Least Favorable Radius

Consequentially,

lim sup

n→∞

ω^min_c,n = lim sup

n→∞

trAn

En|AnΛn−an| ≤ω_c^min (2.1.143)

∗=v, k= 1 : The minimum bias given in Theorem1.3.9(b) reads ω_v,n^min= E_n(Λ_n)₊−1

(2.1.144) where

lim sup

n→∞

En(Λn)+≤lim sup

n→∞

En|Λn| ≤lim sup

n→∞

pEn(Λn)² (2.1.145)

=√

I <∞ (2.1.146)

Since L_n(Λ_n)−→w L(Λ) by assumption (2.1.103), an application of Vitali’s theorem yields

n→∞lim En(Λn)+= E(Λ)+ (2.1.147)

i.e., (2.1.106) holds. ////

Remark 2.1.12 (a) As shown at the beginning of the proof of Theorem 2.1.11, conditions (2.1.102) and (2.1.103) imply the convergence of the corresponding Fisher information

Iθ_n= Eθ_nΛθ_nΛ^τ_θ

n→EθΛθΛ^τ_θ=Iθ (2.1.148) (b)In view of the approximation used in the proof of Lemma 6.4.4 of Rieder (1994) it might even be possible to abandon assumption (2.1.102).

(c)By the convergences stated in the previous theorem we at once get E|˜η_θn|²= trA_θn−r_n²b²_θ

n −→trA_θ−r²b²_θ= E|˜η_θ|² (2.1.149) as n→ ∞. That is, the trace of the asymptotic covariance is continuous in θ∈Θ . (d) In view of Ruckdeschel (2005b) a generalization of Proposition 2.1.11 to arbitrary D∈R^p×k with rkD=p seems to be in reach. ////

2.2 Least Favorable Radius

Given a neighborhood radius, we can now determine the optimally robust ICs via the implicit equations stated in Subsections 1.3.3and 1.3.4. But, in most applica-tions the neighborhood radius is unknown or unknown except to belong to some radius interval, respectively. That is, the radius appears as a one-dimensional nui-sance parameter of the infinitesimal neighborhood models introduced in Section1.2.

InRieder et al.(2001) we numerically solve the implicit equations for location, scale and linear regression models and calculate the increase of the maximum asymptotic risk over the minimax asymptotic risk in case that the optimally robust estimator for the false radius is used. More precisely, we determine theinefficiency- the limit

36 Supplements to the Asymptotic Theory of Robustness

of the ratio of sample sizes such as to achieve the same accuracy asymptotically;

i.e., in case of the MSE, we consider

relMSE(˜ηr₀, r) =maxMSE(˜ηr₀, r)

maxMSE(˜ηr, r) (2.2.1)

where

maxMSE(˜ηr₀, r) = E|η˜r₀|²+r²ω_∗(˜ηr₀)² ∗=c, v (2.2.2) That is, the maximum asymptotic MSE of the AL estimator with IC ˜η_r0 which is optimal for an infinitesimal neighborhood of radius r0 ∈ [0,∞] , is evaluated over an infinitesimal neighborhood of another radius r ∈ [0,∞] , and is related to the minimax asymptotic MSE for that radius r. Often, we also use the term subefficiency which means inefficiency minus 1 .

The trace of the asymptotic covariance of the MSE solution ˜ηr is increasing and the standardized asymptotic bias is decreasing in the radius r∈(0,∞) . The proof of the following lemma is based on arguments provided by P. Ruckdeschel.

Lemma 2.2.1 Let η˜r be the solution for radius r ∈ (0,∞) provided by Theo-rem1.3.11. Then, E|˜ηr|² is increasing and ω∗(˜ηr) is decreasing in r.

Proof Given some b ∈ (0,∞) with b ≥ ω^min_∗ , the solutions ˜ηb to the corre-sponding Hampel type problems (1.3.7) are given by Theorems 1.3.7 and 1.3.9.

Obviously, g(b) = ω_∗(˜ηb)² = b² is non-negative, convex and strictly increasing in b; confer also Lemma 1.3.4. Moreover, given t∈[0,1] and b0, b1 ∈[ω_∗^min,∞) , define bt= (1−t)b0+tb1. Then, ηt= (1−t)˜ηb₀+tη˜b₁ ∈Ψ^D₂ with |ηt| ≤bt and

E|˜ηb_t|²≤E|ηt|²≤(1−t) E|˜ηb₀|²+tE|˜ηb₁|² (2.2.3) by the convexity of | · |²; i.e., f(b) = E|˜ηb|² is convex in b. In addition, f(b) is non-negative and strictly decreasing in b. Now, consider the corresponding MSE problem which reads

h(r, b) =f(b) +r²g(b) (2.2.4) and let br be the optimal clipping bound for radius r; i.e.,

h(r, br) = min{h(r, b)|b∈[ω^min_∗ ,∞)} (2.2.5) By the convexity of f, g and h in b, one-sided derivatives in b∈(ω_∗^min,∞) exist and we denote them by f⁰, g⁰ and ∂₂h, respectively. Then, by convexity of h

∂₂h(r, b_r−0)≤0≤∂₂h(r, b_r+ 0) (2.2.6) for br∈(ω_∗^min,∞) which implies

−f⁰(b_r−0)

g⁰(br−0) ≥r²≥ −f⁰(b_r+ 0)

g⁰(br+ 0) (2.2.7)

where the convexity of f and g entails that −f⁰ and 1/g⁰ are non-negative and decreasing. Now, assume r₁< r₂ and ω_∗^min≤b_r1 < b_r2. Then,

r²₂≤ −f⁰(br₂−0)

g⁰(b_r2−0) ≤ −f⁰(br₁+ 0)

g⁰(b_r1+ 0) ≤r²₁ (2.2.8)

2.2 Least Favorable Radius 37

which is a contradiction. Hence, br = ω_∗(˜ηr) is decreasing and consequentially f(br) = E|η˜r|² is increasing in r∈(0,∞) . ////

Remark 2.2.2 As the notation in the proof suggests, the preceding lemma holds more generally, namely for all loss functions of the form

h(α, x) =f(x) +αg(x) (2.2.9)

where α∈(0,∞) and f and g are non-negative, convex and decreasing, respec-tively increasing in x∈R. Hence, it for instance applies to the setup ofRuckdeschel and Rieder(2004); confer also Remark1.3.3. ////

The following lemma shows that the MSE–inefficiency curves attain two relative maxima at the boundaries if we fix some interval for the neighborhood radius r. Lemma 2.2.3 Let r∈[r_l, r_u] with 0< r_l< r_u<∞.

(a)Then,

sup

r∈[r_l,r_u]

relMSE(˜ηs, r)≤maxnE|˜η_s|²

E|˜ηr_l|², ω_∗(˜η_s)² ω_∗(˜ηr_u)²

o ∀s∈[rl, ru] (2.2.10)

and there exists some r0∈[rl, ru] such that E|˜ηr₀|²

E|˜η_rl|² = ω_∗(˜ηr₀)²

ω_∗(˜η_ru)² (2.2.11) Moreover, in case rl= 0 and ru=∞,

inf

s∈[0,∞] sup

r∈[0,∞]

relMSE(˜η_s, r) = E|˜η_r0|²

trDI⁻¹D^τ = ω_∗(˜η_r0)²

(ω^min_∗ )² (2.2.12) (b)It holds,

sup

r∈[rl,ru]

relMSE(˜η_s, r) = relMSE(˜η_s, r_l)∨relMSE(˜η_s, r_u) (2.2.13) for all s∈[r_l, r_u]. Moreover, there exists some r₀∈[r_l, r_u] such that

relMSE(˜η_r0, r_l) = relMSE(˜η_r0, r_u) (2.2.14) and

inf

s∈[rl,r_u] sup

r∈[rl,r_u]

relMSE(˜ηs, r) = relMSE(˜ηr₀, rl) = relMSE(˜ηr₀, ru) (2.2.15) Proof

(a)Let s, r∈[rl, ru] with 0< rl< ru<∞. If ^E_E^|˜_|˜^η_η^s|2

r|² ≤_ω^{ω∗( ˜ηs)2}

∗( ˜η_r)², E|˜η_s|²+r²ω_∗(˜η_s)²≤ ω_∗(˜ηs)²

ω_∗(˜η_r)² E|˜η_r|²+r²ω_∗(˜η_r)²

(2.2.16)

38 Supplements to the Asymptotic Theory of Robustness andRuckdeschel(2005b). Hence, the intermediate value theorem yields some r₀∈ [r_l, r_u] such that equality holds in (2.2.11). In case r_l= 0 and r_u=∞, we obtain

2.2 Least Favorable Radius 39

This leads us to (r₂²−r₁²)b²_s

E|η˜s|²+r₁²b²_s−E|˜ηr₂|²+r₂²b²_r2−E|˜ηr₁|²−r²₁b²_r1 E|˜ηr₁|²+r²₁b²_r1

E|˜ηs|²+r²₁b²_s

E|η˜r₂|²+r²₂b²_r2 ≥0 (2.2.27) which can be simplified to

E|˜η_s|²+r²₂b²_s

E|η˜r₂|²+r²₂b²_r2 − E|˜η_s|²+r²₁b²_s

E|˜ηr₁|²+r²₁b²_r1 ≥0 (2.2.28) i.e., relMSE(˜ηs, r2)≥relMSE(˜ηs, r1) . Analogously, we may verify relMSE(˜ηs, r) is decreasing for r < s. Hence, (2.2.13) holds. Moreover, we have

E|˜η_r2|²+s²b²_r

2−E|˜η_r1|²−s²b²_r

1

= (s²−r²₁)(b²_r

2−b²_r

1) +r²₁(b²_r

2−b²_r

1) + E|˜η_r2|²−E|˜η_r1|² (2.2.29)

= (s²−r²₁)(b²_r

2−b²_r

1) + E|η˜_r2|²+r²₁b²_r

2−E|˜η_r1|²−r₁²b²_r

1 ≥0 (2.2.30) That is,

relMSE(˜ηr₁, s)≤relMSE(˜ηr₂, s) (2.2.31) for s < r1< r2. Analogously, we may show

relMSE(˜ηr₁, s)≤relMSE(˜ηr₂, s) (2.2.32) for s > r₁ > r₂. Thus, using the fact relMSE(˜η_rl, r_l) = 1 = relMSE(˜η_ru, r_u) and the continuity of relMSE(˜η_r, s) in r, which is entailed by the continuity of E|˜ηr|² and br in r (cf. Proposition 2.1.9 and Ruckdeschel (2005b)), the shown monotonicity (2.2.31), respectively (2.2.32) together with the intermediate value

theorem implies (2.2.14) and (2.2.15). ////

Remark 2.2.4 (a)The proof of the preceding lemma is based on arguments pro-vided by P. Ruckdeschel.

(b)If ¯r= 0 (cf. Remark3.1.3(b)), respectively rl≥r¯, there is only one solu-tion ˜η on [0,∞] , respectively [rl, ru] ; confer Propositions2.1.3and 2.1.5. Hence, we have to consider bounded intervals in the previous lemma.

(c)The calculation of some radius r0 such that both boundary values are equal leads to an AL estimator that is radius–minimax; i.e., minimizes the maximum inefficiency over the given radius range. We call this radius r₀ least favorable. In case the true radius is completely unknown, respectively unknown except to belong to some radius interval, we recommend to use this optimally robust estimator.

(d) Ruckdeschel and Rieder (2004) prove that Lemma 2.2.3 (a) holds more generally for a large class of optimization problems of form (1.3.10) where G in addition has to fulfill a certain homogeneity condition; confer Theorem 6.1 (a) (idid.). In particular, by parameterizing ˜η not by the radius r but by the optimal clipping bound b, they show, that the radius–minimax IC for completely unknown radius r is the same for all Lq-risks (q∈[1,∞) ); confer Theorem 6.1 (b) (idid.).

(e)InRieder et al.(2001) and also in this thesis we consider the cases that the radius r is completely unknown, respectively unknown up to a factor of 1/3 or 1/2 ;

40 Supplements to the Asymptotic Theory of Robustness

i.e., any r3 or r2 such that the true radius r certainly would stay within [¹₃r3,3r3] or [¹₂r2,2r2] , respectively. In a second step, we then determine least favorable values of r3 and r2 by maximizing the minimax subefficiencies over [¹₃r3,3r3] and

[¹₂r2,2r2] , respectively. ////

In the following remark we state the conclusions ofRieder et al.(2001), Section 1.6.

Remark 2.2.5 (a) The minimax subefficiency is small. Small in compari-son with the most robust estimators, and small for practical purposes. Consistent estimation of the radius from the data hence seems neither necessary nor worth-while – however, under the provision that the radius–minimax robust estimator is employed.

(b) The least favorable radii are small. This surprising fact seems to confirm Huber(1997), p 61, who distinguishes robustness from diagnostics by its purpose to safeguard against – as opposed to find and identify – deviations from the assumptions; in particular, to safeguard against deviations below or near the limits of detectability. LikeHuber(1997), the small least favorable radii we obtain might question the breakdown literature, which is concerned only with (stability under) large contamination and, at most, (efficiency under) zero contamination. ////

Im Dokument Numerical Contributions to the Asymptotic Theory of Robustness (Seite 105-110)