## Limits and Inferences for Alpha-Stable Variables

### Jensen DR

*Department of Statistics, Virginia Polytechnic Institute and State University, USA*

**Submission: ** October 05, 2017; **Published: ** December 8, 2017

***Corresponding author: ** DR Jensen, Emeritus Professor, Department of Statistics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA, Tel: 540-639-0865; Email: djensen@vt.edu

** How to cite this article: ** Jensen DR. Limits and Inferences for Alpha–Stable Variables. Biostat Biometrics Open Acc J. 2017; 4(1): 555630. DOI: 10.19080/BBOAJ.2017.04.555630

**Abstract**

Distributions having excessive tails are modeled in various venues via a-stable sequences deficit in moments of various orders. Essential statistics are examined under independent vs spherically dependent cases. The former exhibits such critical pathologies as inconsistent sample means. The latter support versions of some classical procedures even without moments as conventionally required. In particular, despite heavy tails, Student's tests for means nonetheless remain exact in level and power. etc. *AMS Subject Classification:* 62E15, 62H15, 62J20.

**Keywords:** Excessive tails; *α-* stable sequences; Consistency; Exact Student's *t*

**Introduction**

Classical statistics rest heavily on means, variances, correlations, skewness and kurtosis, requiring moments to fourth order. To the contrary, probability distributions having excessive tails, often void of first or second moments, arise in a variety of circumstances. These encompass radar tracking, image processing, acoustics, risk management, portfolios in finance, biometrics, and other venues. Supporting references include Bonato & Matteo [1], Cheng & Rachev [2], Kim et al. [3], Kuruoglu & Zerubia [4], Qiou et al. [5], Tsionas & Efthymios [6]. Salient monographs are Arce [7], Chernobai & Rachev [8], Ibragimov et al. [9], and Samorodnitsky & Taqqu [10]. In these settings the classical foundations accordingly must be reworked.

Excessive tails typically are modeled through *α-* stable distributions with index α∈[0,2]. These comprise all limit distributions for standardized partial sums, of which Gaussian central limit theory applies under second moments with ^{α = 2} Despite the circumstances cited, usage has been limited for want of explicit expressions, apart from special cases, for α-stable density and cumulative distribution functions. Nonetheless, signal progress is supported through the use of characteristic functions (* ^{chfs}*). as undertaken in this study. Even here a divide emerges between independent, identically distributed

*(iid)*α-stable variables, or dependent

*α-stable*

*(sαs)*sequences. An outline follows. Notation and technical foundations are provided in Section 2. The findings in Section 3 are twofold: First, that limit properties diverge widely between

*(iid)*and spherically dependent

*(sαs*) sequences; and second, that conventional inferences, though largely lacking in the former, may be validated in large part in the latter. Conclusions are tallied in Section 4.

**Preliminaries**

**Notation:** Spaces include ℝ^{N} as Euclidean N-space. Vectors and matrices are set in bold type; the transpose, inverse, trace,and determinant of A are A', A^{-1}, *tr(A)*, and *|A|* the unit vector in ℝ^{N} is 1_{N} = [1,...., 1]'; and *I _{N}* is the

*(N x N)*identity. Moreover, Diag

*(A*is a block-diagonal array.

_{1}.......A_{k})**Special distributions**

For *Y=[Y _{1},...,Y_{N}]' ϵℝ^{N}*, its distribution, mean, and dispersion matrix are denoted by L

*(y*),

*E(y) =μ*and

*v*(y)=Σ, say, with variance

*var(y) = σ*on 1 ℝ

^{2}^{1}. Specifically, L(Y)=N

_{N}(μΣ) is Gaussian on ℝ

^{N}with parameters (μ,Σ). Distributions on ℝ

^{1}include the χ

^{2}(μ;v,λ) having degrees of freedom and non centrality parameter λ; and the corresponding Student's t

^{2}(μ;v,λ). The

*(chfs)*for

**Y**is the expectation ϕY(t)=E[e

^{it'Y}] with argument t' = [t

_{1},........t

_{N}]; a standard source is Lukacs & Laha [11]. Reference is drawn subsequently to probability density (PD) and cumulative distribution (CD) functions.

**Foundations**

A random process *ℤ={Z _{t};tϵτ} *is spherically invariant if for each

^{N}, the joint

^{(chfs)}of [Z

_{1},...,Z

_{N}] has the form ϕ

_{z}(t)=Ψ(t't) for some function Ψ(.) not depending on N [12,13]. Averages and limits are basic in statistical analysis; for example, notions of consistency in estimation and of large-sample distributions. These are undertaken here without benefit of moments in keeping with excessive tails. Specifically, the sequence ℤ={Z

_{t};tϵ

*τ*} supplies the context for taking limits. In addition, the following is central to this study, where[δ,ε] respectively comprise a location vector and a matrix of scale parameters, the latter taking the value Σ;=I

_{N}in the case of spherical symmetry on ℝ

^{N}

**Definition 1:** L(Z)ϵ S_{N}^{α}(δ,Σ) designates an elliptical *α-*stable law on ℝ^{N} centered at δϵℝ^{N} with scale parameters Σ and stable index αϵ(o,2], having the *(chfs)* ϕZ(t)=exp{it'δ-(t'Σt)α/2}
Each marginal distribution of S_{N}^{α}(δI_{N} onϵ^{1} namely S_{1}^{α}(δ,1), has the*(chfs) *ϕZ_{t}(t)= exp {itδ-/t/^{α}}

**Remark 1:** For αϵ(0,2) these have moments of order up to but excluding α but with moments of all orders at *α =* 2. Included are elliptical Cauchy and Gaussian laws at α = {1,2}, respectively. In addition, the following is central to subsequent developments.

**Lemma 1:** Let *g _{N}*(z;δ,I

_{n}) be the density for N

_{N}(δ,σ

^{2},I

_{n}and

*f*the

^{α}_{N}(z;δ,I_{n})*SαS*density for L(Z)ϵS

^{α}

_{N}(δ,I

_{n}). Then therechfs and

^{pdfs}are related as follows.

Proof: Hartman & Wintner [12] gave a necessary and sufficient condition that the process be spherically invariant, namely, that for each** N** the chfs ϕZ_{(t)} is a scale mixture of N-dimensional spherical Gaussian chfs. This applies in context to give conclusion (i). To continue, is the standard inversion formula from chfs to densities with ^(.) as Lebesgue measure. Accordingly, we invert both sides of the second and third expressions in conclusion (i) to get the density on the left of conclusion (ii). We then recover the right side of conclusion (ii) on reversing the order of integration in the iterated integral found on inverting the third expression of conclusion (i).

**The Principal findings**

**Sequences of iid and** *SαS* **Variables:** Much of classical statistics rests on iid random variables. In the present context it is germane to ask whether spherical *SαS* Variables [Z_{1},....,Z_{N}] might also be independent. To the contrary, for any spherical sequence. Maxwell [14] showed this to be the case if and only if Gaussian. In view of this, it remains to examine the limit properties of iid *SαS* Variables in comparison with spherically dependent *SαS* Variables in ℝ^{N} of critical interest to users. Limit properties of these are shown next to be widely disparate, despite the fact that their marginals coincide. At issue are statistics in order to be iid. A principal finding follows.

**I.Theorem 1**

a. If independent, then

b. The *chfs* of *S _{N}* isand

c.

Conclusions (i) and (ii) follow directly from these. Developments to follow invoke the following principles, first for iid and then for spherical *SαS* sequences.

**Remark 2:** In the *chfs* note that factors in as a scale parameter. For consistency in estimatingS,δ it is necessary and sufficientthat which, by the Levy-Cramer continuity theorem, is the *chfs* of a distribution degenerate at δ Consistency of for then follows using the equivalence of convergence in law to degeneracy, and convergence in probability.

**II.Theorem 2**

**III. Theorem 3**

Proof: Conclusion (i) follows from the second ^{chfs} of Theorem 1(ii) as in Remark 2, and conclusion (ii) directly per se from the third ^{chfs} of Theorem 1(ii). In short, it is seen from Theorem 2 that ^{iid α-stable} variables hold little promise to exhibit even basic properties in data analysis and statistical inference, where even the consistency of δ for requires knowing that 1 2. On the other hand, the next section motivates circumstances for the occurrence of spherical *sas* samples, and sets out to establish useful statistical properties in the analysis of data from these models.

**Properties of spherical **^{SαS} samples

^{SαS}samples

Consider anew a single sample with parameters *(δ,σ ^{2}),* taking in lieu of the conventional iid N

_{t}(δ,σ

^{2}) data. To these ends and identify as the ordinary residuals, with as the projection onto the error space, and . Recall that the two-sided normal-theory test for uses the conventional Student's The following is central to our findings.

**IV. Theorem 4**

Given that we seek the joint distribution of and that of.

(i) The joint distribution of is given by a distribution ℝ^{N+1} on of rank _{N.}

(ii) The marginals are centered at **δ** with scale parameter the latter a distribution on ℝ^{N} of rank n-1 centered at **0** with scale parameters σ^{2}p

(iii) has density with h(μv,s)
as the scaled central chi-squared density having v=(N-1)degrees of freedom, and with *s;a)* as a mixing distribution from Lemma 1.

(iv) The test for is exact in level and power as its normal - theory version, for all

**Proof:** Take Its *chfs* with argument

is with argument V=H's replacing *t*

Conclusion (i) follows on substituting into to givesince p is idempotent, so that as claimed.

Conclusion (ii) follows directly.

Conclusions (iii) and (iv) attribute to Hartman & Wintner[12] through Lemma 1(ii). Specifically, a change of variables μ→e→S^{2} behind the integral on the right of Lemma 1(ii) gives the conditional density for L(S^{2}\s) namely the scaled chi-squared density h(μv,s) depending on ^{s,} so that integrating with respect to *dΨ(s;α)* as in Lemma 1(ii) gives conclusion (iii). In like manner, the change of variables behind the integral in Lemma 1(ii) gives the conditional density for L(T^{2}\s) But this statistic is scale-invariant and thus independent of the mixing distribution Ψ(s;α) so that L(T^{2}\s)=L(T^{2})unconditionally, the latter being its conventional normal-theory distribution

**Remark 3:** Despite the diagonal structure in conclusion (i), (Ž,e) are independent if and only if Gaussian at *α = 2* on applying Maxwell’s (1860) result. The mixtures of Lemma 1 and their consequences deserve further mention. Given l(y) L(Y)=N(δ,σ ^{2}I_{N})the multivariate t-distribution of [Y_{1}/S,....Y_{N}/S] is known to be spherical SαS Cauchy at α = 1. The following gives details regarding these mixtures explicitly for the Cauchy case.

**Corollary 1:** Consider The Cauchy *chfs* and density functions at *α =* 1 may be represented as follows.

(i) has the mixing distribution at v=1 namely, the chi-distribution with density having of freedom

(ii) Beginning with the spherical Cauchy density is

**Proof:** The conclusions follow on specializing the mixing distribution for the multi variate **t** at *v=1* degree of freedom, and its known density at v =1

**Conclusion**

In practice scale mixtures may arise as conditionally iid Gaussian variables subject to scaling in a random environment. Linear models so structured are treated in Zellner [15] under multivariate Student *t* in lieu of Gaussian errors. The present study complements that work, eschewing moments through spherical Cauchy errors having *v* =1 degree of freedom. In summary, we have modeled errors not as ^{iid,} but instead as spherical _{α-stable} errors. The former holds little promise as noted, where even the consistency of for δ requires that 1 <α≤2, yet ^{α} typically is unknown. On the other hand, spherical *α-stable* errors offer a reasonable resolution to open topics in linear inference without moments. Not only is consistent for δ for all αϵ(o,2], but a mixture representation is given for the density of S^{2}/σ^{2} Moreover, a corresponding representation for Student's T^{2} exploits its scale invariance to show that tests using T^{2} are exact in level and power, as for Gaussian errors, for all *L(Z)=S _{N}^{α}(δ1_{N},σ^{2}I_{N}) with 0 ≤ α ≤2.*

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