Estimation of quantiles of the ratio of two correlatedNormals (*)

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Estimation of quantiles of the ratio of two correlated Normals

^(*)

Stima dei quantili del rapporto fra due v.c. Normali correlate Carlotta Galeone^1,2, Angiola Pollastri³

1 Istituto di Statistica Medica e Biometria “G.A. Maccacaro”,Università di Milano

2 Dipartimento di Epidemiologia, Istituto “Mario Negri”, Milano

3 Dipartimento di Metodi Quantitativi per le Scienze Economico-Aziendali, Università di Milano-Bicocca. e-mail: angiola.pollastri@unimib.it

Keywords: ratio of two correlated normal r.v.s, quantiles estimation

1. Distribution of the ratio of two correlated Normals

Let suppose we have a Bivariate Correlated Normal ^{( ,}^{X X}₁ ₂⁾~B.C.N.

2

1 1 1 1 2

2 2 1 2 22

~N ;

X X

   

   

  

   

  

   

 

     

If ( ,Z Z1 2)is a Standardized B.C.N., we can indicate (see Kotz, 2000)

1 2

( , ; ) ( , ; )

P Z h Z k  L h k  . Let us consider the r.v.

1/ 2

W  X X with ^X₂ ^⁰ useful in various scientific fields.

Aroian (1986) gave the following convenient expression for the Distribution Function (DF) of the ratio of two correlated normal r.v.s.

  

² ^'

 

² ^'



Pr W w F w_W( )L a bt(  _w) / 1t_w,b;_w L ( a bt_w) / 1 t_w x b, ;_w where:

1 2 2

2 2

( ) / 1

a    

 

   , ²

2

b 

 , ¹ ²

2

( ) / 1

tw  w  

    , _w't_w(1t_w²)^¹² The moments of W do not exist. The median and the quantiles exist but cannot be obtained analytically. Descriptive aspects of the distribution of the r.v. W were studied in Galeone (2007).

2. Estimation of quantiles of W

Let’s suppose we have drawn a sample of n elements and we obtained the observations

1 2

( ,x x_i _i)(i=1,…,n). We can compute the estimates of the means  and ₁  indicated₂ respectively by x₁ and x . The estimates of ₂  , 1  and  are respectively given by₂

(*) The present paper is financially supported by MURST

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2 2

1 1 1

1

( ) /( 1)

n i i

s x x n





  ²² ² ¹ ²

1

( ) /( 1)

n i i

s x x n





 

2 2

1 1 2 2 1 1 2 2

1 1 1

( )( ) / ( ) ( )

n n n

i i i i

i i i

r x x x x x x x x

  





 









We can estimate F w_W( ) substituting it for the estimates of  ,₁  , ₂  , ₁  and  . The₂

estimate of the DF, indicated by  ( )FW w , can be computed as a function of w using Fortran+IMSL library or MATLAB or other libraries, especially those containing routines regarding the DF of a BCN or other functions which can give the DF.

If we are interested in finding the estimate of the quantile w_p F^¹( )p , we can follow the procedure here reported.

Firstly, we can compute some points( ,w Fi ^ W( ))wi . Then, identify the two values of

 ( )W i

F w closer to p, such that

 ( )W j

F w  p and F ( )W wl  p,

we evaluate the points ( ,w Fi ^W( ))wi with wj wiwl until F ( )W wh  p 

where  0 is the required degree of precision. The value w_h is the estimate of wp. The estimates of the quantiles of the r.v. W are also useful for making inferences about the ratio R ₁/ ₂. F.i., quantile estimates in Galeone (2007) are used in order to construct the interval estimates for R.

References

Aroian L.A. (1986) The distribution of the quotient of two correlated random variables.

Proceedings of the Am. Stat. Ass. Business and Economic Section.

Galeone C. (2007) On the ratio of two Normal r.v., jointly distributed as a Bivariate Normal, PhD Thesis, Università di Milano-Bicocca.

Kotz S., Balakrishnan N., Johnson N.L. (2000) Continuous Multivariate Distributions, Wiley, New York.