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Optimal choice of order statistics under confidence region estimation in case of large samples cko 1 Aleksander Zaigrajew 2 Magdalena Alama-Bu 1 Instytut Matematyki i Fizyki Uniwersytet Technologiczno-Przyrodniczy w Bydgoszczy 2 Wydzia

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Optimal choice of order statistics under confidence region estimation in case of large samples

Magdalena Alama-Bu´ cko1 Aleksander Zaigrajew2

1Instytut Matematyki i Fizyki

Uniwersytet Technologiczno-Przyrodniczy w Bydgoszczy

2Wydział Matematyki i Informatyki

Uniwersytet Mikołaja Kopernika w Toruniu

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Outline

1

Introduction

2

L-statistics

3

Main results

4

Examples

5

References

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X = (X1, X2, . . . , Xn) is a sample from the distribution Fθ ∈ F, F =

  • Fθ : Fθ(u) = F(0,1)

u − θ1 θ2

  • , u ∈ R1, θ1 ∈ R1, θ2 ∈ R1

+

  • F(0,1) is a distribution function of the standard distribution, while f is the density

function 1 − α is a given confidence level

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X = (X1, X2, . . . , Xn) is a sample from the distribution Fθ ∈ F, F =

  • Fθ : Fθ(u) = F(0,1)

u − θ1 θ2

  • , u ∈ R1, θ1 ∈ R1, θ2 ∈ R1

+

  • F(0,1) is a distribution function of the standard distribution, while f is the density

function 1 − α is a given confidence level

A strong confidence region of level 1 − α for θ

B(X) : Rn → B2 such that Pθ(θ ∈ B(X)) = 1 − α

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X = (X1, X2, . . . , Xn) is a sample from the distribution Fθ ∈ F, F =

  • Fθ : Fθ(u) = F(0,1)

u − θ1 θ2

  • , u ∈ R1, θ1 ∈ R1, θ2 ∈ R1

+

  • F(0,1) is a distribution function of the standard distribution, while f is the density

function 1 − α is a given confidence level

A strong confidence region of level 1 − α for θ

B(X) : Rn → B2 such that Pθ(θ ∈ B(X)) = 1 − α

Risk function

R(θ, B) = Eθλ2(B(X)) → min

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Consider the vector T = T(X, θ) = θ1 − t1(X) t2(X) , θ2 t2(X) − 1

  • .

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Consider the vector T = T(X, θ) = θ1 − t1(X) t2(X) , θ2 t2(X) − 1

  • .

If A ∈ B2 is such that Pθ

  • T(X, θ) ∈ A
  • = 1 − α,

then B(X) = (t1(X), t2(X)) + t2(X) A.

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Consider the vector T = T(X, θ) = θ1 − t1(X) t2(X) , θ2 t2(X) − 1

  • .

If A ∈ B2 is such that Pθ

  • T(X, θ) ∈ A
  • = 1 − α,

then B(X) = (t1(X), t2(X)) + t2(X) A. If for any b > 0, a ∈ R: t1(bX + a1n) = bt1(X) + a, t2(bX + a1n) = bt2(X), (1)

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Consider the vector T = T(X, θ) = θ1 − t1(X) t2(X) , θ2 t2(X) − 1

  • .

If A ∈ B2 is such that Pθ

  • T(X, θ) ∈ A
  • = 1 − α,

then B(X) = (t1(X), t2(X)) + t2(X) A. If for any b > 0, a ∈ R: t1(bX + a1n) = bt1(X) + a, t2(bX + a1n) = bt2(X), (1) then Pθ

  • T(X, θ) ∈ A
  • = P(0,1)
  • T(X, (0, 1)) ∈ A
  • = 1 − α,

and distribution of the vector T doesn’t depend on θ ( vector T is a pivot).

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B(X) = (t1(X), t2(X)) + t2(X) A. R(θ, B) = Eθλ2(B(X)) = Eθt2

2(X) · λ2(A) → min

Since vector T is a pivot, then P(0,1)

  • T(X, (0, 1)) ∈ A
  • = 1 − α

⇔

  • A

p(u) du = 1 − α

Einmahl, Mason 1992

Aopt = {u : p(u) zα}, gdzie

  • Aopt

p(u) du = 1 − α

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Asymptotics of Aopt

p(u) - density function of the vector

  • − t1(X)

t2(X), 1 t2(X) − 1

  • Alama-Bu´

cko, Nagajew, Zaigrajew 2006

If for n → ∞ density p(u) is two-dimentional normal distribution N2((0, 0), W), then for n → ∞ the set Aopt = {u : p(u) zα} can be approximated by ellipse A0 = {u ∈ R2 : ϕW(u) z′

α},

z′

α =

1 − α 2π √ det W Then as n → ∞ λ2(Aopt) → λ2(A0) = 2π(− ln(1 − α)) √ det W.

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case k = 2

Zaigraev A., Alama-Bu´ cko M. (2013). On optimal choice of order statistics in large samples for the construction of confidence regions for the location and scale. Metrika 76(4), 577-593 t1(X) = Xk:nF −1(q) − Xm:nF −1(p) F −1(q) − F −1(p) , t2(X) = Xm:n − Xk:n F −1(q) − F −1(p), (2) where Xk:n and Xm:n are central order statistics, that is k n → p, m n → q, 0 < p < q < 1 for example k = [np] + 1, m = [nq] + 1. while F −1(p) = inf{t ∈ R : F(t) p}, 0 < p < 1.

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case any (fixed) k

Consider the case, where t1 and t2 are L-statistics depending on k order statistics. Let 1 < r1 < r2 < ... < rk < n be the numbers od ordered statistics such that ri n → pi, 0 < p1 < p2 < ... < pk < 1 and t1(X) =

k

  • i=1

aiXri :n, t2(X) =

k

  • i=1

biXri :n.

Questions:

(a1, a2, ..., ak) (b1, b2, ..., bk) p1 < p2 < ... < pk to minimise risk function as n → ∞.

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t1(X) =

k

  • i=1

aiXri :n, t2(X) =

k

  • i=1

biXri :n. t1 i t2 satisfy (1) when

k

  • i=1

ai = 1,

k

  • i=1

bi = 0. t1 i t2 are asymptotically unbiased estimators for θ1 i θ2 if

k

  • i=1

aiF −1(pi) = 0,

k

  • i=1

biF −1(pi) = 1.

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t1(X) =

k

  • i=1

aiXri :n, t2(X) =

k

  • i=1

biXri :n. t1 i t2 satisfy (1) when

k

  • i=1

ai = 1,

k

  • i=1

bi = 0. t1 i t2 are asymptotically unbiased estimators for θ1 i θ2 if

k

  • i=1

aiF −1(pi) = 0,

k

  • i=1

biF −1(pi) = 1. Idea: choose t1 and t2 as ABLUE.

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Estimators, (Sarhan & Greenberg, 1962)

ABLUE for θ1 and θ2 are written by the formulas: t1(X) = K2Z1 − K3Z2 ∆ , t2(X) = −K3Z1 + K1Z2 ∆ , where Z1 =

k+1

  • j=1
  • f(uj) − f(uj−1
  • f(uj)Xri :n − f(uj−1)Xrj −1:n
  • pj − pj−1

Z2 =

k+1

  • j=1
  • f(uj)F −1(pj) − f(uj−1)F −1(pj−1)
  • f(uj)Xri :n − f(uj−1)Xrj−1:n
  • pj − pj−1

K1 =

k+1

  • j=1

(f(uj) − f(uj−1)2 pj − pj−1 , K2 =

k+1

  • j=1

(f(uj)uj − f(uj−1)uj−1)2 pj − pj−1 K3 =

k+1

  • j=1

(f(uj) − f(uj−1))(f(uj)uj − f(uj−1)uj−1) pj − pj−1 , ∆ = K1K2 − K 2

3

p0 = 0, pk+1 = 1, f(u0) = f(uk+1) = 0, ui = F −1(pi)

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√ n(t1(X), t2(X) − 1)

d

→ N((0, 0), W) where covariance matrix W can be written as W =     K2 ∆ −K3 ∆ −K3 ∆ K1 ∆     .

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√ n(t1(X), t2(X) − 1)

d

→ N((0, 0), W) where covariance matrix W can be written as W =     K2 ∆ −K3 ∆ −K3 ∆ K1 ∆     .

Lemma

√ n

  • − t1(X)

t2(X), 1 t2(X) − 1

  • d

= √ n(t1(X), t2(X) − 1)

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Then for n → ∞ R(θ, B) = 1 n · Eθt2

2(X) · λ2(Aopt)

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Then for n → ∞ R(θ, B) = 1 n · Eθt2

2(X) · λ2(Aopt) ∼ 1

n · θ2

2 · E(0,1)t2 2(X) · λ2(A0)

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Then for n → ∞ R(θ, B) = 1 n · Eθt2

2(X) · λ2(Aopt) ∼ 1

n · θ2

2 · E(0,1)t2 2(X) · λ2(A0)

∼ 1 n · θ2

2 · 2π(− ln(1 − α))

√ det W

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det W = 1 ∆ ∆ = K1K2 − K 2

3

Then (p∗

1, p∗ 2, ..., p∗ k ) ∈ arg inf det W

⇔ (p∗

1, p∗ 2, ..., p∗ k ) ∈ arg sup ∆(p1, p2, . . . , pk).

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the formula of ∆

For any p1, p2, . . . , pk : ∆(p1, p2, . . . , pk) =

k+1

  • i<j, i,j=1

(AiBj − AjBi)2 (pi − pi−1)(pj − pj−1), where Ai = f(ui) − f(ui−1), Bi = uif(ui) − ui−1f(ui−1). ui = F −1(pi) p0 = 0, pk+1 = 1

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the formula of ∆

For any p1, p2, . . . , pk : ∆(p1, p2, . . . , pk) =

k+1

  • i<j, i,j=1

(AiBj − AjBi)2 (pi − pi−1)(pj − pj−1), where Ai = f(ui) − f(ui−1), Bi = uif(ui) − ui−1f(ui−1). ui = F −1(pi) p0 = 0, pk+1 = 1

Lemma

Larger k ⇒ larger value of sup ∆.

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What about the behaviour of ∆ for p1 → 0

  • r

pk → 1? . it depends on the limit distribution of the properly normed Xmin and Xmax cn(Xmax − dn), an(Xmin − bn) Frechet distribution (H1), Weibull distr. (H2) i Gumbel distr. (H3) attraction to D(Hi), i = 1, 2, 3 : von Mises conditions we analyze behaviour of ∆ at the ends of the support by the von Mises conditions

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F ∈ D(H∗

2) - Weibull distribution

If u+

F < ∞ and

lim

u↑u+

F

(u+

F − u)

f(u) 1 − F(u) = β > 0, then F ∈ D(H∗

2 ).

F ∈ D(H2) - Weibull distribution

If u−

F > −∞ and

lim

u↓u−

F

(u − u−

F ) f(u)

F(u) = γ > 0, then F ∈ D(H2).

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H1-Frechet distribution, H2- Weibull distribution, H3- Gumbel distribution

Theorem

If F ∈ { D(H∗

1 ), D(H∗ 2 ) for β > 2, D(H∗ 3 )}

then for pk → 1 ∆(p1, ...pk) is non-increasing. Then p∗

k < 1.

Theorem

If F ∈ D(H∗

2 )

for β ∈ (0, 2), then for pk → 1 ∆(p1, ...pk) → ∞. Then p∗

k = 1.

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Uniform distribution

U(− 1

2, 1 2)

F ∈ D(H2) (Weibull distribution) with γ = 1 ⇒ p∗

1 = 0

F ∈ D(H∗

2 ) (Weibull distribution) β = 1 ⇒ p∗ k = 1

for any k : ∆(p1, p2, ..., pk) = ∆(p1, pk) t1(X) = Xmin + Xmax 2 , t2(X) = Xmax − Xmin. confidence region is built on the basis of T(X, (0, 1)) = n

  • − t1(X)

t2(X), 1 t2(X) − 1

  • R(θ, B) is of order n−2

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Normal distribution

F ∈ D(H3) (Gumbel distribution) ⇒ p∗

1 > 0

F ∈ D(H∗

3 ) (Gumbel distribution) ⇒ p∗ k < 1

p∗

1, p∗ 2, ..., p∗ k ∈ (0, 1)

confidence region is built on the basis of T(X, (0, 1)) = √ n

  • − t1(X)

t2(X), 1 t2(X) − 1

  • R(θ, B) is of order n−1

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k = 2

p∗

1 = 0.133, p∗ 2 = 0.867

max ∆ = ∆(p∗

1 , p∗ 2 ) = 0.813

t1(X) = 0.5(X[0.133n]+1:n + X[0.867n]+1:n), t2(X) = 0.44951(X[0.867n]+1:n − X[0.133n]+1:n)

k = 3

p∗

1 = 0.083, p∗ 2 = 0.500, p∗ 3 = 0.917

max ∆ = ∆(p∗

1 , p∗ 2 , p∗ 3 ) = 1.10536

t1(X) = 0.224(X[0.083n]+1:n + X[0.917n]+1:n) + 0.552XMe t2(X) = 0.36097(X[0.917n]+1:n − X[0.083n]+1:n)

k = 4

p∗

1 = 0.045, p∗ 2 = 0.245, p∗ 3 = 0.755, p∗ 4 = 0.955

max ∆ = ∆(p∗

1 , p∗ 2 , p∗ 3 , p∗ 4 ) = 1.3652

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Laplace distribution

F ∈ D(H3) (Gumbel distribution) ⇒ p∗

1 > 0

F ∈ D(H∗

3 ) (Gumbel distribution)

⇒ p∗

k < 1

p∗

1, p∗ 2, ..., p∗ k ∈ (0, 1)

t1(X) = X[0.5n]+1:n = XMe confidence region is built on the basis of T(X, (0, 1)) = √ n

  • − t1(X)

t2(X), 1 t2(X) − 1

  • R(θ, B) is of order n−1

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k = 2:

p∗

1 = 0.102, p∗ 2 = 0.5 albo p∗ 1 = 0.5, p∗ 2 = 0.898.

max ∆ = ∆(p∗

1 , p∗ 2 ) = 0.3238

t1(X) = X[0.5n]+1:n = XMe t2(X) = 0.6291(XMe − X[0.102n]+1) albo t2(X) = 0.6291(X[0.989n]+1 − XMe)

k = 3

p∗

1 = 0.102, p∗ 2 = 0.5, p∗ 3 = 0.898

max ∆ = ∆(p∗

1 , p∗ 2 , p∗ 3 ) = 0.6476

t1(X) = X[0.5n]+1:n = XMe t2(X) = 0.3145(X[0.898n]+1:n − X[0.102n]+1:n)

k = 4

p∗

1 = 0.037, p∗ 2 = 0.181, p∗ 3 = 0.5, p∗ 4 = 0.898

p∗

1 = 0.102, p∗ 2 = 0.5, p∗ 3 = 0.819, p∗ 4 = 0.963

max ∆ = ∆(p∗

1 , p∗ 2 , p∗ 3 , p∗ 4 ) = 0.73393

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Exponential distribution

F ∈ D(H2) (Weibul distribution) with γ = 1 ⇒ p1 = 0 F ∈ D(H∗

3 ) (Gumbel distribution)

⇒ pk < 1 p2, ..., pk ∈ (0, 1), t1(X) = X1:n = XMin confidence region is built on the basis of T(X, (0, 1)) =

  • − nt1(X)

t2(X), √ n( 1 t2(X) − 1)

  • R(θ, B) is of order n−3/2

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k = 2

p∗

1 = 0, p∗ 2 = 0.797

t1(X) = X1:n = Xmin t2(X) = 0.627(X[0.797n]+1:n − Xmin)

k = 3

p∗

1 = 0, p∗ 2 = 0.639, p∗ 3 = 0.927

t1 = Xmin t2 = −0.701X1:n + 0.523X[0.639n]+1:n + 0.178X[0.927n]+1:n

k = 4

p∗

1 = 0, p∗ 2 = 0.529, p∗ 3 = 0.829, p∗ 4 = 0.965

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References

  • M. Alama-Bu´

cko, A.V. Nagaev, A. Zaigraev (2006), Asymptotic analysis of minimum volume confidence regions for location-scale families, Applicationes Mathematicae 33,1-20

  • M. Alama-Bu´

cko, A. Zaigraev (2006), Asymptotics of the optimal confidence region for location and scale basing on two order statistics Statistical Methods of Estimation and Testing Hypotheses. Perm University, Perm, 49-65

  • M. Alama-Bu´

cko, A. Zaigraev (2013), On optimal choice of order statistics in large samples for the construction of confidence regions for the location and scale, Metrika, 76, 4, 577-593 Barry C. Arnold, N. Balakrishnan, H. N. Nagaraja (1992), A First Course in Order

  • Statistics. Wiley Series in Probability and Mathematical Statistics. Wiley, New York

.

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J.H.J. Einmahl, D.M.Mason (1992), Generalized quantile processes, Ann. Stat., 20 1062-1078. H.A.David, H.N.Nagaraja (2003), Order Statistics. Wiley, New York

  • F. Mosteller (1946) On some useful "inefficient" statistics, Ann. Math. Statist., 17

377-408.

  • J. Ogawa (1998) Optimal spacing of the selected sample quantiles for the joint

estimation of the location and scale parameters of a symmetric distribution, J. Stat.

  • Plan. and Inf., 70, 345-360.

A.E. Sarhan, B.G. Greenberg (eds.), (1962) Contributions to Order Statistics. Wiley, New York

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THANK YOU FOR YOUR ATTENTION !

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