Estimation of the quantile function using Bernstein-Durrmeyer ...

Estimation of the quantile function using Bernstein-Durrmeyer polynomials

Andrey Pepelyshev Ansgar Steland Ewaryst Rafajlowic

The work is supported by the German Federal Ministry of the Environment, Nature and Nuclear Safety, grant 0325226.

Hamburg February 20, 2013

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Bernstein-Durrmeyer quatile estimation

Contents

Contents

Bernstein-Durrmeyer operator Properties of the BD estimator of the distribution function Properties of the BD estimator of the quantile function The adaptive choice of N Application in photovoltaics

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Bernstein-Durrmeyer quatile estimation

Bernstein-Durrmeyer operator

Bernstein-Durrmeyer operator

The BD operator DN (u(x)) has the kernel form 1

Z DN (u(x)) =

KN (x, z)u(z)dz 0

where KN (x, z) = (N + 1)

PN

i=0

(N )

Bi

DN (u(x)) = (N + 1)

(N )

(x)Bi

N X

(N )

ai Bi

(z), or

(x)

i=0

whereZ ai = (N )

Bi 3/23

1

(N )

u(x)Bi

(x)dx,

0

(x) =

N! xi (1 i!(N −i)!

− x)N −i .

A. Pepelyshev, A. Steland

Bernstein-Durrmeyer quatile estimation

Bernstein-Durrmeyer operator

BD operator for deterministic functions

Derriennic (1981) shown that DN (u) converges uniformly,

sup |DN (u(x)) − u(x)| ≤ 2ωu (N −1/2 ), x∈[0,1]

where ωu denotes the modulus of continuity of u. Chen and Ditzian (1991) established the fact that

ku(x) − DN (u(x))kp = O(N −δ/2 ), if and only if

(u) =

inf

a0 ,...,aN

n o ku(x) − a0 − a1 x − · · · − aN xN kp = O(N −δ ),

as N → ∞ for some δ ∈ (0, 2) and p ∈ {2, ∞}, where
1/p R kf kp = I |f (x)|p dx . 4/23

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Bernstein-Durrmeyer quatile estimation

Bernstein-Durrmeyer operator

BD operator for the distribution function

Let Fm (x) be the empirical d.f. for a sample X1 , . . . , Xm ∼ F[0,1] with density f and m

1 X fem,N (x) = KN (Xj , x), Fem,N (x) := m j=1

Z

x

fem,N (t)dt 0

Theorem. (Ciesielski, 1988) If F ∈ C[0, 1], then

Pr(kFem,N (x) − F k∞ → 0 as m, N → ∞) = 1. If f ∈ Lp [0, 1] and m = bN β c for some β ∈ (0, 0.5), then

Pr(kfem,N (x) − f kp = o(1) as N → ∞) = 1. 5/23

A. Pepelyshev, A. Steland

Bernstein-Durrmeyer quatile estimation

Bernstein-Durrmeyer operator

BD operator for the quantile function

Let Qm (x) be the empirical q.f. for a sample X1 , . . . , Xm . Version 1: N X (N ) em,N (x) := DN (Qm (x)) = (N + 1) Q e ai Bi (x), i=0

Z e ai =

1

(N )

Qm (x)Bi

(x)dx,

i = 1, . . . , m,

0

Version 2:

bm,N (x) = (N + 1) Q

N X

(N )

b ai Bi

(x),

i=0

b ai = Note that 6/23

e ai

1 m

m X

(N )

X(j) Bi

j=1

are asymptotically equivalent to A. Pepelyshev, A. Steland

(

j−1 ) m−1

e ai

in mean square (Yang, 1985).

Bernstein-Durrmeyer quatile estimation

Bernstein-Durrmeyer operator

Properties of the BD estimator Lemma. For e ai it is hold

Var(e ai ) ≤ Proof. Set

1 max Var(X(j) ) (N + 1)2 j=1,...,m R j/m

w ej =

(N ) B (t) dt (j−1)/m i , Pm R l/m (N ) (t) dt l=1 (l−1)/m Bi

j = 1, . . . , m. Then we have that Var(e ai ) =

m Z X l=1

l/m

!2 (N ) Bi (t) dt

(l−1)/m

E

m X

(X(j) −EX(j) )2 w ej .

j=1

Apply the Jensen inequality (Eζ (aζ − Eaζ ))2 ≤ Eζ (aζ − Eaζ )2 , where ζ is a random variable such that P(ζ = j) = wj . 7/23

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Bernstein-Durrmeyer quatile estimation

Bernstein-Durrmeyer operator

Properties of the BD estimator Lemma. For the BD estimator it is hold

em,N (x)) ≤ Cm := max Var(X(j) ) Var(Q j=1,...,m

and

Z

1

em,N (x))dx ≤ Cm . Var(Q 0

If the derivative of Q(x) is continuous, then Cm = O(1/m). Proof. If X1 , . . . , Xm ∼ U (0, 1) then X(j) ∼ β(j, m − j + 1). Note that the variance of the beta distribution β(a, b) is ab . Therefore, it is easy to see that Cm = O(1/m). (a+b)2 (a+b+1) 8/23

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Bernstein-Durrmeyer quatile estimation

Bernstein-Durrmeyer operator

Maximum variance of order statistics

Since Q0 (x) is continuous, then we can write X(j) = Q(U(j) ). Applying the LagrangeTaylor formula for Q(U(j) ) at the point EU(j) = j/(m + 1) = pj , we obtain

X(j) = Q(pj ) + (U(j) − pj )Q0 (ζj ) for some ζj ∈ [min{U(j) , pj }, max{U(j) , pj }]. Then we have

Var(X(j) ) = Var((U(j) − pj )Q0 (ζj )) ≤ max Q02 (x)E(U(j) − pj )2 x

pj (1 − pj ) = max Q02 (x). x m+2

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Bernstein-Durrmeyer quatile estimation

Consistency

MSE and MISE consistency

Let X1 , . . . , Xm be m random variables with common quantile function Q(x) with a continuous derivative on [a, b] ⊂ [0, 1] that satises the Chen-Ditzian condition with p = ∞. Then the Bernstein-Durrmeyer estimator with integral weights, em,N (x), satises Q

em,N (x) − Q(x))2 = O(1/m + N −δ ) max E(Q

x∈[a,b]

and

Z E

b

em,N (x) − Q(x))2 dx = O(1/m + N −δ ) (Q

a

as m → ∞ and N → ∞. 10/23

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Bernstein-Durrmeyer quatile estimation

Consistency

The BD estimator with correction term

  em,N (x) . Let eel (x) = Dl Qm (x) − Q Dene the error-corrected Bernstein-Durrmeyer em,N,l (x) = Q em,N (x) + eel (x). Q

estimator

Theorem. Suppose X1 , X2 , . . . are i.i.d. random variables

with common quantile function Q(x) and distribution function R F (x) satisfying |x|p dF (x) < ∞ for some 1 ≤ p ≤ ∞. Then em,N,l (x) is consistent in the pth the error-corrected estimator Q mean for the true quantile function Q(x), em,N,l − Qkp → 0, kQ as m → ∞ and N, l → ∞, with probability 1, and in the mean, i.e. em,N,l − Qkp → 0, EkQ as m → ∞ and N, l → ∞. 11/23

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Bernstein-Durrmeyer quatile estimation

Adaptive choice of N

Adaptive choice of

N

We modify an algorithm from (Golyandina, Pepelyshev, Steland, 2012) whose idea is based on a balance of the closeness of the BD estimator to the empirical q.f. in terms of the law of iterated logarithms and the stability of the number of modes of the corresponding density estimator. (N )

Note that the Bernstein polynomial Bi (x) can be approximated by the density p of the normal distribution φ((i/N − x)/s), where s = x(1 − x)/N and 2 φ(x) =√(2π)−0.5 e−x /2 . Thus, we can introduce the quantity h = 1/ N which plays the role of the bandwidth. 12/23

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Bernstein-Durrmeyer quatile estimation

Adaptive choice of N

Algorithm of adaptive choice of

N

1] Compute n ¯h = max h ∈ (0, 1] : o bm,d1/t2 e (q)) − q ≤ 1/Rm ∀ t ∈ (0, h) max Fm (Q ∞ q

where Fm (x) is the empirical distribution function, dze is the smallest integer that is larger or equal to z , and √ p Rm = 2 m/ 2 log log m.

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Bernstein-Durrmeyer quatile estimation

Adaptive choice of N

Algorithm of adaptive choice of

N

¯ such that 2] Dene a set {h1 , . . . , hn } ∈ (0, h) maxh∈(0,h) ¯ minj=1,...,n |h − hj | is small and compute the sequence M1 , . . . , Mn , where Mj is the number of local bm,N (x) minimums of the Bernstein-Durrmeyer estimator Q with N = d1/h2j e. ˇ j = min{M1 , M2 , . . . , Mj }, j = 1, . . . , n. 3] Compute M

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Bernstein-Durrmeyer quatile estimation

Adaptive choice of N

Algorithm of adaptive choice of

N

4] Divide the set {h1 , . . . , hn } into groups as follows. Dene ai and bi such that a1 ≤ b1 < a2 ≤ b2 < . . . < ak ≤ bk and ˇi = M ˇ j for all hi , hj ∈ [al , bl ] for l ∈ {1, . . . , k}. M

P 5] Compute b h = ki=1 ai wi , where P b = d1/b wi = (bi − ai )/ kj=1 (bj − aj ), and then set N h2 e.

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Bernstein-Durrmeyer quatile estimation

Adaptive choice of N

Consistency of the BD estimator with

Nˆ

Let X1 , . . . , Xm be m random variables with √the continuously 2 m dierentiable quantile function Q, Rm = √2 log . log m Then the adaptive Bernstein-Durrmeyer estimator b b (q) = D b (Qm (q)) where N b is selected by the above Q m,N N algorithm is consistent as m → ∞. Moreover, we have the a.s. uniform error bound p √ b−1 (x) − F (x)| ≤ 2 log log m/ m sup |Q b m,N −∞ c) is the probability of the lot acceptance for given p, the true proportion of non-conforming items.

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Bernstein-Durrmeyer quatile estimation

Application in photovoltaics

Requirements for the sampling plan

The sampling plan is a minimal solution satisfying the following conditions ( OCn,c (p) ≥ 1 − α for p ≤ AQL, OCn,c (p) ≤ β for p ≥ RQL

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Bernstein-Durrmeyer quatile estimation

Application in photovoltaics

Sampling plans for quality control

If measurements are distributed according to a distribution G(x) with mean a and variance σ 2 , then
√ OCn,c (p) ∼ = 1 − Φ c + nG−1 (1 − p) . The asymptotically optimal sampling plan (n, c) is &
2 ' Φ−1 (α) − Φ−1 (1 − β) n =
2 , F −1 (AQL) − F −1 (RQL) √
n −1 F (AQL) + F −1 (RQL) , c = − 2 where F (x) = G((x − a)/σ) and Φ(x) is the standard normal distribution. 21/23

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Bernstein-Durrmeyer quatile estimation

Application in photovoltaics

Numerical results

Characteristics of estimation of the sampling plan (n, c) for two quantile estimators, α = β = 5%, AQL = 2% and RQL = 5%.

m 200 400 800 1600

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Bernstein-Durrmeyer estimator n c mean std.dev. mean std.dev. 57.9 24.8 13.3 2.3 57.9 19.9 13.5 1.9 55.7 14.0 13.4 1.5 54.3 9.9 13.4 1.0

A. Pepelyshev, A. Steland

Empirical quantile estimator n c mean std.dev. mean std.dev. 84.0 178.3 14.7 7.7 59.3 40.0 13.5 3.8 54.1 23.4 13.2 2.6 53.9 18.2 13.3 1.9

Bernstein-Durrmeyer quatile estimation

Application in photovoltaics

Thank you for your attention!

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