Multivariate descriptive measures for location and scatter are the foundation of multivariate statistics and underpin many methods in the field. The classical notions of multivariate location and scatter are moment-based and are estimated by the sample mean vector and the sample covariance matrix. However, these estimators are not robust and are extremely sensitive to outlier(s). Many efforts have been made to solve this problem, and various estimators for multivariate location and scatter have been proposed. The widely used ones include the M-estimators [1], the minimum volume ellipsoid (MVE) and the minimum covariance determinant (MCD) estimators [2], the S-estimators [3] [4], the τ -estimators [5], the constrained M-estimators [6], and so on.

In addition, various nonparametric estimators for multivariate location and scatter have been developed via depth functions. Generally, a depth function D F ( x ) is a nonnegative real-valued mapping which provides a distribution-based center-outward ordering of points x in ℝ d . Desirable properties for a depth function are: 1) affine invariance (i.e., D F A X + b ( A X + b ) = D F X ( x ) for any nonsingular d × d matrix A and d -vector b ); 2) maximality at “center”; 3) monotonicity relative to the deepest point; 4) vanishing at infinity. Widely used depth functions include the halfspace depth [7], the projection depth [8] [9], the simplicial depth [10], the L p depth [11], and the Mahalanobis depth [8] [12]. The general discussions on depth functions and their applications can be found in [11] [13] [14].

Given a depth function D F ( x ) , the deepest point plays the role of “center” of the distribution F and thus is considered as a multivariate median, denoted by M F . The α depth inner region is defined as I F ( α ) = { x ∈ ℝ d : D F ( x ) ≥ α } , α ≥ 0 , and the pth central region as C F ( p ) = I F ( α ( p ) ) with α ( p ) = sup α { α : P ( I F ( α ) ) ≥ p } , i.e., the smallest α -depth inner region having a probability weight of at least p , 0 ≤ p ≤ 1 . The boundary ∂ I F ( α ) of I F ( α ) is called the α depth contour and can be characterized by the radius function,

r F ( α , u ) = inf { r ≥ 0 : r u ∉ I F ( α ) } , u ∈ S d − 1 = { u ∈ ℝ d : ‖ u ‖ = 1 } ,

when M F is the origin. If the underlying distribution F is continuous, the random depth D F ( X ) is continuous for all the depths mentioned above, and then p = P ( C F ( p ) ) = P ( D F ( X ) ≥ α ( p ) ) = 1 − F D ( α ( p ) ) . Therefore, α ( p ) = F D − 1 ( 1 − p ) and C F ( p ) = I F ( F D − 1 ( 1 − p ) ) .

Utilizing a depth function D F ( x ) , one can define a depth weighted mean as

L ( F ) = ∫ x w 1 ( D F ( x ) ) d F ( x ) / ∫ w 1 ( D F ( x ) ) d F ( x )

and a depth weighted scatter matrix as

S ( F ) = ∫ ( x − L ( F ) ) ( x − L ( F ) ) ′ w 2 ( D F ( x ) ) d F ( x ) / ∫ w 2 ( D F ( x ) ) d F ( x ) ,

where w 1 ( ⋅ ) and w 2 ( ⋅ ) are suitable weight functions and can be different. Conceptionally, the general depth weighted means and scatter matrices include depth trimmed means and trimmed scatter matrices as a special case. However, when studying the properties of the depth weighted means and scatter matrices, some conditions on the weight functions were invoked such that the usual trimmed means and scatter matrices were excluded. For example, to establish the asymptotic distributions of the sample version of L ( F ) , Dümbgen [15], Massé [16], and Zuo, Cui, and He [17] all made the assumption that the weight function w 1 ( t ) is continuously differentiable, which excluded the usual trimmed mean for which the weight function is an indicator function. For the finite sample breakdown point of the sample version of S ( F ) , Zuo and Cui [18] assumed that both weight functions are positive, which ruled out any trimmed scatter matrices.

It is well-known that the univariate trimmed means are not only robust but also highly efficient (much more efficient than the univariate median). It is conjectured that multivariate trimmed means and scatter matrices have the same properties, which motivates this work. In fact, the α depth trimmed means based on the projection depth have been studied by Zuo [19], and the ones based on the halfspace depth have been studied by Donoho and Gasko [20], Massé [21], and Wang [22], with the α depth trimmed means defined as

μ F ( α ) = ∫ I F ( α )   x w 1 ( D F ( x ) ) d F ( x ) / ∫ I F ( α )   w 1 ( D F ( x ) ) d F ( x ) .

However, since the depth value of a point depends on both the depth function and the underlying distribution, it is very hard to choose an appropriate α value for μ F ( α ) in practice, and thus their applications are thwarted. Table 1 lists the theoretical trimming proportions 1 − p for the μ F ( α ) based on the halfspace depth with α = 0.01 when the underlying distributions are the d -dimensional standard normal distribution N d ( 0 , I d × d ) , the d -dimensional t distribution with 5 degrees of freedom M t d ( 5, 0 , I d × d ) , and the d -dimensional spherically symmetric uniform distribution M U d ( 0 , I d × d ) , respectively. The formula for the calculation is: 1 − p = P ( ‖ Z ‖ > − F 1 − 1 ( α ) ) , where random vector Z follows a spherically symmetric distribution centered at 0 , ‖ ⋅ ‖ denotes the Euclidean norm, and F 1 − 1 is its univariate marginal quantile function. The details for the formula can be seen from the proof of Corollary 1 and the following remark.

From the table, we see that the theoretical trimming proportion 1 − p varies enormously from distribution to distribution. It is even harder to choose the α value for many practical problems, in which the underlying distribution is unknown. Therefore, it is necessary to restudy multivariate trimmed means. As for multivariate trimmed scatter matrices, we have not seen any investigation in the literature.

In this paper, we propose and study some new general depth-based trimmed means and trimmed scatter matrices. Their definitions, basic properties, and algorithms are given in Section 2. The asymptotic distributions of their sample versions are established in Section 3. Using the asymptotic distributions, we study the asymptotic efficiencies of the sample trimmed means and the sample trimmed scatter matrices in Section 4. The results show that both the trimmed means and trimmed scatter matrices are highly efficient. Section 5 is devoted to the robustness of the sample trimmed means and the sample trimmed scatter matrices, which is explored through finite-sample breakdown point. The proofs of main results are reserved for the Appendix.

Throughout this paper, matrices are denoted by bold letters and vectors by bold italicized letters. We use uppercase letters to denote distribution functions and their lowercase counterparts to denote density functions if they exist. For example, we denote by F X and f X the cdf and density of a random vector X in ℝ d . When X is a random variable, the quantile function of X is denoted by F X − 1 . Without confusion, we will omit the subscript. The indicator function of a set A is denoted by I A ( ⋅ ) and the transpose of a matrix or a vector v is denoted by v ′ . For an m × k matrix M , v e c ( M ) denotes the m k -dimensional vector formed by stacking the columns of M , and B ⊗ C denotes the Kronecker product of matrices B and C .

Let w 1 ( t ) and w 2 ( t ) be suitable weight functions on [ 0, α F ∗ ] , where α F ∗ = sup x ∈ ℝ d ( D F ( x ) ) . To reduce or eliminate the impacts of outliers, we assume that w 1 ( t ) and w 2 ( t ) are nondecreasing. Given p 1 and p 2 in ( 0,1 ] , we define the general depth-based trimmed mean of trimming proportion 1 − p 1 as

μ F ( p 1 ) = ∫ C F ( p 1 )   x w 1 ( D F ( x ) ) d F ( x ) / ∫ C F ( p 1 )   w 1 ( D F ( x ) ) d F ( x )

and define the general depth-based trimmed scatter matrix of trimming proportion 1 − p 2 using the location measure μ F ( p 1 ) as

Σ F ( p 1 , p 2 ) = ∫ C F ( p 2 )   ( x − μ F ( p 1 ) ) ( x − μ F ( p 1 ) ) ′ w 2 ( D F ( x ) ) d F ( x ) ∫ C F ( p 2 )   w 2 ( D F ( x ) ) d F ( x ) .

As a special case, μ F ( 1 ) = L ( F ) and Σ F ( 1,1 ) = S ( F ) . Furthermore, μ F ( 1 ) = E ( X ) if w 1 ( t ) = 1 on [ 0, α F ∗ ] , and Σ F ( 1,1 ) = C o v ( X ) if w 1 ( t ) = w 2 ( t ) = 1 on [ 0, α F ∗ ] . In general, if p 1 ≠ 1 and p 2 ≠ 1 , μ F ( p 1 ) and Σ F ( p 1 , p 2 ) are much more broadly defined than E ( X ) and C o v ( X ) , respectively. They do not require existence of any moments.

Remark. Unlike L ( F ) and S ( F ) , we separate weighting and trimming in μ F ( p 1 ) and Σ F ( p 1 , p 2 ) with trimming controlled by C F ( p 1 ) and C F ( p 2 ) , and thus weight functions are not affected by trimming. For example, the weight function for the usual trimmed mean can be w 1 ( t ) = 1 on [ 0, α F ∗ ] , which is continuously differentiable.

For simplicity, we will confine attention to the case w 1 ( t ) = w 2 ( t ) : = w ( t ) and p 1 = p 2 : = p , and call μ F ( p ) and Σ F ( p ) the 1 − p proportion or 100 ( 1 − p ) % general depth-based trimmed mean and trimmed scatter matrix, respectively. Given a sample X n = { X 1 , ⋯ , X n } , the sample versions of D F ( x ) , I F ( α ) , C F ( p ) , r F ( α , u ) , μ F ( p ) , and Σ F ( p ) are obtained by replacing F with F n , the empirical distribution function of X n , and are denoted by D n ( x ) , I n ( α ) , C n ( p ) , r n ( α , u ) , μ n ( p ) , and Σ n ( p ) , respectively. For convenience, we will suppress F in D F ( x ) , I F ( α ) , C F ( p ) , and r F ( α , u ) in the following discussions unless it is necessary for clarity. μ F ( p ) , Σ F ( p ) , and their sample versions have the following basic properties. The proofs are relatively straightforward and thus omitted.

Theorem 1. Suppose that the depth function D F ( x ) is affine invariant. Then we have the following results:

1) μ F ( p ) and Σ F ( p ) are affine equivariant, that is, μ F A X + b ( p ) = A μ F X ( p ) + b and Σ F A X + b ( p ) = A Σ F X ( p ) A ′ for any nonsingular d × d matrix A and d -vector b , and so are their sample versions.

2) If F is centrally symmetric about θ ∈ ℝ d and has the first moment, then for any p ∈ ( 0,1 ] , μ F ( p ) = θ (i.e., μ F ( p ) is Fisher consistent for θ ) and E ( μ n ( p ) ) = θ .

3) If X follows an elliptically symmetric distribution F and has the second moment, then Σ F ( p ) = k C o v ( X ) and E ( Σ n ( p ) ) = k n C o v ( X ) , where k and k n are some positive constants.

Computation of μ n ( p ) and Σ n ( p ) is quite easy. The following are the algorithms by R.

1) Compute the depths of all points in the sample. Several R packages are available for depth-based statistical analysis. For the halfspace depth, the projection depth, the L p depth, and the Mahalanobis depth, one can use the command “depth” in the R package “DepthProc” of Zawadzki, Kosiorowski, Slomczynski, Bocian, and Wegrzynkiewicz [23].

2) Order the data points according to their depths decreasingly, denoted by X ( 1 ) , ⋯ , X ( n ) , and get the points in C n ( p ) ,.

3) Compute μ n ( p ) as and compute Σ n ( p ) as.

To use an estimator or a statistic for inferences, its distribution or asymptotic distribution is necessary. To establish the asymptotic distributions of μ n ( p ) and Σ n ( p ) , we invoke the following conditions.

( C 1 ) D ( x ) is affine invariant, and n 1 / 2 ( D n ( x ) − D ( x ) ) = ∫ φ ( x , y ) d v n ( y ) + o p ( 1 ) uniformly in x , where v n = n 1 / 2 ( F n − F ) ;

Then, by Theorem 1 1), both μ n ( p ) − μ F ( p ) and Σ n ( p ) − Σ F ( p ) are invariant for any translation. Thus, we can assume without loss of generality that the origin is the deepest point.

( C 2 ) r ( α , u ) has a continuous partial derivative with respect to α , and n 1 / 2 ( r n ( α , u ) − r ( α , u ) ) = ∫ ψ ( x , α , u ) d v n ( x ) + o p ( 1 ) with ψ ( x , α , u ) being continuous in α ;

( C 3 ) There exists a Donsker class D of sets such that I ( α ) ∈ D and I n ( α ) ∈ D almost surely for any n and α ∈ ( 0, α F ∗ ) ;

( C 4 ) F has a positive continuous density f in some neighborhood of ∂ C ( p ) ;

( C 5 ) w ( 1 ) ( t ) = d d t w ( t ) is continuous and w ( t ) > 0 for t ≥ F D − 1 ( 1 − p ) with p ∈ ( 0,1 ) .

Remark. Uniform weak convergence of n 1 / 2 ( D n ( x ) − D F ( x ) ) has been established for the simplicial depth by Dümbgen [15] and Arcones and Giné [24], for the halfspace depth by Massé [16], for the projection depth and some generalized halfspace depth by Arcones, Cui, and Zuo [25], and thus ( C 1 ) is satisfied for these depth functions. For a fixed α ∈ ( 0, α F ∗ ) , the asymptotic representation of n 1 / 2 ( r n ( α , u ) − r F ( α , u ) ) was derived for the halfspace depth by Nolan [26] and for the projection depth by Zuo [19]. Considering α there as a variable, we see that ( C 2 ) is satisfied by these depth functions. As for Condition ( C 3 ) , we know that I n ( α ) is an ellipsoid for the Mahalanobis depth, a convex set formed by halfspaces for the halfspace depth and projection depth, and a union of some convex sets formed by halfspaces for the simplicial depth. Since {all ellipsoids in ℝ d } is a Vapnik-Červonenkis (VC) class [27] and so is {all sets in ℝ d formed by halfspaces} [24] [28] [29]. Condition ( C 3 ) is satisfied for these depths.

Remark. It can be shown that every Donsker class is a Glivenko-Cantelli class almost surely [27].

Define Δ F ( p ) = ∫ C ( p )   x x ′ w ( D ( x ) ) d F ( x ) ∫ C ( p )   w ( D ( x ) ) d F ( x ) . Then we have the following results.

Theorem 2. Under the conditions ( C 1 ) - ( C 5 ) , for any p ∈ ( 0,1 )

1) μ n ( p ) − μ F ( p ) = 1 n ∑ i = 1 n   l ( X i ) − E ( l ( X i ) ) + o p ( 1 / n ) , and thus

n ( μ n ( p ) − μ F ( p ) ) → d N d ( 0 , C o v ( l ( X ) ) ,

where l ( x ) = ( l 1 ( x ) + l 2 ( x ) + l 3 ( x ) + l 4 ( x ) ) / l 0 ( p ) with

l 0 ( p ) = ∫ C ( p )   w ( D ( x ) ) d F ( x ) ,

l 1 ( x ) = ( x − μ F ( p ) ) w ( D ( x ) ) I C ( p ) ( x ) ,

l 2 ( x ) = ∫ C ( p )   ( y − μ F ( p ) ) w ( 1 ) ( D ( y ) ) φ ( y , x ) d F ( y ) , where w ( 1 ) ( t ) = d d t w ( t ) ,

l 3 ( x ) = ∫ S d − 1 ( r ( F D − 1 ( 1 − p ) , u ) u − μ F ( p ) ) w ( D ( r ( F D − 1 ( 1 − p ) , u ) u ) )                     f ( r ( F D − 1 ( 1 − p ) , u ) u ) | J ( r ( F D − 1 ( 1 − p ) , u ) , u ) | ψ ( x , F D − 1 ( 1 − p ) , u ) d u ,

where S d − 1 = { u ∈ ℝ d : ‖ u ‖ = 1 } and J ( r , u ) is the Jacobian of

the transformation x = r u with r = ‖ x ‖ and u = x ‖ x ‖ , and

l 4 ( x ) = − l 0 ( p ) μ F ( 1 ) ( p ) [ I C ( p ) ( x ) + ∫ S d − 1   f ( r ( F D − 1 ( 1 − p ) , u ) u )                     | J ( r ( F D − 1 ( 1 − p ) , u ) , u ) | ψ ( x , F D − 1 ( 1 − p ) , u ) d u ] ,

where μ F ( 1 ) ( p ) = d d p μ F ( p ) .

2) Σ n ( p ) − Σ F ( p ) = 1 n ∑ i = 1 n   k ( X i ) − E ( k ( X i ) ) + o p ( 1 / n ) , and thus

n ( v e c ( Σ n ( p ) ) − v e c ( Σ F ( p ) ) ) → d N d 2 ( 0 , C o v ( v e c ( k ( X ) ) ) ) ,

where k ( x ) = ( k 1 ( x ) + k 2 ( x ) + k 3 ( x ) + k 4 ( x ) ) / l 0 ( p ) − μ F ( p ) l ′ ( x ) − l ( x ) μ ′ F ( p ) with

k 1 ( x ) = ( x x ′ − Δ F ( p ) ) w ( D ( x ) ) I C ( p ) ( x ) ,

k 2 ( x ) = ∫ C ( p ) ( y y ′ − Δ F ( p ) ) w ( 1 ) ( D ( y ) ) φ ( y , x ) d F ( y ) ,

k 3 ( x ) = ∫ S d − 1 ( r 2 ( F D − 1 ( 1 − p ) , u ) u u ′ − Δ F ( p ) ) w ( D ( r ( F D − 1 ( 1 − p ) , u ) u ) )                       f ( r ( F D − 1 ( 1 − p ) , u ) u ) | J ( r ( F D − 1 ( 1 − p ) , u ) , u ) | ψ ( x , F D − 1 ( 1 − p ) , u ) d u ,

k 4 ( x ) = − l 0 ( p ) Δ F ( 1 ) ( p ) [ I C ( p ) ( x ) + ∫ S d − 1   f ( r ( F D − 1 ( 1 − p ) , u ) u )                       | J ( r ( F D − 1 ( 1 − p ) , u ) , u ) | ψ ( x , F D − 1 ( 1 − p ) , u ) d u ] ,

where Δ F ( 1 ) ( p ) = d d p Δ F ( p ) .

Both C o v ( l ( X ) ) and C o v ( v e c ( k ( X ) ) are quite complicated in general. However, the results are relatively simple when w ( t ) = 1 on [ 0, α F ∗ ] and F is an elliptically symmetric distribution. As an example, we concretize C o v ( l ( X ) ) and C o v ( v e c ( k ( X ) ) for the halfspace depth-based trimmed means and scatter matrices for the very important special case. A continuous random vector X in ℝ d is said to have an elliptically symmetric distribution, denoted by E d ( h ; θ , Λ ) , if it has a density of the form

f ( x ) = | Λ | − 1 / 2 h ( ( x − θ ) ′ Λ − 1 ( x − θ ) ) , x ∈ ℝ d

for a nonnegative function h ( ⋅ ) with ∫ 0 ∞ r d / 2 − 1 h ( r ) d r < ∞ and a positive definite matrix Λ . If the first moment of X exists, E ( X ) = θ . If the second moment of X exists, C o v ( X ) = [ E ( ‖ Λ − 1 / 2 ( X − θ ) ‖ 2 ) / d ] Λ , where ‖ ⋅ ‖ denotes the Euclidean norm. When Λ = I d × d , the d × d identity matrix, X is said to have a spherically symmetric distribution with center θ . If X ~ E d ( h ; θ , Λ ) , Z = Λ − 1 / 2 ( X − θ ) ~ E d ( h ; 0 , I d × d ) . Let U = Z ‖ Z ‖ . Then U has a uniform distribution on S d − 1 = { u ∈ ℝ d : ‖ u ‖ = 1 } , and thus E ( U ) = 0 and C o v ( U ) = E ( U U ′ ) = 1 d I d × d . In addition, U and ‖ Z ‖ are independent. See Fang, Kotz, and Ng [30] for broad discussion about elliptically symmetric distributions.

Because the trimming based on the halfspace depth is a natural extension of the usual univariate trimming, we choose the halfspace depth here. The halfspace depth is first introduced by Tukey [7] and is defined as

H D F ( x ) = inf { P ( H ) : x ∈ H ∈ H } , x ∈ ℝ d ,

where H = {all closed halfspaces}. As a leading depth function, the halfspace depth has been widely used. The finite-sample breakdown point of the halfspace depth was obtained by Donoho and Gasko [20]. Romanazzi [31] derived the influence function of H D F ( x ) . The asymptotic distribution of its sample version H D n ( x ) was established by Massé [16]. Nolan [26] studied the asymptotics of the sample α halfspace depth inner regions. An important feature of the trimming based on the halfspace depth is that it trims off the same proportion of data points in a closed halfspace at each direction normal to the boundary of C n ( p ) and outward with respect to C n ( p ) .

Corollary 1. Suppose that X ~ E d ( h ; θ , Λ ) with a positive continuous density f in a small neighborhood of ∂ C ( p ) and w ( t ) = 1 on [ 0, α F ∗ ] . Let f 1 and F 1 be the univariate marginal density and the marginal distribution function of Z = Λ − 1 / 2 ( X − θ ) . If the trimming is based on the halfspace depth, then for any p ∈ ( 0,1 ) and d ≥ 2 .

1) n ( μ n ( p ) − μ F ( p ) ) → d N d ( 0 , V 1 ) , where V 1 = E [ c 1 2 ( ‖ Z ‖ ) + c 2 2 ( ‖ Z ‖ ) ] Λ d with c 1 ( ‖ z ‖ ) = 1 p ‖ z ‖ I { ‖ z ‖ ≤ F ‖ Z ‖ − 1 ( p ) } ( z ) , and

c 2 ( ‖ z ‖ ) = F ‖ Z ‖ − 1 ( p ) f ‖ Z ‖ ( F ‖ Z ‖ − 1 ( p ) ) ( d − 1 ) p f 1 ( F ‖ Z ‖ − 1 ( p ) ) ∫ 0 π sin d − 2 θ 1 d θ 1 [ 1 − ( F ‖ Z ‖ − 1 ( p ) ‖ z ‖ ) 2 ] ( d − 1 ) / 2 I { ‖ z ‖ > F ‖ Z ‖ − 1 ( p ) } ( z ) .

2) n ( v e c ( Σ n ( p ) ) − v e c ( Σ F ( p ) ) ) → d N d 2 ( 0 , V 2 ) , where V 2 = σ 1 ( I d 2 × d 2 + K d , d ) ( Λ ⊗ Λ ) + σ 2 v e c ( Λ ) v e c ( Λ ) ′ with

σ 1 = E ( c 3 2 ( ‖ Z ‖ ) ) / ( d ( d + 2 ) ) ,

σ 2 = σ 1 + 2 E ( c 3 ( ‖ Z ‖ ) c 4 ( ‖ Z ‖ ) ) / d + E ( c 4 2 ( ‖ Z ‖ ) ) − [ E ( c 3 ( ‖ Z ‖ ) ) / d + E ( c 4 ( ‖ Z ‖ ) ) ] 2 ,

c 3 ( ‖ z ‖ ) = 1 p ‖ z ‖ 2 I { ‖ z ‖ ≤ F ‖ Z ‖ − 1 ( p ) } ( z ) + [ F ‖ Z ‖ − 1 ( p ) ] 2 f ‖ Z ‖ ( F ‖ Z ‖ − 1 ( p ) ) d p f 1 ( F ‖ Z ‖ − 1 ( p ) ) ∫ 0 π sin d θ 1 d θ 1                         [ 1 − ( F ‖ Z ‖ − 1 ( p ) ‖ z ‖ ) 2 ] ( d − 1 ) / 2 ( F ‖ Z ‖ − 1 ( p ) ‖ z ‖ ) I { ‖ z ‖ > F ‖ Z ‖ − 1 ( p ) } ( z ) ,

c 4 ( ‖ z ‖ ) = [ F ‖ Z ‖ − 1 ( p ) ] 2 f ‖ Z ‖ ( F ‖ Z ‖ − 1 ( p ) ) d p f 1 ( F ‖ Z ‖ − 1 ( p ) ) [ ∫ 0 arccos ( F ‖ Z ‖ − 1 ( p ) / ‖ z ‖ ) sin d θ 1 d θ 1 ∫ 0 π sin d θ 1 d θ 1                         − ∫ 0 arccos ( F ‖ Z ‖ − 1 ( p ) / ‖ z ‖ ) sin d − 2 θ 1 d θ 1 ∫ 0 π sin d − 2 θ 1 d θ 1 ] I { ‖ z ‖ > F ‖ Z ‖ − 1 ( p ) } ( z )                         − 1 d p [ F ‖ Z ‖ − 1 ( p ) ] 2 I { ‖ z ‖ ≤ F ‖ Z ‖ − 1 ( p ) } ( z ) ,

and

K d , d being a ( d 2 × d 2 ) -block matrix with ( i , j ) -block being δ j i , which is a d × d matrix with 1 at entry ( j , i ) and 0 elsewhere.

When we evaluate the performance of an estimator, two important criteria are efficiency and robustness. Are μ n ( p ) and Σ n ( p ) efficient? We will answer this question in this section. It is well-known that the sample mean vector X ¯ n and the sample covariance matrix S n are the most efficient unbiased estimators for the population mean vector μ and covariance matrix Σ , respectively, when the population distribution is N d ( μ , Σ ) . It is natural to compare μ n ( p ) with X ¯ n and Σ n ( p ) with S n for efficiency.

Now we explore the robustness of μ n ( p ) and Σ n ( p ) based on the projection depth by finite-sample breakdown point. Donoho and Huber [35] discussed two types of finite-sample breakdown points: replacement breakdown point (RBP) and addition breakdown point (ABP). We employ the finite-sample replacement breakdown point in this section. Generally speaking, the finite-sample replacement breakdown point of an estimator is the minimum fraction of contaminated points (outliers or inliers) in a data set such that the estimator breaks down. Specifically, suppose that any m points in a sample X n = { x 1 , ⋯ , x n } are replaced (contaminated) by m arbitrary points with the contaminated data set denoted by X m n . Then, the finite-sample replacement breakdown point of a location estimator T at X n is defined as

RBP ( T , X n ) = min { m n : sup X m n ‖ T ( X m n ) − T ( X n ) ‖ = ∞ } ,

and the finite-sample replacement breakdown point of a scatter estimator V at X n is defined as

RBP ( V , X n ) = min { m n : trace [ V ( X n ) V ( X m n ) − 1 + V ( X n ) − 1 V ( X m n ) ] = ∞ } .

Essentially, the breakdown of V means that the largest eigenvalue λ 1 ( V ( X m n ) ) of V ( X m n ) can be made arbitrarily large (explosion breakdown) or the smallest eigenvalue λ d ( V ( X m n ) ) of V ( X m n ) can be made arbitrarily close to 0 (implosion breakdown). Usually, explosion breakdown is caused by outliers and implosion breakdown by inliers. However, the breakdown of a location estimator is only caused by outliers unless it is related to some scatter estimator. Before we give our main result, we briefly recall the projection depth.

The projection depth was first introduced by Liu [8] and was thoroughly studied by Zuo [9]. Given a univariate location estimator μ and a univariate scale estimator σ , the outlyingness of a point x in ℝ d with respect to a sample X n = { x 1 , ⋯ , x n } is defined as

O X n ( x ) = sup u ∈ S d − 1 | u ′ x − μ ( u ′ X n ) σ ( u ′ X n ) | ,

where S d − 1 = { u ∈ ℝ d : ‖ u ‖ = 1 } and u ′ X n = { u ′ x 1 , ⋯ , u ′ x n } . Then the projection depth of x with respect to X n is defined as

P D X n ( x ) = 1 / ( 1 + O X n ( x ) ) .

It is clear that P D X n ( x ) depends on μ and σ , and so does any estimator based on P D X n ( x ) . For the best robustness of an estimator based on P D X n ( x ) , the pair ( μ , σ ) = ( Med , MAD k ) is often chosen, where Med is the univariate median and MAD_k is a modified median absolute deviation (MAD): MAD k ( X n ) = Med k { | x i − Med ( X n ) | , i = 1, ⋯ , n } with for k = 0, ⋯ , n − 1 and x ( 1 ) , ⋯ , x ( n ) being the ordered values of X n = { x 1 , ⋯ , x n } in ℝ 1 from the smallest to the largest, where is the ceiling function and the floor function. When k = 0 , MAD k = MAD the usual median absolute deviation. It is well-known that. As for MAD k , Gather and Hilker [36] showed that

when k = c ( X n ) , where c ( X n ) denotes the maximal multiplicity of points in X n .

Unlike the trimming based on the halfspace depth, the trimming based on the projection depth is done based on only the depth values of data points regardless of their directions. This feature can significantly increase the finite-sample breakdown points of μ n ( p ) and Σ n ( p ) . That is why we choose the projection depth here. To make things clear, C n ( p ) , μ n ( p ) , and Σ n ( p ) at X n are denoted by C X n ( p ) , μ X n ( p ) , and Σ X n ( p ) , respectively, in the following discussion. If a data set X n in ℝ d is in general position, that is, no more than d points in X n are contained in any ( d − 1 ) -dimensional hyperplane, we have the following results.

Theorem 4. Let μ n ( p ) and Σ n ( p ) , p ∈ ( 0,1 ) , be the 1 − p proportion trimmed mean and trimmed scatter matrix based on the projection depth with ( μ , σ ) = ( Med , MAD d ) . Suppose that X n is in general position and ‖ z w ( P D n ( z ) ) ‖ → ∞ as ‖ z ‖ → ∞ for any z in ℝ d . Then

1) for n > d , and

2) for n such that.

From this theorem, we see that RBP for all the trimmed means with, i.e., with at least points trimmed off. However, RBP for the only trimmed scatter matrix with, i.e., with points trimmed off. It is well-known that is the upper bound of RBP of any affine equivariant scatter estimators [3]. Also, is the highest RBP of almost all existing affine equivariant location estimators.

The author declares no conflicts of interest regarding the publication of this paper.