A multivariate parallelogram and its application to multivariate trimmed means

A MULTIVARIATE PARALLELOGRAM AND ITS APPLICATION TO MULTIVARIATE TRIMMED MEANS

Jyh-Jen Horng Shiau

and Lin-An Chen

Summary

This paper introduces a multivariate parallelogram that can play the role of the univariate quantile in the location model, and uses it to define a multivariate trimmed mean. It assesses the asymptotic efficiency of the proposed multivariate trimmed mean by its asymptotic variance and by Monte Carlo simulation.

Key words: multivariate parallelogram; quantile; trimmed mean.

1. Introduction

Lety₁, . . . , y_n denote a random sample from a univariate population with distribution function F, and let ˆF be the empirical distribution function obtained from this sample. Let Q(α₁, α₂) = (F−1(α₁), F−1(α₂)) denote the (α₁, α₂)-quantile interval of F and let

ˆ

Q(α1, α2) = ( ˆF−1(α1), ˆF−1(α2)) denote the corresponding sample quantile interval, where

F−1 and ˆF−1 are the inverse functions of F and ˆF, respectively. The sample quantile interval plays a very important role in statistical inference. For example, as a region with a particular coverage probability, the interval is a natural estimator for scale parameters such as the range and interquartile range. With this property, the quantile interval can be used in industrial applications to define a process capability index for process capability assessment, especially for non-normal processes. Also, this interval is routinely used in classifying the observations of a sample into good or bad observations in robust mean estimation, such as for the trimmed mean and Winsorized mean.

Analogues have been proposed for quantiles or order statistics in high dimensions. It is well known that the univariate quantile can be obtained by solving a minimization problem. Breckling & Chambers (1988) and Koltchinskii (1997) generalized the minimization prob-lem for the multivariate case and then defined a multivariate quantile as the minimizer of the problem. Chaudhuri (1996) considered a geometric quantile that uses the geometry of mul-tivariate data clouds. Chakraborty (2001) used a transformation-retransformation technique to introduce a multivariate quantile. However, these approaches do not have obvious settings for defining multivariate regions suitable for constructing descriptive statistics because they lack a natural ordering in multi-dimensional data.

Received November 2001; revised July 2002; accepted September 2002. ∗_{Author to whom correspondence should be addressed.}

1_{Institute of Statistics, National Chiao Tung University, Hsinchu, Taiwan. e-mail: jyhjen@stat.nctu.edu.tw} Acknowledgments. The authors thank the Managing Editor, the Technical Editor, an associate editor and a referee for their valuable comments, which greatly improved the quality of the paper. This research work was partially supported by the National Science Council of the Republic of China Grant Nos NSC90-2118-M-009-008 and NSC90-2118-M-009-016.

In contrast to the above approaches, Chen & Welsh (2002) proposed for the bivariate random vector y a bivariate quantile that partitions R2 into two-dimensional intervals of specified probabilities. In this sense, their approach is a natural extension of the univariate quantile. They first applied an appropriate transformation toy, attempting to make the two co-ordinate components of the transformed random vectorx uncorrelated. Denote x = (x₁, x₂). Then the partition takes the following two steps. (i) Find q₁ such that Pr(x₁ ≤ q₁) = α₁. More specifically, the line x₁ = q₁ divides R2 into two sets, of which one set has cover-age probability α₁ and the other has coverage probability 1− α₁. (ii) Find q₂ such that Pr(x₁≤ q₁, x₂≤ q₂) = α₂. Thus, the line x₂= q₂ divides the set with coverage probability α1 into two subsets such that one has coverage probability α2 and the other has coverage probability α₁− α₂. Then (q₁, q₂) is called the (α₁, α₂)-th bivariate quantile of x in the transformed space. Finally, the bivariate quantile of y is obtained by back-transforming the bivariate quantile ofx. Let ( ˆq₁, ˆq₂) denote the corresponding sample quantile obtained from the data. Note that the distribution of ˆq₂ depends on the distribution of ˆq₁, which makes the asymptotic properties of( ˆq₁, ˆq₂) quite complicated. This approach can be extended to higher dimensional data. However, it would be too complicated to study the asymptotic distribution of the sample multivariate quantile defined in this way, and difficult to use it to make statistical inference from data.

There is some work in the literature on multivariate median estimation. Oja (1983) de-fined the multivariate simplex median by minimizing the sum of volumes of simplices with vertices on the observations; Liu (1988, 1990) introduced the simplicial depth median by maximizing an empirical simplicial depth function. Small (1990) gives an excellent review of these papers.

Chaudhuri (1996) noted that most authors introduced descriptive statistics by merely gen-eralizing the univariate statistics to the multivariate setup, with no clear population analogues for these multivariate descriptive statistics. In other words, descriptive statistics were being defined without the target population parameters to be estimated. Although the approach of Chen & Welsh (2002) defines both the multivariate population parameters and their corre-sponding estimators, it is worth developing alternatives that are easier to use in theoretical study and practical applications. The major purposes of this paper are to define a multivariate parallelogram region as a counterpart of the univariate quantile interval, and to propose a statistic to estimate it. The approach used here considers the same variable transformation as that in Chen & Welsh (2002), but it defines the multivariate quantile through the univari-ate quantile of each coordinunivari-ate of the transformed variable. This avoids the difficulty that we encountered in studying the complicated asymptotic distribution of the Chen & Welsh multivariate quantile.

Inspired by an idea that Huber (1973, 1981) used when constructing a location-scale equivariant studentized M-estimator for location, we introduce multivariate quantile points and use them to construct a multivariate parallelogram. With sample multivariate parallelo-grams, many multivariate descriptive statistics, such as multivariate versions of scale estima-tors, process capability indices, and trimmed means, are easy to construct. In this paper, we study the large-sample properties of the sample multivariate quantile points and the trimmed means constructed by this parallelogram. We compute asymptotic generalized variances of the proposed multivariate trimmed mean and the Cram´er–Rao lower bounds for various multi-variate contaminated normal distributions. The study reveals that the proposed trimmed mean is quite efficient.

Section 2 introduces the multivariate parallelogram and the multivariate quantile points, and gives a large sample representation of the multivariate quantile points. Section 3 presents a multivariate trimmed mean and its large sample representation. Section 4 gives a comparative study of multivariate trimmed means for two different coordinate transformations and the sample mean based on the asymptotic efficiency and Monte Carlo simulations. Proofs of the theorems in this paper are given in Shiau & Chen (2002), which is available at http://www.stat.nctu.edu.tw/TechnicalReports/jyhjen/MultiTrimMean.ps

2. Multivariate parallelograms Consider the multivariate location model

y = µ + v ,

wherey, µ, and v are p × 1 vectors with E(v) = 0 and V(y) = V(v) = . We want to find a subset of the sample space of y with a fixed (either exact or approximate) coverage probability. Apparently, there can be many choices for the shape of this subset. Popular ones include the cube, ellipsoid, parallelogram, trapezoid, etc. If we further require this subset to have the smallest volume, then we can expect different shapes for different distributions. For example, our investigation shows that the ellipsoid is better than the parallelogram when the distribution is bivariate normal while the result is opposite when the distribution is bivariate exponential or chi-squared.

Since  is symmetric positive definite, there exists a non-singular matrix D = 1/2, such that = DDT. The transformed random vector x = D−1y has the model

x = D−1µ + 

with = D−1v. Denote x = (x₁, . . . , x_p). Since V(x) = I, x₁, . . . , x_p are uncorrelated. It is then natural to consider a region formed by the Cartesian product of thep marginal quantile intervals. The multivariate parallelogram is thus obtained by transforming this region back to the y-space. An advantage of choosing the parallelogram as the shape of the region is that asymptotic study of the proposed statistics based on the sample parallelogram is relatively simple compared to other shapes.

Definition 1. For j = 1, . . . , p and 0 < α_j < 1, let ξ_j = F_j−1(α_j) denote the univariate

α_j-th quantile of the random variable x_j.

(a) Define the (α₁, . . . , α_p)-th multivariate quantile point by q(α) = Dξ, where α = (α1, . . . , αp) and ξ = (ξ1, . . . , ξp). Denote q(α1) by q(α) when α1= · · · = αp= α,

where 1 = (1, . . . , 1).

(b) Let α₁ = (α₁₁, . . . , α_p1) and α₂ = (α₁₂, . . . , α_p2) where α_j1 < α_j2. Define the multivariate quantile set by

Q(α1, α2) = {q(α1a1, . . . , αpap): aj = 1, 2, j = 1, . . . , p} , which contains 2p _{quantile points. Define ξ}

jk = Fj−1(αjk). The region R(α1, α2) = {y = Dx: ξj1≤ xj ≤ ξj2, j = 1, 2, . . . , p}

Suppose that we have a random sampley₁, . . . , y_n obeying the following multivariate location model:

y_i = µ + v_i (i = 1, . . . , n) ,

where v₁, . . . , v_n are independent and identically distributed error vectors with mean zero and covariance matrix . Let ˆ denote an estimator of the variance–covariance matrix . Then ˆD = ˆ1/2. Let x_i = ˆD−1y_i, i = 1, . . . , n, denote the transformed random vectors. Denotex_i = (x_1i, . . . , x_pi). Let ˆξ_j1 and ˆξ_j2 denote the α_j1-th and α_j2-th sample quantiles, respectively, based on the transformed observationsx_j1, . . . , x_jn.

Definition 2. Define an estimator of the multivariate quantile point q(α) by ˆq(α) = ˆD ˆξ ,

where ˆξ = (ˆξ1, . . . , ˆξp). Consequently, an estimator of the multivariate quantile set is ˆ

Q(α1, α2) = { ˆq(α1a1, . . . , αpap): aj = 1, 2, j = 1, . . . , p} . The parallel multivariate quantile region is estimated by

ˆR(α1, α2) = {y = ˆDx: ˆξj1≤ xj ≤ ˆξj2, j = 1, . . . , p} . (1) Theorem 1 states that the estimators of the quantile points and the parallelogram defined above are of desired equivariance properties. First, to simplify the notation, we fix the quantile levels α₁= α = (α₁, α₂, . . . , α_p) and α₂ = 1 − α₁ = (1 − α₁, 1 − α₂, . . . , 1 − α_p), and then suppress them from the notation of the statistics under study. Re-denote the estimated multivariate quantile points obtained from the samplesy₁, . . . , y_nandAy₁+ c, . . . , Ay_n+ c by ˆq_α1(y) and ˆq_α1(Ay +c), respectively. We also re-denote D by D(y) and ˆξ_j by ˆξ_j(α_j, y) to indicate which dataset the statistics are based on. For the moment, we let ˆ be more general than the sample covariance matrix. Let ˆ(y) denote a p × p matrix representing a statistic obtained from the random sampley₁, . . . , y_n.

Theorem 1. Let A denote a p×p non-singular matrix and c a p×1 vector. Suppose that the

statistic ˆD satisfies ˆD(Ay + c) = A ˆD(y) and we denote the estimated parallel multivariate

quantile region of (1) by ˆR(α, y). Then (a) ˆq_α

1(Ay + c) = A ˆqα1(y) + c and

(b) ˆR(α₁, Ay + c) = A ˆR(α₁, y) + c.

On the other hand, suppose that the statistic ˆD satisfies ˆD(Ay + c) = −A ˆD(y). Then

(c) ˆq_α

1(Ay + c) = A ˆqα2(y) + c and

(d) ˆR(α₁, Ay + c) = A ˆR(α₂, y) + c.

For the large sample study, the following conditions are assumed for random vector v and sample covariance matrix ˆ. For j = 1, . . . p, let g_j and G_j denote the probability density function (pdf) and the cumulative distribution function (cdf) of the transformed error vectorD−1v, respectively. Let σ_jT denote the j th row ofD−1. Let u, δ ∈ Rp.

(i) The pdf g_j and its derivative are both bounded and bounded away from 0 in a neigh-bourhood ofG_j−1(α_j) for α_j ∈ (0, 1), j = 1, 2, . . . , p.

(ii) n1/2( ˆ − ) = O_p(1).

(iii) There exists θ > 0 such that the pdf of vT(σ_j + δ) is uniformly bounded in a neigh-bourhood ofG_j−1(α) for δ ≤ θ, and the pdf of vT(σ_j+ δ)(vTu)(vTσ_j) is uniformly bounded away from 0 for u = 1 and δ ≤ θ.

(iv) E
(vTσ_j)2 v < ∞.

A representation of the multivariate quantile point is stated in Theorem 2. Theorem 2. Under conditions (i)–(iv),

n1/2_{( ˆq} α− qα) = n−1/2D n  i=1 u_i+ n1/2_{( ˆD − D)ξ + Dn}1/2
_{( ˆD}−1_{− D}−1_{)µ + v}_{+ o} p(1) , where u_i = (u_i1, . . . , u_ip) with u_ij = f_j−1(ξ_j)(α_j − I (x_ji ≤ ξ_j)); v = (v1, . . . , vp) with

v_j = (s_j − σ_j)T_{E(v | x}

j = ξj); fj is the pdf of xj; sjT and σjT are the jth row of ˆD−1 and

D−1, respectively; and I denotes the indicator function.

The asymptotic distribution of the multivariate quantile point completely relies on the asymptotic property of the estimator of the scale matrixD.

Having defined the multivariate quantile point and parallel multivariate quantile region, we now introduce a simple multivariate median.

Definition 3. Define a multivariate median by q(0.5) and let ˆq(0.5) denote its estimator. Corollary 3. Under conditions (i)–(iv), ˆq(0.5) has the following representation:

n1/2
_{ˆq(0.5) − q(0.5)} = n−1/2 1 2D n  i=1 ˜u_i+ Dn1/2_{( ˆD}−1_{− D}−1_{)µ + n}1/2_{( ˆD − D)˜ξ + Dn}1/2_{˜v + o} p(1) , where ˜u_i = ( ˜u_i1, . . . , ˜u_ip), ˜u_ij = f_j−1(˜ξ_j) sgn(x_ji ≤ ˜ξ_j), ˜ξ = (˜ξ1, . . . , ˜ξp), ˜ξj = F_j−1(0.5),

and ˜v = (˜v1, . . . , ˜vp), ˜vj = (sj − σj)TE(v | xj = ˜ξj). Here sgn(A) = 1, if condition A

holds, and −1 otherwise.

Compared to the multivariate medians defined by Liu (1988) and Oja (1983), this defi-nition is much simpler, and the estimate is easier to compute.

3. Multivariate trimmed means by parallelogram

The simplest way to construct a robust multivariate estimator is to take a robust esti-mator for each coordinate. The multivariate trimmed mean of this type has been studied by Gnanadesikan & Kettenring (1972). Unfortunately, this approach does not consider all variables simultaneously so that the estimators thus constructed do not have the equivariance property. We propose a multivariate trimmed mean based on the studentized observations

x_i = ˆD−1y_i, i = 1, . . . , n. Recall that sT

j is the j th row of ˆD−1, j = 1, . . . , p.

Definition 4. The multivariate trimmed mean is ˆµ_t = ˆD ˆm with ˆm = ( ˆm₁, . . . , ˆm_p), where ˆm_j = _n i=1xjiI(ˆξj1≤ xji≤ ˆξj2) _n i=1I(ˆξj1≤ xji≤ ˆξj2) .

Denote the multivariate trimmed mean by ˆµ_t(α) for simplicity if α = α11 = · · · = αp1 = 1− α₁₂= · · · = 1 − α_p2.

Theorem 4. Suppose that the sample scale matrix ˆD is such that ˆD(Ay + c) = A ˆD(y).

Here we denote ˆµ_t by ˆµ_t(y). Then ˆµ_t(Ay + c) = A ˆµ_t(y) + c.

Letσ_jT denote the j th row ofD−1. Let G_j andg_j denote the cdf and pdf of_j = σ_jTv, respectively. Denote η_j = G_j−1(α_j) and η_jk = G_j−1(α_jk). It is seen that ξ_j = σ_jTµ + η_j. Denote δ = (δ₁, δ₂, . . . , δ_p) with δ_j =_ηηj2

j1 jgj(j)dj,φ(j) = jI(ηj1≤ j ≤ ηj2) − δ_j+η_j1( I (_j < η_j1)−α_j1)+η_j2( I (_j > η_j2)−(1−α_j2)), E_vj = E(vT_I(η

j1≤ j ≤ ηj2)), andH = diag(h₁, . . . , h_p), where h_j = 1/(α_j2− α_j1).

Theorem 5. Under conditions (i)–(iv),

n1_/2
ˆµ t − (µ + ˆDH δ)  = DHn−1/2n i=1 (φ_i+ ω) + o_p(1) , where φ_i = (φ(1i), . . . , φ(pi)) and ω = (Ev1(s1− σ1), . . . , Evp(sp− σp)).

Corollary 6. Suppose that E_vj = 0 and G_j are all symmetric about zeros. Denote ˜η_j1 =

G_j−1(α) and ˜η_j2= G_j−1(1 − α). Then n1_/2
ˆµ t(α) − µ  = (1 − 2α)−1_Dn−1/2n i=1 φ0i + op(1) , where φ0i = (φ0(1i), . . . , φ0(pi)) with φ0(j) =      ˜η_{j1 j} < ˜η_j1, _j ˜η_j1≤ _j ≤ ˜η_j2, ˜η_{j2 j} > ˜η_j2, for j = 1, . . . , p, and n1_{/2( ˆµ} t(α) − µ) d → N_p(0, K), where K = (1 − 2α)−2_DTDT_, T = [τ_jk], with τ_jj = _˜ηj2 ˜ηj1 2_g j()d + 2α( ˜ηj2)2, for j = 1, . . . , p , and τ_jk= ˜η_j1˜η_k1Pr(_j < ˜η_j1, _k < ˜η_k1) + ˜η_{j1 ˜ηk2} ˜ηk1 _˜ηj1 −∞kgjk(j, k) djdk + ˜η_j1˜η_k2Pr(_j < ˜η_j1, _k> ˜η_k2) + ˜η_{k1 ˜ηk1} −∞ _˜ηj2 ˜ηj1 _jg_jk(_j, _k) d_jd_k + _˜ηk2 ˜ηk1 _˜ηj2 ˜ηj1 _jkg_jk(_j, _k) d_jd_k+ ˜η_{k2 ∞} ˜ηk2 _˜ηj2 ˜ηj1 _jg_jk(_j, _k) d_jd_k + ˜η_j2˜η_k1Pr(_j > ˜η_j2, _k< ˜η_k1) + ˜η_{j2 ˜ηk2} ˜ηk1 _∞ ˜ηj2 _kg_jk(_j, _k) d_jd_k + ˜η_j2˜η_k2Pr(_j > ˜η_j2, _k> ˜η_k2) ,

Table

Asymptotic generalized variances of estimators and Cram´er–Rao lower bounds (δ = 0.1) Estimate _{σ = 2} ρ = 0.2_{σ = 5 σ = 10 σ = 2} ρ = 0.5_{σ = 5 σ = 10} ¯ y 1.287 3.394 10.89 1.219 3.368 10.89 ˆ µt α = 0.1 1.202 1.364 1.438 1.138 1.314 1.364 α = 0.2 1.282 1.387 1.432 1.199 1.318 1.356 α = 0.3 1.359 1.495 1.531 1.299 1.437 1.453 ˆ µtt α = 0.1 1.203 1.367 1.441 1.153 1.302 1.365 α = 0.2 1.270 1.398 1.437 1.226 1.332 1.377 α = 0.3 1.388 1.483 1.519 1.332 1.414 1.453 C–R 1.152 1.152 1.115 1.017 1.013 0.983

4. Asymptotic efficiency and Monte Carlo study for the trimmed means From Theorem 5, the asymptotic efficiency of the multivariate trimmed mean relies on the performance of the estimator ˆ since the covariance matrix  affects D and φ. It is well known that the sample covariance matrix is efficient as an estimator of the covariance matrix for normal distributions, but becomes less efficient when the error vector  departs from normal distributions. It is then quite natural to suspect that using ˆD may reduce the efficiency of the multivariate trimmed mean when departs from normal distributions. We thus consider a ‘robustified’ version of the multivariate trimmed mean.

Denote the multivariate trimmed mean based on the ordinary sample covariance matrix ˆ by ˆµ_t. Let z_α denote the truncated variable obtained by restricting the random variable z on the interval [F_z−1(α), F_z−1(1 − α)], where F_z−1 is the population quantile function of z. For simplicity, consider the bivariate case. Denote the pdf of v = (v1, v2) by h and the marginal pdfs ofv₁andv₂ byh₁ andh₂, respectively. Similar to the trimmed mean for the location parameter, a robust scale matrix can be defined based on the truncated variablesv_1α andv_2α. Define the trimmed covariance matrix by_α = V(v) = [ϕ_ij] with

ϕ_ii= 1 1− 2α Ci v2 ihi(vi) dvi, ϕ12= 1 a(α) C2 C1 v1v2h(v1, v2) dv1dv2, wherea(α) = Pr(v₁∈ C₁, v₂∈ C₂), C_j = [H_j−1(α), H_j−1(1 − α)], j = 1, 2, and H_j−1 is the population quantile function of the variablev_j. Let ˆ_α be an estimator of _α satisfying assumption (ii). Denote the robustified bivariate trimmed mean based on ˆ_α by ˆµ_tt.

To compare the sample mean ¯y, the trimmed mean ˆµ_t, and the robustified trimmed mean ˆµ_tt, we consider the error vector

v d =

N2(0, R) with probability 1− δ ,

N₂(0, σ2I) with probability δ , where R = 

1 ρ

ρ 1

, (2)

for 0≤ δ ≤ 1. When δ = 0, v has a bivariate normal distribution. Define the generalized variance as the determinant of the covariance matrix. Consider δ = 0.1, ρ = 0.2, 0.5, and σ = 3, 5, 10. Table 1 gives the square-root of the asymptotic generalized variances of ¯y, ˆµ_t, and ˆµ_tt for the cases α = 0.1, 0.2, 0.3. The Cram´er–Rao lower bounds (C-R) for the distributions under study are also included.

Table

Asymptotic generalized variances of the two trimmed means (δ = 0.3, ρ = 0.5, σ = 10)

ˆ

µt µˆtt

α = 0.1 α = 0.2 α = 0.3 α = 0.1 α = 0.2 α = 0.3 ρ = 0.5

σ = 10 1.263 1.299 1.421 1.328 1.311 1.432

The following are some observations from Table 1.

(a) Relatively, the sample mean ¯y has larger asymptotic generalized variances than the two trimmed means. This confirms that ¯y is quite sensitive to the outliers. It also shows that the two trimmed means are fairly robust, as expected.

(b) The two trimmed means have nearly the same efficiency.

(c) When the correlation coefficient ρ gets larger, the asymptotic generalized variances of the two trimmed means get smaller.

(d) Withδ = 0.1, which means that approximately 10% of observations are drawn from a distribution with larger variance, a 10% trimming percentage seems to be reasonable for both of the trimmed means under study. A similar observation has been made for the univariate trimmed means under the location and linear regression models (Ruppert & Carroll, 1980; Chen & Chiang, 1996).

The two trimmed means are quite competitive in the above study. Would the trimmed mean based on the sample covariance matrix ˆ be relatively less efficient than the one based on the trimmed covariance matrix ˆ_α, if the error distribution had more outliers? We compute the asymptotic generalized variances of these two trimmed means for the error distribution of (2) withδ = 0.3 and σ = 10. Table 2 lists the results.

Surprisingly, the generalized variances are all smaller for ˆµ_t than for ˆµ_tt, which indicates that the sample covariance matrix ˆ is a better choice.

To study the trimmed means under the multivariate model with asymmetric error distri-butions, we perform a Monte Carlo simulation for the multivariate location modely = µ + v. Letz = (z₁, z₂) denote a vector of two independent exponential random variables with mean 1. Assume that the error vector v = (v₁, v₂) has the following mixed distribution:

v d

=  Gz with probability 1 − δ ,

σ z with probability δ , where G =  

1− ρ2 _ρ

0 1

.

This design ensures thatv has a zero-mean asymmetric distribution, and has either a covari-ance matrixR with probability (1 − δ) or a covariance matrix σ2I with probability δ. Note that large values ofσ may produce outliers.

In this study, we letµ = (1, 1) and consider the cases δ = 0.1, 0.2, ρ = 0.2, 0.5, 0.8, andσ = 2, 5, 10. The sample size is n = 30. For each case, we simulate 1000 sets of data from the above mixture model. Fori = 1, . . . , 1000, let ˆµ_i stand for the estimate of the i th replicate for the mean estimator ˆµ, where ˆµ can be any of the three mean estimators under study. With 1000 replicates, we compute the averaged mean squared error (AMSE) of the mean estimators defined by

AMSE( ˆµ) = 1 1000 1000_ i=1 ( ˆµ_i− µ)T_{( ˆµ} i− µ) ,

Table

AMSEof the three mean estimators (ρ = 0.2)

Estimate δ = 0.1 δ = 0.2 σ = 2 σ = 5 σ = 10 σ = 2 σ = 5 σ = 10 ¯ y 86.50 237.2 697.7 111.8 376.1 1370 ˆ µt α = 0.1 0.059 0.074 0.111 0.069 0.116 0.311 α = 0.2 0.088 0.098 0.108 0.099 0.112 0.144 α = 0.3 0.120 0.132 0.144 0.132 0.146 0.161 ˆ µtt α = 0.1 0.105 0.128 0.191 0.121 0.228 0.586 α = 0.2 0.165 0.173 0.184 0.180 0.201 0.251 α = 0.3 0.220 0.232 0.247 0.239 0.253 0.265

Table

AMSEof the three mean estimators (ρ = 0.5)

Estimate _{σ = 2} δ = 0.1_{σ = 5 σ = 10 σ = 2} δ = 0.2_{σ = 5 σ = 10} ¯ y 86.89 211.2 765.6 118.9 381.3 1422 ˆ µt α = 0.1 0.073 0.083 0.119 0.085 0.132 0.302 α = 0.2 0.112 0.117 0.124 0.124 0.139 0.163 α = 0.3 0.151 0.160 0.164 0.163 0.182 0.196 ˆ µtt α = 0.1 0.102 0.123 0.189 0.128 0.231 0.563 α = 0.2 0.160 0.162 0.170 0.184 0.204 0.249 α = 0.3 0.214 0.214 0.223 0.236 0.252 0.263

Table

AMSEof the three mean estimators (ρ = 0.8)

Estimate _{σ = 2} δ = 0.1_{σ = 5 σ = 10 σ = 2} δ = 0.2_{σ = 5 σ = 10} ¯ y 85.43 218.3 723.9 101.3 377.7 1320 ˆ µt α = 0.1 0.083 0.100 0.125 0.092 0.150 0.317 α = 0.2 0.130 0.142 0.139 0.142 0.169 0.188 α = 0.3 0.176 0.194 0.193 0.191 0.218 0.226 ˆ µtt α = 0.1 0.097 0.121 0.186 0.118 0.226 0.530 α = 0.2 0.144 0.156 0.157 0.174 0.208 0.241 α = 0.3 0.187 0.203 0.201 0.227 0.250 0.251

We observe the following for the asymmetric distribution from Tables 3, 4, and 5. (a) Both the trimmed means perform better than the sample mean.

(b) Again, ˆµ_t performs better than ˆµ_tt. It seems that the robustified trimmed mean does not benefit from trimming the covariance matrix. This is somewhat surprising. The trimmed mean using the ordinary sample covariance matrix attains good efficiency.

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