عنوان البحث(Papers / Research Title)
An Asymptotic Expansion for the Non-Central F-Distribution
الناشر \ المحرر \ الكاتب (Author / Editor / Publisher)
جنان حمزة فرهود الخناني
Citation Information
جنان,حمزة,فرهود,الخناني ,An Asymptotic Expansion for the Non-Central F-Distribution , Time 5/23/2011 8:34:50 AM : كلية التربية للعلوم الصرفة
وصف الابستركت (Abstract)
The non-central F-distribution is defined by (Henry, 1959; Walkk, 2001)
الوصف الكامل (Full Abstract)
An Asymptotic Expansion for the Non-Central F-Distribution
By Jinan Hamzah Farhood Department of MathematicsCollege of Education2006
Abstract
A new asymptotic expansion is derived for the non-central F-distribution F(F?\m1,m2,?), which is suitable for large , small , r?N and 0<q<1, where is the incomplete Gamma function ratio and . This form has some advantages over previous asymptotic expansions in this region of the parameter space in which Hn depends on all three parameters , and q.
The advantage of this new expansion is that an algorithm based on it can be more easily tuned for particular accuracy requirements and for particular parameter ranges.
1. Introduction
The non-central F-distribution is defined by (Henry, 1959; Walkk, 2001) as the form . If X1 and X2 are independent random variables and X1 is a non central chi- square distribution with m1 degrees of freedom and non centrality parameter and X2 is a central chi-square distribution with degrees of freedom then the variable is said to have a non-central F-Distribution with degrees of freedom (positive integers) and non-central parameter ? ? 0 and we write .
The distribution function is given by , (1)where , is incomplete Beta function. which is the cumulative distribution function (c.d.f.) of non-central F-Distribution
The asymptotic expansion was studied by other researchers who worked in our field which as the following.
The asymptotic expansion for the ratio of two Gamma functions derived by (Fields, 1966; Luke, 1969; Frenzen, 1987).A special case of the asymptotic expansion for a ratio of products of gamma functions derived by (Biihring, 2000). He generalized a formula which was stated by (Dingle, 1973), first proved by (Paris, 1992) and recently reconsidered by (Oliver, 1995).
The special functions and their approximations had been studied by (Luke, 1969). The incomplete laplace integrals: Uniform asymptotic expansion with application to the incomplete beta function studied by (Temme, 1987). Asymptotic expansions of the Coefficients in asymptotic series solutions of linear differential equations, Methods and applications of analysis derived by (Olver, 1994).
The Uniform asymptotic expansions of integrals studied by (Temme, 1995) by using examples of stieltjies work on asymptotic of special functions. A Uniform asymptotic expansion for the Jacobi polynomials with explicit remainder derived by (Wong & Zhang, 1996). The valid asymptotic expansions for the maximum likelihood estimator of the parameter of a stationary, Gaussian, strongly dependent process studied by (Lieberman, Roussean & Zucker, 2003). A uniform asymptotic expansions for incomplete Rieman zeta functions derived by (Dunster, 2004). The uniform asymptotic expansions for hypogeometric functions with large parameters studied by (Daalhuis, 2005).
2. Derivation of an Asymptotic Expansion for the Non Central F-Distribution.We derive an asymptotic expansion of , through two stages:
First Stage:
We shall derive the asymptotic expansion of where , >0, ? and r?N, we start from the Beta function , which has the formula. . (2)Then by using the substitution t=e-u and dt=-e-u du we obtain . (3)And using the fact that we have , (4)where . Now expand in powers of u2 as u??
, (5)Let . The last quality follows by (Didonate & Morris,1992). Where Cn are the expansion coefficients of , and which can be expressed of the generalized Bernoulli polynomials (Luke,1969), .By substitution equation (5) in (4) we get .And using Watson’s Lemma we obtain the asymptotic expansion . (6)
Second Stage:
In this stage we derive the asymptotic expansion of , , (7)when >0, , 0< q <1 and r?N and then transform the expression for it as the same in equation (2) and changing integration terms, to obtain. , (8)where as before , by using (5) we have . (9)Let w=ku, then and , so from (9) we have
, (10)where Q(.,.) is incomplete gamma function ratio.We can proceed by using the recurrence relations for Q(.,.) to express in terms of . This gives . (11)where we have use the formula (6) to cancel out the factors multiplying Q, the other term is a double summation over n and 2n residual terms obtained by expressing in terms of .To obtain the asymptotic expansion we require to reordering this sum. First we write (8) in the form . (12)Integrate the first integral by parts twice as follows : . (13)In the integral in (13) we now subtract the second term in expansion of and add a corresponding integral so that the integral in (13) becomes .(14)
The first of these integrals is then integrated by parts twice producing two further integrated terms evaluated at and an integral of a fourth derivative. In this integral, a further term from the expansion of , is subtracted from the differentiated part and a corresponding integral added on separately. This procedure is continued indefinitely. The separate integrals starting from the ones on the right of (12) and (14) add together to give as in (11) so that , (15)where = ,where in the summation is to be interpreted as largest integer ? as in integer division. The quantities satisfy the simple recurrence formula ,
. (16)
We can express directly in terms of and q, for example, . However, for q close to 1, evaluation Hn in this way can lead to large rounding errors on subtraction, and so is better evaluated from its power series expansion in u. Now, when we substitute the formula (15) in equation (1), we get
.By using the identity , we get
. (17)which is an asymptotic expansion for the non-central F-distribution.
References
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Dunster T M (2004) Uniform asymptotic expansions for incomplete riemam zeta functions. Department of Mathematics and Statistics, state University,U.S. A. http://www.rohan.sdsu.edu/~dunster publications
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