Robustness and Bootstrap for
Markov chains
P. Bertail* ˆJ. Tressou @ ˆS. Clémençon+
*Modal'X, Université Paris Nanterre
+Télécom
ParisTech, UMR CNRS 5141 LTCI, Groupe TSI
@
INRA Paris
Statistiques pour les PDMP , Nancy, 2-3 février 2017
Patrice Bertail (MODAL'X)
er
11er février
février
2017 2017
1 / 38
Outlines
1
Pseudo-regeneration for Markov Chains
Framework Harris recurrence
The atomic case
Approximate renewal sequence : pseudo-regeneration
Regenerative and Approximate regenerative Bootstrap
2
Robustness for Markov chains
Robustness and inuence function for Markov chains
Examples of inuence functions
Main Hypotheses
a Central Limit Theorem and its bootstrap version
Asymptotic validity of the regenerative Bootstrap
3
Two applications : robustied L-statitics and R-statistics
Robustied L-statistics
Linear Rank signed statistics for Markov chains
4
A few references
Patrice Bertail (MODAL'X)
1er février 2017
2 / 38
Pseudo-regeneration for Markov Chains
Outlines
1
Pseudo-regeneration for Markov Chains
Framework Harris recurrence
The atomic case
Approximate renewal sequence : pseudo-regeneration
Regenerative and Approximate regenerative Bootstrap
2
Robustness for Markov chains
Robustness and inuence function for Markov chains
Examples of inuence functions
Main Hypotheses
a Central Limit Theorem and its bootstrap version
Asymptotic validity of the regenerative Bootstrap
3
Two applications : robustied L-statitics and R-statistics
Robustied L-statistics
Linear Rank signed statistics for Markov chains
4
A few references
Patrice Bertail (MODAL'X)
1er février 2017
3 / 38
Pseudo-regeneration for Markov Chains
Framework Harris recurrence
General framework : Harris recurrent Markov chains
X = (Xn )n ∈N , a ψ-irreducible aperiodic time-homogeneous Markov chain,
valued in a (countable generated) measurable space (E , E) with transition
probability Π(x , dy ) and initial distribution ν.
Notations : Pν (respectively, Px for x in E ) the probability measure such
that X0 ∼ ν (resp., conditioned upon X0 = x ), Eν [.] the Pν -expectation
(resp. Ex [.] the Px (.)-expectation). EA [.] denotes the expectation
conditioned on the event {X0 ∈ A}.
Main tool : Regeneration properties of Markov chains.
Refer to the books by Revuz (1984), Meyn and Tweedie (1994),
Thorisson(2000). Series of works by Bertail and Clémençon, PRTF (2004),
Bernoulli(2006), Test(2008), Mat. Method in Statist. (2010) and Bertail,
Clémençon, Tressou Extremes(2008) and Electronic Journal Stat. (2013).
Patrice Bertail (MODAL'X)
1er février 2017
4 / 38
Pseudo-regeneration for Markov Chains
The atomic case
General framework : Harris recurrent Markov chains
Markov chain X is said regenerative when it possesses an accessible
atom, i.e. a measurable set A such that ψ(A) > 0 and Π(x , .) = Π(y , .)
for all x , y in A,
Denote by τA = τA (1) = inf {n ≥ 1, Xn ∈ A} the hitting time on A.
Put also τA (j ) = inf {n > τA (j − 1), Xn ∈ A} , j ≥ 2 for the successive
return times to A,
P
ln = ni=1 I{Xi ∈ A} number of visits of X to the regeneration set A
until time n ,
if τA < ∞ or if there exists 0 < β such that E τβA < ∞, the sample
paths of the chain may be divided into i.i.d. blocks of random length
corresponding to consecutive visits to A, generally called regeneration
cycles :
B1 = (Xτ
A
(1)+1 , ...,
Patrice Bertail (MODAL'X)
Xτ
A
(2) ), ...,
Bj = (Xτ (j )+1 , ...,
A
Xτ (j +1) ), ...
A
1er février 2017
5 / 38
Pseudo-regeneration for Markov Chains
The atomic case
Example 1 : Cramer-Lundberg with a dividend barrier
see for instance the books by Asmussen(2003), Embrechts, Klüppelberg, and
Mikosch(1997)
Number of claims an interval [0, t ] : {N (t ), t ≥ 0, N (0) = 0} : an
homogeneous Poisson process with rate λ,. input times (Tn )n ∈N times
of the claims
Claims sizes Ui , i = 1,....∞ , i.i.d rv's with cdf F .
S (t ) =
NX
(t )
i =1
Ui
Constant premium rate (price per unit of time) c .
Reserve of company with a constant barrier b , over which prot is
redistributed evolves like
X (t ) = (u + ct − S (t )) ∧ b ,
Under the net prot condition, the embedded chain Xn = X (Tn ) is an
atomic Markov chains with an atom at b .
Patrice Bertail (MODAL'X)
1er février 2017
6 / 38
Pseudo-regeneration for Markov Chains
The atomic case
The blocks corresponding to successive visits to b are independent and the ruin is
attained when the min = to the min over regeneration blocks of the embedded
chain is negative, Smith(1955).
4
0
2
Compagny reserves
6
8
X(t) Cramer−Lundberg model with a barrier
0
20
40
60
80
100
time
Figure: Cramér-Lundberg model with a dividend barrier at b, ruin at 0.
Patrice Bertail (MODAL'X)
1er février 2017
7 / 38
Pseudo-regeneration for Markov Chains
The atomic case
Harris chains : Nummelin splitting trick
Denition
A set S ∈ E is said to be small for X if there exist m ∈ N∗ , δ > 0 and a
probability measure Φ supported by S such that, for all x ∈ S , B ∈ E ,
Πm (x , B ) ≥ δΦ(B ).
Some simplications :
m=1 (even if it entails replacing the initial chain X by the chain
{(Xnm , ..., Xn (m +1)−1 )}n ∈N ).
Φ(B ) may be chosen to be the uniform distribution over S
Patrice Bertail (MODAL'X)
1er février 2017
8 / 38
Pseudo-regeneration for Markov Chains
The atomic case
The Nummelin splitting trick
Nummelin (1978), Athreya and Ney (1978) : Any Harris recurrent Markov
chains can be made atomic !
The sample space is expanded so as to dene a sequence (Yn )n ∈N of
independent Bernoulli r.v.'s with parameter δ by dening the joint
distribution Pν,M , whose construction relies on the following randomization
of the transition probability Π each time the chain hits S . If Xn ∈ S and
if Yn = 1 (with probability δ ∈ ]0, 1[), then Xn +1 ∼ Φ,
if Yn = 0, then Xn +1 ∼ (1 − δ)−1 (Π(Xn +1 , .) − δΦ(.)).
AS
= S × {1} is an atom for the bivariate Markov chain (X , Y ), which
inherits all its communication and stochastic stability properties from X
(refer to Chapt. 14 of Meyn and Tweedie (1996).
Patrice Bertail (MODAL'X)
1er février 2017
9 / 38
Pseudo-regeneration for Markov Chains
The atomic case
The Nummelin splitting trick
Assumption : {Π(x , dy )}x ∈E and the initial distribution ν are dominated by
a σ-nite measure λ of reference, so that ν(dy ) = fν (y )λ(dy ) and
Π(x , dy ) = π(x , y )λ(dy ) for all x ∈ E . Φ is absolutely continuous with
respect to λ too, Φ(dy ) = φ(y ).λ(dy ),
∀x ∈ S , π(x , y ) ≥ δφ(y ), λ(dy )-almost surely.
Theoretical Splitting
Given the sample path X (n +1) , Yi 's are independent random variables
and the conditional distribution of Yi is the Bernoulli distribution with
parameter
δφ(Xi +1 )
· I{Xi ∈ S } + δ · I{Xi ∈
/ S }.
π(Xi , Xi +1 )
Patrice Bertail (MODAL'X)
(1)
1er février 2017
10 / 38
Pseudo-regeneration for Markov Chains
Approximate renewal sequence : pseudo-regeneration
The approximate splitting trick
Unfortunately φ is unknown : use a plug-in rule to create articial Bernoulli
r.v.'s with estimated parameters.
Approximate splitting trick algorithm
Compute rst an estimate πbn (x , y ) of the transition density over S 2 ,
such that πbn (x , y ) ≥ bδφ(y ) a.s. for all (x , y ) ∈ S 2
Given the sample path X (n +1) , Ybi 's are independent random
variables and the conditional distribution of Ybi is Bernoulli
distribution with estimated parameters
b
δφ(Xi +1 )
· I{Xi ∈ S } + b
δ · I{Xi ∈
/ S }.
b(Xi , Xi +1 )
π
Split the chain each time Xi ∈ S and Ybi = 1. The corresponding
blocks are "asymptotically" independent.
Patrice Bertail (MODAL'X)
1er février 2017
11 / 38
Pseudo-regeneration for Markov Chains
Approximate renewal sequence : pseudo-regeneration
Some additional notations in the approximate world
Approximate number of regenerations for a given small set
^ln =
X
1≤k ≤n
bk ) ∈ S × {1}}
I{(Xk , Y
Approximate renewal times
bn ) ∈ S × {1}}, for 1 ≤ j ≤ bln − 1,
τbA (j + 1) = inf {n ≥ 1 + τbA (j )/ (Xn , Y
S
S
Practical choice of S : optimize over a class of small sets to obtain the
maximum number of approximate regenerations
Patrice Bertail (MODAL'X)
1er février 2017
12 / 38
Pseudo-regeneration for Markov Chains
Approximate renewal sequence : pseudo-regeneration
Example 2 : Splitting a time series
A time series model exhibiting some nonlinearities and structural changes
both in level and variance : ETAR(1)-ARCH(1) model (Exponential
Threshold AutoRegressive Model with AutoRegressive Conditional
Heteroscedasticity)
Xt +1 = (α1 + α2 e −X )Xt + (1 + βXt2 )1/2 εt +1 ,
2
t
where the noise (εt )t =1,...T are i.i.d with variance σ2 .
Under standard conditions, admits a stationary solutions, but exhibits some
threshold and heteroscedatic behavior. See Tjøstheim(1990).
Patrice Bertail (MODAL'X)
1er février 2017
13 / 38
Pseudo-regeneration for Markov Chains
Approximate renewal sequence : pseudo-regeneration
Figure: Splitting a time-series exhibiting thresholds and conditional
heteroscedasticity, n=200,α1 = 0.60, α2 = 0.45, β = 0.35 and σ2 = 1.
Patrice Bertail (MODAL'X)
1er février 2017
14 / 38
Pseudo-regeneration for Markov Chains
Regenerative and Approximate regenerative Bootstrap
Regenerative and Approximate regenerative Bootstrap
Identify the (pseudo-)blocks B1 , . . . , Bl
n
−1
from the observed trajectory
X0 , . . . , Xn and compute the statistic of interest as
Tb n = T (B1 , . . . , Bl −1 ) and its standard deviation
n
.
Draw sequentially bootstrap data blocks B1∗ , . . . , Bk∗ independently from
the empirical distribution of the blocks B1 , . . . , Bl −1 until the length of
P
the bootstrap series l ∗ (k ) = kj=1 l (Bj∗ ) is larger than n . Let
ln∗ = inf {k ≥ 1, l ∗ (k ) > n }.
From the bootstrap data blocks generated at step 2, reconstruct a
pseudo-trajectory by binding the blocks together, getting the
reconstructed RBB sample path X ∗(n ) = (B1∗ , . . . , Bl∗∗ −1 ) of length n or
n ∗ (when getting rid of the last block.
Compute the bootstrap version of the regenerative blocks estimator :
Tn∗ = T (B1∗ , . . . , Bl∗∗ −1 ) and its standard deviation
b∗n = σ(B1∗ , . . . , Bl∗∗ −1 ).
σ
bn = σ(B1 , . . . , Bl
σ
n
−1 )
n
n
n
n
Patrice Bertail (MODAL'X)
1er février 2017
15 / 38
Pseudo-regeneration for Markov Chains
Regenerative and Approximate regenerative Bootstrap
Regenerative Bootstrap condence intervals
Bootstrap condence intervals (CI) at level 1 − α ∈ (1/2, 1) for the
parameter of interest are obtained by computing the bootstrap root's
∗
∗
quantiles qα/
2 and q1−α/2 , of orders α/2 and 1 − α/2
∗
∗
∗
Basic percentile bootstrap CI : I1BP
−α = [qα/2 , q1−α/2 ].
∗
∗
bn − q ∗
b
Percentile boostrap CI : I1P−α
= 2T
1−α/2 , 2Tn − qα/2
t-Percentile
:
h boostrap CI √
√ i
tP
∗
∗
∗
th
b
bn − t ∗ σ
bn / n , T
b
I1−α = Tn − t1−α/2 σ
α/2 n / n , where tp is the p
h
i
√ bn / σ
b∗n / n .
quantile of the studentized bootstrap root Tn∗ − T
Patrice Bertail (MODAL'X)
1er février 2017
16 / 38
Robustness for Markov chains
Outlines
1
Pseudo-regeneration for Markov Chains
Framework Harris recurrence
The atomic case
Approximate renewal sequence : pseudo-regeneration
Regenerative and Approximate regenerative Bootstrap
2
Robustness for Markov chains
Robustness and inuence function for Markov chains
Examples of inuence functions
Main Hypotheses
a Central Limit Theorem and its bootstrap version
Asymptotic validity of the regenerative Bootstrap
3
Two applications : robustied L-statitics and R-statistics
Robustied L-statistics
Linear Rank signed statistics for Markov chains
4
A few references
Patrice Bertail (MODAL'X)
1er février 2017
17 / 38
Robustness for Markov chains
Robustness and inuence function for Markov chains
Inuence function for Markov chains
In general it is dicult to dene the notion of inuence function for general
time series ( see Martin and Yohai, ) . Pb with the denition of a
contaminated model (Innovative outliers, additive outliers, structural break
etc...) BUT an easier and natural version for Harris Markov Chain. All
functionals of the stationary distribution may be seen as functionals of the
block distribution belonging to the torus on the torus T.
Denition
Let (V, ||.||) be a separable
Banach space. Let T : PT → V be a functional on PT . If, for all L in PT ,
t −1 (T ((1 − t )L + t δb ) − T (L)) has a nite limit as t → 0 for any b ∈ T,
the inuence function is dened by
(Influence function on the torus)
T (1) (b ,
Patrice Bertail (MODAL'X)
T ((1 − t )L + t δb ) − T (L)
.
t →0
t
L) = lim
1er février 2017
(2)
18 / 38
Robustness for Markov chains
Robustness and inuence function for Markov chains
Robustness for Markov chains
Denition
(Gross-error sensivity) A functional T is said to be Markov-robust i
its inuence function T (1) (b , L) is bounded on the torus T. The
gross-error sensitivity to block contamination is then dened as
γ∗ (T , L) = sup ||T (1) (b , L)||.
b ∈T
Patrice Bertail (MODAL'X)
1er février 2017
19 / 38
Robustness for Markov chains
Robustness and inuence function for Markov chains
Fréchet dierentiability on the toraus
Denition
It is said that T is
Fréchet-dierentiable at L0 ∈ PT for a metric d on the Torus, if there
exists a continuous linear operator DTL0 (from on the set of the signed
measures of the form L − L0 in (V, ||.||)) and a function
(1) (., L0 ) : R → (V, ||.||), continuous at 0 with (1) (0, L0 ) = 0 such that :
∀L ∈ PT ,
(Fréchet differentiability on the torus)
T (L) − T (L0 ) = DTL0 (L − L0 ) + R(1) (L, L0 ),
with R(1) (L, L0 ) = d (L, L0 )(1) (d (L, L0 ), L0 ). In addition, T is said to
have a canonical gradient (or inuence function) T (1) (., L0 ) if the
following representation for DTL0 holds : ∀L ∈ PT ,
DTL0 (L − L0 ) =
Z
T (1) (b , L0 )L(db ).
T
Patrice Bertail (MODAL'X)
1er février 2017
20 / 38
Robustness for Markov chains
Examples of inuence functions
Examples of inuence functions
def
µ(f ) = Eµ [f (X )]. Denote by B a r.v. valued in T with distribution L
then
µ(f ) = EL [f (B)] /EL [L(B)] = T (L),
then
T (1) (b , L) =
d
f (b ) − µ(f )L(b )
(T ((1 − t )L + tb )|t =0 =
.
dt
EL [L(B)]
Even if f is bounded this functional is not robust.
M-parameter/ estimators Let θ be the unique solution of the equation :
Eµ [g (X , θ)] = 0,
(3)
where g : R2 → R is of class C 2 which is is equivalent to EL [g(B, θ)] = 0
then the inuence function is given by
Tg(1) (b , L) = −
Patrice Bertail (MODAL'X)
g(b , θ)
h
i,
EL
∂g(B,θ)
∂θ
1er février 2017
21 / 38
Robustness for Markov chains
Examples of inuence functions
Examples of inuence functions
Assume that the stationary distribution has a continuous strictly
increasing cumulative distribution function (cdf) Fµ (x ) = µ(] − ∞, x ])
e α (µ) = F −1 (α). if
and density fµ (x ). Consider the α−quantile T
µ
fµ (Tα (L)) 6= 0, the inuence function (sme arguments as see [10]) is
given here by
T
(1)
α (
b , L) =
PL(b )
i =1 (α − I{bi ≤ Tα (L)})
EL [L(B)]fµ (Tα (L))
Not bounded ! Quantiles are not robust in time series and for Markov
chains : a single contamination (innovative outlier) may create a very
long (outlier) block (before reaching again stationarity).
Needed to truncate both the functional to make it robust to additive
outlier but also long blocks to make it robust to innovative outlier.
Patrice Bertail (MODAL'X)
1er février 2017
22 / 38
Robustness for Markov chains
Main Hypotheses
Hypotheses
A0. Mean squared error (MSE) of the estimator of the transition density
αn = Rn (π^n , π) = E[( sup
(x ,y )∈S 2
|π^n (x , y ) − π(x , y )|)2 ] −→ 0
in Pν probability as n → ∞
A1. the parameters S and φ are chosen so that inf x ∈S φ(x ) > 0
A2. sup(x ,y )∈S 2 π(x , y ) < ∞ and Pν -almost surely
sup sup π^n (x , y ) < ∞
.
A3 (Regenerative case)
n ∈N (x ,y )∈S 2
H(κ)
H(ν, κ)
Patrice Bertail (MODAL'X)
:
:
EA [τκA ] < ∞,
Eν [τκA ] < ∞.
1er février 2017
23 / 38
Robustness for Markov chains
Main Hypotheses
Hypotheses
A4 (General Harris recurrent case)
~
H(κ)
~ κ)
H(ν,
Patrice Bertail (MODAL'X)
:
:
supx ∈S Ex [τκS ] < ∞,
Eν [τκS ] < ∞.
1er février 2017
24 / 38
Robustness for Markov chains
a Central Limit Theorem and its bootstrap version
Plug-in estimators based on blocks
Given an observed path of length n , natural empirical estimates of
parameters T (L) are of course the plug-in estimators T (Ln ) based on the
empirical distribution of the observed regeneration blocks
Ln = (ln − 1)−1
lX
−1
n
j =1
δB ∈ PT
j
or in the general Harris recurrent case, T (Lbn )
Lbn = (bln − 1)
b
ln −1
X
j =1
δBb ,
j
Distance on the Torus : Bounded Lipschitz type metric on PT
dBL (L, L0 ) = sup
f ∈LipT1
Patrice Bertail (MODAL'X)
Z
{
Z
f (b )L(db ) − f (b )L0 (db )},
1er février 2017
(4)
25 / 38
Robustness for Markov chains
a Central Limit Theorem and its bootstrap version
Controlling the distance between empirical block-distribu
Theorem
(Rate bounds for the Lipschitz distance)
e0 (4, ν), we have
H
dBL (Ln , L) = OP (n −1/2 )
Under He0 (4) and
as n → ∞, and
dBL (Ln , Lbn ) = OP (α1n/2 n −1/2 ) = oP (n −1/2 ), as n → ∞.
Patrice Bertail (MODAL'X)
1er février 2017
26 / 38
Robustness for Markov chains
a Central Limit Theorem and its bootstrap version
CLT for Fréchet dierentiable functionals on the torus
Theorem
In the regenerative case, if T : PT → R is
Fréchet dierentiable at L and d (Ln , L) = OP (n −1/2 ) (or
R(1) (Ln , L) = oP (n −1/2 )) as n → ∞, and if EA [τA ] < ∞ and
0 < VarA (T (1) (B1 , L)) < ∞ then, under Pν , we have the convergence
in distribution
(Central Limit Theorem)
ν
ν
n 1/2 (T (Ln ) − T (L)) ⇒ N (0, EA [τA ]VarA (T (1) (B1 , L)), as n → ∞.
In the general Harris case, if the split chain satises the assumptions
above (with A replaced by AS and d (Lbn , L) = OP (n −1/2 )), under the
assumptions of Theorem 4, as n → ∞ we have, under Pν ,
ν
n 1/2 T (Lbn ) − T (L)
Patrice Bertail (MODAL'X)
⇒ N (0, EA [τA ]VarA (T (1) (B1 , L)).
S
S
S
1er février 2017
27 / 38
Robustness for Markov chains
Asymptotic validity of the regenerative Bootstrap
Theorem
Denote by L∗n the empirical
distribution of the bootstrap taken with replacement either from the
regenerative blocks or the approximate blocks. In the regenerative
case, assume that the conditions of Theorem 5 hold for the metric
chosen d then we also have
(Regenerative bootstrap version)
n 1/2 (T (L∗n ) − T (Ln )) ⇒ N (0, EA [τA ]VarA (T (1) (B1 , L)), as n → ∞.
Similarly, in the general Harris positive recurrent case under the
assumptions of Theorem 5, we have the convergence in distribution :
n 1/2 T (L∗n ) − T (Lbn )
Patrice Bertail (MODAL'X)
⇒ N (0, EA [τA ]VarA (T (1) (B1 , L))
S
S
S
as n → ∞.
1er février 2017
28 / 38
Robustness for Markov chains
Asymptotic validity of the regenerative Bootstrap
Non robustness of the Bootstrap for non robust function
The bootstrap distribution is very sensitive to outlier in the framework
of Markov chains.
To illustrate this fact consider waiting times of a M/M/1 process (cf
[1]) with exponential parameters λ1 = 0.6, λ2 = 0.8 with n=500. In this
case, A = {0} (i.e. "empty le") is an atom. The stationary distribution
is a mixture 0.25δ0 + 0.75µ0 , where µ0 is a highly asymmetric
continuous distribution on the half-line R+ .
What happens if at t=250 the server does not work for 20 period (that
is the queue increase with entry with 0.6 exponential distribution).
Patrice Bertail (MODAL'X)
1er février 2017
29 / 38
Asymptotic validity of the regenerative Bootstrap
0.0
0.1
0.2
0.3
0.4
Robustness for Markov chains
−2
0
2
4
6
8
10
True and Bootstrap dist., Mean in a MM1 0.8 0.6
Figure: Comparison of the true distribution of the recentered normalized mean in
a MM1 model, with parameter 0.8 0.6 no outlier, n=500
Patrice Bertail (MODAL'X)
1er février 2017
30 / 38
Asymptotic validity of the regenerative Bootstrap
0.0
0.1
0.2
0.3
0.4
Robustness for Markov chains
−2
0
2
4
6
8
10
True and Bootstrap dist., Mean in a MM1 0.8 0.6
Figure: Comparison of the true distribution of the recentered normalized mean in
a MM1 model, with parameter 0.8 0.6 Innovative outlier for 20 periods, n=500
Patrice Bertail (MODAL'X)
1er février 2017
31 / 38
Asymptotic validity of the regenerative Bootstrap
0.0
0.1
0.2
0.3
0.4
0.5
Robustness for Markov chains
−2
0
2
4
6
8
10
True and Bootstrap dist., Median in a MM1 0.8 0.6
Figure: Comparison of the true distribution of the recentered normalized median
in a MM1 model, with parameter 0.8 0.6 Innovative outlier for 20 periods, n=500
Patrice Bertail (MODAL'X)
1er février 2017
32 / 38
Two applications : robustied L-statitics and R-statistics
Outlines
1
Pseudo-regeneration for Markov Chains
Framework Harris recurrence
The atomic case
Approximate renewal sequence : pseudo-regeneration
Regenerative and Approximate regenerative Bootstrap
2
Robustness for Markov chains
Robustness and inuence function for Markov chains
Examples of inuence functions
Main Hypotheses
a Central Limit Theorem and its bootstrap version
Asymptotic validity of the regenerative Bootstrap
3
Two applications : robustied L-statitics and R-statistics
Robustied L-statistics
Linear Rank signed statistics for Markov chains
4
A few references
Patrice Bertail (MODAL'X)
1er février 2017
33 / 38
Two applications : robustied L-statitics and R-statistics
Robustied L-statistics
Robustied empirical distribution
Let M > 0 and consider the robustied version of the cdf
Fµ (y ) = µ(] − ∞, y ]) given by
E
FL,M (y ) = A
Pτ
i =1 I{Xi ≤ y } I{τA ≤ M } .
EA [τA I{τA ≤ M }]
A
It is easy to see that the inuence function of FL,M (y ) is given by
(1)
FM (b ; y , L) =
Patrice Bertail (MODAL'X)
PL(b )
i =1 (I{bi ≤ y } − Fµ (y )) I{L(b ) ≤ M } for all b ∈ T,
EA [τA I{τA ≤ M }]
1er février 2017
34 / 38
Two applications : robustied L-statitics and R-statistics
Robustied L-statistics
−1
For M ≥ 1, denote by FL,
M (α) the α−quantile of FL,M and by fL,M its
density. In this case, the inuence function is given by : ∀b ∈ T ,
(1)
QL,
M (b ; α, L) =
I{L(b ) ≤ M }
P
L(b )
j =1
−1
α − I{bj ≤ FL,
(α)}
M
−1
EA [τA I{τA ≤ M }]fL,M (FL,
M (α))
.
Dene a robust L-statistics by
TJ ,α,M (L) =
Z1
0
−1
J (u )w (FL,
M (u ))du ,
(5)
where J : (0, 1) → R a continuous score function bounded by MJ and
w : R → R is a continuously dierentiable bounded by Lw . Then
|TJ ,α,M (L) − TJ ,α,∞ (L)|
≤
2MJ Lw
EA [τA I{τA ≤ M }]
Patrice Bertail (MODAL'X)
P(τA ≥ M )1/2 (E[τA ])1/2 (EA [|
τ
X
A
i =1
Xi |])1/2 .
1er février 2017
35 / 38
Two applications : robustied L-statitics and R-statistics
Robustied L-statistics
Bootstrap and Asympt. normality of robustied L-statisti
Theorem
The functional TJ ,α,M (L) is Fréchet dierentiable at L with respect to
the dBL norm and we have
)
n 1/2 (TJ ,α,M (Ln )−TJ ,α,M (L)) ⇒ N (0, EA [τA ]VarA (TJ(1,α,
M (B1 , L)) as n → +∞
with
)
VarA (TJ(1,α,
M (B1 , L)) = VarA
Z ∞
−∞
J (FL,M (v ))FM(1) (b ; v , L)w (1) (v )dv
.
In addition, the (A)RBB applied to the robust L-statistic TJ ,α,M (Ln )
is asymptotically valid.
Patrice Bertail (MODAL'X)
1er février 2017
36 / 38
Robustied L-statistics
0.0
0.1
0.2
0.3
0.4
Two applications : robustied L-statitics and R-statistics
−4
−2
0
2
4
True and Bootstrap dist., Robust Winsorized Mean in a MM1 0.8 0.6
Figure: Comparison of the true distribution of the robustied winzorized mean in
a MM1 model, with parameter 0.8 0.6 Innovative outlier for 20 periods, n=500
Patrice Bertail (MODAL'X)
1er février 2017
37 / 38
Two applications : robustied L-statitics and R-statistics
Linear Rank signed statistics for Markov chains
R-parameter and their empirical counterpart
Statistics associated with the functional
Tφ (L) =
1
EL [L(B)]
LX
(B)


EL 
φ(Fµ+ (|Xi |))sign (Xi )
i =1
where Fµ+ (x ) = Pµ (|X1 | ≤ x ) and the score generating function φ is of
class C 1 .
Horrible formulas but it works... Not possible to control the remainder
with the Bounded Lipschitz distance. Compute the inuence function
and then control directly the remainder with U-statistics arguments.
The plug-in estimator is asymptotically gaussian but not pivotal due to
the dependance structure. In addition, the RBB applied to the
regenerative linear R-statistic is asymptotically valid.
Patrice Bertail (MODAL'X)
1er février 2017
38 / 38
A few references
A few bibliographical references
S. Asmussen
, Springer-Verlag, New York, 2003.
Applied probability and queues
P. Bertail and S. Clémençon, Edgeworth expansions for suitably normalized sample mean statistics of atomic Markov chains,
Prob. Th. Rel. Fields 130 (2004), no. 3, 388414.
P. Bertail and S. Clémençon,
Regenerative-block bootstrap for Markov chains
, Bernoulli 12 (2005), no. 4.
P. Bertail and S. Clémençon, Regeneration-based statistics for Harris recurrent Markov chains, Probability and
Statistics for dependent data (P. Bertail, P. Doukhan, and P. Soulier, eds.), Lecture notes in Statistics,
vol. 187, Springer, 2006, pp. 354.
P. Bertail and S. Clémençon. Sharp bounds for the tails of functionals of Markov chains.
54(3) :505515, 2010.
,
Th. Prob. Appl.
P. Bertail and S. Clémençon. A renewalapproach to U-statistics for Markovian data.Mathematical Methods of
, 20(2) :79115, 2011.
Statistics
R.D. Martin and V.J. Yohai. Inuence functionals for time series.
S.P. Meyn and R.L. Tweedie,
E. Nummelin,
H. Rieder.
Markov chains and stochastic stability
A splitting technique for Harris recurrent chains.
Robust asymptotic statistics
R. Sering.
, 14 :781818, 1986.
, Springer-Verlag, 1996.
, Z. Wahrsch. Verw. Gebiete 43 (1978), 309318.
. 1994.
Approximation Theorems of Mathematical Statistics
Patrice Bertail (MODAL'X)
Ann. Stat.
. 2002.
1er février 2017
39 / 38