Formulation of the . . . Need for Strong . . . What We Do in This Talk Skew Normal . . . Why Cannot We Have a Analysis of the Problem Strongly Consistent Family Scale Invariance Main Result of Skew Normal (and Higher Discussion Proof Order) Distributions Home Page Title Page Thongchai Dumrongpokaphan 1 and Vladik Kreinovich 2 ◭◭ ◮◮ 1 Department of Mathematics, Faculty of Science, ◭ ◮ Chiang Mai University, Thailand, tcd43@hotmail.com 2 Department of Computer Science, University of Texas at El Paso, Page 1 of 18 500 W. University, El Paso, Texas 79968, USA, vladik@utep.edu Go Back Full Screen Close Quit
Formulation of the . . . Need for Strong . . . 1. Formulation of the Problem What We Do in This Talk • Often, the only information that we have about the Skew Normal . . . probability distribution is its first few moments. Analysis of the Problem Scale Invariance • Many statistical techniques requires us to select a sin- Main Result gle distribution. Discussion • It is therefore desirable to select, Proof • out of all possible distributions with these mo- Home Page ments, Title Page • a single “most representative” one. ◭◭ ◮◮ • When we know the first two moments, a natural idea ◭ ◮ is to select a normal distribution. Page 2 of 18 • This selection is strongly consistent in the sense that: Go Back • if a random variable is a sum of several ones, Full Screen • and we select normal distribution for all of them, • then the sum is also normally distributed. Close Quit
Formulation of the . . . Need for Strong . . . 2. Need for Strong Consistency What We Do in This Talk • Often, the random variable of interest has several com- Skew Normal . . . ponents. Analysis of the Problem Scale Invariance • For example, an overall income consists of salaries, pen- Main Result sions, unemployment benefits, interest, etc. Discussion • Each of these categories, in its turn, can be subdivided Proof into more subcategories. Home Page • If for each of these categories, we only know the first Title Page moments, then we can apply the selection: ◭◭ ◮◮ • either to the overall sum, ◭ ◮ • or separately to each term. Page 3 of 18 • It seems reasonable to require that the resulting distri- Go Back bution for the overall sum should be the same. Full Screen Close Quit
Formulation of the . . . Need for Strong . . . 3. What We Do in This Talk What We Do in This Talk • When we know three moments, there is also a widely Skew Normal . . . used selection – a skew-normal distribution. Analysis of the Problem Scale Invariance • However, this selection is not strongly consistent in the Main Result above sense. Discussion • In this talk, we show that this absence of strong con- Proof sistency: Home Page • is not a fault of a specific selection but a general Title Page feature of the problem; ◭◭ ◮◮ • namely, for third and higher order moments, no ◭ ◮ strongly consistent selection is possible. Page 4 of 18 Go Back Full Screen Close Quit
Formulation of the . . . Need for Strong . . . 4. Skew Normal Distributions What We Do in This Talk • In addition to the first two moments µ and M 2 , we may Skew Normal . . . also know the third moment M 3 . Analysis of the Problem Scale Invariance • This can be described by the mean µ , the variance V = def Main Result σ 2 , and the third central moment m 3 = E [( X − µ ) 3 ] . Discussion • There is a widely used selection, called skew normal : Proof ρ ( x ) = 2 � x − η � � α · x − η � Home Page ω · φ · Φ , where ω ω Title Page � x − x 2 1 � � ◭◭ ◮◮ φ ( x ) = 2 π · exp , and Φ( x ) = φ ( t ) dt. √ 2 ◭ ◮ −∞ � 2 α Page 5 of 18 def • Here, µ = η + ω · δ · π , where δ = 1 + α 2 , √ Go Back 1 − 2 δ 2 � 2 /π ) 3 � � , and m 3 = 4 − π · σ 3 · ( δ · σ 2 = ω 2 · Full Screen (1 − 2 δ 2 /π ) 3 / 2 . π 2 Close Quit
Formulation of the . . . Need for Strong . . . 5. Analysis of the Problem What We Do in This Talk • We want to assign, to each triple ( µ, V, m 3 ), a proba- Skew Normal . . . bility distribution ρ ( x, µ, V, m 3 ). Analysis of the Problem Scale Invariance • Let us list the natural properties of this assignment. Main Result • Moments are rarely known exactly, we usually know Discussion them with some accuracy. Proof • It is reasonable to require that if the moments change Home Page slightly, then ρ ( x, µ, V, m 3 ) should not change much. Title Page • In other words, it is reasonable to require that the func- ◭◭ ◮◮ tion ρ ( x, µ, V, m 3 ) is continuous. ◭ ◮ • Comment: in our proof, we will only use that ρ ( x ) is Page 6 of 18 measurable. Go Back • Strong consistency: if X 1 and X 2 are independent, Full Screen X 1 ∼ ρ ( x, µ 1 , V 1 , m 31 ), and X 2 ∼ ρ ( x, µ 2 , V 2 , m 32 ), then Close X 1 + X 2 ∼ ρ ( x, µ 1 + µ 2 , V 1 + V 2 , m 31 + m 32 ) . Quit
Formulation of the . . . Need for Strong . . . 6. Scale Invariance What We Do in This Talk • Numerical values of different quantities depend on the Skew Normal . . . choice of a measuring unit. Analysis of the Problem Scale Invariance • E.g.: income can be described in Baht or in dollars. Main Result • If we change the unit to λ times smaller one, then: Discussion • the actual incomes will not change, Proof • but the numerical values will change x → x ′ = λ · x . Home Page Title Page • If we perform the selection in the original units, then ◭◭ ◮◮ we get ρ ( x, µ, V, m 3 ). • If we simply re-scale x to x ′ = λ · x , then for x ′ , we get ◭ ◮ a new distribution ρ ′ ( x ′ ) = 1 � x ′ � Page 7 of 18 λ · ρ λ , µ, V, m 3 . Go Back Full Screen Close Quit
Formulation of the . . . Need for Strong . . . 7. Scale Invariance (cont-d) What We Do in This Talk • If we re-scale ρ ( x, µ, V, m 3 ), we get Skew Normal . . . Analysis of the Problem ρ ′ ( x ′ ) = 1 � x ′ � λ · ρ λ , µ, V, m 3 . Scale Invariance Main Result • We should get the exact same distribution if we make Discussion a selection after the re-scaling, i.e., for Proof Home Page µ ′ = λ · µ, V ′ = λ 2 · V, 3 = λ 3 · m 3 . m ′ Title Page • In the new units, we get ρ ( x ′ , λ · µ, λ 2 · V, λ 3 · m 3 ). ◭◭ ◮◮ • A natural requirement is that the resulting selection ◭ ◮ should be the same: Page 8 of 18 1 � x ′ � = ρ ( x ′ , λ · µ, λ 2 · V, λ 3 · m 3 ) . λ · ρ λ , µ, V, m 3 Go Back Full Screen Close Quit
Formulation of the . . . Need for Strong . . . 8. Definitions What We Do in This Talk • We say that a tuple ( µ, V, m 3 ) is possible if there exists Skew Normal . . . a distr. with mean µ , variance V , and moment m 3 . Analysis of the Problem Scale Invariance • By a 3-selection , we mean a measurable mapping Main Result ρ ( x, µ, V, m 3 ) defined for all possible tuples. Discussion • We say that a 3-selection is strongly consistent if X i ∼ Proof ρ ( x, µ i , V i , m 3 i ) for independent X i implies Home Page X 1 + X 2 ∼ ρ ( x, µ 1 + µ 2 , V 1 + V 2 , m 31 + m 32 ) . Title Page ◭◭ ◮◮ • We say that a 3-selection is scale-invariant if for every possible tuple ( µ, V, m 3 ), for every λ > 0 and x ′ : ◭ ◮ Page 9 of 18 1 � x ′ � = ρ ( x ′ , λ · µ, λ 2 · V, λ 3 · m 3 ) . λ · ρ λ , µ, V, m 3 Go Back Full Screen Close Quit
Formulation of the . . . Need for Strong . . . 9. Main Result What We Do in This Talk • Proposition. No 3-selection is strongly consistent and Skew Normal . . . scale-invariant. Analysis of the Problem Scale Invariance • A similar result can be formulated for the case when Main Result we also know higher order moments. Discussion • In this case, instead of the original moments, we can Proof consider cumulants κ n . Home Page • Cumulants are terms at i n · t n in the Taylor expansion Title Page n ! of 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. Page 10 of 18 • Cumulants are additive: if X = X 1 + X 2 and X 1 and Go Back X 2 are independent, then κ n ( X ) = κ n ( X 1 ) + κ n ( X 2 ). Full Screen Close Quit
Formulation of the . . . Need for Strong . . . 10. Discussion What We Do in This Talk • Since we cannot make a strongly consistent selection, Skew Normal . . . what should we do? Analysis of the Problem Scale Invariance • min and max are also natural operations in many ap- Main Result plications; for example, in econometrics: Discussion • if there are several ways to invest money with the Proof same level of risk, Home Page • then an investor selects the one that leads to the Title Page largest interest rate. ◭◭ ◮◮ • From this viewpoint, it is reasonable to consider min- ◭ ◮ ima and maxima of normal variables. Page 11 of 18 • In some cases, these minima and maxima are dis- tributed according to the skew normal distribution. Go Back Full Screen • This may be an additional argument in favor of using these distributions. Close Quit
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