'}</mo>}U\text{}\mathrm{<mo>=</mo>}\text{}Z\mathrm{<mo>\text{'}</mo>}{\sum }^{-1}Z$are dimensionless. As to whether $\mathrm{<mo>\{</mo>}{\rho }_{ij}\mathrm{<mo>\}</mo>}$ again might quantify associations for $\alpha \text{}\mathrm{<mo><</mo>}\text{}\mathrm{2,}$ a definitive answer is supplied in the following.

Lemma 1 Let $\mathcal{L}\mathrm{<mo\; stretchy="false">(</mo>}Z\mathrm{<mo\; stretchy="false">)</mo>}\text{}\in \text{}{\text{S}}_n^{\alpha }\mathrm{<mo\; stretchy="false">(</mo>}\delta \mathrm{,}\sum \mathrm{<mo\; stretchy="false">)</mo>.}$ For , the parameters $\mathrm{<mo>\{</mo>}{\rho }_{ij}\text{}\mathrm{<mo>=</mo>}\text{}{\sigma }_{ij}\mathrm{<mo>/</mo><mo\; stretchy="false">(</mo>}{\sigma }_{ii}{\sigma }_{jj}{\mathrm{<mo\; stretchy="false">)</mo>}}^{\frac{1}{2}}\mathrm{<mo>\}</mo>}$serve to quantify degrees of association between $\mathrm{<mo\; stretchy="false">(</mo>}{Z}_i\mathrm{,}{Z}_j\mathrm{<mo\; stretchy="false">)</mo>,}$the extent of their association increasing with

Proof: It suffices to consider $\mathrm{<mo\; stretchy="false">(</mo>}{Z}_1\mathrm{,}{Z}_2\mathrm{<mo\; stretchy="false">)</mo>}$centered at $\mathrm{<mo\; stretchy="false">(</mo>0,}\mathrm{0<mo\; stretchy="false">)</mo>}$ with $\text{S}\text{}\mathrm{<mo>=</mo>}\text{}\left[\begin{array}{cc}1& \rho \\ \rho & 1\end{array}\right]\mathrm{.}$On taking $U\text{}\mathrm{<mo>=</mo>}\text{}\mathrm{<mo\; stretchy="false">(</mo>}{Z}_1\text{}-\text{}{Z}_2\mathrm{<mo\; stretchy="false">)</mo>,}$ $\mathcal{L}\mathrm{<mo\; stretchy="false">(</mo>}U\mathrm{<mo\; stretchy="false">)</mo>}$ clearly is symmetric about 0 with scale parameter ${\sigma }_U\text{}\mathrm{<mo>=</mo>}\text{}\mathrm{2<mo\; stretchy="false">(</mo>1}\text{}-\text{}\rho \mathrm{<mo\; stretchy="false">)</mo>.}$A result of Fefferman et al.²⁶ shows for each $c\text{}\mathrm{<mo>></mo>}\text{}0$that $P\mathrm{<mo\; stretchy="false">(</mo>}U\text{}\in \text{}\mathrm{<mo\; stretchy="false">(</mo>}-c\mathrm{,}c\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}$ is decreasing in ${\sigma }_U$thus increasing in $\rho \mathrm{.}$Equivalently, $P\mathrm{<mo\; stretchy="false">(</mo><mo>|</mo>}{Z}_1\text{}-\text{}{Z}_2\mathrm{<mo>|</mo>}\text{}\le c\mathrm{<mo\; stretchy="false">)</mo>}\text{}\uparrow \text{}1$as $\rho \text{}\uparrow \text{}\mathrm{1,}$identifying the sense in which $\mathrm{<mo\; stretchy="false">(</mo>}{Z}_1\mathrm{,}{Z}_2\mathrm{<mo\; stretchy="false">)</mo>}$become increasingly indistinguishable, thus associated, with increasing values of

Definition 3 For $\mathcal{L}\mathrm{<mo\; stretchy="false">(</mo>}Z\mathrm{<mo\; stretchy="false">)</mo>}\text{}\in \text{}{\text{S}}_n^{\alpha }\mathrm{<mo\; stretchy="false">(</mo>}\delta \mathrm{,}\text{S}\mathrm{<mo\; stretchy="false">)</mo>}$with $\alpha \text{}\mathrm{<mo><</mo>}\text{}\mathrm{2,}$the entities $\mathrm{<mo>\{</mo>}{\rho }_{ij}\text{}\mathrm{<mo>=</mo>}\text{}{\sigma }_{ij}\mathrm{<mo>/</mo><mo\; stretchy="false">(</mo>}{\sigma }_{ii}{\sigma }_{jj}{\mathrm{<mo\; stretchy="false">)</mo>}}^{\frac{1}{2}}\mathrm{<mo>\}</mo>}$are called pseudo–correlation, specifically, $\alpha$-association parameters

The principal findings

Take $\mathcal{L}\mathrm{<mo\; stretchy="false">(</mo>}Y\mathrm{<mo\; stretchy="false">)</mo>}\text{}\in \text{}{\text{S}}_n^{\alpha }\mathrm{<mo\; stretchy="false">(</mo>}X\beta \mathrm{,}{\sigma }^2{I}_n\mathrm{<mo\; stretchy="false">)</mo>}$with $\mathrm{<mo\; stretchy="false">(</mo>}X\beta \mathrm{,}{\sigma }^2{I}_n\mathrm{<mo\; stretchy="false">)</mo>}$as centering and scale parameters, where $\mathrm{<mo>\{</mo>}Y\text{}\in \text{}{ℝ}^n\mathrm{,}X\text{}\in \text{}{\mathbb{F}}_{\text{}n\text{}\times \text{}k}\mathrm{,}\beta \text{}\in \text{}{ℝ}^k\mathrm{<mo>\}</mo>.}$ $O\text{}L\text{}S$solutions $\stackrel{⌢}{\beta }\text{}\mathrm{<mo>=</mo>}\text{}{\mathrm{<mo\; stretchy="false">(</mo>}X\mathrm{<mo>\text{'}</mo>}\text{}X\mathrm{<mo\; stretchy="false">)</mo>}}^{\text{}-1}X\mathrm{<mo>\text{'}</mo>}Y$, as minimally dispersed unbiased linear estimates, are available here only for $\alpha \text{}\mathrm{<mo>=</mo>}\text{}\mathrm{2,}$whereas alternative moment criteria necessarily are subject to moment constraints. Specifically, for scalars $\mathrm{<mo\; stretchy="false">(</mo>}\stackrel{⌢}{\theta }\mathrm{,}\theta \mathrm{<mo\; stretchy="false">)</mo>}\text{}\in \text{}{ℝ}^1$ under loss $L\mathrm{<mo\; stretchy="false">(</mo>}\stackrel{⌢}{\theta }\mathrm{,}\theta \mathrm{<mo\; stretchy="false">)</mo>}\text{}\mathrm{<mo>=</mo>}\text{}\mathrm{<mo>|</mo>}\stackrel{⌢}{\theta }-\text{}\theta \mathrm{<mo>|</mo>,}$the risk $R\mathrm{<mo\; stretchy="false">(</mo>}\stackrel{⌢}{\theta }\mathrm{<mo\; stretchy="false">)</mo>}\text{}\mathrm{<mo>=</mo>}\text{}\text{E}\mathrm{<mo\; stretchy="false">[</mo>}L\mathrm{<mo\; stretchy="false">(</mo>}\stackrel{⌢}{\theta }\mathrm{,}\theta \mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">]</mo>}$ is undefined for $\alpha \text{}\mathrm{<mo><</mo>}\text{}1$ as for Cauchy errors at $\alpha \text{}\mathrm{<mo>=</mo>}\text{}1.$ Moreover, risk functions $\mathrm{<mo>\{</mo>}R\mathrm{<mo\; stretchy="false">(</mo>}\stackrel{⌢}{\theta }\mathrm{<mo\; stretchy="false">)</mo>}\text{}\mathrm{<mo>=</mo>}\text{}\text{E}\mathrm{<mo\; stretchy="false">(</mo><mo>|</mo>}\stackrel{⌢}{\theta }-\text{}\theta {\mathrm{<mo>|</mo>}}^{\kappa }\mathrm{<mo\; stretchy="false">)</mo><mo>\}</mo>}$ are defined but concave for $\mathrm{<mo>\{</mo>}\kappa \text{}\mathrm{<mo><</mo>}\text{}\alpha \text{}\mathrm{<mo><</mo>}\text{}\mathrm{1<mo>\}</mo>,}$ and for $\mathrm{<mo>\{</mo>1}\text{}\mathrm{<mo><</mo>}\text{}\kappa \text{}\mathrm{<mo><</mo>}\text{}\alpha \text{}\le \text{}\mathrm{2<mo>\}</mo>}$ are convex, at issue in attaining global optima. Versions of these apply also for vector parameters; however, minimal risk estimation would require not only knowledge regarding $\alpha \mathrm{,}$but also optimizing algorithms. Instead we seek what might be salvaged from classical linear models under the constraints of $\text{S}\alpha \text{S}$errors. In addition, portions of our findings extend beyond Gauss–Markov theory and $O\text{}L\text{}S$ to include the much larger class of equivariant estimators.

Definition 4 An estimator $\delta \mathrm{<mo\; stretchy="false">(</mo>}Y\mathrm{<mo\; stretchy="false">)</mo>}$ for $\beta \in \text{}{ℝ}^k$ is translation –equivariant if for $\mathrm{<mo>\{</mo>}Y\text{}\to \text{}Y\text{}+\text{}Xb\mathrm{<mo>\}</mo>,}$ then $\mathrm{<mo>\{</mo>}\delta \mathrm{<mo\; stretchy="false">(</mo>}Y\text{}+\text{}Xb\mathrm{<mo\; stretchy="false">)</mo>}\text{}\mathrm{<mo>=</mo>}\text{}\delta \mathrm{<mo\; stretchy="false">(</mo>}Y\mathrm{<mo\; stretchy="false">)</mo>}\text{}+\text{}b\mathrm{<mo>\}</mo>}$ for every $b\text{}\in \text{}{ℝ}^k\mathrm{.}$

On taking $P\text{}\mathrm{<mo>=</mo>}\text{}\mathrm{<mo\; stretchy="false">[</mo>}{I}_n\text{}-\text{}X{\mathrm{<mo\; stretchy="false">(</mo>}X\mathrm{<mo>\text{'}</mo>}\text{}X\mathrm{<mo\; stretchy="false">)</mo>}}^{\text{}-1}X\mathrm{<mo>\text{'}</mo><mo\; stretchy="false">]</mo>,}$ the elements of $e\text{}\mathrm{<mo>=</mo>}\text{}PY$ comprise the observed residuals and $S^2\text{}\mathrm{<mo>=</mo>}\text{}e\mathrm{<mo>\text{'}</mo>}e\mathrm{<mo>/</mo><mo\; stretchy="false">(</mo>}n\text{}-\text{}k\mathrm{<mo\; stretchy="false">)</mo>}$ the residual mean square. Normal–theory tests for $H_0\text{}\mathrm{<mo>:</mo>}\text{}\beta \text{}\mathrm{<mo>=</mo>}\text{}{\beta }_0$against $H_1\text{}\mathrm{<mo>:</mo>}\text{}\beta \text{}\ne \text{}{\beta }_0$utilize $F\text{}\mathrm{<mo>=</mo>}\text{}\mathrm{<mo\; stretchy="false">(</mo>}\stackrel{⌢}{\beta }\text{}-\text{}{\beta }_0{)}^{\prime }X\mathrm{<mo>\text{'}</mo>}\text{}X\mathrm{<mo\; stretchy="false">(</mo>}\stackrel{⌢}{\beta }\text{}-\text{}{\beta }_0\mathrm{<mo\; stretchy="false">)</mo><mo>/</mo>}{S}^2$ having the distribution $F\mathrm{<mo\; stretchy="false">(</mo>}u\mathrm{<mo>;</mo>}k\mathrm{,}n\text{}-\text{}k\mathrm{,}\lambda \mathrm{<mo\; stretchy="false">)</mo>}$ with $\lambda \text{}\mathrm{<mo>=</mo>}\text{}\mathrm{<mo\; stretchy="false">(</mo>}\beta -\text{}{\beta }_0{)}^{\prime }X\mathrm{<mo>\text{'}</mo>}\text{}X\mathrm{<mo\; stretchy="false">(</mo>}\beta \text{}-\text{}{\beta }_0\mathrm{<mo\; stretchy="false">)</mo><mo>/</mo>}{\sigma }^2\mathrm{.}$ We proceed to examine essential properties of ${\text{S}}_n^{\alpha }\mathrm{<mo\; stretchy="false">(</mo>}X\beta \mathrm{,}{\sigma }^2{I}_n\mathrm{<mo\; stretchy="false">)</mo>}$as $\alpha$ranges over $\mathrm{<mo\; stretchy="false">(</mo>}\mathrm{0,}2)\mathrm{,}$where some expressions simplify on taking ${\sigma }^2\text{}\mathrm{<mo>=</mo>}\text{}\mathrm{1,}$then reinstating ${\sigma }^2$as needed. The following properties are fundamental.

Theorem 2 Given $\mathcal{L}\mathrm{<mo\; stretchy="false">(</mo>}Y\mathrm{<mo\; stretchy="false">)</mo>}\text{}\mathrm{<mo>=</mo>}\text{}{\text{S}}_n^{\alpha }\mathrm{<mo\; stretchy="false">(</mo>}X\beta \mathrm{,}{\sigma }^2{I}_n\mathrm{<mo\; stretchy="false">)</mo>,}$consider $\mathrm{<mo\; stretchy="false">[</mo>}\stackrel{⌢}{\beta }\mathrm{,}e\mathrm{<mo\; stretchy="false">]</mo>}$with $e\text{}\mathrm{<mo>=</mo>}\text{}PY$as the residual vector, and $U\text{}\mathrm{<mo>=</mo>}\text{}\mathrm{<mo\; stretchy="false">(</mo>}n\text{}-\text{}k\mathrm{<mo\; stretchy="false">)</mo>}{S}^2\mathrm{<mo>/</mo>}{\sigma }^2\mathrm{.}$Then

(i) $\mathcal{L}\mathrm{<mo\; stretchy="false">(</mo>}\stackrel{⌢}{\beta }\mathrm{,}e\mathrm{<mo\; stretchy="false">)</mo>}\text{}\mathrm{<mo>=</mo>}\text{}{\text{S}}_{n\text{}+\text{}k}^{\alpha }\mathrm{<mo\; stretchy="false">(</mo><mo\; stretchy="false">[</mo>}\beta \mathrm{,}0\mathrm{<mo\; stretchy="false">]</mo>,}\text{S}\mathrm{<mo\; stretchy="false">)</mo>,}$with $\sum \text{}\mathrm{<mo>=</mo>}\text{}{\sigma }^{2}Diag\mathrm{<mo\; stretchy="false">(</mo><mo\; stretchy="false">(</mo>}X\mathrm{<mo>\text{'}</mo>}\text{}X{\mathrm{<mo\; stretchy="false">)</mo>}}^{\text{}-1}\mathrm{,}P\mathrm{<mo\; stretchy="false">)</mo>,}$a distribution on ${ℝ}^{n\text{}+\text{}k}$of rank s

(ii) The marginal’s are $\mathcal{L}\mathrm{<mo\; stretchy="false">(</mo>}\stackrel{⌢}{\beta }\mathrm{<mo\; stretchy="false">)</mo>}\text{}\mathrm{<mo>=</mo>}\text{}{\text{S}}_k^{\alpha }\mathrm{<mo\; stretchy="false">(</mo>}\beta \mathrm{,}{\sigma }^2{\mathrm{<mo\; stretchy="false">(</mo>}X\mathrm{<mo>\text{'}</mo>}\text{}X\mathrm{<mo\; stretchy="false">)</mo>}}^{\text{}-1}\mathrm{<mo\; stretchy="false">)</mo>}$centered at $\beta$with scale parameters ${\sigma }^2{\mathrm{<mo\; stretchy="false">(</mo>}X\mathrm{<mo>\text{'}</mo>}\text{}X\mathrm{<mo\; stretchy="false">)</mo>}}^{\text{}-1}\mathrm{,}$and

(iii) $\mathcal{L}\mathrm{<mo\; stretchy="false">(</mo>}e\mathrm{<mo\; stretchy="false">)</mo>}\text{}\mathrm{<mo>=</mo>}\text{}{\text{S}}_n^{\alpha }\mathrm{<mo\; stretchy="false">(</mo>}0\mathrm{,}{\sigma }^{2}P\mathrm{<mo\; stretchy="false">)</mo>}$on ${ℝ}^n$of rank $n\text{}-\text{}k$centered at 0with scale parameters ${\sigma }^{2}P\mathrm{<mo>;</mo>}$

(iv) $U\text{}\mathrm{<mo>=</mo>}\text{}\mathrm{<mo\; stretchy="false">(</mo>}n\text{}-\text{}k\mathrm{<mo\; stretchy="false">)</mo>}{S}^2\mathrm{<mo>/</mo>}{\sigma }^2$has density $f\mathrm{<mo\; stretchy="false">(</mo>}u\mathrm{<mo>;</mo>}\nu \mathrm{,}\alpha \mathrm{<mo\; stretchy="false">)</mo>}\text{}\mathrm{<mo>=</mo>}\text{}{\displaystyle {\int }_0^{\infty }}h\mathrm{<mo\; stretchy="false">(</mo>}u\mathrm{<mo>;</mo>}\nu \mathrm{,}s\mathrm{<mo\; stretchy="false">)</mo>}d\Psi \mathrm{<mo\; stretchy="false">(</mo>}s\mathrm{<mo>;</mo>}\alpha \mathrm{<mo\; stretchy="false">)</mo>}$with $h\mathrm{<mo\; stretchy="false">(</mo>}u\mathrm{<mo>;</mo>}\nu \mathrm{,}s\mathrm{<mo\; stretchy="false">)</mo>}$as the central chi–squared density on $\nu \text{}\mathrm{<mo>=</mo>}\text{}\mathrm{<mo\; stretchy="false">(</mo>}n\text{}-\text{}k\mathrm{<mo\; stretchy="false">)</mo>}$degrees of freedom, scaled by S, and with $\Psi \mathrm{<mo\; stretchy="false">(</mo>}s\mathrm{<mo>;</mo>}\alpha \mathrm{<mo\; stretchy="false">)</mo>}$as a mixing distribution.

Proof. Let $L\mathrm{<mo>\text{'}</mo>}\text{}\mathrm{<mo>=</mo>}\text{}{\mathrm{<mo\; stretchy="false">(</mo>}X\mathrm{<mo>\text{'}</mo>}\text{}X\mathrm{<mo\; stretchy="false">)</mo>}}^{\text{}-1}\text{}X\mathrm{<mo>\text{'}</mo>}$and $P\text{}\mathrm{<mo>=</mo>}\text{}\left[I_n\text{}-\text{}X{\mathrm{<mo\; stretchy="false">(</mo>}X\mathrm{<mo>\text{'}</mo>}\text{}X\mathrm{<mo\; stretchy="false">)</mo>}}^{\text{}-1}\text{}X\mathrm{<mo>\text{'}</mo>}\right]$to project onto the error space, so that $G\text{}\mathrm{<mo>=</mo>}\text{}\mathrm{<mo\; stretchy="false">[</mo>}L\mathrm{,}P\mathrm{<mo\; stretchy="false">]</mo>}$operates on y to give

$Z\mathrm{<mo>=</mo>}G\mathrm{<mo>\text{'}</mo>}Y\mathrm{<mo>=</mo>}\left[\begin{array}{c}\stackrel{⌢}{\beta }\\ e\end{array}\right]\mathrm{<mo>=</mo>}\left[\begin{array}{c}L\mathrm{<mo>\text{'}</mo>}\\ P\mathrm{<mo>\text{'}</mo>}\end{array}\right]Y\in {ℝ}^{n\text{}+\text{}k}andG\mathrm{<mo>\text{'}</mo>}G\mathrm{<mo>=</mo>}\left[\begin{array}{cc}{\mathrm{<mo\; stretchy="false">(</mo>}X\mathrm{<mo>\text{'}</mo>}\text{}X\mathrm{<mo\; stretchy="false">)</mo>}}^{\text{}-1}& 0\\ 0& P\end{array}\right]\mathrm{,}$ (2)

 the latter of order $\mathrm{<mo\; stretchy="false">[</mo><mo\; stretchy="false">(</mo>}n\text{}+\text{}k\mathrm{<mo\; stretchy="false">)</mo>}\text{}\times \text{}\mathrm{<mo\; stretchy="false">(</mo>}n\text{}+\text{}k\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">]</mo>}$ and rank $n\mathrm{.}$The chf with argument $s\mathrm{<mo>\text{'}</mo>}\text{}\mathrm{<mo>=</mo>}\text{}\mathrm{<mo\; stretchy="false">[</mo>}{s}_1\mathrm{,}\dots \mathrm{,}{s}_{n\text{}+\text{}k}\mathrm{<mo\; stretchy="false">]</mo>}$is $\text{E}\mathrm{<mo\; stretchy="false">[</mo>}\exp \mathrm{<mo\; stretchy="false">(</mo>}\iota s\mathrm{<mo>\text{'}</mo>}Z\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">]</mo>}\text{}\mathrm{<mo>=</mo>}\text{}\text{E}\mathrm{<mo\; stretchy="false">[</mo>}\exp \mathrm{<mo\; stretchy="false">(</mo>}\iota s\mathrm{<mo>\text{'}</mo>}G\mathrm{<mo>\text{'}</mo>}Y\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">]</mo>}\text{}\mathrm{<mo>=</mo>}\text{}\text{E}\mathrm{<mo\; stretchy="false">[</mo>}\exp \mathrm{<mo\; stretchy="false">(</mo>}\iota v\mathrm{<mo>\text{'}</mo>}Y\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">]</mo>}$= ${\varphi }_Y\text{}\mathrm{<mo\; stretchy="false">(</mo>}v\mathrm{<mo\; stretchy="false">)</mo>}$with argument $v\text{}\mathrm{<mo>=</mo>}\text{}Gs$ replacing t, to give conclusion (i). Next partition $s\mathrm{<mo>\text{'}</mo>}\text{}\mathrm{<mo>=</mo>}\text{}\mathrm{<mo\; stretchy="false">[</mo>}{s}_1\mathrm{<mo>\text{'}</mo>,}{s}_2\mathrm{<mo>\text{'}</mo><mo\; stretchy="false">]</mo>}$with $s_1\mathrm{<mo>\text{'}</mo>}\text{}\mathrm{<mo>=</mo>}\text{}\mathrm{<mo\; stretchy="false">[</mo>}{s}_1\mathrm{,}\dots \mathrm{,}{s}_k\mathrm{<mo\; stretchy="false">]</mo>,}$ to obtain = ${\varphi }_Z\text{}\mathrm{<mo\; stretchy="false">(</mo>}s\mathrm{<mo\; stretchy="false">)</mo>}\text{}\mathrm{<mo>=</mo>}\text{}\exp \mathrm{<mo\; stretchy="false">[</mo>}\iota s\mathrm{<mo>\text{'}</mo>}G\mathrm{<mo>\text{'}</mo>}\text{}X\beta -\frac{1}{2}{\mathrm{<mo\; stretchy="false">(</mo>}s\mathrm{<mo>\text{'}</mo>}G\mathrm{<mo>\text{'}</mo>}Gs\mathrm{<mo\; stretchy="false">)</mo>}}^{\frac{\alpha }{2}}\mathrm{<mo\; stretchy="false">]</mo>}$=$\exp \mathrm{<mo\; stretchy="false">[</mo>}\iota s_1\mathrm{<mo>\text{'}</mo>}\beta -\frac{1}{2}{\mathrm{<mo\; stretchy="false">(</mo>}{s}_1\mathrm{<mo>\text{'}</mo><mo\; stretchy="false">(</mo>}X\mathrm{<mo>\text{'}</mo>}\text{}X{\mathrm{<mo\; stretchy="false">)</mo>}}^{\text{}-1}{s}_1+s_2\mathrm{<mo>\text{'}</mo>}P{s}_2\mathrm{<mo\; stretchy="false">)</mo>}}^{\frac{\alpha }{2}}\mathrm{<mo\; stretchy="false">]</mo>.}$ The marginal s of $\stackrel{⌢}{\beta }$ and e follow on setting $s_2\text{}\mathrm{<mo>=</mo>}\text{}0\mathrm{,}$ then $s_1\text{}\mathrm{<mo>=</mo>}\text{}0$in succession, to give conclusions (ii) and (iii). Conclusion (iv) attributes to Hartman et al.²⁴ through Theorem 1. Specifically, a change of variables $u\text{}\to \text{}e\text{}\to \text{}e\mathrm{<mo>\text{'}</mo>}e\text{}\mathrm{<mo>=</mo>}\text{}\mathrm{<mo\; stretchy="false">(</mo>}n\text{}-\text{}k\mathrm{<mo\; stretchy="false">)</mo>}{S}^2$ behind the integral on the right of Theorem 1(iii) gives the conditional density for $\mathcal{L}\mathrm{<mo\; stretchy="false">(</mo><mo\; stretchy="false">(</mo>}n\text{}-\text{}k\mathrm{<mo\; stretchy="false">)</mo>}{S}^2\mathrm{<mo>|</mo>}s\mathrm{<mo\; stretchy="false">)</mo>,}$ namely the scaled chi–squared density $h\mathrm{<mo\; stretchy="false">(</mo>}u\mathrm{<mo>;</mo>}\nu \mathrm{,}s\mathrm{<mo\; stretchy="false">)</mo>}$depending on s so that integrating with respect to $d\Psi \mathrm{<mo\; stretchy="false">(</mo>}s\mathrm{<mo>;</mo>}\alpha \mathrm{<mo\; stretchy="false">)</mo>}$ gives conclusion (iv).

Remark 3 That $\text{S}\text{}\mathrm{<mo>=</mo>}\text{}{\sigma }^{2}Diag\mathrm{<mo\; stretchy="false">(</mo>}X\mathrm{<mo>\text{'}</mo>}\text{}X\mathrm{,}P\mathrm{<mo\; stretchy="false">)</mo>}$ is block–diagonal in conclusion (i), assures under $\text{S}\alpha \text{S}$ errors that $\mathrm{<mo\; stretchy="false">(</mo>}\stackrel{⌢}{\beta }\mathrm{,}e\mathrm{<mo\; stretchy="false">)</mo>}$are $\alpha$–unassociated as in Definition 3, well known to be mutually uncorrelated under second moments.

It remains to reexamine topics in inference under  errors. The following are germane.

Definition 5 An estimator $\stackrel{⌢}{\theta }$ for $\theta \text{}\in \text{}{ℝ}^k$ is said to be linearly median unbiased if and only if the median $\text{med}\mathrm{<mo\; stretchy="false">(</mo>}a\mathrm{<mo>\text{'}</mo>}\stackrel{⌢}{\theta }\mathrm{<mo\; stretchy="false">)</mo>}\text{}\mathrm{<mo>=</mo>}\text{}a\mathrm{<mo>\text{'}</mo>}\theta$for each $a\text{}\in \text{}{ℝ}^k\mathrm{<mo>;</mo>}$ and to be modal unbiased provided that the mode

Definition 6 An estimator $\stackrel{⌢}{\theta }$for $\theta$is said to be more concentrated about $\theta$than provided that $\stackrel{⌢}{\theta }$ $P\mathrm{<mo\; stretchy="false">(</mo><mo\; stretchy="false">(</mo>}\widehat{\theta }\text{}-\text{}\theta \mathrm{<mo\; stretchy="false">)</mo>}\text{}\in \text{}{\text{C}}_0\mathrm{<mo\; stretchy="false">)</mo>}\ge P\mathrm{<mo\; stretchy="false">(</mo><mo\; stretchy="false">(</mo>}\tilde{\theta }\text{}-\text{}\theta \mathrm{<mo\; stretchy="false">)</mo>}\text{}\in \text{}{\text{C}}_0\mathrm{<mo\; stretchy="false">)</mo>}$for every convex set ${\text{C}}_0$ in ${ℝ}^k$ symmetric under reflection about $0\text{}\in \text{}{ℝ}^k\mathrm{.}$

 Essential properties under $\text{S}\alpha \text{S}$ errors include the following.

Theorem 3 For $\mathcal{L}\mathrm{<mo\; stretchy="false">(</mo>}Y\mathrm{<mo\; stretchy="false">)</mo>}\text{}\mathrm{<mo>=</mo>}\text{}{\text{S}}_n^{\alpha }\mathrm{<mo\; stretchy="false">(</mo>}X\beta \mathrm{,}{\sigma }^2{I}_n\mathrm{<mo\; stretchy="false">)</mo>,}$consider properties of the $O\text{}L\text{}S$solutions $\stackrel{⌢}{\beta }\mathrm{<mo>=</mo>}\text{}{\mathrm{<mo\; stretchy="false">(</mo>}X\mathrm{<mo>\text{'}</mo>}\text{}X\mathrm{<mo\; stretchy="false">)</mo>}}^{\text{}-1}X\mathrm{<mo>\text{'}</mo>}Y\mathrm{,}$and of the equivariant estimators $\stackrel{⌢}{\beta }\mathrm{<mo>=</mo>}\text{}\delta \mathrm{<mo\; stretchy="false">(</mo>}Y\mathrm{<mo\; stretchy="false">)</mo>}$ of Definition 4.

1. $\stackrel{⌢}{\beta }$ is unbiased for $\beta$for each $\mathrm{<mo>\{</mo>1}\text{}\mathrm{<mo><</mo>}\text{}\alpha \text{}\le \text{}\mathrm{2<mo>\}</mo><mo>;</mo>}$
2. $\stackrel{⌢}{\beta }$is linearly median unbiased for $\beta \mathrm{<mo>;</mo>}$
3. $\stackrel{⌢}{\beta }$is most concentrated about $\beta \mathrm{<mo>;</mo>}$among all median–unbiased linear estimators;
4. $\stackrel{⌢}{\beta }$ is modal unbiased for $\beta \mathrm{<mo>;</mo>}$
5. ${\stackrel{⌢}{\beta }}_{\text{}N}$is consistent for $\beta \mathrm{<mo>;</mo>}$ in a sequence of identical but dependent experiments$\mathrm{<mo>\{</mo>}{Y}_{\text{}i}\text{}\mathrm{<mo>=</mo>}\text{}X\beta \text{}+\text{}{\text{e}}_i\mathrm{<mo>;</mo>}i\mathrm{<mo>=</mo>1,2,}\dots \mathrm{,}N\mathrm{<mo>\}</mo><mo>;</mo>}$
6. The null distribution of $F\text{}\mathrm{<mo>=</mo>}\text{}\mathrm{<mo\; stretchy="false">(</mo>}\stackrel{⌢}{\beta }\text{}-\text{}{\beta }_0{)}^{\prime }X\mathrm{<mo>\text{'}</mo>}\text{}X\mathrm{<mo\; stretchy="false">(</mo>}\stackrel{⌢}{\beta }\text{}-\text{}{\beta }_0\mathrm{<mo\; stretchy="false">)</mo><mo>/</mo>}{S}^2$has exactly its normal–theory form; the power increases with increasing $\lambda \text{}\mathrm{<mo>=</mo>}\text{}\mathrm{<mo\; stretchy="false">(</mo>}\beta -\text{}{\beta }_0{)}^{\prime }X\mathrm{<mo>\text{'}</mo>}\text{}X\mathrm{<mo\; stretchy="false">(</mo>}\beta \text{}-\text{}{\beta }_0\mathrm{<mo\; stretchy="false">)</mo><mo>/</mo>}{\sigma }^2\mathrm{<mo>;</mo>}$and such tests are unbiased;
7. $\stackrel{⌢}{\beta }$is most concentrated about $\beta$among all modal–unbiased linear estimators.

$\stackrel{⌢}{\beta }$is most concentrated about $\beta$among all equivariant estimators $\stackrel{⌢}{\beta }\text{}\mathrm{<mo>=</mo>}\text{}\delta \mathrm{<mo\; stretchy="false">(</mo>}Y\mathrm{<mo\; stretchy="false">)</mo>.}$

Proof. Conclusions (i)–(vi) carry over from reference Jensen DR²⁷ without benefit of moments, regardless of membership in the $\text{S}\alpha \text{S}$class. To consider concentration properties of modal–unbiased estimators, begin with ${\varphi }_Y\text{}\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\text{}\mathrm{<mo>=</mo>}\text{}\exp \mathrm{<mo\; stretchy="false">[</mo>}\iota t\mathrm{<mo>\text{'}</mo>}X\beta -\frac{1}{2}{\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo>\text{'}</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}}^{\frac{\alpha }{2}}\mathrm{<mo\; stretchy="false">]</mo>,}$and consider $\tilde{\beta }\text{}\mathrm{<mo>=</mo>}L\mathrm{<mo>\text{'}</mo>}Y$with $L\mathrm{<mo>\text{'}</mo>}\text{}\mathrm{<mo>=</mo>}\text{}\mathrm{<mo\; stretchy="false">[</mo><mo\; stretchy="false">(</mo>}X\mathrm{<mo>\text{'}</mo>}\text{}X{\mathrm{<mo\; stretchy="false">)</mo>}}^{\text{}-1}X\mathrm{<mo>\text{'}</mo>,}G\mathrm{<mo>\text{'}</mo><mo\; stretchy="false">]</mo>,}$so that

${\varphi }_{\tilde{\beta }}\text{}\mathrm{<mo\; stretchy="false">(</mo>}s\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}\exp \mathrm{<mo\; stretchy="false">[</mo>}\iota s\mathrm{<mo>\text{'}</mo>}L\mathrm{<mo>\text{'}</mo>}X\beta -\frac{1}{2}{\mathrm{<mo\; stretchy="false">(</mo>}s\mathrm{<mo>\text{'}</mo>}L\mathrm{<mo>\text{'}</mo>}Ls\mathrm{<mo\; stretchy="false">)</mo>}}^{\frac{\alpha }{2}}\mathrm{<mo\; stretchy="false">]</mo><mo>;</mo>}$

$s\mathrm{<mo>\text{'}</mo>}L\mathrm{<mo>\text{'}</mo>}X\beta \mathrm{<mo>=</mo>}s\mathrm{<mo>\text{'}</mo><mo\; stretchy="false">[</mo><mo\; stretchy="false">(</mo>}X\mathrm{<mo>\text{'}</mo>}\text{}X{\mathrm{<mo\; stretchy="false">)</mo>}}^{\text{}-1}X\mathrm{<mo>\text{'}</mo>,}G\mathrm{<mo>\text{'}</mo><mo\; stretchy="false">]</mo>}X\beta \mathrm{.}$

 That $\stackrel{⌢}{\beta }$ should have mode at $\beta \mathrm{,}$ it is necessary that $s\mathrm{<mo>\text{'}</mo>}L\mathrm{<mo>\text{'}</mo>}X\beta \text{}\mathrm{<mo>=</mo>}\text{}s\mathrm{<mo>\text{'}</mo>}\beta \mathrm{,}$ i.e. $G\mathrm{<mo>\text{'}</mo>}X\text{}\mathrm{<mo>=</mo>}\text{}0\mathrm{.}$ accordingly, ${\varphi }_{\stackrel{⌢}{\beta }}\text{}\mathrm{<mo\; stretchy="false">(</mo>}s\mathrm{<mo\; stretchy="false">)</mo>}\text{}\mathrm{<mo>=</mo>}\text{}\exp \mathrm{<mo\; stretchy="false">[</mo>}\iota s\mathrm{<mo>\text{'}</mo>}\beta -\frac{1}{2}{\mathrm{<mo\; stretchy="false">[</mo>}s\mathrm{<mo>\text{'}</mo>}\Omega s\mathrm{<mo\; stretchy="false">]</mo>}}^{\frac{\alpha }{2}}\mathrm{<mo\; stretchy="false">]</mo>,}$ with $\Omega \text{}\mathrm{<mo>=</mo>}\text{}L\mathrm{<mo>\text{'}</mo>}L\text{}\mathrm{<mo>=</mo>}\text{}\mathrm{<mo\; stretchy="false">[</mo><mo\; stretchy="false">(</mo>}X\mathrm{<mo>\text{'}</mo>}\text{}X{\mathrm{<mo\; stretchy="false">)</mo>}}^{\text{}-1}\text{}+\text{}G\mathrm{<mo>\text{'}</mo>}G\mathrm{<mo\; stretchy="false">]</mo>.}$Clearly the matrix $\mathrm{<mo\; stretchy="false">[</mo>}L\mathrm{<mo>\text{'}</mo>}L\text{}-\text{}{\mathrm{<mo\; stretchy="false">(</mo>}X\mathrm{<mo>\text{'}</mo>}\text{}X\mathrm{<mo\; stretchy="false">)</mo>}}^{\text{}-1}\mathrm{<mo\; stretchy="false">]</mo>}\text{}\mathrm{<mo>=</mo>}\text{}G\mathrm{<mo>\text{'}</mo>}G$is positive semi definite, giving conclusion (vii) from Jensen.²⁸ Conclusion (viii) follows from Theorem 2.7 of Burk et al.²⁹ since  distributions are unimodal from Theorem 1(iv).

Spherical multivariate t errors on v degrees of freedom trace to Zellner³⁰ to include Cauchy errors at $\nu \text{}\mathrm{<mo>=</mo>}\text{}\mathrm{1,}$ equivalently, at $\alpha \text{}\mathrm{<mo>=</mo>}\text{}1$ in the class $\text{S}\alpha \text{S}\mathrm{.}$ Specializing from Theorem 1(ii), the spherical Cauchy chf is ${\varphi }_Z\text{}\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\text{}\mathrm{<mo>=</mo>}\text{}\exp \mathrm{<mo\; stretchy="false">[</mo>}it\mathrm{<mo>\text{'}</mo>}\text{d}\text{}-\text{}\frac{1}{2}{\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo>\text{'}</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}}^{\frac{1}{2}}\mathrm{<mo\; stretchy="false">]</mo>.}$ Recast in terms of linear inference, we have the following specialization of Theorems 1 and 2.

Corollary 1 Under the conditions of Theorems and 2, the following properties hold under spherical Cauchy errors.

1. The spherical Cauchy density on ${ℝ}^n$ at $\alpha \text{}\mathrm{<mo>=</mo>}\text{}1$ is

$$f_n^1\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo>;</mo>}\delta \mathrm{,}{I}_n\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}{\displaystyle {\int }_0^{\infty }}{g}_n\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo>;</mo>}\delta \mathrm{,}{s}^{-2}{I}_n\mathrm{<mo\; stretchy="false">)</mo>}d\Psi \mathrm{<mo\; stretchy="false">(</mo>}s\mathrm{<mo>;</mo>1<mo\; stretchy="false">)</mo>}$$

$$\mathrm{<mo>=</mo>}c\mathrm{<mo\; stretchy="false">(</mo>}n\mathrm{<mo\; stretchy="false">)</mo>}{\left[1+\mathrm{<mo\; stretchy="false">(</mo>}z\text{}-\text{}\delta {)}^{\prime }\mathrm{<mo\; stretchy="false">(</mo>}z\text{}-\text{}\delta \mathrm{<mo\; stretchy="false">)</mo>}\right]}^{-\frac{n+1}{2}}$$

$$c\mathrm{<mo\; stretchy="false">(</mo>}n\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}\Gamma \mathrm{<mo\; stretchy="false">(</mo>}\frac{n\text{}+\text{}1}{2}\mathrm{<mo\; stretchy="false">)</mo><mo>/</mo>}{\pi }^{\frac{n\text{}+\text{}1}{2}}$$

where $d\Psi \mathrm{<mo\; stretchy="false">(</mo>}s\mathrm{<mo>;</mo>1<mo\; stretchy="false">)</mo>}\text{}\mathrm{<mo>=</mo>}\text{}{e}^{-\frac{s^2}{2}}\mathrm{<mo>/</mo><mo\; stretchy="false">(</mo>2}\pi {\mathrm{<mo\; stretchy="false">)</mo>}}^{\frac{1}{2}}\mathrm{,}$the mixing $\chi \mathrm{<mo\; stretchy="false">(</mo>}s\mathrm{<mo>;</mo>1<mo\; stretchy="false">)</mo>}$density.

2. The elliptical Cauchy density for $\stackrel{⌢}{\beta }$ on ${ℝ}^k$ is

$f_k^1\mathrm{<mo\; stretchy="false">(</mo>}\stackrel{⌢}{\beta }\mathrm{<mo>;</mo>}\beta \mathrm{,}X\mathrm{<mo>\text{'}</mo>}\text{}X\mathrm{<mo\; stretchy="false">)</mo>}\text{}\mathrm{<mo>=</mo>}\text{}c\mathrm{<mo\; stretchy="false">(</mo>}k\mathrm{<mo\; stretchy="false">)</mo>}{\left[1+\mathrm{<mo\; stretchy="false">(</mo>}\stackrel{⌢}{\beta }\text{}-\text{}\beta {)}^{\prime }X\mathrm{<mo>\text{'}</mo>}\text{}X\mathrm{<mo\; stretchy="false">(</mo>}\stackrel{⌢}{\beta }\text{}-\text{}\beta \mathrm{<mo\; stretchy="false">)</mo>}\right]}^{-\frac{k+1}{2}}$ (3)

Proof. The multivariate $t$distribution on ${ℝ}^n$ is that of $\mathrm{<mo>\{</mo>}{T}_i\text{}\mathrm{<mo>=</mo>}\text{}{Y}_{\text{}i}\mathrm{<mo>/</mo>}S\mathrm{<mo>;</mo>}1\text{}\le \text{}i\text{}\le \text{}n\mathrm{<mo>\}</mo>}$ from $\mathcal{L}\mathrm{<mo\; stretchy="false">(</mo>}Y\mathrm{<mo\; stretchy="false">)</mo>}\text{}\mathrm{<mo>=</mo>}\text{}{\text{N}}_n\mathrm{<mo\; stretchy="false">(</mo>}\delta \mathrm{,}{\sigma }^2{I}_n\mathrm{<mo\; stretchy="false">)</mo>,}$ with $S$ as a sample standard deviation on $\nu$ degrees of freedom, known to be spherical Cauchy at $\nu \text{}\mathrm{<mo>=</mo>}\text{}1.$ This gives conclusion (i) on specializing the conventional multivariate  density. Conclusion (ii) follows directly on specializing Theorem 2(ii) at $\alpha \text{}\mathrm{<mo>=</mo>}\text{}1$

The viability $\mathrm{<mo\; stretchy="false">(</mo>}{Y}_{\text{}i}\mathrm{<mo\; stretchy="false">)</mo>}$ for each of $n\text{}\mathrm{<mo>=</mo>}\text{}13$ biological specimens was recorded after storage under additives $X_{i1}$ and $X_{i2}$ as listed in Table 1;³¹ Walpole RE & Myers RH.³¹ The model is $\mathrm{<mo>\{</mo>}{Y}_{\text{}i}\text{}\mathrm{<mo>=</mo>}\text{}{\beta }_0\text{}+\text{}{\beta }_1{X}_{i1}\text{}+\text{}{\beta }_2{X}_{i2}\text{}+\text{}{\epsilon }_i\mathrm{<mo>\}</mo>,}$where the errors are taken to be spherical Cauchy. The conventional $O\text{}L\text{}S$ solutions are ${\stackrel{⌢}{\beta }}_0\text{}\mathrm{<mo>=</mo>}\text{}\mathrm{36.094,}$ $\widehat{{\beta }_1}\text{}\mathrm{<mo>=</mo>}\text{}\mathrm{1.031,}$ ${\stackrel{⌢}{\beta }}_2\text{}\mathrm{<mo>=</mo>}\text{}-\mathrm{1.870,}$ as elements of $\stackrel{⌢}{\beta }\mathrm{<mo>=</mo>}\text{}\mathrm{<mo\; stretchy="false">[</mo>}{\stackrel{⌢}{\beta }}_0\mathrm{,}{\stackrel{⌢}{\beta }}_1\mathrm{,}{\stackrel{⌢}{\beta }}_2{]}^{\prime }\mathrm{.}$The matrix $X\mathrm{<mo>\text{'}</mo>}\text{}X\mathrm{,}$its inverse ${\mathrm{<mo\; stretchy="false">(</mo>}X\mathrm{<mo>\text{'}</mo>}\text{}X\mathrm{<mo\; stretchy="false">)</mo>}}^{\text{}-1}\mathrm{,}$and the transition of the latter into its $\alpha$-association form of Definition 3 are given respectively by

$${\left[\begin{array}{ccc}13& 59.43& 81.82\\ 59.43& 394.7255& 360.6621\\ 81.82& 360.6621& 576.7264\end{array}\right]}^{-1}\text{}\mathrm{<mo>=</mo>}\text{}\left[\begin{array}{ccc}1.0114& -0.0494& -0.1126\\ -0.0494& 0.0083& 0.0018\\ -0.1126& 0.0018& 0.0166\end{array}\right]\text{}\to \text{}\left[\begin{array}{ccc}1& -0.5392& -0.8690\\ -0.5392& 1& 0.1533\\ -0.8690& 0.1533& 1\end{array}\right]\mathrm{.}$$

 The following properties are evident.

1. The elliptical Cauchy density for $\stackrel{⌢}{\beta }$ is given by equation (3) with $k\text{}\mathrm{<mo>=</mo>}\text{}3$and $X\mathrm{<mo>\text{'}</mo>}\text{}X$as listed for these data.
2. The solution $\stackrel{⌢}{\beta }$ is both linear median–unbiased and modal–unbiased, and among all such estimators is most concentrated about $\beta .$
3. The normal–theory confidence set $\mathrm{<mo>\{</mo>}\beta \text{}\in \text{}\mathrm{<mo\; stretchy="false">(</mo>}\stackrel{⌢}{\beta }\text{}-\text{}\beta {)}^{\prime }X\mathrm{<mo>\text{'}</mo>}\text{}X\mathrm{<mo\; stretchy="false">(</mo>}\stackrel{⌢}{\beta }\text{}-\text{}\beta \mathrm{<mo\; stretchy="false">)</mo>}\text{}\le \text{}{S}^2{c}_{\gamma }\mathrm{<mo>\}</mo>}$ holds exactly with confidence coefficient $1\text{}-\text{}\gamma \text{}\mathrm{<mo>=</mo>}\text{}\mathrm{0.95,}$ where $S^2\text{}\mathrm{<mo>=</mo>}\text{}4.001$ is the residual mean square on $\nu \text{}\mathrm{<mo>=</mo>}\text{}10$ degrees of freedom, and $c_{\gamma }\text{}\mathrm{<mo>=</mo>}\text{}3.71$is the upper 0.95 percentile for $F\mathrm{<mo\; stretchy="false">(</mo>3,10,0<mo\; stretchy="false">)</mo>.}$
4. As correlations are undefined, elements of the $\alpha$–association matrix nonetheless do serve to quantify the degrees of association among $\mathrm{<mo\; stretchy="false">[</mo>}{\stackrel{⌢}{\beta }}_0\mathrm{,}{\stackrel{⌢}{\beta }}_1\mathrm{,}{\stackrel{⌢}{\beta }}_2\mathrm{<mo\; stretchy="false">]</mo>}$ as in Definition 3, on taking $\alpha \text{}\mathrm{<mo>=</mo>}\text{}1$ in Lemma 1.
5. In particular, ${\stackrel{⌢}{\beta }}_0$ is negatively associated with $\mathrm{<mo\; stretchy="false">(</mo>}{\stackrel{⌢}{\beta }}_1\mathrm{,}{\stackrel{⌢}{\beta }}_2\mathrm{<mo\; stretchy="false">)</mo>,}$ whereas $\mathrm{<mo\; stretchy="false">(</mo>}{\stackrel{⌢}{\beta }}_1\mathrm{,}{\stackrel{⌢}{\beta }}_2\mathrm{<mo\; stretchy="false">)</mo>}$ are themselves positively associated (Table 1).

This study offers further insight into the class $\text{S}\alpha \text{S}$comprising the spherical $\alpha$–stable laws as limit distributions under conditions for central limit theory. In addition to their essential properties, expanded here to include representations for density functions, this study focuses on models of type $\mathrm{<mo>\{</mo>}Y\text{}\mathrm{<mo>=</mo>}\text{}X\beta \text{}+\text{}\text{e}\mathrm{<mo>\}</mo>}$when devoid of moments undergirding the classical theory. Recall that normal–theory procedures routinely are applied in practice as large–sample approximations in distributions attracted to Gaussian laws. Specifically, Berry–Esséen bounds on rates of convergence to Gaussian limits are given Jensen,^32,33 with special reference to linear models in Jensen.^34,35 Results here validate corresponding large–sample approximations for distributions attracted to  laws as cited in references.^17–21 Of similar importance are rates of convergence to stable limits as in Paulauskas.²² By showing that many standard properties carry over in essence under significantly weakened assumptions, this study gives further credence to the widely and correctly held view that Gauss–Markov estimation and normal theory inferences extend considerably beyond the confines of the classical theory.

The preceding study has developed exclusively around spherically dependent $\text{S}\alpha \text{S}$errors, as alternative to $\text{i}\text{}\text{i}\text{}\text{d}$stable errors. This choice is prompted by discrepancies encountered in the simplest case $\mathrm{<mo>\{</mo>}{Z}_i\text{}\to \text{}{Z}_i\text{}+\text{}\delta \mathrm{<mo>;</mo>}i\text{}\mathrm{<mo>=</mo>}\text{}\mathrm{1,2,}\dots \mathrm{,}N\mathrm{<mo>\}</mo>}$ with common location parameter. Essential details from Jensen¹⁴ may be summarized as follows. To distinguish the disparate properties of $\text{i}\text{}\text{i}\text{}\text{d}$vs spherical $\text{S}\alpha \text{S}$models, sequences $\mathbb{Z}\text{}\mathrm{<mo>=</mo>}\text{}\mathrm{<mo>\{</mo>}{Z}_1\mathrm{,}{Z}_2\mathrm{,}{Z}_3\mathrm{,}\dots \mathrm{<mo>\}</mo>}$are fundamental in order to take limits. Of significance is that averages of $\text{S}\alpha \text{S}$sequences with $\alpha \text{}\mathrm{<mo><</mo>}\text{}2$may be inconsistent for $\text{i}\text{}\text{i}\text{}\text{d}$sequences but consistent under $\text{S}\alpha \text{S}$symmetry. Accordingly, let ${\mathcal{L}}_{\infty }\mathrm{<mo\; stretchy="false">(</mo>}{Z}_N\mathrm{<mo\; stretchy="false">)</mo>}\text{}\mathrm{<mo>=</mo>}\text{}\mathrm{liminf}\mathcal{L}\mathrm{<mo\; stretchy="false">(</mo>}{Z}_N\mathrm{<mo\; stretchy="false">)</mo>.}$Essentials follow.

Lemma 2 Given $\mathbb{Z}\text{}\mathrm{<mo>=</mo>}\text{}\mathrm{<mo>\{</mo>}{Z}_1\mathrm{,}{Z}_2\mathrm{,}{Z}_3\mathrm{,}\dots \mathrm{<mo>\}</mo>,}$consider the case that $Z\mathrm{<mo>\text{'}</mo>}\text{}\mathrm{<mo>=</mo>}\text{}\mathrm{<mo\; stretchy="false">[</mo>}{Z}_1\mathrm{,}\dots \mathrm{,}{Z}_N\mathrm{<mo\; stretchy="false">]</mo>}$either are $\text{i}\text{}\text{i}\text{}\text{d}$ ${\text{S}}_1^{\alpha }\mathrm{<mo\; stretchy="false">(</mo>}\delta \mathrm{,}\mathrm{1<mo\; stretchy="false">)</mo>,}$ with chf ${\varphi }_{Z_i}\text{}\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\text{}\mathrm{<mo>=</mo>}\text{}\exp \mathrm{<mo>\{</mo>}\iota t\delta -\mathrm{<mo>|</mo>}t{\mathrm{<mo>|</mo>}}^{\alpha }\mathrm{<mo>\}</mo>,}$or are $\text{S}\alpha \text{S}$ on ${ℝ}^N$with chf ${\varphi }_Z\text{}\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\text{}\mathrm{<mo>=</mo>}\text{}\exp \mathrm{<mo>\{</mo>}\iota \delta t\mathrm{<mo>\text{'}</mo>}{1}_{\text{}N}\text{}-\text{}{\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo>\text{'}</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}}^{\frac{\alpha }{2}}\mathrm{<mo>\}</mo>.}$ Let $S_N\text{}\mathrm{<mo>=</mo>}\text{}\mathrm{<mo\; stretchy="false">(</mo>}{Z}_1+\dots +Z_N\mathrm{<mo\; stretchy="false">)</mo>}$and ${\overline{Z}}_{{}_N}\text{}\mathrm{<mo>=</mo>}\text{}{N}^{-1}{S}_N\mathrm{,}$ and consider the standardized variables $U_N\text{}\mathrm{<mo>=</mo>}\text{}{N}^{\frac{1}{2}}\mathrm{<mo\; stretchy="false">(</mo>}{\overline{Z}}_{{}_N}\text{}-\text{}\delta \mathrm{<mo\; stretchy="false">)</mo>.}$

1. Consistent and inconsistent properties of for ${\overline{Z}}_{{}_N}$sequences are as follow.

 For $0\text{}\mathrm{<mo><</mo>}\text{}\alpha \text{}\mathrm{<mo><</mo>}\text{}\mathrm{1<mo>:</mo>}$ ${\varphi }_{{\overline{Z}}_{{}_N}}\text{}\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\text{}\mathrm{<mo>=</mo>}\text{}{e}^{\iota t\delta \text{}-\text{}{N}^{\epsilon }\mathrm{<mo>|</mo>}t{\mathrm{<mo>|</mo>}}^{\alpha }}$for $\epsilon \text{}\mathrm{<mo>></mo>}\text{}\mathrm{0,}$so that ${\overline{Z}}_{{}_N}$is inconsistent for $\delta \mathrm{.}$

 For $\alpha \text{}\mathrm{<mo>=</mo>}\text{}\mathrm{1,}$ ${\varphi }_{{\overline{Z}}_{{}_N}}\text{}\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\text{}\mathrm{<mo>=</mo>}\text{}{e}^{\iota t\delta \text{}-\text{}\mathrm{<mo>|</mo>}t{\mathrm{<mo>|</mo>}}^{\alpha }}\text{}\equiv \text{}{\varphi }_{Z_i}\text{}\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>,}$so that ${\overline{Z}}_{{}_N}$is inconsistent for

 For $1\text{}\mathrm{<mo><</mo>}\text{}\alpha \text{}\le \text{}\mathrm{2,}$ ${\varphi }_{{\overline{Z}}_{{}_N}}\text{}\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}\text{}\mathrm{<mo>=</mo>}\text{}{e}^{\iota t\delta \text{}-\text{}{N}^{-\epsilon }\mathrm{<mo>|</mo>}t{\mathrm{<mo>|</mo>}}^{\alpha }}$for $\epsilon \text{}\mathrm{<mo>></mo>}\text{}\mathrm{0,}$so that ${\overline{Z}}_{{}_N}$is consistent for $\delta \mathrm{.}$

2. For $SaS$ sequences ${\overline{Z}}_{{}_N}$ is consistent for $\delta \mathrm{.}$ for every $0\text{}\mathrm{<mo><</mo>}\text{}\alpha \text{}\le \text{}2.$
3. For iid sequences with $0\text{}\mathrm{<mo><</mo>}\text{}\alpha \text{}\mathrm{<mo><</mo>}\text{}\mathrm{2,}$ ${\mathcal{L}}_{\infty }\mathrm{<mo\; stretchy="false">(</mo>}{U}_N\mathrm{<mo\; stretchy="false">)</mo>}$diverges to an improper distribution.
4. For $\text{S}\alpha \text{S}$sequences $\mathrm{liminf}\mathcal{L}\mathrm{<mo\; stretchy="false">(</mo>}{U}_N\mathrm{<mo\; stretchy="false">)</mo>}\text{}\equiv \text{}\mathcal{L}\mathrm{<mo\; stretchy="false">(</mo>}{Z}_i\mathrm{<mo\; stretchy="false">)</mo>,}$the limit being identical to each component.