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University of New Brunswick CANADA 1 DENSITY OF THE RATIO OF TWO NORMAL RANDOM VARIABLES by T. Pham-Gia* and N. Turkkan Universite de Moncton and E. Marchand, University of New Brunswick CANADA Abstract: We give the exact closed form expression of the density of X12/ X , where X1 and X 2 are normal random variables, in terms of Hermite or Confluent Hypergeometric functions. All cases will be considered: standardized and non standardized variables, independent or correlated variables. Several new applications are presented , and relationships with mixtures of normal distributions are given. Keywords and Phrases: Normal, Bivariate normal, Ratio, Hermite function, Kummer confluent hypergeometric function, Integral representation, Finite sampling AMS Classification: 62E15, 62N05. 1. INTRODUCTION The density of of WXY= / , where X and Y are normal random variables, has attracted the interest of several researchers, as early as 1930, since it was encountered in some basic problems in Statistics. Although the cases where both X and Y are standard normal, or standard bivariate normal, are fairly simple, general cases are much more complex. Geary (1930) was the first to investigate this question and Fieller (1932 ) presented another approach to evaluate this density. In the sixties, two important papers ( Marsaglia ( 1965) and Hinkley (1969)) both addressed this concern, with different viewpoints, however. Recently, demand for this expression resurface in a number of important applications, and an active exchange on the Internet Mathforum ( Startz (1997), Ward (1997) and Marsaglia (2001)), has rekindled the need for a convenient expression of this distribution . Springer ( 1984, p. 139), using Mellin Transform methods, has presented a method to obtain this density, in terms of an infinite series, but the inversion of the Mellin transform in the complex plane naturally requires some advanced computation, that is not always easy to handle. Although this method does guarantee the results, much numerical analysis has to be performed, and the result is not a closed form expression. In this article we will use special mathematical functions, the Hermite function, which is a generalization of the Hermite polynomials, and Kummer’s confluent hypergeometric function, to give a convenient closed form expression for the density of X /Y . This expression can also be obtained using a completely different approach, based on conditional expectation. Some particular cases have, however, simpler expressions, expressed as common functions. In section 2, we will first recall the Hermite function Hzν ( ) , and give its basic properties, and its integral representation when ν is a negative real number. In sections 3 and 4, the density of W is given for different, and exhausting cases. Section 5 discusses the interesting shapes that the density of W can take, while section 5 presents some numerical applications, with __________________________________________________________________________ *) Research partially supported by NSERC grant A9249 ( Canada). 2 related discussions. Finally, section 6 gives another look at the problem, from a cumulative distribution viewpoint, and puts it in its wider context of continuous and discrete mixture of normal distributions. 2. THE HERMITE FUNCTION and THE POWER-QUADRATIC EXPONENTIAL FAMILY Although Hermite polynomials are well utilized in Statistics, for instance in the Gram- Charlier expansion of a density, the Hermite function has only timid encounters with distribution theory ( Pham-Gia (1992)). The use of special functions, already very widespread in Mathematical Physics, is gaining ground in Statistics, where they provide powerful tools to complement the classical common functions. Dickey ( 1983) has championed such a use, already a few decades ago. The Hermite function with parameter ν , Hzν ( ) , can be derived from the parabolic ν /2 2 cylinder function Dν by the relation Hzνν( )= 2 exp( z / 2) Dz ( 2), where Dν itself is related to the ν − th derivative of the function exp(−z 2 / 2) by the relation: ν d 2 Dz( )=− ( 1)ν exp( z2/2 / 4) ( e−z ) , ν >1. ν dzν For any value of ν , Hν can be directly defined by the infinite series ( Lebedev ( 1972, p. 289)): ∞ n 1(1)(()/2)−Γm −ν m Hzν ()= ∑ (2) z, (1) 2(Γ−ν )m=0 m ! with the Gamma function for negative values obtained by repeatedly applying the relation: Γ−(νν ) =Γ− ( / 2) Γ ((1 − ν ) / 2) /[2ν +1 πν ], > 0 and Γ(1/2)(3/2)−Γ =−π . When ν is a positive integer, we have the corresponding Hermite polynomial, and with ν < 0 , Hzν ( ) has an integral representation of the form: ∞ 1 2 Hz()= e−−ttz2(1) t −+ν dt, (2) ν ∫ Γ−()ν 0 which shows that Hxν ( ) is a positive function on the whole real line. The Hermite function Hz−2 () is of particular interest in this article. We have: ∞ 2 Hz()= tedt−−ttz2 , (3) −2 ∫ 0 with H −2 (0)= π / 2, and from the general relation ν +1 ∞ 2 2 Hz()=+ zet−−+t (1)2νν ( t z 2(1)/2 ) − dt, ν < 0 ( Lebedev (1972, p.297) ν ∫ Γ−(/2)ν 0 2 zet∞ −t we have: Hz()= dt. −2 ∫ 223/2 2(0 tz+ ) An important identity relates the Hermite function to Kummer’s classical confluent hypergeometric function of first kind, 11F , defined by: 3 ∞ (,)α kzk 11Fz(,;)αγ = ∑ . , γ ≠−−0, 1, 2,.., , with the ascending factorial, or Pochhamer k=0 (,)γ kk ! coefficients, (α ,kkk )=+αα ( 1)...( α +−=Γ+Γ 1) ( α ) / ( α ), and (α ,0)= 1. It is: 12z Hz( )=−−− 2ν πν [ . F ( /2;1/2; z22 ) . F ((1 ν )/2;3/2; z ) (4) ν 1−νν11 11 ΓΓ−() () 22 2.2 As presented in Pham-Gia (1994), the power-quadratic exponential family of distributions has the property that its hazard rates can be ordered under certain conditions. Further properties on the ratios of two members of this family are presented in Pham-Gia and Turkkan ( 2004). DEFINITION: The family PQE(γ ;δε , ) of distributions consists of positive continuous random variables with densities of the form : f (;;,)tCtttγδε=−+ (;,) γδεγ exp[( δ ε2 )], where the domain of the parameters is as follows: (;γ δε ,)∈>−−∞<<∞> { γ 1; δ , ε 0}U {1;0,0}γ >−δε > = . We write XPQE~ (γ ;δε , ) , and have the following special cases: 1. Case γ ;,δε≠ 0 : ft(;γ ;δε , ) = Ct(γδε ; , )γ exp(−+ [ δ tt ε2 ]) ,t ≥ 0 . We have the general power-quadratic density, where the normalizing constant C(γ ;δε , ) has a complex expression ν − ε 2 of the form : C(;γδε ,)= , δ Γ+(1).(γ H ) ν 2 ε δ with ν =−(γ + 1) < 0 , and where H ()is the value of the Hermite function with ν 2 ε δ parameter ν at (). 2 ε 2. Case γ = 0 : a) δ ,ε ≠ 0 : We have the normal density, N(,µ σ 2 ), truncated from below at 0, 2 21− 2 denoted NTr (,µ σ ), with εσ= (2 ) and δ =−µσ/ . Hence, f (tC ;0;δε , )=−+ (0; δε , )exp[ ( δ tt ε2 )], t ≥ 0 , δδ2 with C(0;δε , )=− exp( ) /[ π / ε .(1 −Φ ( )], (5) 4ε 2ε where Φ is the cumulative distribution function of the standard normal. We also have δ CH(0;δε , )= ε / ( ) by using the relation Hz()=−Φπ [1 ( z 2)]. (6) −1 2 ε −1 2 b) If δ = 0 , ε > 0 , we have the half normal distribution, denoted , NH (0,σ ) , defined for t ≥ 0 , with variance σ 21= (2ε )− , and f (tC ;0;0,εεε )= (0;0, ) exp(−≥ tt2 ), 0 ε with C(0;0,ε )= 2 . π 4 c) If ε = 0 but δ > 0 , then the distribution is exponential, of the form f (tt ;0,δ ,0)= δδ exp(− ) ,0t ≥ . 3. Case δ = 0 : For γ >−1,γε ≠ 0, > 0 , we have the Rayleigh distribution , denoted Ray(,)γ ε , of the form f (tCtt ;γ ,0,εγε )=− ( ;0, )γ exp( ε2 ) , t ≥ 0 with 1−ν ε −ν /2Γ() C(;0,)γε= 2 , where ν = −+(γ 1) , if we take into consideration the fact that 2()(1/2)ν Γ−ν Γ 2(1/2)ν Γ H (0) = . In particular, when γ = 2 , we have the Maxwell distribution of the well- ν 1−ν Γ() 2 3 4ε 2 known form : f (tttt ;γε ; )=−<22 exp( ε ),0 . π 4. Case ε = 0 : For γ >−1,γδ ≠ 0, > 0 , we have the common gamma density in two δ γ +1 parameters, of the form. f (tCtt ;γ ;δγδδ ,0)= ( ; ,0)γ exp(− ) ,0t ≥ , with C(;γδ ,0)= , Γ+(1)γ denotedXGa ~ (γ + 1,1/δ ) , if the density of XGa~ (λ ,δ ) is given its standard expression ft( ;λδ , )=− tλλ−1 exp( t / δ ) /[ δ Γ≥ ( λ )], t 0 , λ,δ > 0. 3. RATIO OF TWO INDEPENDENT NORMAL VARIABLES 3.1 Works toward the derivation of the density of W =X/Y , where X and Y are normal variables, dependent or independent, have generally followed the approach of finding Ua1 + the cumulative distribution of the ratio W* = , where Ui , i =1,2, are standard Ub2 + µ µ normal variables with the same correlation coefficient, and a = X and b = Y , with σ X σ Y σ PW()(*)≤= t PW ≤Y t ( Geary (1930), Fieller (1932), Hinkley(1969) ). But Marsaglia σ X (1965) required further that Ui are independent, and hence, a and b now have specific values containing the correlation coefficient. More specifically, we have: 1 µ ρµ µ a =−()XY and b = Y . (7) 2 1− ρ σσXY σ Y Section 6 gives the relation between W and W* in this case. Naturally, the tabulated values of the standard bivariate normal distribution, 12∞∞ xxyy22−+γ Lhk(,;ρ )=− exp dxdy are used in the expression of the 22∫∫ 2(1πρ−− )hk 2(1 ρ ) cumulative distribution function, by the first three authors, while Marsaglia (1965) also qx h h expressed it in terms of Nicholson (1943) V function Vhq(, )= ∫∫ϕϕ ()() x ydxdy, whereϕ 00 is the standard normal density.
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