[PPT] - Sharp bounds on the expectations of linear combinations of k th PowerPoint Presentation

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Sharp bounds on the expectations of linear combinations of kth records expressed in the Gini mean difference units 1/14

Sharp bounds on the expectations

f linear combinations of kth records

expressed in the Gini mean difference units

Pawe l Marcin Kozyra and Tomasz Rychlik Institute of Matematics, Polish Academy of Sciences ´ Sniadeckich 8, 00 656 Warsaw, Poland e-mails: pawel m kozyra@wp.pl and trychlik@impan.pl XLII Conference on Mathematical Statistics B¸ edlewo November 28 - December 2, 2016

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Sharp bounds on the expectations of linear combinations of kth records expressed in the Gini mean difference units 2/14

kth records:

X1, X2, . . . — i.i.d. with some continuous common distribution function F and finite mean µ. R1,k, R2,k, . . . – kth record values based on X1, X2, . . .

def

= increasing subsequence of non-decreasing sequence of kth maxima in X1, X2, . . ..

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Sharp bounds on the expectations of linear combinations of kth records expressed in the Gini mean difference units 3/14

Problem:

Provide sharp bounds for E n

i=1 ci(Ri,k − µ

∆

where c = (c1, . . . , cn) ∈ Rn — arbitrarily fixed vector of combination coefficients, and ∆ = E|X1 − X2| = E(X2:2 − X1:2) the Gini mean difference of F.

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Sharp bounds on the expectations of linear combinations of kth records expressed in the Gini mean difference units 4/14

Solution:

inf

0<u<1 Ψc,k(u) ≤ E

n

i=1 ci(Ri,k − µ

∆

≤ sup

0<u<1

Ψc,k(u), where Ψc,k(u) = 1 2u (1 − u)k−1

n−1

i=0

 

n

j=i+1

cj   [−k ln(1 − u)]i i! −

n

j=1

cj.

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Sharp bounds on the expectations of linear combinations of kth records expressed in the Gini mean difference units 5/14

Idea of proof:

Gn,k(F(x)) — distribution function of Rn,k, where Gn,k(u) = 1 − (1 − u)k

n−1

i=0

[−k ln(1 − u)]i i! , 0 < u < 1.

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Sharp bounds on the expectations of linear combinations of kth records expressed in the Gini mean difference units 6/14

Expectation of kth record spacing:

E(Ri+1,k − Ri,k) = ∞

−∞

x Gi+1,k(F(dx)) − ∞

−∞

x Gi,k(F(dx)) = ∞

−∞

x ((Gi+1,k − Gi,k) ◦ F) (dx) = x

Gi+1,k (F(x)) − Gi,k (F(x))
∞

−∞

− ∞

−∞

[Gi+1,k (F(x)) − Gi,k (F(x))] dx = ∞

−∞

1 − F(x)

k

− k ln(1 − F(x))

i i! dx =

0<F(x)<1
1 − F(x)

k

− k ln(1 − F(x))

i i! dx.

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Sharp bounds on the expectations of linear combinations of kth records expressed in the Gini mean difference units 7/14

Expectation of standard spacing + useful observations:

Similarly, E(Xl+1:k − Xl:k) =

0<F(x)<1

k l

F l(x)
1 − F(x)

k−ldx.

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Sharp bounds on the expectations of linear combinations of kth records expressed in the Gini mean difference units 7/14

Expectation of standard spacing + useful observations:

Similarly, E(Xl+1:k − Xl:k) =

0<F(x)<1

k l

F l(x)
1 − F(x)

k−ldx. In particular ∆ = E(X2:2 − X1:2) =

0<F(x)<1

2F(x)[1 − F(x)]dx.

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Sharp bounds on the expectations of linear combinations of kth records expressed in the Gini mean difference units 7/14

Expectation of standard spacing + useful observations:

Similarly, E(Xl+1:k − Xl:k) =

0<F(x)<1

k l

F l(x)
1 − F(x)

k−ldx. In particular ∆ = E(X2:2 − X1:2) =

0<F(x)<1

2F(x)[1 − F(x)]dx. Also, R1,k = X1:k and µ = E

1

k

j=1 Xj:k

.

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Sharp bounds on the expectations of linear combinations of kth records expressed in the Gini mean difference units 8/14

Spacing representations:

E(Rn,k − µ) = E  Rn,k − R1,k − 1 k

k

j=1

(Xj:k − X1:k)   = E  

n−1

i=1
Ri+1,k − Ri,k
− 1

k

j=2

j−1

l=1

(Xl+1:k − Xl:k)   = E n−1

i=1
Ri+1,k − Ri,k
−

k−1

l=1

k − l k (Xl+1:k − Xl:k)

,

and

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Sharp bounds on the expectations of linear combinations of kth records expressed in the Gini mean difference units 8/14

Spacing representations:

E(Rn,k − µ) = E  Rn,k − R1,k − 1 k

k

j=1

(Xj:k − X1:k)   = E  

n−1

i=1
Ri+1,k − Ri,k
− 1

k

j=2

j−1

l=1

(Xl+1:k − Xl:k)   = E n−1

i=1
Ri+1,k − Ri,k
−

k−1

l=1

k − l k (Xl+1:k − Xl:k)

,

and E

n

i=1

ci(Rn,k− µ)=E

n

i=1

ci n−1

i=1
Ri+1,k −Ri,k
−

k−1

l=1

k−l k (Xl+1:k −Xl:k)

.

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Sharp bounds on the expectations of linear combinations of kth records expressed in the Gini mean difference units 9/14

Bounds:

By the integral representations of spacing expectations E

n

i=1

ci(Rn,k − µ) =

0<F(x)<1

Ψc,k(F(x))2F(x)[1 − F(x)]dx ≤ sup

0<u=F(x)<1

Ψc,k(u) · ∆

≥

inf

0<u=F(x)<1 Ψc,k(u) · ∆

.

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Sharp bounds on the expectations of linear combinations of kth records expressed in the Gini mean difference units 10/14

Upper bound attainability:

0<F(x)<1

Ψc,k(F(x))2F(x)[1 − F(x)]dx = sup

0<u=F(x)<1

Ψc,k(u) · ∆ when u0 = arg sup Ψc,k(u) ∈ (0, 1): F(x) = either 0 or 1 or u0, i.e. for two-point distributions Fu0 = u0δa + (1 − u0)δb for any a < b.

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Sharp bounds on the expectations of linear combinations of kth records expressed in the Gini mean difference units 10/14

Upper bound attainability:

0<F(x)<1

Ψc,k(F(x))2F(x)[1 − F(x)]dx = sup

0<u=F(x)<1

Ψc,k(u) · ∆ when u0 = arg sup Ψc,k(u) ∈ (0, 1): F(x) = either 0 or 1 or u0, i.e. for two-point distributions Fu0 = u0δa + (1 − u0)δb for any a < b. Bound attainability for continuous parent distributions: in the limit, by continuous approximation of two-point distribution.

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Sharp bounds on the expectations of linear combinations of kth records expressed in the Gini mean difference units 10/14

Upper bound attainability:

0<F(x)<1

Ψc,k(F(x))2F(x)[1 − F(x)]dx = sup

0<u=F(x)<1

Ψc,k(u) · ∆ when u0 = arg sup Ψc,k(u) ∈ (0, 1): F(x) = either 0 or 1 or u0, i.e. for two-point distributions Fu0 = u0δa + (1 − u0)δb for any a < b. Bound attainability for continuous parent distributions: in the limit, by continuous approximation of two-point distribution. Cases sup Ψc,k(u) = either limuց0 Ψc,k(u) or limuր1 Ψc,k(u): double limit: continuous approximation of two-point distribution Fu, and u ց 0 (u ր 1, respectively).

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Sharp bounds on the expectations of linear combinations of kth records expressed in the Gini mean difference units 10/14

Upper bound attainability:

0<F(x)<1

Ψc,k(F(x))2F(x)[1 − F(x)]dx = sup

0<u=F(x)<1

Ψc,k(u) · ∆ when u0 = arg sup Ψc,k(u) ∈ (0, 1): F(x) = either 0 or 1 or u0, i.e. for two-point distributions Fu0 = u0δa + (1 − u0)δb for any a < b. Bound attainability for continuous parent distributions: in the limit, by continuous approximation of two-point distribution. Cases sup Ψc,k(u) = either limuց0 Ψc,k(u) or limuր1 Ψc,k(u): double limit: continuous approximation of two-point distribution Fu, and u ց 0 (u ր 1, respectively). Similarly for lower bounds.

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Sharp bounds on the expectations of linear combinations of kth records expressed in the Gini mean difference units 11/14

Special cases: single kth record Rn,k:

Bounds: extremes of Ψn,k(u) = 1 2u

(1 − u)k−1

n−1

i=0
− k ln(1 − u)

i i! − 1

.

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Sharp bounds on the expectations of linear combinations of kth records expressed in the Gini mean difference units 11/14

Special cases: single kth record Rn,k:

Bounds: extremes of Ψn,k(u) = 1 2u

(1 − u)k−1

n−1

i=0
− k ln(1 − u)

i i! − 1

.

Upper bounds: k = 1 ⇒ Ψn,k(1−) = +∞ k ≥ 2 and n = 2 ⇒ Ψn,k(0+) = 1 2, k = 2 and n ≥ 3 ⇒ Ψn,k(unique solution to Ψ′

n,k(u) = 0) > 1

2 , k ≥ 3 and n ≥ 3 ⇒ Ψn,k(smaller of 2 solutions to Ψ′

n,k(u) = 0) > 1

2.

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Sharp bounds on the expectations of linear combinations of kth records expressed in the Gini mean difference units 12/14

Special cases: single kth record Rn,k cont.:

Lower bounds: k = 1 ⇒ Ψn,k(0+) = 1 2 k = 2 and n ≥ 2 ⇒ Ψn,k(1−) = −1 2, k ≥ 3 and n = 2 ⇒ Ψn,k(unique solution to Ψ′

n,k(u) = 0) < −1

2, k ≥ 3 and n ≥ 3 ⇒ Ψn,k(greater of 2 solutions to Ψ′

n,k(u)=0)<−1

2.

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Sharp bounds on the expectations of linear combinations of kth records expressed in the Gini mean difference units 13/14

Special cases: kth record differences:

Only most complicated case: upper bound for Rn,k − Rm,k when k ≥ 4, m ≥ 2, n ≥ m + 3: For Ψm,n;k(u) = (1 − u)k−1 2u

n−1

i=m
− k ln(1 − u)

i i! , the derivative has either one zero 0 < u1 < 1 or three zeros 0 < u1 < u2 < u3 < 1, and then sup Ψm,n;k(u) = Ψm,n;k(u1), sup Ψm,n;k(u) = max{Ψm,n;k(u1), Ψm,n;k(u3)}, respectively.

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Sharp bounds on the expectations of linear combinations of kth records expressed in the Gini mean difference units 14/14

Ψm,n;k(u) = 1 2u (1 − u)k−1

n−1

i=m
− k ln(1 − u)

i i! = ψm,n;k(u) 2u .

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Sharp bounds on the expectations of linear combinations of kth records expressed in the Gini mean difference units 14/14

Ψm,n;k(u) = 1 2u (1 − u)k−1

n−1

i=m
− k ln(1 − u)

i i! = ψm,n;k(u) 2u . ψ′

m,n;k(u) x=−k ln(1−u)

= ψ′

m,n;k(x) = exp

−k − 2

k x

×
k

xm−1 (m − 1)! +

n−2

i=m

xi i! − (k − 1) xn−1 (n − 1)!

.

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Sharp bounds on the expectations of linear combinations of kth records expressed in the Gini mean difference units 14/14

Ψm,n;k(u) = 1 2u (1 − u)k−1

n−1

i=m
− k ln(1 − u)

i i! = ψm,n;k(u) 2u . ψ′

m,n;k(u) x=−k ln(1−u)

= ψ′

m,n;k(x) = exp

−k − 2

k x

×
k

xm−1 (m − 1)! +

n−2

i=m

xi i! − (k − 1) xn−1 (n − 1)!

.

First derivative is +−.

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Sharp bounds on the expectations of linear combinations of kth records expressed in the Gini mean difference units 14/14

Ψm,n;k(u) = 1 2u (1 − u)k−1

n−1

i=m
− k ln(1 − u)

i i! = ψm,n;k(u) 2u . ψ′

m,n;k(u) x=−k ln(1−u)

= ψ′

m,n;k(x) = exp

−k − 2

k x

×
k

xm−1 (m − 1)! +

n−2

i=m

xi i! − (k − 1) xn−1 (n − 1)!

.

First derivative is +−. ψ′′

m,n;k(x) = exp

−k − 3

k x k2 xm−2 (m − 2)! − k(k − 3) xm−1 (m − 1)! +

n−3

i=m

2xi i! − (k2 − 2) xn−2 (n − 2)! + (k − 1)(k − 2) xn−1 (n − 1)!

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Sharp bounds on the expectations of linear combinations of kth records expressed in the Gini mean difference units 14/14