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Biometrics & Biostatistics International Journal

Review Article Volume 7 Issue 3

Linear inference under alpha–stable errors

Donald R Jensen

Virginia Polytechnic Institute and State University, USA

Correspondence: Donald R Jensen, Professor Emeritus, Virginia Polytechnic Institute and State University, Blacksburg VA, 24061, USA, Tel 540 639 0865

Received: May 02, 2018 | Published: May 23, 2018

Citation: Jensen DR. Linear inference under alpha–stable errors. Biom Biostat Int J. 2018;7(3):205–210. DOI: 10.15406/bbij.2018.07.00210

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Abstract

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 α –stable measures on n with stability index α(0,2], arising in turn through multivariate central limit theory devoid of the second moments required for Gaussian limits. This study revisits linear inference under α–stable errors, focusing on aspects to be salvaged from the classical theory even without moments. Critical entities include Ordinary Least Squares (OLS) solutions, residuals, and conventional F ratios in inference. Closure properties are seen in that OLS solutions and residual vectors under α –stable errors also have α–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 α –stable symmetry.

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

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

Introduction

Models here are {Y=Xβ+ε}with error vector εn.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 include1–6 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 {Sn(0,Σ)}consisting of elliptically contoured distributions in ncentered at 0 with scale parameters Σ. 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 αstable (SαS)distributions in n with index α(0,2]. These comprise the limit distributions of standardized vector sums, specifically, Gaussian limits at  Cauchy limits at α=2, and corresponding stable limits otherwise. These distributions are contained in the class {Sn(0,In)},thus sharing its essential geometric features, but instead are deficient in moments usually ascribed to {Sn(0,In)}.Despite the venues cited, αstable 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 (chfs), on critical representations for these, and on the inversion of the latter in order to represent the α-stable densities themselves. Even here a divide emerges between independent, identically distributed α-stable sequences, and dependent SαS 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 α(0,2], 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 SαS Distributions as subsections. The principal sections following these address Linear Models under SαS Errors, with a separate subsection on Models Having Cauchy Errors, and Conclusions. Collateral topics are contained for completeness in Appendix A.

Preliminaries

Notation

Spaces of note include nas Euclidean nspace, with Snas the real symmetric (n×n) matrices and S+n as their positive definite varieties. Vectors and matrices are set in bold type; the transpose, inverse, trace, and determinant of are A', A1, tr(A), and |A|; the unit vector in n is ; and Inis the (n×n)identity.

Moreover, Diag(A1,,Ak)is a block-diagonal array, and Σ12is the spectral square root of

Special distributions

Given Y=[Y1,,Yn]n,its distribution, expected value, and dispersion matrix are designated as L(Y), E(Y)=μ, and V(Y)=,with variance Var(Y)=σ2on 1.Specifically, L(Y)=Nn(m,)is Gaussian on nwith parameters (μ,).Distributions on 1of note include the χ2(u;ν,λ)and related χ(u;ν,λ)distributions, together with the Snedecor -Fisher F(u;ν1,ν2,λ),these having (ν,ν1,ν2)as degrees of freedom and λa noncentrality parameter. The characteristic function (chf)for Ynis the expectation ϕY(t)=E[eιt'Y]with argument t'=[t1,,tn] and ι=1;a standard source is Lukacs & Laha.15 Attention is drawn subsequently to probability density (pdf) and cumulative distribution (cdf) functions. Moreover, the class {L(Z)Sn(0,)}consists of elliptically contoured distributions in ncentered at 0 and having chf’s of type ϕZ(t)=ψ(t'St). We adopt the following.

Definition 1 A distribution P on n is said to be monotone unimodal about 0nif for every ynand every convex set C symmetric about 0n,P[C+ky] is no increasing in k[0,).See reference.16

Central limit theory

For iidvectors {Z1,Z2,Z3,}in n,let ˉZN=N1[Z1++ZN],and consider limit distributions of type {L(cˉZN)=liminfL(cˉZN)}for suitably chosen C. On specializing from the elliptical class Sn(d,)having location-scale parameters (d,S),we consider αstable limit distributions as follow on identifying L(cˉZN)with

Definition 2 Let L(Z)Sαn(d,)designate an elliptical α-Stable law on ncentered at dnwith scale parameters and stable index α(0,2],having the chf ϕZ(t)=exp{ιt'd12(t'St)α2}. Each marginal distribution of Sαn(δ1n,In) on 1, namely Sα1(δ,1),has the chf ϕZi(t)=exp{ιtδ12|t|α}.Let SαS={Sαn(d,S);(d,)(nn)}designate the class of all such distributions.

Remark 1 L(Z)is of full rank and has a density in nif and only if is of full rank in S+n;otherwise L(Z)is concentrated in a subspace of nof dimension equal to the rank of

To continue, designate by Dαthe domain of attraction of each element Zi in {Z1,Z2,Z3,} in n having liminfL(cˉZN)in SαS.That is, their chfs satisfy {liminfϕcˉZN(t)=exp[ιt'd12(t'St)α2]}when scaled suitably. Specifically, the distributions D2 attracted to Gaussian limits comprise all distributions L(Zi)in nhaving second moments. More generally, domains of attraction to distributions in SαShave 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 ΦZ(t)=exp[ιt'd12(t'St)α2] has elliptical contours derives from the spherical chf ϕU(t)=exp[ιt'q12(t't)α2] through the transformation a

Essentials for SαSdistributions

As noted, closed expressions for SαSdensities are known in selected cases only, to be complemented by results to follow. Here gn(u;d,)is the Gaussian density on n having parameters (δ,), and fαn(μ;δ,) is the provisional SαS density corresponding to ϕZ(t)=exp[ιt'd12(t'St)α2]. The following properties are essential.

Theorem 1 Let L(Z)Sαn(d,S)have the chf ϕZ(t)=exp[ιt'd12(t'St)α2] and density function fαn(z;δ,) if defined. Then the following properties hold.

  1. For nonsingular, L(Z)Sαn(δ,) is absolutely continuous in n, having a density function fαn(z;δ,);
  2. The Gaussian mixture ϕZ(t)=0eιt'dt'St/2sdΨ(s;α) holds with Ψ(s;α)as a mixing on 1;
  3. The Gaussian mixture fαn(z;δ,)=0gn(z;δ,s1)dΨ(s;α) holds with Ψ(s;α)as a mixing cdf as before;
  4. L(Z)Sαn(δ,) is monotone unimodal with mode at d, for each α(0,2);
  5. Let T(Z)=Uk be scale-invariant; then for L(Z)Sαn(δ,), the distribution L(U)is identical to its normal-theory form under L(Z)=Nn(z;δ,).

Proof: Conclusion (i) is Theorem 6.5.4 of Press.23 Conclusion (ii) invokes a result of Hartman et al.24 namely, the process {Zt;t=1,2,}is spherically invariant if and only if, for each n andZ=[Z1,,Zn],the chf ϕZ(t) is a scale mixture of spherical Gaussian s on n, to give conclusion (ii) on transforming from spherical to elliptical symmetry. To continue, fZ(z)=(2π)nneit'zϕZ(t)Λ(dt)is the standard inversion formula from s to densities in n with Λ() as Lebesgue measure, so that from conclusion (ii) we recover

fαn(μ;δ,In)=1(2π)nkeit'x0eit'δt't/2sdΨ(s;α)Λ(dt).(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 0gn(Z;δ,/s)dΨ(s;α), that the change of variables ZU=T(Z) behind the integral is independent of Ψ(s;α) since T(Z) is scale-invariant independently of s to give conclusion (v).

It remains to reconsider degrees of association in SαS distributions, as distinct from the classical second-moment correlation parameters {ρij=σij/(σiiσjj)12}. For L(Z)Sαk(δ,) with α<2, the elements of s serve instead as scale parameters, since U=12Z and U'U=Z'1Zare dimensionless. As to whether {ρij} again might quantify associations for α<2, a definitive answer is supplied in the following.

Lemma 1 Let L(Z)Sαn(δ,). For , the parameters {ρij=σij/(σiiσjj)12}serve to quantify degrees of association between (Zi,Zj),the extent of their association increasing with

Proof: It suffices to consider (Z1,Z2)centered at (0,0) with S=[1ρρ1].On taking U=(Z1Z2), L(U) clearly is symmetric about 0 with scale parameter σU=2(1ρ).A result of Fefferman et al.26 shows for each c>0that P(U(c,c)) is decreasing in σUthus increasing in ρ.Equivalently, P(|Z1Z2|c)1as ρ1,identifying the sense in which (Z1,Z2)become increasingly indistinguishable, thus associated, with increasing values of

Definition 3 For L(Z)Sαn(δ,S)with α<2,the entities {ρij=σij/(σiiσjj)12}are called pseudo–correlation, specifically, α-association parameters

Linear models under errors

The principal findings

Take L(Y)Sαn(Xβ,σ2In)with (Xβ,σ2In)as centering and scale parameters, where {Yn,XFn×k,βk}. OLSsolutions β=(X'X)1X'Y, as minimally dispersed unbiased linear estimates, are available here only for α=2,whereas alternative moment criteria necessarily are subject to moment constraints. Specifically, for scalars (θ,θ)1 under loss L(θ,θ)=|θθ|,the risk R(θ)=E[L(θ,θ)] is undefined for α<1 as for Cauchy errors at α=1. Moreover, risk functions {R(θ)=E(|θθ|κ)} are defined but concave for {κ<α<1}, and for {1<κ<α2} 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 α,but also optimizing algorithms. Instead we seek what might be salvaged from classical linear models under the constraints of SαSerrors. In addition, portions of our findings extend beyond Gauss–Markov theory and OLS to include the much larger class of equivariant estimators.

Definition 4 An estimator δ(Y) for βk is translation –equivariant if for {YY+Xb}, then {δ(Y+Xb)=δ(Y)+b} for every bk.

On taking P=[InX(X'X)1X'], the elements of e=PY comprise the observed residuals and S2=e'e/(nk) the residual mean square. Normal–theory tests for H0:β=β0against H1:ββ0utilize F=(ββ0)X'X(ββ0)/S2 having the distribution F(u;k,nk,λ) with λ=(ββ0)X'X(ββ0)/σ2. We proceed to examine essential properties of Sαn(Xβ,σ2In)as αranges over (0,2),where some expressions simplify on taking σ2=1,then reinstating σ2as needed. The following properties are fundamental.

Theorem 2 Given L(Y)=Sαn(Xβ,σ2In),consider [β,e]with e=PYas the residual vector, and U=(nk)S2/σ2.Then

(i) L(β,e)=Sαn+k([β,0],S),with =σ2Diag((X'X)1,P),a distribution on n+kof rank s

(ii) The marginal’s are L(β)=Sαk(β,σ2(X'X)1)centered at βwith scale parameters σ2(X'X)1,and

(iii) L(e)=Sαn(0,σ2P)on nof rank nkcentered at 0with scale parameters σ2P;

(iv) U=(nk)S2/σ2has density f(u;ν,α)=0h(u;ν,s)dΨ(s;α)with h(u;ν,s)as the central chi–squared density on ν=(nk)degrees of freedom, scaled by S, and with Ψ(s;α)as a mixing distribution.

Proof. Let L'=(X'X)1X'and P=[InX(X'X)1X']to project onto the error space, so that G=[L,P]operates on y to give

Z=G'Y=[βe]=[L'P']Yn+kandG'G=[(X'X)100P], (2)

 the latter of order [(n+k)×(n+k)] and rank n.The chf with argument s'=[s1,,sn+k]is E[exp(ιs'Z)]=E[exp(ιs'G'Y)]=E[exp(ιv'Y)]= ϕY(v)with argument v=Gs replacing t, to give conclusion (i). Next partition s'=[s1',s2']with s1'=[s1,,sk], to obtain = ϕZ(s)=exp[ιs'G'Xβ12(s'G'Gs)α2]=exp[ιs1'β12(s1'(X'X)1s1+s2'Ps2)α2]. The marginal s of β and e follow on setting s2=0, then s1=0in succession, to give conclusions (ii) and (iii). Conclusion (iv) attributes to Hartman et al.24 through Theorem 1. Specifically, a change of variables uee'e=(nk)S2 behind the integral on the right of Theorem 1(iii) gives the conditional density for L((nk)S2|s), namely the scaled chi–squared density h(u;ν,s)depending on s so that integrating with respect to dΨ(s;α) gives conclusion (iv).

Remark 3 That S=σ2Diag(X'X,P) is block–diagonal in conclusion (i), assures under SαS errors that (β,e)are α–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 θ for θk is said to be linearly median unbiased if and only if the median med(a'θ)=a'θfor each ak; and to be modal unbiased provided that the mode

Definition 6 An estimator θfor θis said to be more concentrated about θthan provided that θ P((ˆθθ)C0)P((˜θθ)C0)for every convex set C0 in k symmetric under reflection about 0k.

 Essential properties under SαS errors include the following.

Theorem 3 For L(Y)=Sαn(Xβ,σ2In),consider properties of the OLSsolutions β=(X'X)1X'Y,and of the equivariant estimators β=δ(Y) of Definition 4.

  1. β is unbiased for βfor each {1<α2};
  2. βis linearly median unbiased for β;
  3. βis most concentrated about β;among all median–unbiased linear estimators;
  4. β is modal unbiased for β;
  5. βNis consistent for β; in a sequence of identical but dependent experiments{Yi=Xβ+ei;i=1,2,,N};
  6. The null distribution of F=(ββ0)X'X(ββ0)/S2has exactly its normal–theory form; the power increases with increasing λ=(ββ0)X'X(ββ0)/σ2;and such tests are unbiased;
  7. βis most concentrated about βamong all modal–unbiased linear estimators.

βis most concentrated about βamong all equivariant estimators β=δ(Y).

Proof. Conclusions (i)–(vi) carry over from reference Jensen DR27 without benefit of moments, regardless of membership in the SαSclass. To consider concentration properties of modal–unbiased estimators, begin with ϕY(t)=exp[ιt'Xβ12(t't)α2],and consider ˜β=L'Ywith L'=[(X'X)1X',G'],so that

ϕ˜β(s)=exp[ιs'L'Xβ12(s'L'Ls)α2];

s'L'Xβ=s'[(X'X)1X',G']Xβ.

 That β should have mode at β, it is necessary that s'L'Xβ=s'β, i.e. G'X=0. accordingly, ϕβ(s)=exp[ιs'β12[s'Ωs]α2], with Ω=L'L=[(X'X)1+G'G].Clearly the matrix [L'L(X'X)1]=G'Gis 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 cauchy errors

Spherical multivariate t errors on v degrees of freedom trace to Zellner30 to include Cauchy errors at ν=1, equivalently, at α=1 in the class SαS. Specializing from Theorem 1(ii), the spherical Cauchy chf is ϕZ(t)=exp[it'd12(t't)12]. 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 n at α=1 is
  2. f1n(z;δ,In)=0gn(z;δ,s2In)dΨ(s;1)

    =c(n)[1+(zδ)(zδ)]n+12

    c(n)=Γ(n+12)/πn+12

    where dΨ(s;1)=es22/(2π)12,the mixing χ(s;1)density.

  3. The elliptical Cauchy density for β on k is

f1k(β;β,X'X)=c(k)[1+(ββ)X'X(ββ)]k+12 (3)

Proof. The multivariate tdistribution on n is that of {Ti=Yi/S;1in} from L(Y)=Nn(δ,σ2In), with S as a sample standard deviation on ν degrees of freedom, known to be spherical Cauchy at ν=1. This gives conclusion (i) on specializing the conventional multivariate  density. Conclusion (ii) follows directly on specializing Theorem 2(ii) at α=1

Case study

The viability (Yi) for each of n=13 biological specimens was recorded after storage under additives Xi1 and Xi2 as listed in Table 1;31 Walpole RE & Myers RH.31 The model is {Yi=β0+β1Xi1+β2Xi2+εi},where the errors are taken to be spherical Cauchy. The conventional OLS solutions are β0=36.094, ^β1=1.031, β2=1.870, as elements of β=[β0,β1,β2].The matrix X'X,its inverse (X'X)1,and the transition of the latter into its α-association form of Definition 3 are given respectively by

[1359.4381.8259.43394.7255360.662181.82360.6621576.7264]1=[1.01140.04940.11260.04940.00830.00180.11260.00180.0166][10.53920.86900.539210.15330.86900.15331].

 The following properties are evident.

  1. The elliptical Cauchy density for β is given by equation (3) with k=3and X'Xas listed for these data.
  2. The solution β is both linear median–unbiased and modal–unbiased, and among all such estimators is most concentrated about β.
  3. The normal–theory confidence set {β(ββ)X'X(ββ)S2cγ} holds exactly with confidence coefficient 1γ=0.95, where S2=4.001 is the residual mean square on ν=10 degrees of freedom, and cγ=3.71is the upper 0.95 percentile for F(3,10,0).
  4. As correlations are undefined, elements of the α–association matrix nonetheless do serve to quantify the degrees of association among [β0,β1,β2] as in Definition 3, on taking α=1 in Lemma 1.
  5. In particular, β0 is negatively associated with (β1,β2), whereas (β1,β2) are themselves positively associated (Table 1).

Yi

Xi1

Xi2

Yi

Xi1

Xi2

0.5

1.74

3.3

31.2

6.32

5.42

0.9

6.22

8.41

38.4

10.52

4.63

0.4

1.19

11.6

26.7

1.22

5.85

0.4

4.1

6.62

25.9

6.32

8.72

0

4.08

4.42

25.2

4.15

7.6

0.7

10.15

4.83

35.7

1.72

3.12

0.5

1.7

5.3

 

 

 

Table 1 The viability (Yi) of n=13 biological specimens after storage under additives Xi1 and Xi2

Summary and discussion

This study offers further insight into the class SαScomprising the spherical α–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 {Y=Xβ+e}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.

A appendix

The preceding study has developed exclusively around spherically dependent SαSerrors, as alternative to iidstable errors. This choice is prompted by discrepancies encountered in the simplest case {ZiZi+δ;i=1,2,,N} with common location parameter. Essential details from Jensen14 may be summarized as follows. To distinguish the disparate properties of iidvs spherical SαSmodels, sequences ={Z1,Z2,Z3,}are fundamental in order to take limits. Of significance is that averages of SαSsequences with α<2may be inconsistent for iidsequences but consistent under SαSsymmetry. Accordingly, let L(ZN)=liminfL(ZN).Essentials follow.

Lemma 2 Given ={Z1,Z2,Z3,},consider the case that Z'=[Z1,,ZN]either are iid Sα1(δ,1), with chf ϕZi(t)=exp{ιtδ|t|α},or are SαS on Nwith chf ϕZ(t)=exp{ιδt'1N(t't)α2}. Let SN=(Z1++ZN)and ˉZN=N1SN, and consider the standardized variables UN=N12(ˉZNδ).

  1. Consistent and inconsistent properties of for ˉZNsequences are as follow.
  2.  For 0<α<1: ϕˉZN(t)=eιtδNε|t|αfor ε>0,so that ˉZNis inconsistent for δ.

     For α=1, ϕˉZN(t)=eιtδ|t|αϕZi(t),so that ˉZNis inconsistent for

     For 1<α2, ϕˉZN(t)=eιtδNε|t|αfor ε>0,so that ˉZNis consistent for δ.

  3. For SaS sequences ˉZN is consistent for δ. for every 0<α2.
  4. For iid sequences with 0<α<2, L(UN)diverges to an improper distribution.
  5. For SαSsequences liminfL(UN)L(Zi),the limit being identical to each component.

Acknowledgement

None.

Conflict of interest

Authors declare that there is no conflict of interest.

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