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عنوان البحث(Papers / Research Title)


The Non-Central t-Distribution and How Finding Asymptotic Expansion for this Distribution


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جنان حمزة فرهود الخناني

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جنان,حمزة,فرهود,الخناني ,The Non-Central t-Distribution and How Finding Asymptotic Expansion for this Distribution , Time 5/23/2011 8:39:42 AM : كلية التربية للعلوم الصرفة

وصف الابستركت (Abstract)


If X and Y are independent random variables and X is a normal distribution with mean ? and variance 1 and Y is a central chi-square with v2 degrees of freedom

الوصف الكامل (Full Abstract)


The Non-Central t-Distribution and How Finding Asymptotic Expansion for this Distribution
 
 By
Jinan Hamzah Farhood Department of MathematicsCollege of Education
 
Abstract
We introduce in this paper the incomplete beta function , the non-central t-distribution and derive an asymptotic expansion of this distribution ,where t is a variable has a non-central t-distribution with v2 degrees of freedom (positive integer)and non-central parameter  (real),   is the incomplete Gamma function ratio , , F is said to have a non-central F-distribution with v1 ,v2  degrees of freedom (positive integer) ,  ,   and   are signs differing between cases with positive or negative t as well as odd or even r in the summation .
 
Introduction
The non-central t-distribution   is defined by (Henry, 1959; Walkk, 2001) and it is studied by other researchers .The handbook of mathematical functions introduced by (Abramowitz & Stegun ,1970) whose defined the incomplete beta function while the computation of the incomplete gamma function ratios and their inverse studied by (Didonato & Morris ,1986) .
Owing to the wide variation in behavior in different regions of the parameter space , efficient code to evaluate  involves a number of different subroutines for different parts of this parameter space .In this paper we shall confine our interest to a subdomain of the parameter space in which   is large ,  is small and q is close to 1 . Indeed if    ?1 , then   varies most rapidly as q approaches 1 . This region has to be treated very carefully . Asymptotic expansions suitable for this subdomain have been derived by (Temme,1987) . These asymptotic expansions have the form ?An/Zn,where Z is either   or   (Didonato & Morris ,1992) , and in which the expansion coefficients An depend on all three parameters  ,  and q .
The expansion to be described here has the same general form ,but the expansion coefficients An depend only on   and q . The advantage of this new expansion is that it is cleaner and that an algorithm based on it can be more easily tuned for particular accuracy requirements and for particular parameter ranges .
 
 We introduce the asymptotic and asymptotic expansion for different subjects which used by many mathematicians as the form .
On the asymptotics of the jacobi function and its zeros studied by (Wong & Wang , 1992) while asymptotic and numerical aspects of the non central chi-square distribution derived by(Temme,1993),symbolic integration and asymptotic expansions studied by (Cohen,1995) , uniform asymptotic for the incomplete gamma functions starting from negative values of the parameters derived by (Temme,1996) and numerical Algorithms for uniform Airy- type asymptotic expansions introduced by (Temme,1997) , in the same year asymptotic expansions of the generalized Bessel polynomials derived by (Wong & Zhang,1997) and the asymptotics of a second solution to the jacobi differential equation is also derived by (Wong & Zhang,1997) .
On the high-order coefficients in the uniform asymptotic expansion for the incomplete gamma function introduced by (Dunster,Paris & Cang ,1998) but asymptotic approximations for the jacobi and ultraspherical polynomials, and related functions, methods and applications of analysis studied by (Dunster,1999) while a uniform asymptotic expansion for krawtchouk polynomials derived by (Li & Wong,2000) .   Asymptotic expansions in non-central limit theorems for Quadratic forms derived by (Gotze & Tikhamirov,2001), in the same year asymptotics and mellin-barnes integrals studied by (Paris & Kaminski,2001) while an asymptotic expansions for a ratio of products of gamma functions derived by (Wolfgang,2003) and an introduction in asymptotic analysis introduced by (Simon,2004) .
 
2  Definition of the non-Central t-Distribution ( Henry,1959;walk, 2001).             If  X and  Y are independent random variables and X  is a normal distribution with mean ? and variance 1 and Y is a central chi-square with v2 degrees of freedom then the variable  has a non-central t-distribution with v2 degrees of freedom (positive integer) and non-central parameter ? (real) and we write t ?  . The distribution function is given by
F(t \v2,?)=     ,                                  (1)
Which is the cumulative distribution function (c.d.f.)of non-central t-distribution, while best known for its applications in statistics, it is also widely used in many other fields .Where S1 and S2 are signs differing between cases with positive or negative t as well as odd or even r in the summation .The sign S1 is -1 if r is odd and +1 if it is even while S2 is +1 unless t? 0 and r is even in which case it is -1 ,and = =                          (2)
this equation is called the incomplete beta function such that  and
  is said to have a non-central F-distribution with v1, v2 degrees of freedom (positive integers). 3 Derivation of an Asymptotic Expansions for the non-Central t-Distribution .        We derive an asymptotic expansion of , through two sections .
 
3.1 Asymtotic Expansion of              In this section we shall derive an asymptotic expansion which we shall need later ,  which provides an efficient method of calculating  when  ?  and r?N. We start from the beta function .          = =    .                                  (3)Then by using the substitution u = e-w and du =- e-w dw  we obtain          = .                                                         (4)And using the fact that (1- e-w) = e-w/2 2 we have          =  ,                                           (5)where   We now expand  in powers of w2 as w?  .            =  ? .   (6)Let   hn=   .The last quality follows by (Didonato & Morris ,1992) . Where Cn are the expansion coefficients of   .The coefficients Cn can be expressed in terms of the generalized Bernoulli polynomials (Luke,1969),  By substitution equation (6) in (5) we get           ?         ? .     And using Watson,s Lemma we obtain the asymptotic expansion           ?                                                  (7)
3.2 Asymptotic Expansion of the Incomplete Beta Function          .             In this section we transform the expression for  in (2) in the same way as in equation (3) to obtain               =             (8)where as before   by using (6) we have                 ?                               (9)Let f = zw ,then w =  f and dw =  df , 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                   ? +R ,                                   (11)where we have use the formula (7) 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       We can 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)
and producing two further integrated terms evaluated  at w = -logq . Again integral the integral term in (13) by parts twices and subtract a third term        from the expansion of  and add a corresponding integral on separately . This procedure is continued  indefinitely . The separate integrals stating from the ones on the right of (12) and (14) add together to give   as in (11) so that   ?  ,                       (15)where = and  n/2  in the  summation is to be interpreted as largest  integer ? n/2 as in integer division . The quantities Tn satisfy the simple recurrence formula       .                                                                   (16)
We can express  directly in terms of   and q , for example , T0 =  .However, for q close to 1 , evaluation of Tn  in this way can lead to large roundingerrors on subtraction , and so   is better evaluated from its power series expansion in w . Now , when we substitute the formula (15) in equation (1) ,we get                             Which is an asymptotic expansion for the non-central t-distribution .   References Abramowitz M & Stegun I A (1970) Handbook of mathematical function , Dover            ,New York .Cohen J K (1995) Symbolic integration and asymptotic expansions .http://www.mathematical.journal.com/issue/v514/columns/cohen/52.cohe.mj.pdf. Didonato A R & Morris A H (1986) Computation of the incomplete gamma function    ratios and their inverse . ACM Trans. Math. Software 12 :377-393.  Didonato A R & Morris A H (1992) Significant digit computation of the incomplete            beta function ratios . ACM Trans. Math. Software 18 :360-373.Dunster T M , Paris R B & Cang S (1998) on the high-order coefficients in the  uniform asymptotic expansion for the incomplete gamma function .http://www.roham.sdsu.edu/?asymptotic expansion. Dunster T M (1999) Asymptotic approximations for the Jacobi and ultraspherical                polynomials , and related functions , methods and applications of analysis,6:281-315 .  Gotze F & Tikhamirov A N (2001) Asymptotic expansions in non-central limit      theorems for Quadratic forms . University of Bielefeld  .http://www.mathmatik.uni-bielefeld.de. Henry S (1959) The analysis of variance .John Wiley & Sons. Inc. New York .Li X C & Wong R (2000) A uniform asymptotic expansion for krawtchouk polynomials , J. Approximation Theory  106:155-184. Luke Y L (1969) The special functions and their approximations . Vo1.I,Academic        press, New York ,MR 39:3039 . Paris R B & Kaminski D (2001) Asymptotics and Mellin-Barnes integrals .Cambridge  University Press, Cambridge . Simon J A M (2004) An introduction in asymptotic analysis .http://www.ma.hw.ac.uk/?simonm/ae/ae.pdf. Temme N M (1987) Incomplete laplace integrals : uniform asymptotic expansion with     application to the incomplete beta function . SIAM J .Math .Anal 18: 1638-1663.MR 89f:41036 . Temme N M (1993) Asymptotic and numerical aspects of the non-central chi-square  distribution . Computers in mathematics and applications , 25(5):55-63 . Temme N M (1996) Uniform asymptotic for the incomplete gamma function starting    from negative values of the parameters .http://www.db.cwi.n1/rapporten/abstract.pdf?abstractn1=111. Temme N M (1997) Numerical algorithms for uniform airy – type asymptotic  expansions .http://www.citesseer.ist.psu.edu/temme97numerical.html.
   Walkk C (2001)Hand-Book on statistical distribution for experimentalist lists . University of Stockholm .http://www.physto.se/?walck/suf9601.pdf.
Wolfgang B (2003) An asymptotic expansions for a ratio of products of gamma   functions . Physikalisches Institute ,Universitat Heidelberg , philosophenwey 12.69120 Heidelberg , Germany .http://www.uniric.ac.at/EmIs/journals/IJMMS/volume.24/So16117/200310310.pdf.
Wong R & Wang Q Q (1992) On the asymptotics of the Jacobi function and its zeros   .SIAM J .Math . Anal . 23 :1637-1649 .Wong R & Zhang J M (1997) Asymptotic expansions of the generalized bessel polynomials . J. Comput .Appl .Math . 85:87-112 .Wong R & Zhang J M (1997) The asymptotic of a second solution to the Jacobi differential equation , Integral transforms Spec . Funct . 5:287-308 .

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