Copulas Are Needed Most Efficient . . . Why Clayton and . . . Why Symmetries Why Invariant Functions . . . Clayton & Gumbel Copulas: Not All Physical . . . Why Scalings Can Be . . . A Symmetry-Based Answer to the Main . . . Answer to the . . . Explanation Home Page Title Page Vladik Kreinovich 1 , Hung T. Nguyen 2 , 3 , and Songsak Sriboonchitta 3 ◭◭ ◮◮ ◭ ◮ 1 University of Texas at El Paso, USA 2 Department of Mathematics, New Mexico State University, USA Page 1 of 16 3 Department of Economics, Chiang Mai University, Thailand contact email vladik@utep.edu Go Back Full Screen Close Quit
Copulas Are Needed Most Efficient . . . 1. Copulas Are Needed Why Clayton and . . . • Traditionally, in statistics the dependence between ran- Why Symmetries dom variables η , ν , . . . , is described by correlation . Invariant Functions . . . Not All Physical . . . • A Gaussian joint distribution can be uniquely recon- Why Scalings Can Be . . . structed from correlation and marginals Answer to the Main . . . def def F η ( x ) = P ( η ≤ x ) , F ν ( y ) = P ( ν ≤ y ) . Answer to the . . . Home Page • In many practical situations, e.g., in economics, the distributions are often non-Gaussian . Title Page • For non-Gaussian variables ( η, ν ), in general, marginals ◭◭ ◮◮ and correlation are not sufficient . ◭ ◮ • To reconstruct a joint distribution, we need to know a Page 2 of 16 copula , i.e., by a function C ( u, v ) for which Go Back P ( η ≤ x & ν ≤ y ) = C ( F η ( x ) , F ν ( y )) . Full Screen • Usually, Archimedean copulas are used, i.e., f-s C ( u, v ) = Close ψ ( ψ − 1 ( u ) + ψ − 1 ( v )) for some ψ : [0 , ∞ ) → (0 , 1], ψ ↓ . Quit
Copulas Are Needed Most Efficient . . . 2. Most Efficient Archimedean Copulas Why Clayton and . . . • In econometric applications, the following three classes Why Symmetries of Archimedean copulas turned out to be most efficient: Invariant Functions . . . Not All Physical . . . – the Frank copulas Why Scalings Can Be . . . C ( u, v ) = − 1 � 1 − (1 − exp( − θ · u )) · (1 − exp( − θ · v )) � θ · ln ; Answer to the Main . . . 1 − exp( − θ ) Answer to the . . . – the Clayton copulas Home Page � − 1 /θ ; u − θ + v − θ − 1 � C ( u, v ) = Title Page – the Gumbel copulas ◭◭ ◮◮ �� ( − ln( u )) − θ + ( − ln( v )) − θ − 1 � − 1 /θ � C ( u, v ) = exp . ◭ ◮ Page 3 of 16 • Frank copulas are useful as the only Archimedean cop- ulas which satisfy the natural condition Go Back C ( u, v )+ C ( u, 1 − v )+ C (1 − u, v )+ C (1 − u, 1 − v ) = 1: Full Screen P ( U & V )+ P ( U & ¬ V )+ P ( ¬ U & V )+ P ( ¬ U & ¬ V ) = 1 . Close Quit
Copulas Are Needed Most Efficient . . . 3. Why Clayton and Gumbel Copulas? Why Clayton and . . . • Known fact: there are many different classes of copu- Why Symmetries las. Invariant Functions . . . Not All Physical . . . • Main question: why did the above two classes turned Why Scalings Can Be . . . out to be the most efficient in econometrics? Answer to the Main . . . • Auxiliary question: Answer to the . . . Home Page – What if these copulas are not sufficient? – Which classes should we use? Title Page ◭◭ ◮◮ • What we do: we show that natural symmetry -based ideas answer both questions: ◭ ◮ – symmetry ideas explain the efficiency of the above Page 4 of 16 classes of copulas, and Go Back – symmetry ideas can lead us, if necessary, to more Full Screen general classes of copulas. Close Quit
Copulas Are Needed Most Efficient . . . 4. Why Symmetries Why Clayton and . . . • Symmetries are ubiquitous in modern physics, because Why Symmetries most of our knowledge is based on symmetries. Invariant Functions . . . Not All Physical . . . • Example: we drop a rock, and it always falls down with Why Scalings Can Be . . . an acceleration of 9.81 m/sec 2 . Answer to the Main . . . • Meaning: no matter how we shift or rotate, the funda- Answer to the . . . mental laws of physics do not change. Home Page • This idea is actively used in modern physics. Title Page • Starting with quarks, physical theories are often for- ◭◭ ◮◮ mulated in terms of symmetries (not diff. equations). ◭ ◮ • Fundamental physical equations – Maxwell’s, Schr¨ odinder’s, Page 5 of 16 Einstein’s – can be uniquely derived from symmetries. Go Back • In view of efficiency of symmetries in physics, it is rea- Full Screen sonable to use them in other disciplines as well. Close Quit
Copulas Are Needed Most Efficient . . . 5. Basic Symmetries Why Clayton and . . . • Basic symmetries that come from the fact that the Why Symmetries numerical value of a physical quantity depends: Invariant Functions . . . Not All Physical . . . – on the choice of the measuring unit ( scaling ) and Why Scalings Can Be . . . – on the choice of a starting point ( shift ). Answer to the Main . . . • Scaling: when we replace a unit by the one which is λ Answer to the . . . times smaller, we get new values x ′ = λ · x . Home Page • Example: 2 m is the same as 2 · 100 = 200 cm. Title Page ◭◭ ◮◮ • Shift: when we change a starting point to a one which is s units earlier, we get new values x ′ = x + s . ◭ ◮ • Example: 25 C is the same as 25 + 273.16 = 298.16 K. Page 6 of 16 • Observation: many physical formulas are invariant rel- Go Back ative to these symmetries. Full Screen Close Quit
Copulas Are Needed Most Efficient . . . 6. Example of Invariance Why Clayton and . . . • Example: power law y = x a has the following invari- Why Symmetries ance property: Invariant Functions . . . Not All Physical . . . – if we change a unit in which we measure x , Why Scalings Can Be . . . – then get the exact same formula – provided that we Answer to the Main . . . also appropriately changing a measuring unit for y : Answer to the . . . f ( λ · x ) = µ ( λ ) · f ( x ) . Home Page • Interesting fact: power law is the only scale-invariant Title Page dependence: ◭◭ ◮◮ – differentiating both sides by λ , we get ◭ ◮ x · f ′ ( λ · x ) = µ ′ ( λ ) · f ( x ); Page 7 of 16 – for λ = 1, we get x · d dx = µ 0 · f , hence d f f = µ 0 · dx f x ; Go Back Full Screen – integrating, we get ln( f ) = µ 0 · ln( x )+ C , i.e., power law f ( x ) = e C · x µ 0 . Close Quit
Copulas Are Needed Most Efficient . . . 7. Invariant Functions Corr. to Basic Symmetries Why Clayton and . . . • A diff. f-n f ( x ) is called scale-to-scale invariant if for Why Symmetries every λ , there exists a µ for which f ( λ · x ) = µ · f ( x ). Invariant Functions . . . Not All Physical . . . • Thm. A function f ( x ) is scale-to-scale invariant ⇔ it has the form f ( x ) = A · x a for some A and a . Why Scalings Can Be . . . Answer to the Main . . . • A diff. f-n f ( x ) is called scale-to-shift invariant if for Answer to the . . . every λ , there exists an s for which f ( λ · x ) = f ( x )+ s . Home Page • Thm. A function f ( x ) is scale-to-shift invariant ⇔ it Title Page has the form f ( x ) = A · ln( x ) + b for some A and b . ◭◭ ◮◮ • Thm. A function f ( x ) is shift-to-scale invariant ⇔ it ◭ ◮ has the form f ( x ) = A · exp( k · x ) for some A and k . Page 8 of 16 • Thm. A function f ( x ) is shift-to-shift invariant ⇔ it has the form f ( x ) = A · x + c for some A and c . Go Back Full Screen • A function is called invariant ⇔ it is scale-to-scale, scale-to-shift, shift-to-scale, or shift-to-shift invariant. Close Quit
Copulas Are Needed Most Efficient . . . 8. Not All Physical Dependencies Are Invariant Why Clayton and . . . • Physical dependencies of z on x are often indirect: Why Symmetries Invariant Functions . . . – the quantity x influences some intermediate quan- Not All Physical . . . tity y , and Why Scalings Can Be . . . – the quantity y influences z . Answer to the Main . . . • The resulting mapping x → z is a composition Answer to the . . . z = g ( f ( x )) of mappings y = f ( x ) and z = g ( y ). Home Page • The basic mappings f ( x ) and g ( y ) are often invariant. Title Page • However, the composition of inv. f-ns may not be inv.: ◭◭ ◮◮ e.g., f ( x ) = x a , g ( y ) = exp( y ), z = exp( x a ) is not inv. ◭ ◮ • It is thus reasonable to look for functions which are Page 9 of 16 compositions of two invariant functions. Go Back • If necessary, we can look for compositions of three, etc. Full Screen • Let us apply this approach to our problem of finding appropriate copulas. Close Quit
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