Uniqueness of characterization of distributions by regressions of - - PowerPoint PPT Presentation

Introduction and some known results Equivalent condition Special cases for ℓ = 2

Uniqueness of characterization of distributions by regressions

• f generalized order statistics

Mariusz Bieniek

Institute of Mathematics University of Maria Curie–Skłodowska Lublin, Poland mariusz.bieniek@umcs.lublin.pl

XLII Konferencja Statystyka Matematyczna Będlewo, 2016

Introduction and some known results Equivalent condition Special cases for ℓ = 2

Outline of the talk

1

Introduction and some known results

2

Equivalent condition for uniqueness of characterization

3

Special cases for ℓ = 2

Introduction and some known results Equivalent condition Special cases for ℓ = 2

Generalized order statistics

Fix n 1 and parameters γ1, . . . , γn > 0 Define cn−1 = n

i=1 γi and mi = γi − γi+1 − 1, 1 i n − 1.

Assume that F is absolutely continuous cdf with density f and the support (α, β) Let ¯ F(x) = 1 − F(x) denote the survival function of F The random variables X (r)

∗ , 1 r n, are called generalized order statistics

based on F, if their joint density is f X (1)

∗ ,...,X (n) ∗ (x1, . . . , xn) = cn−1

n−1

• i=1

¯

F(xi)

mi f (xi)

• ¯

F(xn)

γn−1f (xn),

for −∞ < x1 . . . xn < +∞

Introduction and some known results Equivalent condition Special cases for ℓ = 2

Generalized order statistics

Fix n 1 and parameters γ1, . . . , γn > 0 Define cn−1 = n

i=1 γi and mi = γi − γi+1 − 1, 1 i n − 1.

Assume that F is absolutely continuous cdf with density f and the support (α, β) Let ¯ F(x) = 1 − F(x) denote the survival function of F The random variables X (r)

∗ , 1 r n, are called generalized order statistics

based on F, if their joint density is f X (1)

∗ ,...,X (n) ∗ (x1, . . . , xn) = cn−1

n−1

• i=1

¯

F(xi)

mi f (xi)

• ¯

F(xn)

γn−1f (xn),

for −∞ < x1 . . . xn < +∞

Introduction and some known results Equivalent condition Special cases for ℓ = 2

Generalized order statistics

Fix n 1 and parameters γ1, . . . , γn > 0 Define cn−1 = n

i=1 γi and mi = γi − γi+1 − 1, 1 i n − 1.

Assume that F is absolutely continuous cdf with density f and the support (α, β) Let ¯ F(x) = 1 − F(x) denote the survival function of F The random variables X (r)

∗ , 1 r n, are called generalized order statistics

based on F, if their joint density is f X (1)

∗ ,...,X (n) ∗ (x1, . . . , xn) = cn−1

n−1

• i=1

¯

F(xi)

mi f (xi)

• ¯

F(xn)

γn−1f (xn),

for −∞ < x1 . . . xn < +∞

Introduction and some known results Equivalent condition Special cases for ℓ = 2

Generalized order statistics

Fix n 1 and parameters γ1, . . . , γn > 0 Define cn−1 = n

i=1 γi and mi = γi − γi+1 − 1, 1 i n − 1.

Assume that F is absolutely continuous cdf with density f and the support (α, β) Let ¯ F(x) = 1 − F(x) denote the survival function of F The random variables X (r)

∗ , 1 r n, are called generalized order statistics

based on F, if their joint density is f X (1)

∗ ,...,X (n) ∗ (x1, . . . , xn) = cn−1

n−1

• i=1

¯

F(xi)

mi f (xi)

• ¯

F(xn)

γn−1f (xn),

for −∞ < x1 . . . xn < +∞

Introduction and some known results Equivalent condition Special cases for ℓ = 2

Generalized order statistics

Fix n 1 and parameters γ1, . . . , γn > 0 Define cn−1 = n

i=1 γi and mi = γi − γi+1 − 1, 1 i n − 1.

Assume that F is absolutely continuous cdf with density f and the support (α, β) Let ¯ F(x) = 1 − F(x) denote the survival function of F The random variables X (r)

∗ , 1 r n, are called generalized order statistics

based on F, if their joint density is f X (1)

∗ ,...,X (n) ∗ (x1, . . . , xn) = cn−1

n−1

• i=1

¯

F(xi)

mi f (xi)

• ¯

F(xn)

γn−1f (xn),

for −∞ < x1 . . . xn < +∞

Introduction and some known results Equivalent condition Special cases for ℓ = 2

Special cases

γi = n − i + 1, 1 i n — order statistics X1:n . . . Xn:n γi = 1 — record values γi = k — kth record values progressively censored order statistics

Introduction and some known results Equivalent condition Special cases for ℓ = 2

Special cases

γi = n − i + 1, 1 i n — order statistics X1:n . . . Xn:n γi = 1 — record values γi = k — kth record values progressively censored order statistics

Introduction and some known results Equivalent condition Special cases for ℓ = 2

Special cases

γi = n − i + 1, 1 i n — order statistics X1:n . . . Xn:n γi = 1 — record values γi = k — kth record values progressively censored order statistics

Introduction and some known results Equivalent condition Special cases for ℓ = 2

Special cases

γi = n − i + 1, 1 i n — order statistics X1:n . . . Xn:n γi = 1 — record values γi = k — kth record values progressively censored order statistics

Introduction and some known results Equivalent condition Special cases for ℓ = 2

Statement of the problem

Fix r, ℓ 1 and parameters γ1, . . . , γr+ℓ > 0 Assume that F is absolutely continuous cdf with the support (α, β) Consider generalized order statistics X (1)

∗ , . . . , X (r) ∗ , . . . , X (r+ℓ) ∗

with parameters γ1, . . . , γr+ℓ based on F Assume that function h : (α, β) → R is strictly increasing and continuous If E

• h

X (r+ℓ)

∗

• < ∞, then we define the regression function

ξ(x) = E

• h

X (r+ℓ)

∗

X (r)

∗

= x

• ,

x ∈ (α, β) Problem: Does ξ determine F uniquely? What is the expression for F in terms of ξ and h?

Introduction and some known results Equivalent condition Special cases for ℓ = 2

Statement of the problem

Fix r, ℓ 1 and parameters γ1, . . . , γr+ℓ > 0 Assume that F is absolutely continuous cdf with the support (α, β) Consider generalized order statistics X (1)

∗ , . . . , X (r) ∗ , . . . , X (r+ℓ) ∗

with parameters γ1, . . . , γr+ℓ based on F Assume that function h : (α, β) → R is strictly increasing and continuous If E

• h

X (r+ℓ)

∗

• < ∞, then we define the regression function

ξ(x) = E

• h

X (r+ℓ)

∗

X (r)

∗

= x

• ,

x ∈ (α, β) Problem: Does ξ determine F uniquely? What is the expression for F in terms of ξ and h?

Introduction and some known results Equivalent condition Special cases for ℓ = 2

Statement of the problem

Fix r, ℓ 1 and parameters γ1, . . . , γr+ℓ > 0 Assume that F is absolutely continuous cdf with the support (α, β) Consider generalized order statistics X (1)

∗ , . . . , X (r) ∗ , . . . , X (r+ℓ) ∗

with parameters γ1, . . . , γr+ℓ based on F Assume that function h : (α, β) → R is strictly increasing and continuous If E

• h

X (r+ℓ)

∗

• < ∞, then we define the regression function

ξ(x) = E

• h

X (r+ℓ)

∗

X (r)

∗

= x

• ,

x ∈ (α, β) Problem: Does ξ determine F uniquely? What is the expression for F in terms of ξ and h?

Introduction and some known results Equivalent condition Special cases for ℓ = 2

Statement of the problem

Fix r, ℓ 1 and parameters γ1, . . . , γr+ℓ > 0 Assume that F is absolutely continuous cdf with the support (α, β) Consider generalized order statistics X (1)

∗ , . . . , X (r) ∗ , . . . , X (r+ℓ) ∗

with parameters γ1, . . . , γr+ℓ based on F Assume that function h : (α, β) → R is strictly increasing and continuous If E

• h

X (r+ℓ)

∗

• < ∞, then we define the regression function

ξ(x) = E

• h

X (r+ℓ)

∗

X (r)

∗

= x

• ,

x ∈ (α, β) Problem: Does ξ determine F uniquely? What is the expression for F in terms of ξ and h?

Introduction and some known results Equivalent condition Special cases for ℓ = 2

Statement of the problem

Fix r, ℓ 1 and parameters γ1, . . . , γr+ℓ > 0 Assume that F is absolutely continuous cdf with the support (α, β) Consider generalized order statistics X (1)

∗ , . . . , X (r) ∗ , . . . , X (r+ℓ) ∗

with parameters γ1, . . . , γr+ℓ based on F Assume that function h : (α, β) → R is strictly increasing and continuous If E

• h

X (r+ℓ)

∗

• < ∞, then we define the regression function

ξ(x) = E

• h

X (r+ℓ)

∗

X (r)

∗

= x

• ,

x ∈ (α, β) Problem: Does ξ determine F uniquely? What is the expression for F in terms of ξ and h?

Introduction and some known results Equivalent condition Special cases for ℓ = 2

Statement of the problem

Fix r, ℓ 1 and parameters γ1, . . . , γr+ℓ > 0 Assume that F is absolutely continuous cdf with the support (α, β) Consider generalized order statistics X (1)

∗ , . . . , X (r) ∗ , . . . , X (r+ℓ) ∗

with parameters γ1, . . . , γr+ℓ based on F Assume that function h : (α, β) → R is strictly increasing and continuous If E

• h

X (r+ℓ)

∗

• < ∞, then we define the regression function

ξ(x) = E

• h

X (r+ℓ)

∗

X (r)

∗

= x

• ,

x ∈ (α, β) Problem: Does ξ determine F uniquely? What is the expression for F in terms of ξ and h?

Introduction and some known results Equivalent condition Special cases for ℓ = 2

Known results: adjacent case ℓ = 1

If ξ(x) = E

• h

X (r+1)

∗

X (r)

∗

= x

• ,

x ∈ (α, β), then (denoting ¯ F = 1 − F) ξ(x) = γr+1 ¯ F(x)γr+1

β

x

h(y) ¯ F(y)γr+1−1f (y)dy Then ξ is also increasing. If F has continuous density, then ξ is differentiable. Denoting H = − log ¯ F we have

• ξ′(x) = γr+1 [ξ(x) − h(x)] H′(x),

h(x) < ξ(x), x ∈ (α, β).

Introduction and some known results Equivalent condition Special cases for ℓ = 2

Known results: adjacent case ℓ = 1

If ξ(x) = E

• h

X (r+1)

∗

X (r)

∗

= x

• ,

x ∈ (α, β), then (denoting ¯ F = 1 − F) ξ(x) = γr+1 ¯ F(x)γr+1

β

x

h(y) ¯ F(y)γr+1−1f (y)dy Then ξ is also increasing. If F has continuous density, then ξ is differentiable. Denoting H = − log ¯ F we have

• ξ′(x) = γr+1 [ξ(x) − h(x)] H′(x),

h(x) < ξ(x), x ∈ (α, β).

Introduction and some known results Equivalent condition Special cases for ℓ = 2

Known results: adjacent case ℓ = 1

If ξ(x) = E

• h

X (r+1)

∗

X (r)

∗

= x

• ,

x ∈ (α, β), then (denoting ¯ F = 1 − F) ξ(x) = γr+1 ¯ F(x)γr+1

β

x

h(y) ¯ F(y)γr+1−1f (y)dy Then ξ is also increasing. If F has continuous density, then ξ is differentiable. Denoting H = − log ¯ F we have

• ξ′(x) = γr+1 [ξ(x) − h(x)] H′(x),

h(x) < ξ(x), x ∈ (α, β).

Introduction and some known results Equivalent condition Special cases for ℓ = 2

Known results: adjacent case ℓ = 1

If ξ(x) = E

• h

X (r+1)

∗

X (r)

∗

= x

• ,

x ∈ (α, β), then (denoting ¯ F = 1 − F) ξ(x) = γr+1 ¯ F(x)γr+1

β

x

h(y) ¯ F(y)γr+1−1f (y)dy Then ξ is also increasing. If F has continuous density, then ξ is differentiable. Denoting H = − log ¯ F we have

• ξ′(x) = γr+1 [ξ(x) − h(x)] H′(x),

h(x) < ξ(x), x ∈ (α, β).

Introduction and some known results Equivalent condition Special cases for ℓ = 2

Known results: adjacent case ℓ = 1 (cntd)

Therefore if ξ(x) = E

• h

X (r+1)

∗

X (r)

∗

= x

• ,

x ∈ (α, β), then F is uniquely determined by F(x) = 1 − exp

• −

x

α

ξ′(y)dy ξ(y) − h(y)

• ,

x ∈ (α, β). In particular if h(x) = x and ξ(x) = ax + b for some a > 0, then

0 < a < 1 and F is power df a = 1 and F is exponential df a > 1 and F is Pareto df

References: e.g. Ferguson (1967), Franco, Ruiz (1995, 1996, 1999), Cramer, Kamps, Keseling (2004)

Introduction and some known results Equivalent condition Special cases for ℓ = 2

Known results: adjacent case ℓ = 1 (cntd)

Therefore if ξ(x) = E

• h

X (r+1)

∗

X (r)

∗

= x

• ,

x ∈ (α, β), then F is uniquely determined by F(x) = 1 − exp

• −

x

α

ξ′(y)dy ξ(y) − h(y)

• ,

x ∈ (α, β). In particular if h(x) = x and ξ(x) = ax + b for some a > 0, then

0 < a < 1 and F is power df a = 1 and F is exponential df a > 1 and F is Pareto df

References: e.g. Ferguson (1967), Franco, Ruiz (1995, 1996, 1999), Cramer, Kamps, Keseling (2004)

Introduction and some known results Equivalent condition Special cases for ℓ = 2

Known results: adjacent case ℓ = 1 (cntd)

Therefore if ξ(x) = E

• h

X (r+1)

∗

X (r)

∗

= x

• ,

x ∈ (α, β), then F is uniquely determined by F(x) = 1 − exp

• −

x

α

ξ′(y)dy ξ(y) − h(y)

• ,

x ∈ (α, β). In particular if h(x) = x and ξ(x) = ax + b for some a > 0, then

0 < a < 1 and F is power df a = 1 and F is exponential df a > 1 and F is Pareto df

References: e.g. Ferguson (1967), Franco, Ruiz (1995, 1996, 1999), Cramer, Kamps, Keseling (2004)

Introduction and some known results Equivalent condition Special cases for ℓ = 2

Known results: non–adjacent case, linearity of regression

Assume that ℓ 1 and h(x) = x and ξ(x) = ax + b. Then using Rao–Shanbhag lemma on integrated Cauchy functional equation we get that the condition E

• X (r+ℓ)

∗

• X (r)

∗

= x

• = ax + b

again implies

0 < a < 1 and F is power df a = 1 and F is exponential df a > 1 and F is Pareto df

References: Dembińska, Wesołowski (1998, 2000), Bieniek, Szynal (2003), Cramer, Kamps, Keseling (2004)

Introduction and some known results Equivalent condition Special cases for ℓ = 2

Known results: non–adjacent case, linearity of regression

Assume that ℓ 1 and h(x) = x and ξ(x) = ax + b. Then using Rao–Shanbhag lemma on integrated Cauchy functional equation we get that the condition E

• X (r+ℓ)

∗

• X (r)

∗

= x

• = ax + b

again implies

0 < a < 1 and F is power df a = 1 and F is exponential df a > 1 and F is Pareto df

References: Dembińska, Wesołowski (1998, 2000), Bieniek, Szynal (2003), Cramer, Kamps, Keseling (2004)

Introduction and some known results Equivalent condition Special cases for ℓ = 2

Known results: non–adjacent case, linearity of regression

Assume that ℓ 1 and h(x) = x and ξ(x) = ax + b. Then using Rao–Shanbhag lemma on integrated Cauchy functional equation we get that the condition E

• X (r+ℓ)

∗

• X (r)

∗

= x

• = ax + b

again implies

0 < a < 1 and F is power df a = 1 and F is exponential df a > 1 and F is Pareto df

References: Dembińska, Wesołowski (1998, 2000), Bieniek, Szynal (2003), Cramer, Kamps, Keseling (2004)

Introduction and some known results Equivalent condition Special cases for ℓ = 2

Known results: non–adjacent case, linearity of regression

Assume that ℓ 1 and h(x) = x and ξ(x) = ax + b. Then using Rao–Shanbhag lemma on integrated Cauchy functional equation we get that the condition E

• X (r+ℓ)

∗

• X (r)

∗

= x

• = ax + b

again implies

0 < a < 1 and F is power df a = 1 and F is exponential df a > 1 and F is Pareto df

References: Dembińska, Wesołowski (1998, 2000), Bieniek, Szynal (2003), Cramer, Kamps, Keseling (2004)

Introduction and some known results Equivalent condition Special cases for ℓ = 2

Known results: non–adjacent case, two regressions known

If two regressions ξ1(x) = E

• h

X (r+ℓ)

∗

X (r)

∗

= x

• ,

ξ2(x) = E

• h

X (r+ℓ)

∗

X (r+1)

∗

= x

• ,

are known Then F is uniquely determined by F(x) = 1 − exp

• −

1 γr+1

x

α

dξ1(y) ξ1(y) − ξ2(y)

• ,

x ∈ (α, β). References: Bieniek (2009)

Introduction and some known results Equivalent condition Special cases for ℓ = 2

Known results: non–adjacent case, two regressions known

If two regressions ξ1(x) = E

• h

X (r+ℓ)

∗

X (r)

∗

= x

• ,

ξ2(x) = E

• h

X (r+ℓ)

∗

X (r+1)

∗

= x

• ,

are known Then F is uniquely determined by F(x) = 1 − exp

• −

1 γr+1

x

α

dξ1(y) ξ1(y) − ξ2(y)

• ,

x ∈ (α, β). References: Bieniek (2009)

Introduction and some known results Equivalent condition Special cases for ℓ = 2

Known results: non–adjacent case, two regressions known

If two regressions ξ1(x) = E

• h

X (r+ℓ)

∗

X (r)

∗

= x

• ,

ξ2(x) = E

• h

X (r+ℓ)

∗

X (r+1)

∗

= x

• ,

are known Then F is uniquely determined by F(x) = 1 − exp

• −

1 γr+1

x

α

dξ1(y) ξ1(y) − ξ2(y)

• ,

x ∈ (α, β). References: Bieniek (2009)

Introduction and some known results Equivalent condition Special cases for ℓ = 2

The key idea for ℓ = 2

Assume that E

• h

X (r+2)

∗

X (r)

∗

= x

• = ξ(x),

x ∈ (α, β) Define ϕ(x) = E

• h

X (r+2)

∗

X (r+1)

∗

= x

• ,

x ∈ (α, β) We apply the Markov property of GOS from continuous distributions combined with the tower property of conditional expectations. Therefore E

• ϕ

X (r+1)

∗

X (r)

∗

= x

• = ξ(x),

x ∈ (α, β). and h(x) < ϕ(x) < ξ(x), x ∈ (α, β).

Introduction and some known results Equivalent condition Special cases for ℓ = 2

The key idea for ℓ = 2

Assume that E

• h

X (r+2)

∗

X (r)

∗

= x

• = ξ(x),

x ∈ (α, β) Define ϕ(x) = E

• h

X (r+2)

∗

X (r+1)

∗

= x

• ,

x ∈ (α, β) We apply the Markov property of GOS from continuous distributions combined with the tower property of conditional expectations. Therefore E

• ϕ

X (r+1)

∗

X (r)

∗

= x

• = ξ(x),

x ∈ (α, β). and h(x) < ϕ(x) < ξ(x), x ∈ (α, β).

Introduction and some known results Equivalent condition Special cases for ℓ = 2

The key idea for ℓ = 2

Assume that E

• h

X (r+2)

∗

X (r)

∗

= x

• = ξ(x),

x ∈ (α, β) Define ϕ(x) = E

• h

X (r+2)

∗

X (r+1)

∗

= x

• ,

x ∈ (α, β) We apply the Markov property of GOS from continuous distributions combined with the tower property of conditional expectations. Therefore E

• ϕ

X (r+1)

∗

X (r)

∗

= x

• = ξ(x),

x ∈ (α, β). and h(x) < ϕ(x) < ξ(x), x ∈ (α, β).

Introduction and some known results Equivalent condition Special cases for ℓ = 2

The key idea for ℓ = 2 (cntd)

Assume that f = F ′ is continuous on (α, β) Then ξ and ϕ are differentiable and

• ϕ′(x) = γr+2[ϕ(x) − h(x)]H′(x)

ξ′(x) = γr+1[ξ(x) − ϕ(x)]H′(x) Therefore ϕ is a solution the problem ϕ′(x) = γr+2 γr+1 ϕ(x) − h(x) ξ(x) − ϕ(x)ξ′(x) (A) with the “initial” condition h(x) < ϕ(x) < ξ(x), x ∈ (α, β).

Introduction and some known results Equivalent condition Special cases for ℓ = 2

The key idea for ℓ = 2 (cntd)

Assume that f = F ′ is continuous on (α, β) Then ξ and ϕ are differentiable and

• ϕ′(x) = γr+2[ϕ(x) − h(x)]H′(x)

ξ′(x) = γr+1[ξ(x) − ϕ(x)]H′(x) Therefore ϕ is a solution the problem ϕ′(x) = γr+2 γr+1 ϕ(x) − h(x) ξ(x) − ϕ(x)ξ′(x) (A) with the “initial” condition h(x) < ϕ(x) < ξ(x), x ∈ (α, β).

Introduction and some known results Equivalent condition Special cases for ℓ = 2

The key idea for ℓ = 2 (cntd)

Assume that f = F ′ is continuous on (α, β) Then ξ and ϕ are differentiable and

• ϕ′(x) = γr+2[ϕ(x) − h(x)]H′(x)

ξ′(x) = γr+1[ξ(x) − ϕ(x)]H′(x) Therefore ϕ is a solution the problem ϕ′(x) = γr+2 γr+1 ϕ(x) − h(x) ξ(x) − ϕ(x)ξ′(x) (A) with the “initial” condition h(x) < ϕ(x) < ξ(x), x ∈ (α, β).

Introduction and some known results Equivalent condition Special cases for ℓ = 2

Equivalent condition for uniqueness: ℓ = 2

Theorem The regression E

• h

X (r+2)

∗

X (r)

∗

= x

• = ξ(x), characterizes F uniquely if and only

if the problem (A) has the unique solution ϕ. Then F(x) = 1 − exp(−H(x)), where H is given in one of the equivalent forms H(x) = 1 γr+2

x

α

ϕ′(y)dy ϕ(y) − h(y) = 1 γr+1

x

α

ξ′(y)dy ξ(y) − ϕ(y) =

x

α

η′(y)dy ξ(y) − h(y), where η(y) = ξ(y) γr+1 + ϕ(y) γr+2 .

Introduction and some known results Equivalent condition Special cases for ℓ = 2

Equivalent condition for uniqueness: ℓ = 2

Theorem The regression E

• h

X (r+2)

∗

X (r)

∗

= x

• = ξ(x), characterizes F uniquely if and only

if the problem (A) has the unique solution ϕ. Then F(x) = 1 − exp(−H(x)), where H is given in one of the equivalent forms H(x) = 1 γr+2

x

α

ϕ′(y)dy ϕ(y) − h(y) = 1 γr+1

x

α

ξ′(y)dy ξ(y) − ϕ(y) =

x

α

η′(y)dy ξ(y) − h(y), where η(y) = ξ(y) γr+1 + ϕ(y) γr+2 .

Introduction and some known results Equivalent condition Special cases for ℓ = 2

Equivalent condition—arbitrary ℓ 2

The condition E

• h

X (r+ℓ)

∗

X (r)

∗

= x

• = ξ(x) with ξ differentiable,

characterizes F with continuous density uniquely, if and only if the problem (system of ODEs)

    

y′

i = γr+i+1

γr+1 yi − yi+1 ξ(x) − y1 ξ′(x) for 1 i ℓ − 1 h < yℓ−1 < . . . < y1 < ξ

• n (α, β)

with yℓ = h has unique solution (ϕℓ−1, . . . , ϕ1). Then F(x) = 1 − exp

• −

x

α

η′(y)dy ξ(y) − h(y)

• ,

x ∈ (α, β) where η(y) = ξ(y) γr+1 +

ℓ−1

• i=1

ϕi(y) γr+i+1

Introduction and some known results Equivalent condition Special cases for ℓ = 2

Equivalent condition—arbitrary ℓ 2

The condition E

• h

X (r+ℓ)

∗

X (r)

∗

= x

• = ξ(x) with ξ differentiable,

characterizes F with continuous density uniquely, if and only if the problem (system of ODEs)

    

y′

i = γr+i+1

γr+1 yi − yi+1 ξ(x) − y1 ξ′(x) for 1 i ℓ − 1 h < yℓ−1 < . . . < y1 < ξ

• n (α, β)

with yℓ = h has unique solution (ϕℓ−1, . . . , ϕ1). Then F(x) = 1 − exp

• −

x

α

η′(y)dy ξ(y) − h(y)

• ,

x ∈ (α, β) where η(y) = ξ(y) γr+1 +

ℓ−1

• i=1

ϕi(y) γr+i+1

Introduction and some known results Equivalent condition Special cases for ℓ = 2

Uniqueness for ℓ = 2 and h(β) < ∞

Theorem If h(β) < ∞, then ξ(β) = h(β) and the regression ξ(x) = E

• h

X (r+2)

∗

X (r)

∗

= x

• ,

x ∈ (α, β), characterizes F with continuous density uniquely. Remark If h(β) = ∞, then also ξ(x) = ∞

Introduction and some known results Equivalent condition Special cases for ℓ = 2

Uniqueness for ℓ = 2 and h(β) < ∞

Theorem If h(β) < ∞, then ξ(β) = h(β) and the regression ξ(x) = E

• h

X (r+2)

∗

X (r)

∗

= x

• ,

x ∈ (α, β), characterizes F with continuous density uniquely. Remark If h(β) = ∞, then also ξ(x) = ∞

Introduction and some known results Equivalent condition Special cases for ℓ = 2

Linearity of regression for ℓ = 2 (new proof)

Assume that h(x) = x and ξ(x) = ax + b, so that E

• X (r+2)

∗

• X (r)

∗

= x

• = ax + b,

x ∈ (α, β), where a > 0. To find F we need to solve the problem

    

y′ = aγ y − x ax + b − y x < y < ax + b, x ∈ (α, β), (A) where γ = γr+2 γr+1

Introduction and some known results Equivalent condition Special cases for ℓ = 2

Linearity of regression for ℓ = 2 (new proof)

Assume that h(x) = x and ξ(x) = ax + b, so that E

• X (r+2)

∗

• X (r)

∗

= x

• = ax + b,

x ∈ (α, β), where a > 0. To find F we need to solve the problem

    

y′ = aγ y − x ax + b − y x < y < ax + b, x ∈ (α, β), (A) where γ = γr+2 γr+1

Introduction and some known results Equivalent condition Special cases for ℓ = 2

The case 0 < a < 1

The problem (A) has exactly one linear solution ϕ(x) = cx + d, where c is the unique solution in (a, 1) to the quadratic equation c2 + a(γ − 1)c − aγ = 0, and d = 1 − c 1 − ab ∈ (0, b). But if a ∈ (0, 1), then β < ∞, so h(β) < ∞, and the linear solution is the unique solution to (A) Therefore E

• X (r+2)

∗

• X (r+1)

∗

= x

• = cx + d,

x ∈ (α, β), where c ∈ (0, 1), so F is power distribution function

Introduction and some known results Equivalent condition Special cases for ℓ = 2

The case 0 < a < 1

The problem (A) has exactly one linear solution ϕ(x) = cx + d, where c is the unique solution in (a, 1) to the quadratic equation c2 + a(γ − 1)c − aγ = 0, and d = 1 − c 1 − ab ∈ (0, b). But if a ∈ (0, 1), then β < ∞, so h(β) < ∞, and the linear solution is the unique solution to (A) Therefore E

• X (r+2)

∗

• X (r+1)

∗

= x

• = cx + d,

x ∈ (α, β), where c ∈ (0, 1), so F is power distribution function

Introduction and some known results Equivalent condition Special cases for ℓ = 2

The case 0 < a < 1

The problem (A) has exactly one linear solution ϕ(x) = cx + d, where c is the unique solution in (a, 1) to the quadratic equation c2 + a(γ − 1)c − aγ = 0, and d = 1 − c 1 − ab ∈ (0, b). But if a ∈ (0, 1), then β < ∞, so h(β) < ∞, and the linear solution is the unique solution to (A) Therefore E

• X (r+2)

∗

• X (r+1)

∗

= x

• = cx + d,

x ∈ (α, β), where c ∈ (0, 1), so F is power distribution function

Introduction and some known results Equivalent condition Special cases for ℓ = 2

The case a 1

If a 1, then β = ∞ and the problem (A) has exactly one linear solution ϕ(x) = cx + d, where

if a = 1, then c = 1 and d =

b 1+γ .

if a > 1, then c is the unique solution to c2 + a(γ − 1)c − aγ = 0 in (1, a) and d = c−1

a−1b ∈ (0, b).

The uniqueness of ϕ follows from the next lemma Lemma If y is any solution to (A) with a 1 such that y(α) = ϕ(α), and c is defined above, then for z = (y′ − c)2 we have z is strictly increasing z(α) > 0 and lim

x→∞ z(x) = 0.

Introduction and some known results Equivalent condition Special cases for ℓ = 2

The case a 1

If a 1, then β = ∞ and the problem (A) has exactly one linear solution ϕ(x) = cx + d, where

if a = 1, then c = 1 and d =

b 1+γ .

if a > 1, then c is the unique solution to c2 + a(γ − 1)c − aγ = 0 in (1, a) and d = c−1

a−1b ∈ (0, b).

The uniqueness of ϕ follows from the next lemma Lemma If y is any solution to (A) with a 1 such that y(α) = ϕ(α), and c is defined above, then for z = (y′ − c)2 we have z is strictly increasing z(α) > 0 and lim

x→∞ z(x) = 0.

Introduction and some known results Equivalent condition Special cases for ℓ = 2

The case a 1

If a 1, then β = ∞ and the problem (A) has exactly one linear solution ϕ(x) = cx + d, where

if a = 1, then c = 1 and d =

b 1+γ .

if a > 1, then c is the unique solution to c2 + a(γ − 1)c − aγ = 0 in (1, a) and d = c−1

a−1b ∈ (0, b).

The uniqueness of ϕ follows from the next lemma Lemma If y is any solution to (A) with a 1 such that y(α) = ϕ(α), and c is defined above, then for z = (y′ − c)2 we have z is strictly increasing z(α) > 0 and lim

x→∞ z(x) = 0.

Introduction and some known results Equivalent condition Special cases for ℓ = 2

The case a 1

If a 1, then β = ∞ and the problem (A) has exactly one linear solution ϕ(x) = cx + d, where

if a = 1, then c = 1 and d =

b 1+γ .

if a > 1, then c is the unique solution to c2 + a(γ − 1)c − aγ = 0 in (1, a) and d = c−1

a−1b ∈ (0, b).

The uniqueness of ϕ follows from the next lemma Lemma If y is any solution to (A) with a 1 such that y(α) = ϕ(α), and c is defined above, then for z = (y′ − c)2 we have z is strictly increasing z(α) > 0 and lim

x→∞ z(x) = 0.

Introduction and some known results Equivalent condition Special cases for ℓ = 2

The case a 1

If a 1, then β = ∞ and the problem (A) has exactly one linear solution ϕ(x) = cx + d, where

if a = 1, then c = 1 and d =

b 1+γ .

if a > 1, then c is the unique solution to c2 + a(γ − 1)c − aγ = 0 in (1, a) and d = c−1

a−1b ∈ (0, b).

The uniqueness of ϕ follows from the next lemma Lemma If y is any solution to (A) with a 1 such that y(α) = ϕ(α), and c is defined above, then for z = (y′ − c)2 we have z is strictly increasing z(α) > 0 and lim

x→∞ z(x) = 0.

Introduction and some known results Equivalent condition Special cases for ℓ = 2

The case a 1

Again the condition E

• X (r+2)

∗

• X (r)

∗

= x

• = ax + b,

x ∈ (α, ∞), with a 1 implies E

• X (r+2)

∗

• X (r+1)

∗

= x

• = cx + d,

x ∈ (α, ∞), where

if a = 1 then c = 1 and F is exponential distribution if a > 1 then c ∈ (1, a) is uniquely determined, and F is Pareto distribution

Introduction and some known results Equivalent condition Special cases for ℓ = 2

The case a 1

Again the condition E

• X (r+2)

∗

• X (r)

∗

= x

• = ax + b,

x ∈ (α, ∞), with a 1 implies E

• X (r+2)

∗

• X (r+1)

∗

= x

• = cx + d,

x ∈ (α, ∞), where

if a = 1 then c = 1 and F is exponential distribution if a > 1 then c ∈ (1, a) is uniquely determined, and F is Pareto distribution

Introduction and some known results Equivalent condition Special cases for ℓ = 2

The case a 1

Again the condition E

• X (r+2)

∗

• X (r)

∗

= x

• = ax + b,

x ∈ (α, ∞), with a 1 implies E

• X (r+2)

∗

• X (r+1)

∗

= x

• = cx + d,

x ∈ (α, ∞), where

if a = 1 then c = 1 and F is exponential distribution if a > 1 then c ∈ (1, a) is uniquely determined, and F is Pareto distribution