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Adaptive estimation of survival function in the convolution model on R + Gwenna elle MABON CREST - ENSAE & Universit e Paris Descartes April, 20th 2016 G. MABON (CREST & MAP5) Survival function in the convolution model April,

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Adaptive estimation of survival function in the convolution model on R+

Gwenna¨ elle MABON

CREST - ENSAE & Universit´ e Paris Descartes

April, 20th 2016

  • G. MABON (CREST & MAP5)

Survival function in the convolution model April, 20th 2016 1 / 15

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Framework Motivation : additive processes

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

− → Application to back calculation problems in AIDS research

Groeneboom 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) .

  • G. MABON (CREST & MAP5)

Survival function in the convolution model April, 20th 2016 2 / 15

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Framework Motivation : additive processes

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

− → Application to back calculation problems in AIDS research

Groeneboom 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) .

  • G. MABON (CREST & MAP5)

Survival function in the convolution model April, 20th 2016 2 / 15

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Framework Motivation : additive processes

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

− → Application to back calculation problems in AIDS research

Groeneboom 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) .

  • G. MABON (CREST & MAP5)

Survival function in the convolution model April, 20th 2016 2 / 15

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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, survival

function SY .

(Xi)i

| = (Yi)i, Zi ∼ h, survival function SZ. Target: estimation of SX when the Zi’s are observed and g is known.

  • G. MABON (CREST & MAP5)

Survival function in the convolution model April, 20th 2016 3 / 15

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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.

ESX − ˆ SX,m2 ≤ ?

Build a model selection procedure `

a la Birg´ e and Massart.

  • m = arg min

m∈M γn(ˆ

SX,m) + pen(m).

  • G. MABON (CREST & MAP5)

Survival function in the convolution model April, 20th 2016 4 / 15

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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.

ESX − ˆ SX,m2 ≤ ?

Build a model selection procedure `

a la Birg´ e and Massart.

  • m = arg min

m∈M γn(ˆ

SX,m) + pen(m).

  • G. MABON (CREST & MAP5)

Survival function in the convolution model April, 20th 2016 4 / 15

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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.

ESX − ˆ SX,m2 ≤ ?

Build a model selection procedure `

a la Birg´ e and Massart.

  • m = arg min

m∈M γn(ˆ

SX,m) + pen(m).

  • G. MABON (CREST & MAP5)

Survival function in the convolution model April, 20th 2016 4 / 15

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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.

ESX − ˆ SX,m2 ≤ ?

Build a model selection procedure `

a la Birg´ e and Massart.

  • m = arg min

m∈M γn(ˆ

SX,m) + pen(m).

  • G. MABON (CREST & MAP5)

Survival function in the convolution model April, 20th 2016 4 / 15

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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−y

f (x) dx

  • g(y)1y≥01z−y≥0 dy

+ +∞ f (x) dx

  • g(y)1y≥01z−y≤0 dy

= z SX(z − y)g(y) dy + SY (z). SZ(z) = SX ⋆ g(z) + SY (z) (2)

  • G. MABON (CREST & MAP5)

Survival function in the convolution model April, 20th 2016 5 / 15

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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−y

f (x) dx

  • g(y)1y≥01z−y≥0 dy

+ +∞ f (x) dx

  • g(y)1y≥01z−y≤0 dy

= z SX(z − y)g(y) dy + SY (z). SZ(z) = SX ⋆ g(z) + SY (z) (2)

  • G. MABON (CREST & MAP5)

Survival function in the convolution model April, 20th 2016 5 / 15

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Survival function estimation Laguerre procedure For R+-supported functions, the convolution product writes

SX ⋆ g(z) = z SX(u)g(z − u) du =

∞

  • k=0

∞

  • j=0

ak(SX)aj(g) z ϕk(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

  • j=0

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

What makes the Laguerre basis relevant is the relation

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

  • G. MABON (CREST & MAP5)

Survival function in the convolution model April, 20th 2016 6 / 15

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Survival function estimation Laguerre procedure For R+-supported functions, the convolution product writes

SX ⋆ g(z) = z SX(u)g(z − u) du =

∞

  • k=0

∞

  • j=0

ak(SX)aj(g) z ϕk(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

  • j=0

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

What makes the Laguerre basis relevant is the relation

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

  • G. MABON (CREST & MAP5)

Survival function in the convolution model April, 20th 2016 6 / 15

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Survival function estimation Laguerre procedure For R+-supported functions, the convolution product writes

SX ⋆ g(z) = z SX(u)g(z − u) du =

∞

  • k=0

∞

  • j=0

ak(SX)aj(g) z ϕk(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

  • j=0

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

What makes the Laguerre basis relevant is the relation

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

  • G. MABON (CREST & MAP5)

Survival function in the convolution model April, 20th 2016 6 / 15

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Survival function estimation Laguerre procedure It yields

SX ⋆ g(z) = 1 √ 2

∞

  • k=0

ϕk(z)

  • ak(SX)a0(g) +

k

  • l=0
  • 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,

  • therwise.

(3)

  • G. MABON (CREST & MAP5)

Survival function in the convolution model April, 20th 2016 7 / 15

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Survival function estimation Laguerre procedure It yields

SX ⋆ g(z) = 1 √ 2

∞

  • k=0

ϕk(z)

  • ak(SX)a0(g) +

k

  • l=0
  • 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,

  • therwise.

(3)

  • G. MABON (CREST & MAP5)

Survival function in the convolution model April, 20th 2016 7 / 15

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Survival function estimation Laguerre procedure It yields

SX ⋆ g(z) = 1 √ 2

∞

  • k=0

ϕk(z)

  • ak(SX)a0(g) +

k

  • l=0
  • 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,

  • therwise.

(3)

  • G. MABON (CREST & MAP5)

Survival function in the convolution model April, 20th 2016 7 / 15

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Survival function estimation Laguerre procedure It yields

SX ⋆ g(z) = 1 √ 2

∞

  • k=0

ϕk(z)

  • ak(SX)a0(g) +

k

  • l=0
  • 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,

  • therwise.

(3)

  • G. MABON (CREST & MAP5)

Survival function in the convolution model April, 20th 2016 7 / 15

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Survival function estimation Laguerre procedure Gm is a lower triangular matrix and is invertible iff the coefficients of

the diagonal are different from 0. a0(g) = √ 2

  • R+ g(u)e−u du =

√ 2E[e−Y ] > 0.

It yields

  • SX,m = G−1

m

  • SZ,m −

SY ,m

  • Remark:

ak(SZ) =

  • R+ SZ(u)ϕk(u) du =
  • R+ ϕk(u)

+∞

u

h(v) dv

  • du

=

  • R+

v ϕk(u) du

  • h(v) dv = E [Φk(Z1)]

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

0 ϕk(u) du.

  • G. MABON (CREST & MAP5)

Survival function in the convolution model April, 20th 2016 8 / 15

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Survival function estimation Laguerre procedure Gm is a lower triangular matrix and is invertible iff the coefficients of

the diagonal are different from 0. a0(g) = √ 2

  • R+ g(u)e−u du =

√ 2E[e−Y ] > 0.

It yields

  • SX,m = G−1

m

  • SZ,m −

SY ,m

  • Remark:

ak(SZ) =

  • R+ SZ(u)ϕk(u) du =
  • R+ ϕk(u)

+∞

u

h(v) dv

  • du

=

  • R+

v ϕk(u) du

  • h(v) dv = E [Φk(Z1)]

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

0 ϕk(u) du.

  • G. MABON (CREST & MAP5)

Survival function in the convolution model April, 20th 2016 8 / 15

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Survival function estimation Laguerre procedure Gm is a lower triangular matrix and is invertible iff the coefficients of

the diagonal are different from 0. a0(g) = √ 2

  • R+ g(u)e−u du =

√ 2E[e−Y ] > 0.

It yields

  • SX,m = G−1

m

  • SZ,m −

SY ,m

  • Remark:

ak(SZ) =

  • R+ SZ(u)ϕk(u) du =
  • R+ ϕk(u)

+∞

u

h(v) dv

  • du

=

  • R+

v ϕk(u) du

  • h(v) dv = E [Φk(Z1)]

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

0 ϕk(u) du.

  • G. MABON (CREST & MAP5)

Survival function in the convolution model April, 20th 2016 8 / 15

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Survival function estimation Laguerre procedure Let Sm = span{ϕk}k∈{0,...,m−1} and consider SX,m the projection of

SX on Sm SX,m(x) =

m−1

  • k=0

ak(SX)ϕk(x). (4) Definition (Projection estimator)

  • SX,m(x) =

m−1

  • k=0

ˆ akϕk(x) (5)

t(ˆ

a0, . . . , ˆ am−1) = ˆ

  • SX,m

and

  • SZ,m = t(ˆ

a0(Z), . . . , ˆ am−1(Z)) with

  • SX,m = G−1

m

  • SZ,m −

SY ,m

  • and

ˆ ak(Z) = 1 n

n

  • i=1

Φk(Zi),

  • G. MABON (CREST & MAP5)

Survival function in the convolution model April, 20th 2016 9 / 15

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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 ESX − SX,m2 ≤ SX − SX,m2 + 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.

  • G. MABON (CREST & MAP5)

Survival function in the convolution model April, 20th 2016 10 / 15

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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 ESX − SX,m2 ≤ SX − SX,m2 + 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.

  • G. MABON (CREST & MAP5)

Survival function in the convolution model April, 20th 2016 10 / 15

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Model selection

Goal: define an empirical version of the upper bound on the MISE SX − SX,m2 + E[Z1] n ̺2(G−1

m )

− → Approximation of the bias term by

− SX,m2

− → Approximation of the variance term by

pen(m) = κE[Z1] n ̺2 G−1

m

  • log n
  • G. MABON (CREST & MAP5)

Survival function in the convolution model April, 20th 2016 11 / 15

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Model selection

Goal: define an empirical version of the upper bound on the MISE SX − SX,m2 + E[Z1] n ̺2(G−1

m )

− → Approximation of the bias term by

− SX,m2

− → Approximation of the variance term by

pen(m) = κE[Z1] n ̺2 G−1

m

  • log n
  • G. MABON (CREST & MAP5)

Survival function in the convolution model April, 20th 2016 11 / 15

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SLIDE 27

Model selection

Goal: define an empirical version of the upper bound on the MISE SX − SX,m2 + E[Z1] n ̺2(G−1

m )

− → Approximation of the bias term by

− SX,m2

− → Approximation of the variance term by

pen(m) = κE[Z1] n ̺2 G−1

m

  • log n
  • G. MABON (CREST & MAP5)

Survival function in the convolution model April, 20th 2016 11 / 15

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Model selection Theorem

(B1) Mn =

  • 1 ≤ m ≤ n, ̺2

G−1

m

  • log n ≤ Cn
  • , where C > 0.

(B2) 0 < E[Z 3

1 ] < ∞.

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 = argmin

m∈Mn

  • −

SX,m2 + pen(m)

  • ,

with pen(m) = κE[Z1] n ̺2 G−1

m

  • log n, then there exists a positive

numerical constant κ ≥ κ0 such that ESX − SX,

m2 ≤ 4

inf

m∈Mn

  • SX − SX,m2 + pen(m)
  • + C

n , where C is a constant depending on E[Z 3

1 ].

  • G. MABON (CREST & MAP5)

Survival function in the convolution model April, 20th 2016 12 / 15

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Model selection Theorem

(B1) Mn =

  • 1 ≤ m ≤ n, ̺2

G−1

m

  • log n ≤ Cn
  • , where C > 0.

(B2) 0 < E[Z 3

1 ] < ∞.

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 = argmin

m∈Mn

  • −

SX,m2 + pen(m)

  • ,

with pen(m) = κE[Z1] n ̺2 G−1

m

  • log n, then there exists a positive

numerical constant κ ≥ κ0 such that ESX − SX,

m2 ≤ 4

inf

m∈Mn

  • SX − SX,m2 + pen(m)
  • + C

n , where C is a constant depending on E[Z 3

1 ].

  • G. MABON (CREST & MAP5)

Survival function in the convolution model April, 20th 2016 12 / 15

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Model selection Theorem

Corollary (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 = argmin

m∈Mn

  • −

SX,m2 + pen(m)

  • pen(m) = 2κ¯

Zn n ̺2 G−1

m

  • log n

where ¯ Zn = 1 n

n

  • i=1

Zi, then there exists a positive numerical constant κ ≥ κ0 such that ESX − SX,

m2 ≤ 4

inf

m∈Mn

  • SX − SX,m2 + pen(m)
  • + C

n where C is a constant depending on E[Z1], E[Z 2

1 ], E[Z 3 1 ].

  • G. MABON (CREST & MAP5)

Survival function in the convolution model April, 20th 2016 13 / 15

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SLIDE 31

Conclusion Extensions Estimation of the survival function in a global setting on R+. Related works: ◮ Estimation of the density → Mabon (2014). ◮ Estimation of linear functionals of the density (c.d.f, pointwise

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

Perspectives: ◮ Estimation when g is unknown → work in progress. ◮ Goodness-of-fit test.

Thank you for your attention.

  • G. MABON (CREST & MAP5)

Survival function in the convolution model April, 20th 2016 14 / 15

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SLIDE 32

Conclusion Extensions Estimation of the survival function in a global setting on R+. Related works: ◮ Estimation of the density → Mabon (2014). ◮ Estimation of linear functionals of the density (c.d.f, pointwise

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

Perspectives: ◮ Estimation when g is unknown → work in progress. ◮ Goodness-of-fit test.

Thank you for your attention.

  • G. MABON (CREST & MAP5)

Survival function in the convolution model April, 20th 2016 14 / 15

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SLIDE 33

Conclusion Bibliography ◮ van Es, B. (2011). Combining kernel estimators in the uniform

deconvolution problem. Statistica Neerlandica, 65(3):275–296.

◮ Groeneboom, P. and Jongbloed, G. (2003). Density estimation in the

uniform deconvolution model. Statistica Neerlandica, 57(1):136–157.

◮ Jirak, M, Meister, A. and Reiß, M. (2014). Adaptive function

estimation in nonparametric regression with one-sided errors. The Annals of Statistics, 42(5):1970–2002.

◮ Jongbloed, G. (1998). Exponential deconvolution: two asymptotically

equivalent estimators. Statistica Neerlandica, 52(1):6–17.

◮ Mabon, G. (2014). Adaptive deconvolution on the nonnegative real

  • line. preprint MAP5 2014-33. In revision

◮ Mabon, G. (2015). Adaptive deconvolution of linear functionals on

the nonnegative real line. preprint MAP5 2015-24. In revision

◮ Reiß & Selk (2015). Efficient estimation of functionals in

nonparametric boundary models. To appear in Bernoulli.

  • G. MABON (CREST & MAP5)

Survival function in the convolution model April, 20th 2016 15 / 15