This chapter contains various results about the Poisson model which is defined in Section 4.1. In Section4.2 we specialize the solutions to the asymptotic MSE problem (1.3.5) given in Theorem1.3.11to this model and investigate the continuity and the smoothness of the Lagrange multipliers contained in the optimally robust ICs. We then derive a normal approximation for these multipliers and show that they can also be approximated by the Lagrange multipliers arising in the optimally robust ICs in case of the binomial model. In the subsequent section (Section4.3) we present numerical results for the least favorable radii and the corresponding MSE–inefficiencies in case of the Poisson model. Finally, we verify that we can construct the optimally robust estimator by means of the one-step construction (cf.
Section4.4) and describe how one can use ourRpackageROptEst(cf. Section4.5).
We conclude this chapter with a small simulation study to demonstrate the use of robust estimation in case of the Poisson model; confer Section4.6.
4.1 Introduction
The Poisson model with unknown mean is, P =
Pois(θ)
θ∈(0,∞) (4.1.1)
where
Pois(θ)({y}) =θy
y! exp(−θ) y∈N0 (4.1.2)
Remark 4.1.1 The Poisson model (4.1.1) forms an exponential family with respect to the counting measure on N0, since
θy
y! exp(−θ) = 1
y!exp{ylogθ−θ} (4.1.3) confer also Example 1.5.12 ofLehmann and Casella (1998). With the notation of Lemma 2.3.6 we obtain, ζ(θ) = logθ, β(θ) = θ, T(y) = y and h(y) = (y!)−1 which leads to Jζ =θ−1, EθT =θ and VarθT =θ. ////
110 Poisson Model
Lemma 4.1.2 The Poisson model (4.1.1) is L2 differentiable at θ∈(0,∞) with L2 derivative Λθ and Fisher information!Poisson model Iθ given by
Λθ(y) =θ−1y−1 Iθ=θ−1 (4.1.4) Proof A consequence of Lemma2.3.6(a) in connection with Remark4.1.1. ////
Remark 4.1.3 (a) There is quite a number of examples where a Poisson distri-bution fits well like radioactive disintegrations, chromosome interchanges in cells, bacteria and blood counts or catches in fishery; confer Section VI.7 ofFeller(1968) and Section 1 ofCadigan and Chen(2001). Hence, robust estimation in the Poisson model is an important topic. This is also confirmed by the small simulation study which is presented at the end of this chapter (cf. Section4.6).
(b)Similarly to the binomial model (cf. Remark3.1.3(c)), we do not consider robust estimation in Poisson generalized linear models. For an example we refer to
Rieder(1996). ////
4.2 Optimally Robust Influence Curves
4.2.1 Contamination Neighborhoods
4.2.1.1 Mean Square Error Solution
For some given D ∈R\ {0} we can rewrite the unique MSE optimal IC ˜ηc,r for infinitesimal contamination neighborhoods (1.2.4) and radius r∈(0,∞) supplied by Theorem1.3.7(a) and Theorem1.3.11(b) as
˜
ηc,r(y) =Ar(Λ(y)−zr) minn
1, cr
|Λ(y)−zr| o
(4.2.1) where
0 = E(Λ−zr) minn 1, cr
|Λ−zr| o
(4.2.2) D=ArE|Λ−zr|min
|Λ−zr|, cr (4.2.3) and
r2cr=E |Λ−zr| −cr
+ (4.2.4)
For r=∞ Theorem1.3.7(b) yields
˜
ηc,∞(y) =ωcminsign(D)h
I(y > M)−I(y < M) +βI(y=M)i
(4.2.5) where
β=h
P(y < M)−P(y > M)i.
P(y=M) (4.2.6)
with any M = med(y) and ˜ηc,∞ achieves the minimum bias ωminc = |D|θ
E|y−M| (4.2.7)
4.2 Optimally Robust Influence Curves 111
confer also Remark1.3.8.
For a plot of the optimally robust ICs in case θ= 5 and for different values of r see Figure 4.1.
Remark 4.2.1 (a) Like in the binomial model, it might be necessary to extend the optimally robust ICs to R\N0. For more details see Remark3.2.1.
(b)Since med(Λ) is non-unique for some θ ∈(0,∞) and the assumptions of Proposition 2.1.3hold, we get zr is non-unique in case r≥r¯ ( ¯r <∞) where the lower case radius ¯r is defined in (2.1.12). For a plot of the lower case radius ¯r for θ∈(0,10] confer Figure4.2. The upper peaks correspond to values of θ for which the median of Λ is non-unique.
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
0 5 10 15 20
−5 0 5 10 15
y
optimally robust IC
● r = 0.00 r = 0.01 r = 0.10 r = ∞
Figure 4.1: Optimally Robust ICs for Pois (5) in case of contamination neigh-borhoods with radius r= 0,0.01,0.10,∞.
112 Poisson Model
0 2 4 6 8 10
0.0 0.5 1.0 1.5 2.0
θ
r
Figure 4.2: Lower case radius ¯r for θ∈(0,10] in case of contamination neigh-borhoods.
4.2.1.2 Continuity and Uniqueness of Lagrange Multipliers
The continuity of the Lagrange multipliers Ar, and br in r, stated in Proposi-tion2.1.9, is visualized in Figure4.3. Since med(Λ) is non-unique for θ≈3.67206 , there is a whole interval of valid centering constants ar for r ≥r¯≈1.443 . The boundaries of this interval can be determined via (2.1.17) and are given in Fig-ure 4.3. In contrast to the standardized asymptotic bias br and the asymptotic variance Ar−r2b2r, which seem to be non-differentiable at some values of r, the maximum asymptotic MSE Ar seems to be very smooth in r.
Similarly to the binomial model the centering constant ar =Arzr can have dis-continuity points coinciding with those values of θ for which the median of Λ is non-unique. This fact is illustrated in Figure4.4where we choose a relatively large radius (r= 2.0 ) to demonstrate the extreme case. In addition, we again see that the Lagrange multipliers and hence the standardized asymptotic bias, the asymptotic variance and the maximum asymptotic MSE are continuous (cf. Theorem 2.1.11) but are not necessarily smooth functions in the parameter θ. ////
4.2 Optimally Robust Influence Curves 113
0.0 0.5 1.0 1.5 2.0
5 10 15 20 25 30 35 40
radius maximum asymptotic MSE = Ar
0.0 0.5 1.0 1.5 2.0
−3.0
−2.5
−2.0
−1.5
−1.0
−0.5 0.0
radius optimal centering constant ar=Arzr
0.0 0.5 1.0 1.5 2.0
3 4 5 6 7
radius standardized bias br=Arcr
0.0 0.5 1.0 1.5 2.0
4.0 4.5 5.0 5.5 6.0 6.5
radius asymptotic variance = Ar−r2br2
θ =3.672 θ =4.5 θ =5
Figure 4.3: Continuity in the radius r of the Lagrange multipliers contained in the MSE optimal ICs for r∈(0,2.0] and θ= 3.672,4.5,5.0 in case of contamination neighborhoods (∗=c).
114 Poisson Model maximum asymptotic MSE = A2
r = 2.0
optimal centering constant a2=A2z2 r = 2.0
0 2 4 6 8 10 standardized bias b2=A2c2
r = 2.0 asymptotic variance = A2−22b22
r = 2.0
Pois(θ)
Figure 4.4: Continuity in the parameter θ of the Lagrange multipliers contained in the MSE optimal ICs for θ ∈ (0,10] in case of contamination neighborhoods (∗=c) with radius r= 2.0 .
4.2 Optimally Robust Influence Curves 115
4.2.1.3 Normal Approximation
Similar to the binomial model there is a normal approximation for the Lagrange multipliers contained in the optimally robust ICs where the corresponding optimally robust IC ˜η1.locc,r in case of one-dimensional normal location is specified in (3.2.9)–
(3.2.11). The lower case solution can be read off from (3.2.12) and attains minimum bias ωcmin,1.loc=pπ
2 .
Lemma 4.2.2 Let D= 1. It holds,
θ→∞lim θ−1Ar=A1.locr lim
θ→∞
√
θ zr=zr1.loc= 0 lim
θ→∞
√
θ cr=c1.locr (4.2.8) for all r∈(0,∞) and
θ→∞lim θ−1/2ωcmin=ωcmin,1.loc (4.2.9) Proof We have √
θΛ(y) =θ−1/2(y−θ) (4.2.10) By the convolution property of the Poisson distribution we can regard Pois (θ) as the N-fold convolution of Pois (θ/N) . Hence, by the central limit theorem of Lindeberg-L´evy L(√
θΛ) −→w N(0,1) as θ → ∞ where E√
θΛ = 0 and E √
θΛ)2 = 1 for all θ ∈ (0,∞) . Thus, we can apply Theorem 2.4.1 and
ob-tain (4.2.8) and (4.2.9). ////
Remark 4.2.3 (a)The convergence of the standardized optimal clipping bounds
√θ zr, respectively of the standardized bias terms θ−1/2ωc(˜ηc,r) are illustrated in Figure 4.5 and Figure 4.6, respectively. In case r = ∞, the discontinuity points of the centering constant zr coincide with those values of θ for which the median of Λ is non-unique. At these non-uniqueness points the standardized infinitesimal bias terms attain local minima in case r =∞. These results are analogously to the binomial model; confer Remark3.2.3(b).
(b) The convergence of the standardized maximum asymptotic MSE, respec-tively the MSE–inefficiencies is visualized in Figure 4.7 and Subsection 4.3.1, re-spectively.
(c)Figure4.8shows the MSE–inefficiency of the normal approximated IC. That is, we used the Lagrange multipliers A1.locr θ, zr1.loc/√
θ and c1.locr /√
θ instead of the optimal Ar, zr and cr; confer Lemma4.2.2. To make sure that the resulting function is indeed an IC (with respect to the Poisson model), we additionally cen-tered and standardized this function. The results are similar to the binomial model (cf. Subsubsection3.2.1.3). For small to moderate radii we get very small, respec-tively small efficiency losses. However, for large radii (r >¯r) the subefficiency can become quite large where we get the smallest MSE–inefficiencies for θ≤log(2) and for such values of θ at which med(Λ) is non-unique. A possible explanation for
this behavior is given in Remark3.2.3(c). ////
116 Poisson Model
0 2 4 6 8 10
−0.04
−0.03
−0.02
−0.01 0.00
θ θzr
r = 0.1
0 2 4 6 8 10
−0.20
−0.15
−0.10
−0.05 0.00
θ θzr
r = 0.5
0 2 4 6 8 10
−0.4
−0.3
−0.2
−0.1 0.0
θ
θzr r = 1.0
0 2 4 6 8 10
−0.8
−0.6
−0.4
−0.2 0.0 0.2 0.4
θ θzr
r = ∞
Pois(θ) 1.loc
Figure 4.5: Normal approximation of the standardized centering constant √ θ zr for θ∈(0,10] in case of contamination neighborhoods (∗=c) with radius r= 0.1,0.25,0.5,∞.
4.2 Optimally Robust Influence Curves 117
0 2 4 6 8 10
2.1 2.2 2.3 2.4 2.5 2.6
θ θ−0.5ωc(η~ c, r)
r = 0.1
0 2 4 6 8 10
1.35 1.40 1.45 1.50
θ θ−0.5ωc(η~ c, r)
r = 0.5
0 2 4 6 8 10
1.20 1.25 1.30 1.35 1.40
θ θ−0.5ωc(η~ c, r)
r = 1.0
0 2 4 6 8 10
1.20 1.25 1.30 1.35 1.40
θ θ−0.5ωc(η~ c, r)
r = ∞
Pois(θ) 1.loc
Figure 4.6: Normal approximation of the standardized infinitesimal bias terms θ−1/2ωc(˜ηc,r) for θ∈(0,10] in case of contamination neighborhoods (∗=c) with radius r= 0.1,0.5,1.0,∞.
118 Poisson Model
0 2 4 6 8 10
1.06 1.07 1.08 1.09 1.10 1.11
θ
θ−1Ar r = 0.1
0 2 4 6 8 10
1.64 1.65 1.66 1.67 1.68 1.69 1.70
θ θ−1Ar
r = 0.5
0 2 4 6 8 10
2.90 2.95 3.00 3.05 3.10 3.15
θ θ−1Ar
r = 1.0
0 2 4 6 8 10
7.5 8.0 8.5 9.0
θ θ−1Ar
r = 2.0
Pois(θ) 1.loc
Figure 4.7: Normal approximation of the standardized maximum asymptotic MSE θ−1Ar for θ∈(0,10] in case of contamination neighborhoods (∗=c) with radius r= 0.1,0.5,1.0,2.0 .
4.2 Optimally Robust Influence Curves 119
0 2 4 6 8 10
1.000 1.005 1.010 1.015
θ
MSE−inefficiency
r = 0.1
0 2 4 6 8 10
1.000 1.001 1.002 1.003 1.004
θ
MSE−inefficiency
r = 0.25
0 2 4 6 8 10
1.000 1.005 1.010 1.015 1.020 1.025 1.030
θ
MSE−inefficiency r = 0.5
0 2 4 6 8 10
1.0 1.1 1.2 1.3 1.4 1.5
θ
MSE−inefficiency
r = 2.0
MSE−inefficiency of normal approximated IC
Figure 4.8: MSE–inefficiency of the normal approximated IC for θ ∈ (0,10]
in case of contamination neighborhoods (∗ = c) with radius r = 0.1,0.25,0.5,2.0 .
120 Poisson Model
4.2.1.4 Poisson Approximation
The Lagrange multipliers contained in the optimally robust ICs in case of the Pois-son model are also approximated by the corresponding Lagrange multipliers in case of the binomial model. This is stated in the following lemma. To distinguish be-tween the two models, we introduce the additional superscripts “ bin ” and “ pois ”, respectively.
Lemma 4.2.4 Let D = 1. Consider the binomial model (∗=c) with size m∈ N and probability of success pm ∈ (0,1) and the Poisson model (∗ = c) with parameter θ∈(0,∞) where mpm→θ∈(0,∞) as m→ ∞. Then,
m→∞lim γm2Abinr =Apoisr lim
m→∞γm−1cbinr =cpoisr (4.2.11) for all r∈(0,∞) where γm= 1−pm
m and
m→∞lim γmωmin,binc =ωcmin,pois (4.2.12) If med(Λpois) is unique, respectively med(Λpois) non-unique and r <r¯, then also
m→∞lim γm−1zbinr =zrpois (4.2.13) Proof We have
γm−1Λbin(y) = y−mpm
mpm
(4.2.14) By the Poisson approximation of the binomial distribution and mpm →θ we get L(γm−1Λbin)−→w L(Λpois) as m→ ∞ where Eγm−1Λbin= 0 for all m∈N and
E γm−1Λbin)2= 1−pmpm
m −→θ−1= E Λpois2
as m→ ∞ (4.2.15)
Hence, we can apply Theorem2.4.1which yields (4.2.11)–(4.2.13). ////
Remark 4.2.5 (a) If the median of Λpois is non-unique, the standardized cen-tering constant λ−1mzbinr does not necessarily converge. As we see in Figure 4.9, different choices of pm lead to different accumulation points. In case pm=θ/m, the median of γm−1Λbin is identical to the maximum median of Λpois. However, if the median of Λpois is unique, respectively non-unique and r <¯r, then the preced-ing lemma implies (4.2.13) for all r∈[0,∞] ; an example is given in Figure4.10.
(b) The convergences of the standardized infinitesimal bias terms γmωc(˜ηc,rbin) and the standardized minimax asymptotic MSE γm2Ar are visualized in Figure4.11 and 4.12, respectively. An example for the Poisson approximation of the MSE–
inefficiencies is given in Section4.3.
(c) Figure 4.13 shows the MSE–inefficiency of the Poisson approximated IC.
That is, we used the Lagrange multipliers Apoisr /γm2 , zrpoisγm and cpoisr γm instead of the optimal Abinr , zbinr and cbinr ; confer Lemma 4.2.4. To make sure that the resulting function is indeed an IC (with respect to the binomial model), we addi-tionally centered and standardized this function. As in case of the normal approxi-mation the Poisson approxiapproxi-mation seem to get worse for increasing radius which is
4.2 Optimally Robust Influence Curves 121
contrary to the results in case of total variation neighborhoods where the quality of the approximation increases with increasing radius; confer Subsubsection 4.2.2.4.
////
20 40 60 80 100 120
−0.15
−0.10
−0.05 0.00 0.05 0.10
size of Binomial distribution
γm−1 z∞bin pm= θ m
pm=(θ +m−1) m
pm=(θ −m−0.5) m
pm=(θ −m−0.9) m
Pois(θ)
Figure 4.9: Poisson approximation of the standardized lower case centering con-stant γm−1z∞bin for m∈[5,120] and θ= 4.670909 (median of Λpois non-unique) in case of contamination neighborhoods (∗=c); confer also Remark4.2.5(a)
122 Poisson Model
Figure 4.10: Poisson approximation of the standardized centering constant γm−1zrbin for m∈[8,100] , pm=θ/m and θ= 4.5 in case of con-tamination neighborhoods (∗=c) with radius r= 0.1,0.5,1.0,∞.
4.2 Optimally Robust Influence Curves 123
Figure 4.11: Poisson approximation of the standardized infinitesimal bias terms γmωc(˜ηbinc,r) form∈[8,100] ,pm=θ/m andθ= 4.5 in case of con-tamination neighborhoods (∗=c) with radius r= 0.1,0.5,1.0,∞.
124 Poisson Model
Figure 4.12: Poisson approximation of the standardized maximum asymptotic MSEγm2Abinr form∈[8,100] ,pm=θ/m and θ= 4.5 for contam-ination neighborhoods (∗=c) with radius r= 0.1,0.5,1.0,2.0 .
4.2 Optimally Robust Influence Curves 125
MSE−inefficiency of Poisson approximated IC
Figure 4.13: MSE–inefficiency of the Poisson approximated IC for m∈[8,100] , pm =θ/m and θ = 4.5 in case of contamination neighborhoods (∗=c) with radius r= 0.1,0.5,1.0,2.0 .
126 Poisson Model
4.2.2 Total Variation Neighborhoods
4.2.2.1 Mean Square Error Solution
Second, we consider the Poisson model (4.1.1) with infinitesimal total variation neighborhoods (1.2.5). The optimally robust IC ˜ηv,r supplied by Theorem1.3.9(a) and Theorem1.3.11(c) for some given D∈R\ {0} can be rewritten as
˜
ηv,r(y) =Ar
gr∨Λ(y)∧(gr+cr)
(4.2.16) where
0 = E gr−Λ
+−E Λ−(gr+cr)
+ (4.2.17)
D=ArEΛ
gr∨Λ∧(gr+cr)
(4.2.18) and
r2cr= E gr−Λ
+ (4.2.19)
For r=∞ Theorem1.3.9(b) provides
˜
ηv,∞(y) =ωminv sign(D)
P(y < θ)
P(y6=θ)I(y > θ)−P(y > θ)
P(y6=θ)I(y < θ)
(4.2.20) with minimum bias
ωminv = |D|θ E(y−θ)+
(4.2.21) For a plot of the optimally robust ICs in case θ= 5 and for different values of r see Figure4.14.
Remark 4.2.6 (a)Similarly to the binomial model it might be necessary to extend the optimally robust ICs to R\N0. For more details see Remark3.2.1.
(b) The map θ 7→r¯ has discontinuity points at θ ∈ N since we may choose θ ∈ (0,∞) such that θ is arbitrarily close to some integer value in N, the gap γ= infPθ
|Λ|
|Λ|>0 becomes arbitrarily small, respectively M and the lower case radius ¯r defined in (2.1.34) and (2.1.33) become arbitrarily large. However, if we choose θ∈(0,∞) such that θ∈N we get γ= 1/θ (=γ1=−γ2) . For a plot of the lower case radius ¯r for θ∈(0,10] see Figure4.15. ////
4.2 Optimally Robust Influence Curves 127
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
0 5 10 15 20
−5 0 5 10 15
y
optimally robust IC
● r = 0.00 r = 0.01 r = 0.10 r = ∞
Figure 4.14: Optimally Robust ICs for Pois (5) in case of total variation neigh-borhoods (∗=v) with radius r= 0,0.01,0.10,∞.
0 2 4 6 8 10
0.0 0.5 1.0 1.5 2.0 2.5 3.0
θ
r
●
● ● ● ● ● ● ● ● ●
Figure 4.15: Lower case radius ¯r for θ∈(0,10] in case of total variation neigh-borhoods (∗=v).
128 Poisson Model
4.2.2.2 Continuity and Uniqueness of Lagrange Multipliers
The Lagrange multipliers contained in the optimally robust ICs are unique and continuous in the radius r as well as in the parameter θ; confer Subsection2.1.4, Proposition2.1.9and Theorem2.1.11, respectively. This is visualized in Figure4.16 and Figure4.17, respectively. In Figure4.17we use theR functionpoints(cf. R Development Core Team(2005)) to illustrate that the lower clipping bound and the asymptotic variance indeed have no jumps but attain values between the local ex-trema, too. But, these Lagrange multipliers and hence the standardized asymptotic bias br, the asymptotic variance Ar−r2b2r and the maximum asymptotic MSE Ar are not necessarily smooth functions in r, respectively θ. Only the maximum asymptotic MSE Ar seems to be smooth in r. Moreover, in case r=∞ the lower as well as the upper clipping bound are discontinuous for θ∈N; confer Figure4.18.
4.2 Optimally Robust Influence Curves 129
0.0 0.5 1.0 1.5 2.0
0 20 40 60 80 100 120 140
radius maximum asymptotic MSE = Ar
0.0 0.5 1.0 1.5 2.0
−3.5
−3.0
−2.5
radius optimal lower bound Argr
0.0 0.5 1.0 1.5 2.0
5 6 7 8 9 10
radius standardized bias br=Arcr
0.0 0.5 1.0 1.5 2.0
4.0 4.5 5.0 5.5 6.0 6.5 7.0
radius asymptotic variance = Ar−r2br2
θ =3.672 θ =4.5 θ =5
Figure 4.16: Continuity in the radius r of the Lagrange multipliers contained in the MSE optimal ICs for r∈(0,2.0] and θ= 3.672,4.5,5.0 in case of total variation neighborhoods (∗=v).
130 Poisson Model
0 2 4 6 8 10
0 50 100 150 200 250
θ maximum asymptotic MSE = A2
r = 2.0
0 2 4 6 8 10
−4
−3
−2
−1 0
θ
optimal lower bound A2g2 r = 2.0
0 2 4 6 8 10
1 2 3 4 5 6 7 8
θ standardized bias b2=A2c2
r = 2.0
0 2 4 6 8 10
0 5 10 15
θ asymptotic variance = A2−22b22
r = 2.0
Pois(θ)
Figure 4.17: Continuity in the parameter θ of the Lagrange multipliers con-tained in the MSE optimal ICs for θ ∈ (0,10] in case of total variation neighborhoods (∗=v) with radius r= 2.0 .