1. INTRODUCTION AND DISCUSSION OF THE RESULTS
1.1. Efficiency and Peakedness of Estimators
A number of problems in econometrics and statistics involve comparisons of estimators' performance. The present paper focuses on comparisons of linear estimators under heavy-tailedness and obtains characterizations of optimal linear estimators for heavy-tailed data.
Let
and
be two estimators of a population parameter θ ∈ R. In the case when
, i = 1,2, are unbiased for θ and have finite second moments, their comparisons are traditionally based on quadratic loss functions leading to comparisons of the variances
, i = 1,2 :
is preferred to
if
(in other words, if
is more efficient than
).
This approach breaks down, however, in the case of heavy-tailed estimators
for which variances do not exist, and one has to rely on loss functions more general than quadratic ones. In the case of an increasing loss function U : R+ = [0,∞) → R,
is preferred to
if (provided that the expectations exist)
(in the efficiency literature, it is common to consider loss functions that satisfy additional assumptions of boundedness). Orderings of estimators based on comparisons (1.1) are, of course, dependent on the choice of the loss functions U.
A natural approach to comparison of performance of estimators is to order them by the likelihood of observing their large deviations from the true parameter (see, e.g., Hwang, 1985; Andrews and Phillips, 1987; and references therein). This approach corresponds to the choice of indicator functions Uε(x) = I(x > ε), ε > 0, in (1.1) and is closely related to the concept of peakedness of random variables (r.v.s) introduced by Birnbaum (1948).
DEFINITION 1.1 (Birnbaum, 1948). An r.v. X is more peaked about θ ∈ R than is Y if P(|X − θ| > ε) ≤ P(|Y − θ| > ε) for all ε ≥ 0. If this inequality is strict whenever the two probabilities are not both zero or both one, then the r.v. X is said to be strictly more peaked about θ than is Y. In case θ = 0, X is simply said to be (strictly) more peaked than Y.
The following definition provides a peakedness-based analogue of the concept of efficiency for estimators that will be explored throughout the paper.
DEFINITION 1.2. The estimator
is said to be more efficient than
in the sense of peakedness (P-more efficient than
for short) if
is strictly more peaked about θ than is
.
The property of being P-less efficient is defined in a similar way. Roughly speaking,
is P-more efficient than
if the distribution of
is more concentrated about the true parameter θ than is that of
. As follows from well-known properties of first-order stochastic dominance (see Shaked and Shanthikumar, 1994, pp. 3–4; and Remark 3.2 in this paper), if
is P-more efficient than
, then comparisons (1.1) are independent of the choice of U and hold for any increasing loss function.
Comparisons of estimators are closely related to the analysis of the problem of whether having more data improves performance of an estimator of a population parameter. Indeed, obviously, an increase in the sample size always improves performance of the estimator
of a population parameter θ if
is P-more efficient than
for all n ≥ 1. In contrast, having larger samples is disadvantageous for performance of the estimator if P-efficiency of
decreases with n. Increasing P-efficiency is the basis for the following definition of monotone consistency, the concept studied by, e.g., Proschan (1965), Tong (1994), and Jensen (1997).
DEFINITION 1.3. A weakly consistent estimator
of a population parameter θ is said to exhibit monotone consistency for θ if
is P-more efficient than
for all n ≥ 1 and, thus,
converges to zero monotonically in n for all ε > 0.
1.2. Objectives and Key Results
The present paper focuses on the analysis of efficiency and peakedness properties of linear estimators under heavy-tailedness assumptions and on the study of efficiency and monotone consistency properties of the sample mean for heavy-tailed data. The main results show that peakedness and majorization properties of log-concavely distributed r.v.s established by Proschan (1965) continue to hold for convolutions of α-symmetric distributions with α > 1 (Theorems 3.1 and 3.3 in Section 3). However, these properties are reversed in the case of convolutions of α-symmetric distributions with α < 1 (Theorems 3.2 and 3.4).
Consider the model
where μ is the location parameter and εi, i = 1,2,…, are possibly heavy-tailed and not necessarily independent r.v.s (specifically, (1.2) is assumed to satisfy one of conditions (A1)–(A4), (A2′), or (B1)–(B4) introduced in Section 2; μ is the mode of
,
, for linearly unimodal distributions satisfying (A1)–(A3), (A2′), or (B1)–(B3) and is the population mean in the case of finite first moments: EXi = μ, i = 1,2,…).
Among other results, we show that the sample mean
is the best linear unbiased estimator of the population mean μ in the sense of P-efficiency for not extremely heavy-tailed populations (see Theorem 3.1). However, according to our results, P-efficiency of the sample mean is smallest among all linear estimators
with weights ai ≥ 0, i = 1,…, n,
, of the location parameter μ in the case of extremely heavy-tailed data (Theorem 3.2). The preceeding results imply that P-efficiency of Xn is increasing in n for not extremely heavy-tailed populations and, thus, an increase in the sample size always improves performance of the sample mean in such a case. In the case of data from extremely heavy-tailed populations, P-efficiency of the sample mean is, however, decreasing in the sample size n.
Convolutions of α-symmetric distributions considered in the paper exhibit both heavy-tailedness and dependence (see Section 2). In particular, they contain, as subclasses, convolutions of certain models with common shocks affecting all heavy-tailed risks in addition to spherical distributions that are α-symmetric with α = 2. Spherical distributions, in turn, include such examples as Kotz type, multinormal, logistic, and multivariate α-stable distributions. In addition, they include a subclass of mixtures of normal distributions and also multivariate t-distributions that were used in the literature to model heavy-tailedness phenomena with dependence and finite moments up to a certain order. The results in the paper are also obtained for skewed stable distributions (such as, e.g., extremely heavy-tailed Lévy distributions with α = ½ concentrated on the positive semiaxis) and for r.v.s with nonidentical one-dimensional distributions.
The law of large numbers (LLN) provides conditions (such as existence of first moments) under which the sample mean Xn converges in probability to the population mean μ: P(|Xn − μ| > ε) → 0 for all ε > 0. However, as discussed in, e.g., Proschan (1965), the LLN does not imply that P(|Xn − μ| > ε) decreases monotonically as the sample size n increases. From the results in Proschan (1965, see Rmk. 3.1) it follows that P(|Xn − μ| > ε) converges to zero monotonically in the case of independent and identically distributed (i.i.d.) symmetric r.v.s Xi with log-concave distributions that are, as discussed in the next section, extremely light-tailed. The results in the present paper imply that monotone decrease of P(|Xn − μ| > ε), where μ is the location parameter, continues to hold in the case of convolutions of α-symmetric distributions with α > 1. On the other hand, according to the results in this paper, the tail probabilities P(|Xn − μ| > ε), where μ is the location parameter, monotonically diverge from zero in the case of convolutions of α-symmetric distributions with α < 1 that have infinite first moments. According to these results and their more general analogues established in the paper, the sample mean and other linear estimators of the location parameter perform poorly under extreme heavy-tailedness. Therefore, more robust statistical procedures, such as those based on sample medians, must be employed in such a setting.
The results obtained in the paper have applications in the study of robustness of models of firm growth theory for firms that can invest in information about their markets, value at risk analysis, optimal strategies for a multiproduct monopolist, and also that of inheritance models in mathematical evolutionary theory (see Ibragimov, 2004a, 2004b, 2004c, 2004d, 2005).
1.3. Organization of the Paper
The paper is organized as follows. Section 2 introduces the classes of distributions considered throughout the paper and discusses their structure and the main properties. Section 3 presents the main results of the paper on efficiency properties of linear estimators for convolutions of α-symmetric distributions. Section 4 contains the proofs of the results obtained.
2. NOTATIONS AND DISTRIBUTIONAL ASSUMPTIONS
An r.v. X with density f : R → R and the convex distribution support Ω = {x ∈ R : f (x) > 0} is said to be log-concavely distributed if log f (x) is concave in x ∈ Ω, i.e., if for all x1,x2 ∈ Ω, and any λ ∈ [0,1], f (λx1 + (1 − λ)x2) ≥ (f (x1))λ(f (x2))1−λ (see An, 1998). A distribution is called log-concave if its density f satisfies the preceding inequalities. Examples of log-concave distributions include the normal distribution, the uniform density, the exponential density, the gamma distribution Γ(α,β) with the shape parameter α ≥ 1, the beta distribution
with a ≥ 1 and b ≥ 1; and the Weibull distribution
with the shape parameter α ≥ 1.
If an r.v. X is log-concavely distributed, then its density has at most an exponential tail; i.e., f (x) = O(exp(−λx)) for some λ > 0, as x → ∞ and all the power moments E|X|γ, γ > 0, of the r.v. exist (see An, 1998, Cor. 1). The reader is referred to Karlin (1968), Marshall and Olkin (1979), and An (1998) for a survey of many other properties of log-concave distributions.
For 0 < α ≤ 2, σ > 0, β ∈ [−1,1], and μ ∈ R, we denote by Sα(σ,β,μ) the stable distribution with the characteristic exponent (index of stability) α, the scale parameter σ, the symmetry index (skewness parameter) β, and the location parameter μ. That is, Sα(σ,β,μ) is the distribution of an r.v. X with the characteristic function
x ∈ R, where i2 = −1 and sign(x) is the sign of x defined by sign(x) = 1 if x > 0, sign(0) = 0, and sign(x) = −1 otherwise. In what follows, we write X ∼ Sα(σ,β,μ), if the r.v. X has the stable distribution Sα(σ,β,μ) and write
if the distribution of X is symmetric and log-concave (
stands for “log-concave”).
A closed form expression for the density f (x) of the distribution Sα(σ,β,μ) is available in the following cases (and only in those cases): α = 2 (Gaussian distributions); α = 1 and β = 0 (Cauchy distributions with densities f (x) = σ/[π(σ2 + (x − μ)2)]); α = ½ and β ± 1 (Lévy distributions that have densities f (x) = (σ/(2π))1/2exp(−σ/(2x))x−3/2, x ≥ 0; f (x) = 0, x < 0, where σ > 0, and their shifted versions). Degenerate distributions correspond to the limiting case α = 0.
The index of stability α characterizes the heaviness (the rate of decay) of the tails of stable distributions Sα(σ,β,μ). The distribution of a stable r.v. X ∼ Sα(σ,β,μ) with α ∈ (0,2) obeys power law P(|X| > x) ∼ x−α, and thus the p-th absolute moments E|X|p of X are finite if p < α and are infinite otherwise. The symmetry index β characterizes the skewness of the distribution. The stable distributions with β = 0 are symmetric about the location parameter μ. The stable distributions with β = ±1 and α ∈ (0,1) (and only they) are one-sided; the support of these distributions is the semiaxis [μ,∞) for β = 1 and is (−∞,μ] for β = −1 (in particular, the Lévy distribution with μ = 0 is concentrated on the positive semiaxis for β = 1 and on the negative semiaxis for β = −1). In the case α > 1 the location parameter μ is the mean of the distribution Sα(σ,β,μ). The scale parameter σ is a generalization of the concept of standard deviation; it coincides with the latter in the special case of Gaussian distributions (α = 2).
Distributions Sα(σ,β,μ) with μ = 0 for α ≠ 1 and β ≠ 0 for α = 1 are called strictly stable. If Xi ∼ Sα(σ,β,μ), α ∈ (0,2], are i.i.d. strictly stable r.v.s, then, for all ai ≥ 0, i = 1,…, n, such that
, one has
For a detailed review of properties of stable distributions the reader is referred to, e.g., the monographs by Zolotarev (1986) and Uchaikin and Zolotarev (1999).
According to the definition introduced by Cambanis, Keener, and Simons (1983), an n-dimensional distribution is called α-symmetric if its characteristic function (c.f.) can be written as
, where φ : R+ → R is a continuous function (with φ(0) = 1) and α > 0. An important property of α-symmetric distributions is that, similar to strictly stable laws, they satisfy property (2.1). The number α is called the index, and the function φ is called the c.f. generator of the α-symmetric distribution. The class of α-symmetric distributions contains, as a subclass, spherical distributions corresponding to the case α = 2 (see Fang, Kotz, and Ng, 1990, p. 184). Spherical distributions, in turn, include such examples as Kotz type, multinormal, multivariate t, and multivariate spherically symmetric α-stable distributions (Fang et al., 1990, Ch. 3). Spherically symmetric stable distributions have c.f.s
, 0 < γ ≤ 2, and are, thus, examples of α-symmetric distributions with α = 2 and the c.f. generator φ(x) = exp(−xγ).
For any 0 < α ≤ 2, the class of α-symmetric distributions includes distributions of risks X1,…, Xn that have the common factor representation
where Yi ∼ Sα(σ,0,0) are i.i.d. symmetric stable r.v.s with σ > 0 and the index of stability α and Z ≥ 0 is a nonnegative r.v. independent of Yi
's (see Bretagnolle, Dacunha-Castelle, and Krivine, 1966; Fang et al., 1990, p. 197). Although the dependence structure in model (2.2) alone is restrictive, convolutions of such vectors provide a natural framework for modeling of random environments with different common shocks Z, such as macroeconomic or political ones, that affect all risks Xi (see Andrews, 2003). In the case Z = 1 (a.s.), model (2.2) represents vectors with i.i.d. symmetric stable components that have c.f.s
that are particular cases of c.f.s of α-symmetric distributions with the generator φ(x) = exp(−λxα).
According to the results in Bretagnolle et al. (1966) and Kuritsyn and Shestakov (1984), the function exp(−(|t1|α + |t2|α)1/α) is a c.f. of two α-symmetric r.v.s for all α ≥ 1 (the generator of the function is φ(u) = exp(−u)). Zastavnyi (1993) demonstrates that the class of more than two α-symmetric r.v.s with α > 2 consists of degenerate variables (so that their c.f. generator φ(u) = 1). For further review of properties and examples of α-symmetric distributions the reader is referred to Fang et al. (1990, Ch. 7) and Gneiting (1998).
Convolutions of α-symmetric distributions exhibit both heavy-tailedness in marginals and dependence among them. Both the classes of convolutions of α-symmetric distributions with α < 1 and those with α > 1 can be used to model heavy-tailedness of an arbitrary order in marginals. For instance, the class of convolutions of models (2.2) with α < 1 has extremely heavy-tailed marginal distributions with infinite means. On the other hand, convolutions of such models with 1 < α ≤ 2 can have marginals with power moments finite up to a certain positive order (or finite exponential moments) depending on the choice of the r.v.s Z. For instance, convolutions of models (2.2) with 1 < α < 2 and E|Z| < ∞ have finite means but infinite variances; however, marginals of such convolutions have infinite means if the r.v.s Z satisfy E|Z| = ∞. Moments E|ZYi|p, p > 0, of marginals in models (2.2) with α = 2 (that correspond to Gaussian r.v.s Yi) are finite if and only if E|Z|p < ∞. In particular, all marginal power moments in models (2.2) with α = 2 are finite if E|Z|p < ∞ for all p > 0. Similarly, marginals of spherically symmetric (i.e., 2-symmetric) distributions range from extremely heavy-tailed to extremely light-tailed ones. For example, marginal moments of spherically symmetric α-stable distributions with c.f.s
, 0 < γ < 2, are finite if and only if their order is less than γ. Marginal moments of a multivariate t-distribution with k degrees of freedom that is an example of a spherical distribution are finite if and only if the order of the moments is less than k.
Let Φ stand for the class of c.f. generators φ such that φ(0) = 1, limt→∞ φ(t) = 0, and the function φ′(t) is concave. In what follows, we consider the following distributional assumptions, (A1)–(A4) and (B1)–(B4).
Let r ∈ (0,2).
(A1) The random vector (X1 − μ,…, Xn − μ) is a sum of i.i.d. random vectors (Y1j,…,Ynj), j = 1,…, k, where (Y1j,…,Ynj) has an absolutely continuous α-symmetric distribution with the c.f. generator φj ∈ Φ and the index αj ∈ (r,2].
(A2) The random vector (X1 − μ,…, Xn − μ) is a sum of i.i.d. random vectors (Y1j,…,Ynj) = (ZjV1j,…, ZjVnj), j = 1,…, k, where Vij ∼ Sαj(σj,0,0), i = 1,…, n, j = 1,…, k, with σj > 0 and αj ∈ (r,2] and Zj are positive absolutely continuous r.v.s independent of Vij.
(A3) The random vector (X1 − μ,…, Xn − μ) has an α-symmetric distribution with a continuous c.f. generator φ : R+ → R and the index α ∈ (r,2].
(A4) (X1,…, Xn) = (ZV1,…, ZVn), where Vi, i = 1,…, n, are i.i.d. r.v.s such that V1 ∼ Sα(σ,β,μ) for some σ > 0, β ∈ [−1,1], and α ∈ (r,2], with β = 0 for α = 1, and Z is a positive r.v. independent of Vi's.
We will also need the following assumption, (A2′), which is more general than Assumption (A2) with r = 1.
(A2′) The random vector (X1 − μ,…, Xn − μ) is a sum of i.i.d. random vectors (Y1j,…,Ynj) = (ZjV1j,…, ZjVnj), j = 1,…, k, where
or Vij ∼ Sαj(σj,0,0), i = 1,…, n, j = 1,…, k, with σj > 0 and αj ∈ (1,2] and Zj are positive absolutely continuous r.v.s independent of Vij.
The following distributional assumptions, (B1)–(B4), involve conditions that are the opposite of those in (A1)–(A4). Let r ∈ (0,2].
(B1) The random vector (X1 − μ,…, Xn − μ) is a sum of i.i.d. random vectors (Y1j,…,Ynj), j = 1,…, k, where (Y1j,…,Ynj) has an absolutely continuous α-symmetric distribution with the c.f. generator φj ∈ Φ and the index αj ∈ (0,r).
(B2) The random vector (X1 − μ,…, Xn − μ) is a sum of i.i.d. random vectors (Y1j,…,Ynj) = (ZjV1j,…, ZjVnj), j = 1,…, k, where Vij ∼ Sαj(σj,0,0), i = 1,…, n, j = 1,…, k, with σj > 0 and αj ∈ (0,r) and Zj are positive absolutely continuous r.v.s independent of Vij.
(B3) The random vector (X1 − μ,…, Xn − μ) has an α-symmetric distribution with a continuous c.f. generator φ : R+ → R and the index α ∈ (0,r).
(B4) (X1,…, Xn) = (ZV1,…, ZVn), where Vi, i = 1,…, n, are i.i.d. r.v.s such that V1 ∼ Sα(σ,β,μ) for some σ > 0, β ∈ [−1,1] and α ∈ (0,r), with β = 0 for α = 1, and Z is a positive r.v. independent of Vi's.
If a random vector (X1,…, Xn) satisfies one of Assumptions (A1), (A2), (A2′), (A3), or (B1)–(B3), then (X1 − μ,…, Xn − μ) is linearly unimodal about 0 = (0,…,0); i.e., the linear combinations
are symmetric and unimodal about zero for all (a1,…, an) ∈ Rn and, thus, μ is the mode of
in the case
(see Dharmadhikari and Joag-Dev, 1988, Def. 2.3; and the proof of the results in the next section). Similarly, linear unimodality (not necessarily about 0) also holds for random vectors (X1,…, Xn) that satisfy Assumption (A4) or (B4).
1As follows from the proof of the results in the paper, the assumption φj ∈ Φ in conditions (A1) and (B1) can be replaced by the assumption that the vectors (Y1j,…,Ynj), j = 1,…, k, are linearly unimodal about 0. The latter assumption is equivalent to the condition that
, j = 1,…, k, are c.f.s of three-dimensional spherically symmetric distributions; see Theorem 4.3 in Gneiting (1998).
It is easy to see that if X1,…, Xn satisfy (2.1) with α < 1, then E|Xn| > E|X1|, which, evidently, cannot hold in the case E|X1| < ∞. Consequently, first moments of such r.v.s are infinite. It is not difficult to see that this implies that marginal first moments of r.v.s X1,…, Xn satisfying one of Assumptions (B1)–(B4) with r ≤ 1 are infinite.
The indices of stability αj and the scale parameters σj in Assumptions (A2) and (B2) are different among the vectors (Y1j,…,Ynj). A linear combination of independent stable r.v.s with the same characteristic exponent α also has a stable distribution with the same α. However, in general, this does not hold true in the case of convolutions of stable distributions with different indices of stability. Therefore, the class of random vectors (X1,…, Xn) satisfying Assumption (A2) (resp., Assumption (B2)) with Zj = Z, where Z is a positive absolutely continuous r.v. independent of symmetric stable r.v.s Vij, is wider than the class of random vectors (X1,…, Xn) satisfying Assumption (A4) (resp., Assumption (B4)) with β = 0.
3. MAIN RESULTS: EFFICIENCY PROPERTIES OF LINEAR ESTIMATORS UNDER HEAVY-TAILEDNESS AND DEPENDENCE
In what follows, for a vector c ∈ Rn, we denote by c[1] ≥ ··· ≥ c[n] its components in decreasing order. A vector a ∈ Rn is said to be majorized by a vector b ∈ Rn, written a [pr ] b, if
, k = 1,…, n − 1, and
. The relation a [pr ] b implies that the components of the vector a are more diverse than those of b (see Marshall and Olkin, 1979). In this context, it is easy to see that the following relations hold:
for all a ∈ R+n. In particular,
A function φ : A → R defined on A ⊆ Rn is called Schur-convex (resp., Schur-concave) on A if (a [pr ] b) ⇒ (φ(a) ≤ φ(b)) (resp. (a [pr ] b) ⇒ (φ(a) ≥ φ(b))) for all a,b ∈ A. If, in addition, φ(a) < φ(b) (resp., φ(a) > φ(b)) whenever a [pr ] b and a is not a permutation of b, then φ is said to be strictly Schur-convex (resp., strictly Schur-concave) on A.
In what follows, given r.v.s X1,…, Xn satisfying (1.2) with the location parameter μ, and weights a = (a1,…, an) ∈ R+n, we denote by
the linear estimator
of μ and by ψ(a,ε), ε > 0, its tail probability
. We also denote by
the simplex
.
Theorem 3.1, which follows, concerns efficiency comparisons for linear estimators in the case of convolutions of α-symmetric distributions with α > 1. It shows that, for convolutions of α-symmetric distributions with α > 1, the sample mean is the best linear unbiased estimator of the population mean in the sense of P-efficiency. In addition, according to the theorem, the sample mean exhibits monotone consistency under such distributional assumptions.
THEOREM 3.1. Let μ ∈ R. Suppose that, for n ≥ 1, the r.v.s X1,…, Xn satisfy Assumption (A2′) or one of Assumptions (A1), (A3), or (A4) with r = 1. Then the following conclusions hold.
(i) Let
. The linear estimator
is P-more efficient than
if a [pr ] b and a is not a permutation of b (equivalently, ψ(a,ε) is strictly Schur-convex in a = (a1,…, an) ∈ R+n for all ε > 0).
(ii) The sample mean
is P-more efficient than any other linear unbiased estimator
,
. In particular, Xn exhibits monotone consistency for μ and P(|Xn − μ| > ε) converges to zero strictly monotonically in n for all ε > 0.
According to the following theorem, the conclusions of Theorem 3.1 are reversed for convolutions of α-symmetric distributions with α < 1. In this case, peakedness of the sample mean about the location parameter decreases with the sample size. In addition, under the preceding distributional assumptions, P-efficiency of the sample mean is smallest among all linear estimators
with
.
THEOREM 3.2. Let μ ∈ R. Suppose that, for n ≥ 1, the r.v.s X1,…, Xn satisfy one of Assumptions (B1)–(B4) with r = 1. Then the following conclusions hold.
(i) Let
. The linear estimator
is P-less efficient than
if a [pr ] b and a is not a permutation of b (equivalently, ψ(a,ε) is strictly Schur-concave in a = (a1,…, an) ∈ R+n for all ε > 0).
(ii) The sample mean
is P-less efficient than any other linear estimator
with
. In particular, P-efficiency of Xn decreases with n, i.e., P(|Xn+1 − μ| > ε) > P(|Xn − μ| > ε) > P(|X1 − μ| > ε) for all n ≥ 1 and all ε > 0.
The following theorem shows that efficiency comparisons for linear estimators for populations with distributions satisfying one of Assumptions (A1)–(A4) are of the same type as in Theorem 3.1 with respect to the comparisons between the powers of the components of the vectors of weights of the combinations. Theorem 3.3 also provides concentration inequalities for linear estimators in the case of such distributions that refine and complement the efficiency and peakedness comparisons implied by Theorem 3.1.
THEOREM 3.3. Suppose that, for n ≥ 1, the r.v.s X1,…, Xn satisfy one of Assumptions (A1)–(A4) with r ∈ (0,2). Then the following conclusions hold.
(i) Let μ = 0 and a,b ∈ R+n. Then
is strictly more peaked than
if (a1r,…, anr) [pr ] (b1r,…, bnr) and (a1r,…, anr) is not a permutation of (b1r,…, bnr) (equivalently, ψ(a,ε) is strictly Schur-convex in (a1r,…, anr) ∈ R+n for all ε > 0).
(ii) Let μ ∈ R. The linear estimators
,
, satisfy the following concentration inequalities for all
, with strict right-hand-side inequality if a = (a1,a2,…,an) is not a permutation of (1,0,…,0) and strict left-hand-side inequality if a ≠ (1/n,1/n,…,1/n).
As follows from Theorem 3.4, the efficiency properties of linear estimators in Theorem 3.3 are reversed in the case of populations with distributions satisfying one of Assumptions (B1)–(B4), in particular, for convolutions of α-symmetric distributions with α < r. The concentration inequalities in Theorem 3.4 refine and complement the efficiency orderings for linear estimators given by Theorem 3.2.
THEOREM 3.4. Suppose that, for n ≥ 1, the r.v.s X1,…, Xn satisfy one of Assumptions (B1)–(B4) with r ∈ (0,2]. Then the following conclusions hold.
(i) Let μ = 0 and a,b ∈ R+n. Then
is strictly more peaked than
if (a1r,…, anr) [pr ] (b1r,…, bnr) and (a1r,…, anr) is not a permutation of (b1r,…, bnr) (equivalently, ψ(a,ε) is strictly Schur-concave in (a1r,…, anr) ∈ R+n for all ε > 0).
(ii) Let μ ∈ R. The linear estimators
,
, satisfy the following concentration inequalities for all
, with strict left-hand-side inequality if a = (a1,…, an) is not a permutation of (1,0,…,0) and strict right-hand-side inequality if a ≠ (1/n,1/n,…,1/n).
The following proposition provides analogues of the results in this section for linear estimators
for not necessarily identically distributed r.v.s Wi (a certain ordering in the components of the vector a is necessary for the extensions of the majorization results in this section to the case of nonidentically distributed r.v.s Wi because Schur-convexity and Schur-concavity of a function f (a) in a imply its symmetry in the components of a). Let σ1,…, σn > 0 and μ1,…, μn < 0.
PROPOSITION 3.1. Let Wi ∼ Sα(σi,β,μi), α ∈ (0,2], β ∈ [−1,1], β = 0 for α = 1, be independent nonidentically distributed stable r.v.s. Then the following conclusions hold.
(i) The function
is strictly Schur-concave in a = (a1,…, an) ∈ Rn if α > 1, σ1 ≥ ··· ≥ σn > 0, and μ1 ≤ ··· ≤ μn ≤ 0 and is strictly Schur-convex in a = (a1,…, an) ∈ Rn if α < 1, σn ≥ ··· ≥ σ1 > 0, and 0 ≥ μ1 ≥ ··· ≥ μn.
(ii) Let μi = μ, i = 1,…, n. Theorems 3.1 and 3.3 hold (in the same range of parameters r and α) for
if σ1 ≥ ··· ≥ σn > 0. Theorems 3.2 and 3.4 hold for
if σn ≥ ··· ≥ σ1 > 0.
The following corollary provides analogues of the results in the paper for convolutions of stable distributions with different location (and scale) parameters.
COROLLARY 3.1. Let X1,…, Xn be i.i.d. r.v.s with a common distribution that is a convolution of stable distributions Sαj(σj,0,μj), j = 1,…, k, with different scale parameters σj > 0 and different location parameters μj ∈ R such that
, where Yij ∼ Sαj(σj,0,μj), are independent stable r.v.s. Then the function
is strictly Schur-convex in a = (a1,…, an) ∈ R+n for all ε > 0.
Remark 3.1. Theorem 3.1 provides generalizations of the results of Proschan (1965), who showed that the tail probabilities
are Schur-convex in a = (a1,…, an) ∈ R+n for all ε > 0 for random samples X1,…, Xn from symmetric log-concavely distributed populations
.
2Proschan (1965) notes that similar majorization orderings also hold for (twofold) convolutions of log-concave distributions with symmetric Cauchy distributions and shows that peakedness comparisons implied by them are reversed for n = 2k, vectors a = (1/n, 1/n,…,1/n) ∈ Rn with identical components and certain transforms of symmetric Cauchy r.v.s.
Remark 3.2. It is well known that if r.v.s X and Y are such that P(X > x) ≤ P(Y > x) for all x ∈ R, then EU(X) ≤ EU(Y) for all increasing functions U : R → R for which the expectations exist (see, e.g., Shaked and Shanthikumar, 1994, pp. 3–4). This fact and Theorems 3.1–3.4 imply corresponding results concerning majorization properties of expectations of loss functions of linear estimators under heavy-tailedness. For instance, we get that if U : R+ → R is an increasing function, then, assuming existence of the expectations, the function
, a ∈ R+n is Schur-convex in (a1r,…, anr) under the assumptions of Theorem 3.3 and is Schur-concave in (a1r,…, anr) under the assumptions of Theorem 3.4. We also get that the function φ(a), a ∈ R+n is Schur-concave in (a12,…, an2) under Assumptions (B1)–(B4) with r = 2. These results complement those in Efron (1969) and Eaton (1970) (see also Marshall and Olkin, 1979, pp. 361–365), who studied classes of functions U : R → R and r.v.s X1,…, Xn for which Schur-concavity of φ(a), a ∈ R+n in (a12,…, an2) holds. Further, we obtain that φ(a) is Schur-convex in a ∈ R+n under the assumptions of Theorem 3.1 and is Schur-concave in a ∈ R+n under the assumptions of Theorem 3.2. Because E|Xi| = ∞ for r.v.s X1,…, Xn satisfying the assumptions of Theorem 3.2 (see Section 2), we get that, in the case of such r.v.s and increasing convex loss functions U : R+ → R, the expectations
are infinite for all a ∈ R+n,
. Therefore, the last result does not contradict the well-known fact that (see Marshall and Olkin, 1979, p. 361) the function
is Schur-convex in (a1,…, an) ∈ R for all i.i.d. r.v.s Y1,…,Yn and convex functions f : R → R as it might seem at first sight.
Remark 3.3. Similar to the proof of Proposition 3.1 in Section 4, one can also obtain analogues of the results in the present section in the case of distributions with dependent and not necessarily identically distributed marginals, including convolutions of shifted and scaled α-symmetric distributions.
4. PROOFS
In the proofs that follow, we provide the complete argument for the main majorization results that provide a reversal of those available in the literature, namely, for Theorems 3.2 and 3.4. The proof of Theorem 3.3 that gives the results on Schur-convexity of the tail probabilities of linear combinations of r.v.s follows the same lines as that of Theorem 3.4, with respective changes in the signs of inequalities. We also provide the complete proof of Theorem 3.2 in the case of Assumption (A2′) because, in this case, it is not implied by Theorem 3.3 alone but needs to combine the results in that theorem with those for log-concave distributions in Proschan (1965).
Proof of Theorems 3.3 and 3.4. Let r,α ∈ (0,2], σ > 0, β ∈ [−1,1], β = 0 for α = 1 and let a = (a1,…, an) ∈ R+n and b = (b1,…, bn) ∈ R+n be such that (a1r,…, anr) [pr ] (b1r,…, bnr) and (a1r,…, anr) is not a permutation of (b1r,…, bnr) (clearly,
and
). Let (X1,…, Xn) be a random vector satisfying one of the Assumptions (A3), (A4), (B3), or (B4) with μ = 0. As follows from the discussion in Section 2, property (2.1) holds for Xi, i = 1,…, n. Consequently, if c = (c1,…, cn) ∈ R+n,
, and ε > 0, then
According to Proposition 3.C.1.a in Marshall and Olkin (1979), the function
is strictly Schur-convex in (c1,…, cn) ∈ R+n if α > 1 and is strictly Schur-concave in (c1,…, cn) ∈ R+n if α < 1. Therefore, we have
, if α/r > 1, and
, if α/r < 1. This, together with (4.1), implies that ψ(a,ε) < ψ(b,ε) if either (A3) or (A4) is satisfied and ψ(a,ε) > ψ(b,ε) if either (B3) or (B4) is satisfied. Consequently, part (i) of Theorem 3.3 holds if (X1,…, Xn) satisfies (A3) or (A4), and part (i) of Theorem 3.4 holds for (X1,…, Xn) satisfying (B3) or (B4).
Now let μ ∈ R and suppose that the random vector (X1 − μ,…, Xn − μ) is a sum of i.i.d. random vectors (Y1j,…,Ynj), j = 1,…, k, satisfying the assumptions in (B1) or (B2). By Theorem 3.4 for distributions satisfying (B3) or (B4), for j = 1,…, k, the r.v.
is strictly more peaked than
; i.e., for all ε > 0 and all j = 1,…, k,
As indicated at the end of Section 2, the r.v.s
, j = 1,…, k, and
, j = 1,…, k, are symmetric and unimodal if one of the conditions (A1), (A2), (B1), or (B2) is satisfied. In the case of (A1) and (B1) this easily follows from a result due to R. Askey; see Theorem 4.1 in Gneiting (1998). In the case of Assumptions (A2) and (B2), symmetry and unimodality of
, j = 1,…, k, and
follows from symmetry and unimodality of
and
implied by Theorem 2.7.6 in Zolotarev (1986, p. 134) and Theorem 1.6 in Dharmadhikari and Joag-Dev (1988, p. 13), the definition of unimodality and conditioning arguments.
From Lemma in Birnbaum (1948) and its proof it follows that if X1,X2 and Y1,Y2 are independent absolutely continuous symmetric unimodal r.v.s such that, for i = 1,2, Xi is more peaked than Yi, and one of the two peakedness comparisons is strict, then X1 + X2 is strictly more peaked than Y1 + Y2. This, together with (4.2) and symmetry and unimodality of
and
, j = 1,…, k, implies, by induction on k (see also Birnbaum, 1948, Thm. 1; Shaked and Shanthikumar, 1994, Thm. 2.C.3), that
for ε > 0. Therefore, part (i) of Theorem 3.4 holds if either (B1) or (B2) is satisfied. Part (i) of Theorem 3.3 for random vectors (X1,…, Xn) satisfying one of conditions (A1) or (A2) with μ ∈ R might be proved in a completely similar way, with the reversal of inequality signs in (4.2). Parts (ii) of Theorems 3.3 and 3.4 follow from their parts (i) and majorization comparisons (3.1). █
Proof of Theorems 3.1 and 3.2. Theorems 3.3 and 3.4 imply that part (i) of Theorem 3.2 holds if one of Assumptions (B1)–(B4) with r = 1 is satisfied, and part (i) of Theorem 3.1 holds if one of Assumptions (A1)–(A4) with r = 1 is satisfied. Let us prove that part (i) of Theorem 3.1 holds under Assumption (A2′). Let vectors
and
be such that a [pr ] b and a is not a permutation of b. Suppose that the vector of r.v.s (X1 − μ,…, Xn − μ) is a sum of i.i.d. random vectors (Y1j,…,Ynj) = (ZjV1j,…, ZjVnj), j = 1,…, k, such that, for i = 1,…, n and j = 1,…, k,
or Vij ∼ Sαj(σj,0,0), where σj > 0 and αj ∈ (1,2], and Zj are absolutely continuous positive r.v.s independent of Vij. From part (i) of Theorem 3.3 and the results in Proschan (1965) it follows that, for j = 1,…, k, the r.v.
is strictly more peaked than
. Furthermore, similar to the proof of Theorems 3.3 and 3.4, from Theorem 2.7.6 in Zolotarev (1986, p. 134) and Theorems 1.6 and 1.10 in Dharmadhikari and Joag-Dev (1988, pp. 13 and 20), together with the definition of unimodality and conditioning arguments, it follows that the r.v.s
and
, j = 0,1,…, k, are symmetric and unimodal. As in the proof of Theorems 3.3 and 3.4, by Lemma in Birnbaum (1948) and its proof and induction, this implies that
is strictly more peaked than
. This completes the proof of part (i) of Theorem 3.1.
As is easy to see, under the assumptions of Theorem 3.1, the characteristic function E exp(itXn) of Xn converges to E exp(itμ) as n → ∞ for all t ∈ R; i.e., Xn is weakly consistent for μ. This, together with parts (i) of Theorems 3.1 and 3.2 and majorization comparisons (3.1) and (3.2), implies parts (ii) of the theorems. █
Proof of Proposition 3.1 and Corollary 3.1. Under the assumptions of Proposition 3.1, according to (2.1),
, where Q1 ∼ Sα(1,β,0). Consequently, the function ϑ(a,ε) in part (i) of the proposition satisfies
and, if μi = 0, then
. By Theorem 3.A.4 in Marshall and Olkin (1979), the function
is strictly Schur-convex in (c1,…, cn) ∈ R+n if α > 1 and σ1 ≥ ··· ≥ σn > 0 and is strictly Schur-concave in (c1,…, cn) ∈ R+n if α < 1 and σn ≥ ··· ≥ σ1 ≥ 0. In addition, by the same theorem,
is Schur-convex in (c1,…, cn) ∈ R+n if 0 ≥ μ1 ≥ ··· ≥ μn and is Schur-concave in (c1,…, cn) ∈ R+n if 0 ≥ μn ≥ ··· ≥ μ1. Similar to the proof of Theorems 3.1 and 3.2, the preceding result implies that Proposition 3.1 holds. Because the sum
is fixed under majorization comparisons, the results in Theorems 3.1 and 3.2 imply that, under the assumptions of Proposition 3.1, the function
is strictly Schur-convex in a = (a1,…, an) ∈ R+n if αj < 1, j = 1,…, k, and is strictly Schur-concave in a = (a1,…, an) ∈ R+n if αj > 1, j = 1,…, k. Consequently, Corollary 3.1 holds. █
