Estimation of the quantile function using Bernstein-Durrmeyer polynomials

Transcription

1 Estimation of the quantile function using Bernstein-Durrmeyer polynomials Andrey Pepelyshev Ansgar Steland Ewaryst Rafaj lowic The work is supported by the German Federal Ministry of the Environment, Nature and Nuclear Safety, grant Hamburg February 20, /23 A. Pepelyshev, A. Steland Bernstein-Durrmeyer quatile estimation

2 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 2/23 A. Pepelyshev, A. Steland Bernstein-Durrmeyer quatile estimation

3 Bernstein-Durrmeyer operator Bernstein-Durrmeyer operator The BD operator D N (u(x)) has the kernel form D N (u(x)) = 1 where K N (x, z) = (N + 1) N where a i = K N (x, z)u(z)dz i=0 B(N) i D N (u(x)) = (N + 1) u(x)b (N) i (x)dx, B (N) i (x) = N! i!(n i)! xi (1 x) N i. N i=0 (x)b (N) (z), or i a i B (N) i (x) 3/23 A. Pepelyshev, A. Steland Bernstein-Durrmeyer quatile estimation

4 Bernstein-Durrmeyer operator BD operator for deterministic functions Derriennic (1981) shown that D N (u) converges uniformly, sup D N (u(x)) u(x) 2ω u (N 1/2 ), x [0,1] where ω u denotes the modulus of continuity of u. Chen and Ditzian (1991) established the fact that if and only if ɛ(u) = u(x) D N (u(x)) p = O(N δ/2 ), { } inf u(x) a 0 a 1 x a N x N p = O(N δ ), a 0,...,a N as N for some δ (0, 2) and p {2, }, where f p = ( I f(x) p dx ) 1/p. 4/23 A. Pepelyshev, A. Steland Bernstein-Durrmeyer quatile estimation

5 Bernstein-Durrmeyer operator BD operator for the distribution function Let F m (x) be the empirical d.f. for a sample X 1,..., X m F [0,1] with density f and f m,n (x) = 1 m m K N (X j, x), Fm,N (x) := x j=1 0 f m,n (t)dt Theorem. (Ciesielski, 1988) If F C[0, 1], then Pr( F m,n (x) F 0 as m, N ) = 1. If f L p [0, 1] and m = N β for some β (0, 0.5), then Pr( f m,n (x) f p = o(1) as N ) = 1. 5/23 A. Pepelyshev, A. Steland Bernstein-Durrmeyer quatile estimation

6 Bernstein-Durrmeyer operator BD operator for the quantile function Let Q m (x) be the empirical q.f. for a sample X 1,..., X m. Version 1: N Q m,n (x) := D N (Q m (x)) = (N + 1) ã i B (N) i (x), Version 2: ã i = 1 0 i=0 Q m (x)b (N) i (x)dx, i = 1,..., m, Q m,n (x) = (N + 1) â i = 1 m m j=1 N i=0 â i B (N) i (x), X (j) B (N) i ( j 1 m 1 ) Note that ã i are asymptotically equivalent to ã i in mean square (Yang, 1985). 6/23 A. Pepelyshev, A. Steland Bernstein-Durrmeyer quatile estimation

7 Bernstein-Durrmeyer operator Properties of the BD estimator Lemma. For ã i it is hold Proof. Set Var(ã i ) l=1 w j = (l 1)/m 1 (N + 1) 2 max j=1,...,m Var(X (j)) j/m (j 1)/m B(N) i (t) dt m l/m l=1 (l 1)/m B(N) i (t) dt, j = 1,..., m. Then we have that ( m 2 l/m m Var(ã i ) = B (N) i (t) dt) E (X (j) EX (j) ) 2 w j. Apply the Jensen inequality (E ζ (a ζ Ea ζ )) 2 E ζ (a ζ Ea ζ ) 2, where ζ is a random variable such that P(ζ = j) = w j. 7/23 A. Pepelyshev, A. Steland Bernstein-Durrmeyer quatile estimation j=1

8 Bernstein-Durrmeyer operator Properties of the BD estimator Lemma. For the BD estimator it is hold Var( Q m,n (x)) C m := max j=1,...,m Var(X (j)) and 1 Var( Q m,n (x))dx C m. 0 If the derivative of Q(x) is continuous, then C m = O(1/m). Proof. If X 1,..., X m U(0, 1) then X (j) β(j, m j + 1). Note that the variance of the beta distribution β(a, b) is ab (a+b) 2 (a+b+1). Therefore, it is easy to see that C m = O(1/m). 8/23 A. Pepelyshev, A. Steland Bernstein-Durrmeyer quatile estimation

9 Bernstein-Durrmeyer operator Maximum variance of order statistics Since Q (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) = p j, we obtain X (j) = Q(p j ) + (U (j) p j )Q (ζ j ) for some ζ j [min{u (j), p j }, max{u (j), p j }]. Then we have Var(X (j) ) = Var((U (j) p j )Q (ζ j )) max Q 2 (x)e(u (j) p j ) 2 x = p j(1 p j ) m + 2 max Q 2 (x). x 9/23 A. Pepelyshev, A. Steland Bernstein-Durrmeyer quatile estimation

10 Consistency MSE and MISE consistency Let X 1,..., X m 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, Q m,n (x), satises and max E( Q m,n (x) Q(x)) 2 = O(1/m + N δ ) x [a,b] E b a as m and N. ( Q m,n (x) Q(x)) 2 dx = O(1/m + N δ ) 10/23 A. Pepelyshev, A. Steland Bernstein-Durrmeyer quatile estimation

11 Consistency The BD estimator with correction term Let ẽ l (x) = D l (Q m (x) Q ) m,n (x). Dene the error-corrected Bernstein-Durrmeyer estimator Q m,n,l (x) = Q m,n (x) + ẽ l (x). Theorem. Suppose X 1, X 2,... are i.i.d. random variables with common quantile function Q(x) and distribution function F (x) satisfying x p df (x) < for some 1 p. Then the error-corrected estimator Q m,n,l (x) is consistent in the pth mean for the true quantile function Q(x), Q m,n,l Q p 0, as m and N, l, with probability 1, and in the mean, i.e. E Q m,n,l Q p 0, as m and N, l. 11/23 A. Pepelyshev, A. Steland Bernstein-Durrmeyer quatile estimation

12 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. Note that the Bernstein polynomial B (N) i (x) can be approximated by the density of the normal distribution φ((i/n x)/s), where s = x(1 x)/n and φ(x) = (2π) 0.5 e x2 /2. Thus, we can introduce the quantity h = 1/ N which plays the role of the bandwidth. 12/23 A. Pepelyshev, A. Steland Bernstein-Durrmeyer quatile estimation

13 Adaptive choice of N Algorithm of adaptive choice of N 1] Compute { h=max h (0, 1] : max Fm ( Q m, 1/t q 2 (q)) q } 1/R m t (0, h) where F m (x) is the empirical distribution function, z is the smallest integer that is larger or equal to z, and R m = 2 m/ 2 log log m. 13/23 A. Pepelyshev, A. Steland Bernstein-Durrmeyer quatile estimation

14 Adaptive choice of N Algorithm of adaptive choice of N 2] Dene a set {h 1,..., h n } (0, h) such that max h (0, h) min j=1,...,n h h j is small and compute the sequence M 1,..., M n, where M j is the number of local minimums of the Bernstein-Durrmeyer estimator Q m,n (x) with N = 1/h 2 j. 3] Compute ˇM j = min{m 1, M 2,..., M j }, j = 1,..., n. 14/23 A. Pepelyshev, A. Steland Bernstein-Durrmeyer quatile estimation

15 Adaptive choice of N Algorithm of adaptive choice of N 4] Divide the set {h 1,..., h n } into groups as follows. Dene a i and b i such that a 1 b 1 < a 2 b 2 <... < a k b k and ˇM i = ˇM j for all h i, h j [a l, b l ] for l {1,..., k}. 5] Compute ĥ = k i=1 a iw i, where w i = (b i a i )/ k j=1 (b j a j ), and then set N = 1/ĥ2. 15/23 A. Pepelyshev, A. Steland Bernstein-Durrmeyer quatile estimation

16 Adaptive choice of N Consistency of the BD estimator with ˆN Let X 1,..., X m be m random variables with the continuously dierentiable quantile function Q, R m = 2 m 2 log log m. Then the adaptive Bernstein-Durrmeyer estimator Q N(q) m, = D N(Q m (q)) where N is selected by the above algorithm is consistent as m. Moreover, we have the a.s. uniform error bound 1 sup Q <x< m, N (x) F (x) 2 log log m/ m 1 for the estimator Q (x) of the distribution function F (x), m, N and the a.s. uniform error bound sup Q N(q) m, Q(q) 2 2 log log m/ m q for the quantile estimator Q m, N(x). 16/23 A. Pepelyshev, A. Steland Bernstein-Durrmeyer quatile estimation

17 Invariance principle Adaptive choice of N Theorem. If R m is chosen such that Rm 1 = o(m 1/2 ), then for the Bernstein-Durrmeyer polynomial estimator with data-adaptive selection of the degree N we have { 1 m[ Q m, N (x) F (x)] : < x < } {B0 (F (x)) : < x < }, as m, in the Skorohod space D(R; R), where B 0 (t) = W t tw 1, t [0, 1], is a Brownian bridge process. 17/23 A. Pepelyshev, A. Steland Bernstein-Durrmeyer quatile estimation

18 Application in photovoltaics Application in photovoltaics Parameters of acceptance sampling for the quality control AQL, the acceptable quality level RQL, the rejectable quality level α, the producer risk β, the consumer risk The sampling plan n, the number of items to be checked from a lot c, the critical value for the T-statistic 18/23 A. Pepelyshev, A. Steland Bernstein-Durrmeyer quatile estimation

19 Application in photovoltaics The meaning of parameters The lot is accepted if and only if T n > c. The operational characteristic OC n,c (p) = P(T n > c) is the probability of the lot acceptance for given p, the true proportion of non-conforming items. 19/23 A. Pepelyshev, A. Steland Bernstein-Durrmeyer quatile estimation

20 Application in photovoltaics Requirements for the sampling plan The sampling plan is a minimal solution satisfying the following conditions { OC n,c (p) 1 α for p AQL, OC n,c (p) β for p RQL 20/23 A. Pepelyshev, A. Steland Bernstein-Durrmeyer quatile estimation

21 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 OC n,c (p) = 1 Φ ( c + ng 1 (1 p) ). The asymptotically optimal sampling plan (n, c) is ( Φ 1 (α) Φ 1 (1 β) ) 2 n = ( F 1 (AQL) F 1 (RQL) ) 2, n ( c = F 1 (AQL) + F 1 (RQL) ), 2 where F (x) = G((x a)/σ) and Φ(x) is the standard normal distribution. 21/23 A. Pepelyshev, A. Steland Bernstein-Durrmeyer quatile estimation

22 Numerical results Application in photovoltaics Characteristics of estimation of the sampling plan (n, c) for two quantile estimators, α = β = 5%, AQL = 2% and RQL = 5%. Bernstein-Durrmeyer estimator Empirical quantile estimator n c n c m mean std.dev. mean std.dev. mean std.dev. mean std.dev /23 A. Pepelyshev, A. Steland Bernstein-Durrmeyer quatile estimation

23 Application in photovoltaics Thank you for your attention! 23/23 A. Pepelyshev, A. Steland Bernstein-Durrmeyer quatile estimation