Distribution of the product of two normal variables. A ...

Distribution of the product of two normal variables. A state of the Art

Am?ilcar Oliveira 2,3 Teresa Oliveira 2,3 Antonio Seijas-Mac?ias 1,3

1Department of Economics. Universidade da Corun~a (Spain) 2Department of Sciences and Technology. Universidade Aberta (Lisbon), Portugal.

3Center of Statistics and Applications, University of Lisbon (Portugal).

A.Oliveira - T.Oliveira - A.Mac?ias

September, 2018

Product Two Normal Variables

September, 2018 1 / 21

Outline

1 INTRODUCTION 2 FIRST APPROACHES 3 ROHATGI'S THEOREM 4 COMPUTATIONAL TECHNIQUES 5 RECENT ADVANCES

A.Oliveira - T.Oliveira - A.Mac?ias

Product Two Normal Variables

September, 2018 2 / 21

Introduction

INTRODUCTION

Normal distribution: the most common in Theory of Probability.

Applications to the real world: biology, psychology, physics, economics,... .

Density

function

(PDF): f (x) =

1 22

exp

-

(x -?)2 22

,where

? is the mean and

is

the

standard deviation (2 is the variance).

Distribution

function

(CDF):

F (x)

=

1 2

1 + erf

x -? 2

erf(t) =

2

t 0

e-y 2

dy

.

, where the error function is:

Normal Distribution N(0, 1)

Abraham de Moivre (1667-1754)

Carl F. Gauss (1777-1855)

A.Oliveira - T.Oliveira - A.Mac?ias

Product Two Normal Variables

September, 2018 3 / 21

Introduction

INTRODUCTION

Several distributions are derived from normal distribution: Chi-square or t distribution are the most famous.

Relation with other distributions (exponential, uniform, ...) is known.

Let X and Y be two normally distributed variables with means ?x and ?y and variances x2, y2, Sum X + Y is normally distributed with mean ?x + ?y and variance x2 + y2, when there is no correlation. When there exists correlation (), variance of the sum is x2 + y2 + 2x y . The product of two variables was not be able to characterize like the sum and remains like an open problem.

A.Oliveira - T.Oliveira - A.Mac?ias

Product Two Normal Variables

September, 2018 4 / 21

FIRST APPROACHES

First Historical Approach

Wishart and Bartlett (1932): The product of two independent normal variables is directly proportional to a second class Bessel function with a zero-order pure imaginary argument [WB32]

Craig (1936): Let be two normal variables X N(?x , x ) and Y N(?y , y ), and

correlation coefficient xy and the inverse of the variation coefficient: rx =

ry

=

?y y

.

Then

we

could

deduce

the

moment-generating

function.[Cra36]

?x x

and

exp

(rx2+ry2-2xy rx ry )t2+2rx ry t 2(1-(1+xy )t)(1-(1-xy )t))

Mxy (t) = ((1 - (1 + xy )t)(1 - (1 - xy )t))1/2

(1)

The product of two normal variables might be a non-normal distribution

Skewness is (-2 2, +2 2), maximum kurtosis value is 12

The function of density of the product is proportional to a Bessel function and its graph is asymptotical at zero.

A.Oliveira - T.Oliveira - A.Mac?ias

Product Two Normal Variables

September, 2018 5 / 21

FIRST APPROACHES

Figure: Examples of the product of two Normal Variables with = 0 Craig (red dashed) and MonteCarlo Simulation (blue)

A.Oliveira - T.Oliveira - A.Mac?ias

Product Two Normal Variables

September, 2018 6 / 21

FIRST APPROACHES

Advances in 50's in 20th Century

Aroian (1947): Type III Pearson function or Gram-Charlier Type A series ([Aro47]) .

Limitations: = 0, the Type III Pearson requires ?x = 0 or ?y = 0, Gram-Charlier approach has a very limited range of applicability.

Advantages: There is no discontinuity at zero.

Theorem ([ATC78], p. 167)

Let X and Y be two normally distributed variables with mean ?x , ?y , variances x2, y2 and

correlation coefficient .

Let be rx

=

?x x

and ry

=

?y y

.

Distribution function of Z =

xy x y

is

1 1

FZ (z) = 2 + 0 (z, rx , ry , , t)dt,

(2)

where (z, rx , ry , , t) =

1 t

1 G

exp

-

(H

+4rx

ry

)t 2 +(1-2 2G 2

)Ht

G +I 2

1/2

sin A

-

G -I 2

1/2

cos A

, with

A=

t

y

-

r1r2I -Ht2 G2

I = 1 + (1 - 2)t2.

, G 2 = (1 + (1 - 2)t2)2 + 42t2, H = r12 + r22 - 2r1r2 and

A.Oliveira - T.Oliveira - A.Mac?ias

Product Two Normal Variables

September, 2018 7 / 21

FIRST APPROACHES

Figure: Examples of the Product of two normal variables no correlated: Gram-Charlier (green - pointed), Pearson Type III (red - dashed) y MonteCarlo simulation (blue)

A.Oliveira - T.Oliveira - A.Mac?ias

Product Two Normal Variables

September, 2018 8 / 21

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