adaptive estimation of survival function in the convolution model...

Adaptive estimation of survival function in theconvolution model on R+

Gwennaelle MABON

CREST - ENSAE & Universite Paris Descartes

April, 20th 2016

G. MABON (CREST & MAP5) Survival function in the convolution model April, 20th 2016 1 / 15

Framework Motivation : additive processes

Motivations: one-sided error in convolution models (a.k.a. additivemeasurement errors).

−→ Application to back calculation problems in AIDS researchGroeneboom and Wellner (1992), van Es et al. (1998), Jongbloed (1998),Groeneboom and Jongbloed (2003).

−→ Application in finance (nonparametric regression)Jirak, Meister and Reiß (2014), Reiß & Selk (2015) .

Statistical model

� We study the following model:

Zi = Xi + Yi , i = 1, . . . , n, (1)

� Xi ’s i.i.d. nonnegative variables with unknown density f ,survival function SX .

� Yi ’s i.i.d. nonnegative variables with known density g , survivalfunction SY .

� (Xi )i |= (Yi )i , Zi ∼ h, survival function SZ .

Target: estimation of SX when the Zi ’s are observed and g is known.

Statistical model Steps

� Assumptions: SX , SY and g belong to L2(R+).

� Find an appropriate orthonormal basis of L2(R+) ,(ϕk)k≥0,

SX (x) =∑k≥0

ak(SX )ϕk(x).

ak(SX ): k-th component of SX in the orthonormal basis.

� Study the MISE of the estimator in this basis.

E‖SX − SX ,m‖2 ≤ ?

� Build a model selection procedure a la Birge and Massart.

m = arg minm∈M

γn(SX ,m) + pen(m).

SX (x) =∑k≥0

ak(SX )ϕk(x).

E‖SX − SX ,m‖2 ≤ ?

m = arg minm∈M

SX (x) =∑k≥0

ak(SX )ϕk(x).

E‖SX − SX ,m‖2 ≤ ?

m = arg minm∈M

SX (x) =∑k≥0

ak(SX )ϕk(x).

E‖SX − SX ,m‖2 ≤ ?

m = arg minm∈M

Survival function estimation Convolution equation

Let z ≥ 0, by definition SZ (z) = P(Z > z), we get

SZ (z) = P(X + Y > z) =

∫∫1x+y>z f (x)1x≥0 g(y)1y≥0 dx dy

∫ (∫ +∞

z−yf (x) dx

)g(y)1y≥01z−y≥0 dy

∫ (∫ +∞

0f (x) dx

)g(y)1y≥01z−y≤0 dy

0SX (z − y)g(y) dy + SY (z).

SZ (z) = SX ? g(z) + SY (z) (2)

Survival function estimation Convolution equation

Let z ≥ 0, by definition SZ (z) = P(Z > z), we get

SZ (z) = P(X + Y > z) =

∫∫1x+y>z f (x)1x≥0 g(y)1y≥0 dx dy

∫ (∫ +∞

z−yf (x) dx

)g(y)1y≥01z−y≥0 dy

∫ (∫ +∞

0f (x) dx

)g(y)1y≥01z−y≤0 dy

0SX (z − y)g(y) dy + SY (z).

SZ (z) = SX ? g(z) + SY (z) (2)

Survival function estimation Laguerre procedure

� For R+-supported functions, the convolution product writes

SX ? g(z) =

0SX (u)g(z − u)du

=∞∑k=0

∞∑j=0

ak(SX )aj(g)

0ϕk(u)ϕj(z − u) du.

� We introduce the Laguerre basis defined for k ∈ N, x ≥ 0, by

ϕk(x) =√

2Lk(2x)e−x with Lk(x) =k∑

)(−x)j

The (ϕk)k ’s form an orthonormal basis of L2(R+).

� What makes the Laguerre basis relevant is the relation∫ x

0ϕk(u)ϕj(x − u)du = 2−1/2 (ϕk+j(x)− ϕk+j+1(x)) .

SX ? g(z) =

0SX (u)g(z − u)du

=∞∑k=0

∞∑j=0

ak(SX )aj(g)

ϕk(x) =√

)(−x)j

SX ? g(z) =

0SX (u)g(z − u)du

=∞∑k=0

∞∑j=0

ak(SX )aj(g)

ϕk(x) =√

)(−x)j

� It yields

SX ? g(z)

=1√2

∞∑k=0

ϕk(z)(

ak(SX )a0(g) +k∑

(ak−l(g)− ak−l−1(g)

)al(SX )

� Equation implies (2)

SX ? g(z) = SZ (z)− SY (z) =∑k≥0

(ak(SZ )− ak(SY ))ϕk(z)

� We obtain for any m that

Gm~SX ,m = ~SZ ,m − ~SY ,m

~S•,m = t(a0(S•), . . . , am−1(S•)).� Gm is the lower triangular Toeplitz matrix with elements

Gm =1√2

a0(g) if i = j ,

ai−j(g)− ai−j−1(g) if j < i ,

0 otherwise.

� It yields

SX ? g(z)

=1√2

∞∑k=0

ϕk(z)(

ak(SX )a0(g) +k∑

)al(SX )

SX ? g(z) = SZ (z)− SY (z) =∑k≥0

Gm =1√2

a0(g) if i = j ,

0 otherwise.

� It yields

SX ? g(z)

=1√2

∞∑k=0

ϕk(z)(

ak(SX )a0(g) +k∑

)al(SX )

SX ? g(z) = SZ (z)− SY (z) =∑k≥0

Gm =1√2

a0(g) if i = j ,

0 otherwise.

� It yields

SX ? g(z)

=1√2

∞∑k=0

ϕk(z)(

ak(SX )a0(g) +k∑

)al(SX )

SX ? g(z) = SZ (z)− SY (z) =∑k≥0

Gm =1√2

a0(g) if i = j ,

0 otherwise.

� Gm is a lower triangular matrix and is invertible iff the coefficients ofthe diagonal are different from 0.

a0(g) =√

g(u)e−u du =√

2E[e−Y ] > 0.

� It yields

~SX ,m = G−1m

(~SZ ,m − ~SY ,m

)� Remark:

ak(SZ ) =

SZ (u)ϕk(u) du =

ϕk(u)

(∫ +∞

uh(v) dv

(∫ v

0ϕk(u) du

)h(v) dv = E [Φk(Z1)]

with Φk a primitive of ϕk defined as Φk(x) =∫ x

0 ϕk(u) du.

a0(g) =√

g(u)e−u du =√

2E[e−Y ] > 0.

� It yields

~SX ,m = G−1m

(~SZ ,m − ~SY ,m

)� Remark:

ak(SZ ) =

SZ (u)ϕk(u) du =

ϕk(u)

(∫ +∞

uh(v) dv

(∫ v

0ϕk(u) du

)h(v) dv = E [Φk(Z1)]

0 ϕk(u) du.

a0(g) =√

g(u)e−u du =√

2E[e−Y ] > 0.

� It yields

~SX ,m = G−1m

(~SZ ,m − ~SY ,m

)� Remark:

ak(SZ ) =

SZ (u)ϕk(u) du =

ϕk(u)

(∫ +∞

uh(v) dv

(∫ v

0ϕk(u) du

)h(v) dv = E [Φk(Z1)]

0 ϕk(u) du.

� Let Sm = span{ϕk}k∈{0,...,m−1} and consider SX ,m the projection ofSX on Sm

SX ,m(x) =m−1∑k=0

ak(SX )ϕk(x). (4)

Definition (Projection estimator)

SX ,m(x) =m−1∑k=0

akϕk(x) (5)

t(a0, . . . , am−1) = ~SX ,m and ~SZ ,m = t(a0(Z ), . . . , am−1(Z ))

with ~SX ,m = G−1m

(~SZ ,m − ~SY ,m

)and ak(Z ) =

n∑i=1

Φk(Zi ),

Survival function estimation Upper bounds

Proposition (M.(2014))

If SX and g ∈ L2(R+) and E[Z1] <∞, for Gm defined by (3) and SX ,m

defined by (5), the following result holds

E‖SX − SX ,m‖2 ≤ ‖SX − SX ,m‖2 +E[Z1]

n%2(G−1

m ). (6)

%2 (A) is the largest eigenvalue of a matrix tAA in absolute value.

Consequence

� m plays the same role as a bandwith parameter.

m too small ⇒ dominant bias.m too big ⇒ dominant variance.

� Choose m to have a trade-off between the bias and the variance.

Survival function estimation Upper bounds

Proposition (M.(2014))

If SX and g ∈ L2(R+) and E[Z1] <∞, for Gm defined by (3) and SX ,m

defined by (5), the following result holds

E‖SX − SX ,m‖2 ≤ ‖SX − SX ,m‖2 +E[Z1]

n%2(G−1

m ). (6)

%2 (A) is the largest eigenvalue of a matrix tAA in absolute value.

Consequence

� m plays the same role as a bandwith parameter.

m too small ⇒ dominant bias.m too big ⇒ dominant variance.

� Choose m to have a trade-off between the bias and the variance.

Model selection

Goal: define an empirical version of the upper bound on the MISE

‖SX − SX ,m‖2 +E[Z1]

n%2(G−1

−→ Approximation of the bias term by

−‖SX ,m‖2

−→ Approximation of the variance term by

pen(m) =κE[Z1]

n%2(G−1

)log n

Model selection

‖SX − SX ,m‖2 +E[Z1]

n%2(G−1

−‖SX ,m‖2

pen(m) =κE[Z1]

n%2(G−1

)log n

Model selection

‖SX − SX ,m‖2 +E[Z1]

n%2(G−1

−‖SX ,m‖2

pen(m) =κE[Z1]

n%2(G−1

)log n

Model selection Theorem

(B1) Mn ={

1 ≤ m ≤ n, %2(G−1

)log n ≤ Cn

}, where C > 0.

(B2) 0 < E[Z 31 ] <∞.

Theorem (M.(2014))

If SX and g ∈ L2(R+), let us suppose that (B1)-(B2) are true. Let SX ,m

be defined by (5) and

m = argminm∈Mn

{−‖SX ,m‖2 + pen(m)

with pen(m) =κE[Z1]

n%2(G−1

)log n, then there exists a positive

numerical constant κ ≥ κ0 such that

E‖SX − SX ,m‖2 ≤ 4 infm∈Mn

{‖SX − SX ,m‖2 + pen(m)

where C is a constant depending on E[Z 31 ].

(B1) Mn ={

1 ≤ m ≤ n, %2(G−1

)log n ≤ Cn

}, where C > 0.

(B2) 0 < E[Z 31 ] <∞.

Theorem (M.(2014))

m = argminm∈Mn

{−‖SX ,m‖2 + pen(m)

with pen(m) =κE[Z1]

n%2(G−1

)log n, then there exists a positive

numerical constant κ ≥ κ0 such that

{‖SX − SX ,m‖2 + pen(m)

where C is a constant depending on E[Z 31 ].

Corollary (M.(2014))

m = argminm∈Mn

{−‖SX ,m‖2 + pen(m)

pen(m) =2κZn

n%2(G−1

)log n where Zn =

n∑i=1

then there exists a positive numerical constant κ ≥ κ0 such that

{‖SX − SX ,m‖2 + pen(m)

where C is a constant depending on E[Z1], E[Z 21 ], E[Z 3

Conclusion Extensions

� Estimation of the survival function in a global setting on R+.

� Related works:

I Estimation of the density → Mabon (2014).I Estimation of linear functionals of the density (c.d.f, pointwise

estimation of the density, Laplace transform) → Mabon (2015).

� Perspectives:

I Estimation when g is unknown → work in progress.I Goodness-of-fit test.

Thank you for your attention.

Conclusion Extensions

� Estimation of the survival function in a global setting on R+.

� Related works:

I Estimation of the density → Mabon (2014).I Estimation of linear functionals of the density (c.d.f, pointwise

estimation of the density, Laplace transform) → Mabon (2015).

� Perspectives:

I Estimation when g is unknown → work in progress.I Goodness-of-fit test.

Thank you for your attention.

Conclusion Bibliography

I van Es, B. (2011). Combining kernel estimators in the uniformdeconvolution problem. Statistica Neerlandica, 65(3):275–296.

I Groeneboom, P. and Jongbloed, G. (2003). Density estimation in theuniform deconvolution model. Statistica Neerlandica, 57(1):136–157.

I Jirak, M, Meister, A. and Reiß, M. (2014). Adaptive functionestimation in nonparametric regression with one-sided errors. TheAnnals of Statistics, 42(5):1970–2002.

I Jongbloed, G. (1998). Exponential deconvolution: two asymptoticallyequivalent estimators. Statistica Neerlandica, 52(1):6–17.

I Mabon, G. (2014). Adaptive deconvolution on the nonnegative realline. preprint MAP5 2014-33. In revision

I Mabon, G. (2015). Adaptive deconvolution of linear functionals onthe nonnegative real line. preprint MAP5 2015-24. In revision

I Reiß & Selk (2015). Efficient estimation of functionals innonparametric boundary models. To appear in Bernoulli.

adaptive estimation of survival function in the convolution model...

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