Quasi logistic distributions and Gaussian scale mixing G erard - PowerPoint PPT Presentation

Quasi logistic distributions and Gaussian scale mixing G´ erard Letac, Universit´ e Paul Sabatier, Toulouse Luminy , June 2020. G´ erard Letac, Universit´ e Paul Sabatier, Toulouse Quasi logistic distributions and Gaussian scale mixing

Program ◮ The Gaussian scale mixing, the logistic and Kolmogorov-Smirnov distributions, and the anonymous physicist question ◮ The quasi logistic distributions ◮ The quasi Kolmogorov Smirnov distributions ◮ More about Gaussian scale mixing in several dimensions ◮ The L 2 approximation G´ erard Letac, Universit´ e Paul Sabatier, Toulouse Quasi logistic distributions and Gaussian scale mixing

Foreword This is an elementary lecture on the unfashoniable distribution theory: but you can pick exercises from it for your undergraduate classes.... G´ erard Letac, Universit´ e Paul Sabatier, Toulouse Quasi logistic distributions and Gaussian scale mixing

Gaussian scale mixing If Z ∼ N (0 , I n ) is independent of the positive definite random matrix V then √ the distribution of X = V Z is called a Gaussian scaled mixing distribution. Example: n = 1 and V ∼ 1 2 δ 1 + 1 2 δ 4 G´ erard Letac, Universit´ e Paul Sabatier, Toulouse Quasi logistic distributions and Gaussian scale mixing

But the law of V can be continuous Example n = 1 V ∼ e − v / 2 1 (0 , ∞ ) ( v ) dv / 2 is an exponential distribution of mean 2 √ independent of Z ∼ N (0 , 1) implies that X = V Z has the bilateral density e −| x | / 2 . Indeed √ 1 V Z | V )) = E ( e − t 2 V 2 / 2 ) = E ( e itX ) = E ( E ( e it 1 + t 2 . G´ erard Letac, Universit´ e Paul Sabatier, Toulouse Quasi logistic distributions and Gaussian scale mixing

After all, this is just a multiplicative deconvolution ? For n = 1 2 log | X | = log Z 2 + log V which means that if we wonder if the law of X is a Gaussian scale mixing we have just to check whether or not its Mellin transform 2 ) of Z 2 is the M X 2 ( s ) divided by the Mellin transform 2 s Γ(1 + s Mellin transform M V ( s ) of some random variable V ? G´ erard Letac, Universit´ e Paul Sabatier, Toulouse Quasi logistic distributions and Gaussian scale mixing

The beautiful example of Edwards-Mallows- Monahan- Stefanski These statisticians have observed in 1973 and 1990 that if 1 Pr( X < x ) = 1 + e − x has the logistic distribution then this law is a Gaussian mixing, √ with Y = V having the Kolmogorov-Smirnov distribution ∞ ( − 1) n − 1 e − 2 n 2 y 2 � Pr( Y < y ) = 2 n =1 (Think of this distribution function of V : the fact that it is increasing is not obvious!) G´ erard Letac, Universit´ e Paul Sabatier, Toulouse Quasi logistic distributions and Gaussian scale mixing

The anonymous physicist On a mathematical site he has asked for the probability measure µ a , b ( dv ) such that for 0 < a < b e − sv µ a , b dv = b sinh a √ s � ∞ a sinh b √ s 0 Since Kolmolmogorov Smirnov is more or less µ 0 , b what about a little generalization on Edwards- Mallows- Monahan- Stefanski? And a little generalization of the logistic distribution? G´ erard Letac, Universit´ e Paul Sabatier, Toulouse Quasi logistic distributions and Gaussian scale mixing

The quasi logistic distributions They are densities proportional to e x 1 2(cosh x + θ ) = e 2 x + 2 θ e x + 1 with θ > − 1 . The case θ = 1 is the logistic one. The shape of the curve ressembles to the normal curve, but the asymptotic is rather e −| x | rather than e − x 2 / 2 . For our purposes of the day, we concentrate to the case − 1 < θ = cos a < 1 with 0 < a < π. The next theorem lists their properties (however the case θ > 1 remains interesting in itself). G´ erard Letac, Universit´ e Paul Sabatier, Toulouse Quasi logistic distributions and Gaussian scale mixing

Quasi logistic laws of parameter θ = cos a : properties Theorem 1: Let 0 < a < π and s ∈ ( − 1 , 1) . 1. We have � ∞ e sx dx sin π s × sin as π 2(cosh x + cos a ) = sin a . (1) −∞ 2. In particular if X ∼ sin a dx a 2(cosh x + cos a ) has the quasi logistic distribution of parameter θ = cos a then for real t we have sin π s × sin as π s sinh π t × sinh at π t E ( e sX ) = E ( e itx ) = , as at G´ erard Letac, Universit´ e Paul Sabatier, Toulouse Quasi logistic distributions and Gaussian scale mixing

Other properties of the QL laws 1 1. The variance of X ∼ − X and the fourth moment are E ( X 2 ) = 1 E ( X 4 ) = 1 3( π 2 − a 2 ) , 15( π 2 − a 2 )(7 π 2 − 3 a 2 ) . 2. The distribution function of X is a Arc cotan e x + cos a F ( x ) = Pr( X < x ) = 1 − 1 sin a and the quantile function Q ( p ) defined for p ∈ (0 , 1) by F ( Q ( p )) = p is equal to sin pa Q ( p ) = log sin(1 − p ) a . G´ erard Letac, Universit´ e Paul Sabatier, Toulouse Quasi logistic distributions and Gaussian scale mixing

Other properties of the QL laws 2 X is infinitely divisible. In particular its L´ evy measure is e −| x | / a − e −| x | /π (1 − e −| x | /π )(1 − e −| x | / a ) × dx ν ( dx ) = | x | � with R min(1 , | x | ) ν ( dx ) = ∞ . G´ erard Letac, Universit´ e Paul Sabatier, Toulouse Quasi logistic distributions and Gaussian scale mixing

Other properties of the QL laws 3 1. If ( ǫ n ) n ≥ 1 are Bernoulli iid rv such that Pr( ǫ n ) = a 2 /π 2 and if ( Y n ) n ≥ 1 are iid rv with bilateral exponential density e −| y | / 2 then ∞ Y n � X ∼ ǫ n n . n =1 2. The Mellin transform of | X | is for s > 0 ∞ ( − 1) n − 1 sin na × 1 E ( | X | s ) = 2Γ(1 + s ) � n s na n =1 G´ erard Letac, Universit´ e Paul Sabatier, Toulouse Quasi logistic distributions and Gaussian scale mixing

Comments about the Laplace transform � ∞ � ∞ e sx dx z s dz sin π s × sin as π s 2(cosh x + cos a ) = z 2 + 2 cos az + 1 = as −∞ 0 is not so easy, the simplest proof uses the residues calculus along the contour iR γ R γ ǫ l 1 R l 2 G´ erard Letac, Universit´ e Paul Sabatier, Toulouse Quasi logistic distributions and Gaussian scale mixing

Comments about the factorization 1 ∞ ∞ 1 − z 2 1 + z 2 sin π z � � sinh π z � � � � = , = . (2) n 2 n 2 π z π z n =1 n =1 For 0 < a < b the second formula of (2) one leads to: ∞ � 1 + a 2 t 2 � b sinh π at � n 2 a sinh π bt = (3) 1 + b 2 t 2 n 2 n =1 G´ erard Letac, Universit´ e Paul Sabatier, Toulouse Quasi logistic distributions and Gaussian scale mixing

Comments about the factorization 2 Let us consider the simple identity for 0 < a < b : 1 + a 2 t 2 1 + b 2 t 2 = a 2 b 2 + (1 − a 2 1 b 2 ) (4) 1 + b 2 t 2 If Y ∼ e −| y | dy / 2 is a bilateral exponential random variable, we have E ( e itY ) = 1 / (1 + t 2 ). If ǫ is a Bernoulli random variable such that Pr( ǫ = 0) = 1 − Pr( ǫ = 1) = a 2 / b 2 and if ǫ and Y are independent, then E ( e itb ǫ Y ) = (1 + a 2 t 2 ) / (1 + b 2 t 2 ) . From this observation and from (3) we get that if ( ǫ n ) n ≥ 1 and ( Y n ) n ≥ 1 are independent with ǫ n ∼ ǫ and Y n ∼ Y we have that ∞ Y n � X = b ǫ n n n =1 satisfies E ( e itX ) = b sinh at a sinh bt . G´ erard Letac, Universit´ e Paul Sabatier, Toulouse Quasi logistic distributions and Gaussian scale mixing

Comments about the Mellin transform If we assume that s > 0 we have � ∞ x s sin a E ( | X | s ) = cosh x + cos adx a 0 � ∞ x s e − x 2 sin a = 1 + 2 e − x cos a + e − 2 x dx a 0 � ∞ 1 � 1 1 � x s = 1 + e − x − ia − dx 1 + e − x + ia ia 0 � ∞ ∞ 1 ( − 1) n − 1 ( e ina − e − ina ) � e − nx x s dx = ia 0 n =1 . ∞ ( − 1) n − 1 sin na × 1 E ( | X | s ) = 2Γ(1 + s ) � n s na n =1 G´ erard Letac, Universit´ e Paul Sabatier, Toulouse Quasi logistic distributions and Gaussian scale mixing

Now the quasi Kolmogorov Smirnov laws Theorem 2. Given 0 < a < b , we denote q = a / b . There exists a probability µ a , b ( dv ) on (0 , ∞ ) such that e − sv µ a , b ( dv ) = b sinh( a √ s ) � ∞ a sinh( b √ s ) . (5) 0 More specifically If ( ǫ n ) ∞ n =1 and ( W n ) ∞ n =1 are Bernoulli and exponential independent random variables: Pr( ǫ n = 0) = 1 − Pr( ǫ n = 1) = q 2 , W n ∼ e − w 1 (0 , ∞ ) ( w ) dw n 2 . Then π 2 we denote V ∼ � ∞ n =1 ǫ n W n b 2 V ∼ µ a , b and V ∼ µ π q ,π . G´ erard Letac, Universit´ e Paul Sabatier, Toulouse Quasi logistic distributions and Gaussian scale mixing

The density of the quasi Kolmogorov Smirnov laws The density of V is ∞ g ( v ) = 2 ( − 1) n − 1 sin( n π q ) × ne − n 2 v � π q n =1 In particular ∞ √ V ) s ) = 2Γ(1 + s ( − 1) n − 1 sin( n π q ) × 1 � E (( 2) (6) n s n π q n =1 G´ erard Letac, Universit´ e Paul Sabatier, Toulouse Quasi logistic distributions and Gaussian scale mixing