Why Cannot We Have a Analysis of the Problem Strongly Consistent - - PowerPoint PPT Presentation

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Why Cannot We Have a Strongly Consistent Family

• f Skew Normal (and Higher

Order) Distributions

Thongchai Dumrongpokaphan1 and Vladik Kreinovich2

1Department of Mathematics, Faculty of Science,

Chiang Mai University, Thailand, tcd43@hotmail.com

2Department of Computer Science, University of Texas at El Paso,

500 W. University, El Paso, Texas 79968, USA, vladik@utep.edu

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1. Formulation of the Problem

• Often, the only information that we have about the

probability distribution is its first few moments.

• Many statistical techniques requires us to select a sin-

gle distribution.

• It is therefore desirable to select,
• out of all possible distributions with these mo-

ments,

• a single “most representative” one.
• When we know the first two moments, a natural idea

is to select a normal distribution.

• This selection is strongly consistent in the sense that:
• if a random variable is a sum of several ones,
• and we select normal distribution for all of them,
• then the sum is also normally distributed.

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2. Need for Strong Consistency

• Often, the random variable of interest has several com-

ponents.

• For example, an overall income consists of salaries, pen-

sions, unemployment benefits, interest, etc.

• Each of these categories, in its turn, can be subdivided

into more subcategories.

• If for each of these categories, we only know the first

moments, then we can apply the selection:

• either to the overall sum,
• or separately to each term.
• It seems reasonable to require that the resulting distri-

bution for the overall sum should be the same.

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3. What We Do in This Talk

• When we know three moments, there is also a widely

used selection – a skew-normal distribution.

• However, this selection is not strongly consistent in the

above sense.

• In this talk, we show that this absence of strong con-

sistency:

• is not a fault of a specific selection but a general

feature of the problem;

• namely, for third and higher order moments, no

strongly consistent selection is possible.

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4. Skew Normal Distributions

• In addition to the first two moments µ and M2, we may

also know the third moment M3.

• This can be described by the mean µ, the variance V =

σ2, and the third central moment m3

def

= E[(X − µ)3].

• There is a widely used selection, called skew normal:

ρ(x) = 2 ω · φ x − η ω

• · Φ
• α · x − η

ω

• , where

φ(x) = 1 √ 2π · exp

• −x2

2

• , and Φ(x) =

x

−∞

φ(t) dt.

• Here, µ = η + ω · δ ·
• 2

π, where δ

def

= α √ 1 + α2, σ2 = ω2·

• 1 − 2δ2

π

• , and m3 = 4 − π

2 ·σ3· (δ ·

• 2/π)3

(1 − 2δ2/π)3/2.

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5. Analysis of the Problem

• We want to assign, to each triple (µ, V, m3), a proba-

bility distribution ρ(x, µ, V, m3).

• Let us list the natural properties of this assignment.
• Moments are rarely known exactly, we usually know

them with some accuracy.

• It is reasonable to require that if the moments change

slightly, then ρ(x, µ, V, m3) should not change much.

• In other words, it is reasonable to require that the func-

tion ρ(x, µ, V, m3) is continuous.

• Comment: in our proof, we will only use that ρ(x) is

measurable.

• Strong consistency:

if X1 and X2 are independent, X1 ∼ ρ(x, µ1, V1, m31), and X2 ∼ ρ(x, µ2, V2, m32), then X1 + X2 ∼ ρ(x, µ1 + µ2, V1 + V2, m31 + m32).

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6. Scale Invariance

• Numerical values of different quantities depend on the

choice of a measuring unit.

• E.g.: income can be described in Baht or in dollars.
• If we change the unit to λ times smaller one, then:
• the actual incomes will not change,
• but the numerical values will change x → x′ = λ·x.
• If we perform the selection in the original units, then

we get ρ(x, µ, V, m3).

• If we simply re-scale x to x′ = λ · x, then for x′, we get

a new distribution ρ′(x′) = 1 λ · ρ x′ λ , µ, V, m3

• .

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7. Scale Invariance (cont-d)

• If we re-scale ρ(x, µ, V, m3), we get

ρ′(x′) = 1 λ · ρ x′ λ , µ, V, m3

• .
• We should get the exact same distribution if we make

a selection after the re-scaling, i.e., for µ′ = λ · µ, V ′ = λ2 · V, m′

3 = λ3 · m3.

• In the new units, we get ρ(x′, λ · µ, λ2 · V, λ3 · m3).
• A natural requirement is that the resulting selection

should be the same: 1 λ · ρ x′ λ , µ, V, m3

• = ρ(x′, λ · µ, λ2 · V, λ3 · m3).

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8. Definitions

• We say that a tuple (µ, V, m3) is possible if there exists

a distr. with mean µ, variance V , and moment m3.

• By a 3-selection, we mean a measurable mapping

ρ(x, µ, V, m3) defined for all possible tuples.

• We say that a 3-selection is strongly consistent if Xi ∼

ρ(x, µi, Vi, m3i) for independent Xi implies X1 + X2 ∼ ρ(x, µ1 + µ2, V1 + V2, m31 + m32).

• We say that a 3-selection is scale-invariant if for every

possible tuple (µ, V, m3), for every λ > 0 and x′: 1 λ · ρ x′ λ , µ, V, m3

• = ρ(x′, λ · µ, λ2 · V, λ3 · m3).

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9. Main Result

• Proposition. No 3-selection is strongly consistent and

scale-invariant.

• A similar result can be formulated for the case when

we also know higher order moments.

• In this case, instead of the original moments, we can

consider cumulants κn.

• Cumulants are terms at in · tn

n! in the Taylor expansion

• f ln(E[exp(i · t · X)]).
• For n = 1, n = 2, and n = 3, we get exactly the mean,

the variance, and the central third moment.

• Cumulants are additive: if X = X1 + X2 and X1 and

X2 are independent, then κn(X) = κn(X1) + κn(X2).

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10. Discussion

• Since we cannot make a strongly consistent selection,

what should we do?

• min and max are also natural operations in many ap-

plications; for example, in econometrics:

• if there are several ways to invest money with the

same level of risk,

• then an investor selects the one that leads to the

largest interest rate.

• From this viewpoint, it is reasonable to consider min-

ima and maxima of normal variables.

• In some cases, these minima and maxima are dis-

tributed according to the skew normal distribution.

• This may be an additional argument in favor of using

these distributions.

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11. Proof

• For sums of independent random variables X = X1 +

X2, it is convenient to use characteristic functions χX(ω)

def

= E[exp(i·ω·X)] for which χX(ω) = χX1(ω)·χX2(ω).

• For characteristic functions χ(ω, µ, V, m3), strong con-

sistency takes the form: χ(ω, µ1 + µ2, V1 + V2, m31 + m32) = χ(ω, µ1, V1, m31) · χ(ω2, µ2, V, m32).

• This requirement becomes even simpler if we take log-

arithm of both sides: for ℓ

def

= ln(χ): ℓ(ω, µ1 + µ2, V1 + V2, m31 + m32) = ℓ(ω, µ1, V1, m31) + ℓ(ω2, µ2, V, m32).

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12. Proof (cont-d)

• It is known that the only measurable functions with

this additivity property are linear functions, so ℓ(ω, µ, V, m3) = µ·ℓ1(ω)+V ·ℓ2(ω)+m3·ℓ3(ω) for some ℓi(ω).

• Let us now use the scale invariance requirement.
• When we replace x with x′ = λ · x, then

χX′(ω) = χX(λ · ω).

• Thus re-scaled χ(λ·ω, µ, V, m3) should be equal to what

we get from re-scaled moments: χ(ω, λ·µ, λ2·V, λ3·m3): χ(λ · ω, µ, V, m3) = χ(ω, λ · µ, λ2 · V, λ3 · m3).

• Their logarithms should also be equal:

ℓ(λ · ω, µ, V, m3) = ℓ(ω, λ · µ, λ2 · V, λ3 · m3).

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13. Proof (cont-d)

• Substituting the above linear expression for the func-

tion ℓ(ω, µ, V, m3) into this equality, we conclude that µ · ℓ1(λ · ω) + V · ℓ2(λ · ω) + m3 · ℓ3(λ · ω) = λ · µ · ℓ1(ω) + λ2 · V · ℓ2(ω) + λ3 · m3 · ℓ3(ω).

• This

equality must hold for all possible triples (µ, V, m3).

• Thus, the coefficient at µ, V , and m3 on both sides

must coincide.

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14. Proof (final)

• By equating coefficients at µ, we conclude that

ℓ1(λ · ω) = λ · ℓ1(ω).

• In particular, for ω = 1, we conclude that ℓ1(λ) =

λ · ℓ1(1), i.e., that ℓ1(ω) = c1 · ω for some constant c1.

• By equating coefficients at V and m3, we similarly get

ℓ2(ω) = c2 · ω2 and ℓ3(ω) = c3 · ω3.

• Thus, ℓ(ω, µ, V, m3) = c1 ·µ·ω +c2 ·V ·ω2 +c3 ·m3 ·ω3,

and χ(ω, µ, V, m3) = exp(c1 · µ · ω + c2 · V · ω2 + c3 · m3 · ω3).

• However, the Fourier transform of the above expression

is, in general, not an everywhere non-negative function.

• Thus, it cannot serve as a probability density function.

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15. Comment

• If we only consider two moments, then the above proof

leads to the characteristic function χ(ω, µ, V ) = exp(c1 · µ · ω + c2 · V · ω2).

• This characteristic function describes the Gaussian dis-

tribution.

• Thus, we have, in effect proven the following auxiliary

result.

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16. Auxiliary Definitions

• By a 2-selection, we mean a measurable mapping

ρ(x, µ, V ) defined for all possible tuples.

• We say that a 2-selection is strongly consistent if Xi ∼

ρ(x, µi, Vi) for independent Xi implies X1 + X2 ∼ ρ(x, µ1 + µ2, V1 + V2).

• We say that a 3-selection is scale-invariant if for every

possible tuple (µ, V ), for every λ > 0 and x′: 1 λ · ρ x′ λ , µ, V

• = ρ(x′, λ · µ, λ2 · V ).
• Proposition.

Every strongly consistent and scale- invariant 2-selection assigns:

• to each possible tuple (µ, V ),
• Gaussian distribution with mean µ and variance V .

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17. Acknowledgments This work was supported in part:

• by the National Science Foundation grants:
• HRD-0734825 and HRD-1242122

(Cyber-ShARE Center of Excellence) and

• DUE-0926721, and
• by an award from Prudential Foundation.