Hypergeometric Functions:
Application in Distribution Theory
Dr. Pooja Singh,
Maharaja Surajmal Institute of Technology (Affiliated to GGSIPU), Delhi
ABSTRACT: Hypergeometric functions are generalized from exponential functions. There are functions which can also be evaluated analytically and expressed in form of hypergeometric function. In this paper, a unified approach to hypergeometric functions is given to derive the probability density function and corresponding cumulative distribution function of the noncentral F variate.
Key words and phrases: Hypergeometric functions; distribution theory; chi-square Distribution, Non-centrality Parameter.
I) Introduction
The hypergeometric function is a special function encountered in a variety of application. Higher-order transcendental functions are generalized from hypergeometric functions. These functions are applied in different subjects like theoretical physics, and also functional i n computers as Maple and Mathematica. They are also broadly explored in the area of statistical/econometric distribution theory.
The purpose of this paper is to understand importance of collection of hypergeometric function in different fields especially initiating economists to the wide class of hypergeometric functions. The paper is organized as follows. In Section II, the generalized hypergeometric series is explained with some of its properties. In Sections 3 and 4, some famous special cases are discussed which will be used in our important results. In Section 5, application to distribution theory is given. It leads to the derivation of the exact cumulative distribution function of the noncentral F variate.
An appendix is attached which summarizes notational abbreviations and function names.
T he paper is of an introductory nature. Much of the content is new unpublished formulae which are integrated with the mathematical literature. The subject was developed by the t hree volumes edited b y Erdélyi ( 1953, 1955). The hypergeometric functions are classified in A b r a m o w i t z a n d s t e g u n ( 1 9 7 2 ) , Carlson (1976), G a s p e r a n d R e h m a n ( 1 9 9 0 ) . For integrals involving such functions, see Chandel, Agarwal and Kumar(1992) Choi, Hasanov and Turaev(2012), Exton(1973), Joshi and Pandey(2013), Saran(1955), Seth and Sindhu(2005), Usha and Shoukat(2012). For the theory, we referred to Whittaker and Watson (1927), Erdélyi (1953, 1955) for a more comprehensive proof refer to Anderson(1984), Slater (1966), Luke (1969), Olver (1974), Mathai (1993). Statistics/econometric theories was explained
extensively in Cox and Hinkley(1974),
Craig(1936), Feller(1971), Hardle and
Linton(1994), Muellbauer(1983). Further
applications in statistics/econometrics and economic theory a r e suggested throughout. Here, Hypergeometric function is applied effectively to Distribution Theory.
II) The generalized Hypergeometric Series
Before introducing the hypergeometric function, we define the Pochhammer’s symbol
If we substitute in eqn 1 we are left with
If we multiply and divide eqn (2) by then we get
So if we set
The general Hypergeometric function is given by
The and
are called the numerator and denominator
parameters respectively and is called the argument. We deal within this paper will concern eqn (4) but for the most part we will deal with specific cases of the general hypergeometric function. For instance,
If we take in eqn (4) we get
If we let and and by using eqn (1)
.
which is the binomial expansion.
Some instant result follows from eqn (4), when one of the parameter of is non negative integer, then
Also,
And interchanging elements separated by commas is feasible as multiplication is commutative.
But, interchanging of semicolons (i.e, between and ) is not allowed as division is not commutative.
Also, if , then
An important property that we will need to use is the convergence criteria of the hypergeometric functions depending on the values of and . The radius of convergence of a series of variable is defined as a value such that the series converges if and diverges if , where , in the case , is the centre of the disc convergence. For hypergeometric function, provided and are not non negative integers for any , the relevant convergence criteria stated below can be derived using the ratio test, which determines the absolute convergence of the series using the limit of the ratio of two consective terms.
a) If , then the ratio of coefficients of in the taylor series of the hypergeometric function
tends to as ; so the radius of
convergence is , so that the series converges for all values of . Hence is entire. In
particular, the radius of convergence for and is .
b) If , the ratio of coefficients of tends to as , so the radius of convergence is , so that series converges only if . In particular, the radius of convergence for is
.
c) If , the ratio of coefficients of tends to as , so the radius of convergence is
, so that the series does not converge for any value of .
We will seek approximation to the relevant hypergeometric function for within radii of convergence. For , as given by Luke (1975), there is a restriction for convergence on the unit disc, the series only converges absolutely at if
So that the selection of values for and must reflect that.
III)The Hypergeometric Function
The Guass hypergeometric function
where is in the radius of convergence of the series
Integral over
The first result is a representation of in terms of beta integral,
In terms of more familiar quantities, the hypergeometric function is
..(14)
This expression can be obtained by expanding by binomial theorem and integrating
termwise.
If we substitute the variable this will give,
The special case produces,
where we applied
To eliminate , we put to obtain,
Also,
where
This equation can also be obtained by using binomial theorem and integrating term by term. For the series is finite with terms in it, and it can also be derived by successive integration (by parts),
If , on rearranging the above equation in such a way that negative term follows every two
consecutive positive term then we get for . See Wittaker and Watson(1927) for proof. This is the expansion of log function in infinite series. Series is absolutely convergent if and conditionally convergent if
where is arbitrary.
is infinite when and or and .
So with these two exceptions, series expansion will give a finite value. For convergence of the hypergeometric series, we must have . General formula for analytic continuation of Gauss Series is given in the volumes of Erdelyi (1955)
IV)Kummer’s Confluent Hypergeometric
Function
The confluent hypergeometric function denoted by
is defined by,
Following two formulas of Kummer are useful for our results.
where a is an arbitrary.
.…(24)
The exponential function is the elementary example of the hypergeometric series. All the functions studied here can be considered as generalization of elementary transcendental function; Further, special cases arise when compared with Poisson process as discussed in Hardle and Linton (1994). We also have integral representation,
where is the modified Bessel function of the first kind with order .
The incomplete gamma functions arise from Euler’s integral for the gamma function,
By decomposing it into an integral from 0 to ∞,
This derivation shows that integrals of elementary function leads to a geometric function.
Other special case is standard normal cumulative distribution function.
where is signum function,
As we know Kummer’s function satisfies a basic relation,
This equation was derived by Sentana(1995) with help of Leibniz formula of fractional integral. In view this eqn (28) can be rewritten as
where is
standard normal density function.
V) Application in Distribution Theory
If follows a non central Chisquared distribution with d degree of freedom and non central parameter then
or ……(31)
In most of the following notation p is fixed and will not be explicitly stated in the notation. Also, it is useful to set .
The formula for the pdf involves Bessel Function which can be limiting behavior,
as
Consider,
Cox and Minkley(1974) had given the definition of equ (32) and further equations follows from eqn(26). When the above distribution reduces to
Equation (32) can be rewritten as,
with the weights
is from Poisson density.
We will get corresponding cdf by term wise integration of equation (32) as,
We must recall that , so if , then .
In addition, if is independent from U, then , with noncentral F distribution with degree of freedom and in denominator with noncentral parameter .
This gives the first definition and next one follows
from using beta function where
. The very last line follows from eqn (18)
The cumulative distribution function may be found by
……..(38)
with
Here we have applied the definition of Incomplete beta function and in the last step we have applied the eqn (18)
Conclusion & Extensions
Hypergeometric functions are able to occur in fractional calculus [e.g. Cox and Hinkley (1974)]. The nature of implementation of these functions is in data (e.g. fractionally integrated) of time series and further area of economics. Given the determination of unemployment and price rises, this relation seems to have significance for economist. We had derived the exact cumulative distribution function by using Bessel function, incomplete gamma, Gauss hypergeometric function or other relevant functions.
A final statement on hypergeometric functions. They have now become so important in many areas of applied mathematics that they can be found in many computer packages, including ones allowing symbolic manipulations like Maple and
Mathematica. A major advantage they have is their parsimonious generality, and their ability to give explicit answers to problems. It is hoped that this paper has made the case for their potential in quantitative economics.
The extension for the content of this paper is in at least three possible ways. Firstly, Meijer’s G and Fox’s functions are special functions of generalized hypergeometric function. These functions will help us in analytical manipulation of functions and its argument. Secondly, in this paper we had discussed hypergeometric function of one variable but we can extend it to more than one variable. We can rewrite hypergeometric function for two variables, instead of single variable functions. Thirdly, it is assumed for convenience that z is a scalar but we can pursue without it. We can define hypergeometric functions even if we have the argument as a square matrix. If we define a matrix function whose output is a scalar, we get the type of hypergeometric functions used in multivariate distribution theory.
References:
[1] M. Abramowitz, I.A Stegun, Handbook of mathematical functions, Dover publications, New York (1972).
[2] T.W. Anderson, An introduction to multivariate statistical analysis (2nd ed.), John Wiley & sons, New York (1984).
[3] B.C. Carlson, The need for a new classification of double hypergeometric series, Proc. Nat. Acad. Sci, 56 (1976),221-224.
[4] R.C.S. Chandel, R.D. Agarwal,H. Kumar, Hypergeometric functions of four variables and their integral representation, The Mathematics Education Journal, 26 (1992),74-94.
[5] J. Choi, A. Hasanov,M. Turaev, Integral representation for Srivastava’s Hypergeometric function Hb, Journal of
Korean Society Mathematics Education,19 (2012), 137-145.
[6] D.R.Cox, D.V. Hinkley, Theoretical Statistics, Chapman and Hall, London, (1974).
[7] C.C. Craig, “On the frequency function of xy”, Annals of Mathematical Statistics, 7 (1936), 1-15.
[8] A. Erdélyi, Higher transcendental functions, volumes 1-2, Mc.Graw-Hill, N.Y(1953).
[9] A. Erdélyi, Higher transcendental functions, volume 3, Mc.Graw-Hill, N.Y. (1955).
[10] H. Exton, Some Integral representation and transformations of hypergeometric function of four variables, Bull. Amer. Math. Soc., 14 (1973),132-140.
[11] W. Feller, An introduction to probability theory and its applications (2nd ed.), John Wiley & sons, New York(1971).
[13] W.Härdle,O. Linton, “Applied nonparametric methods”, in R.F. Engle and D.L. McFadden (eds.), Handbook of Econometrics, Amsterdam(North-Holland) (1994).
[14] S. Joshi,R.M. Pandey, An integral involving Gauss Hypergeometric Function of the Series, International Journal of Scientific and Innovative Mathematical Research, 1 (2013) 117-120.
[15] Y.L.Luke, The special functions and their approximations, volumes 1-2, Academic press, New York(1969).
[16] A.M. Mathai, A handbook of generalized functions for statistical and physical sciences, Oxford University Press, Oxford(1993).
[17] J. Muellbauer, “Surprises in the consumption function”, Economic Journal, 93 (1983), 34-50.
[18] F.W.J. Olver, Asymptotics and special functions, Academic Press, New York(1974).
[19] S. Saran, Integarls associated with hypergeometric functions of these variables, National Institute of Science of India, 21,No. 2 (1955) 83-90.
[20] J.P.L. Seth, B.S. Sidhu, Multivariate integral representation suggested by Laguerre and Jacobi polynomials of matrix argument, Vihnana Prasad Anusandhan Patrika, 48, No.2 (2005),171-219.
[21] L.J. Slater, Generalized hypergeometric functions, Cambridge University Press, Cambridge(1966).
[22] B.Usha, A. Shoukat, An Integral containing Hypergeometric function, Advances in Computational Mathematics and its Applications, 2, No.2 (2012), 263-266.
[23] E.T. Whittaker, G.N.Watson, A course of modern analysis (4th ed.), 15th printing 1988, Cambridge University Press, Cambridge(1927).
APPENDIX : Special notational and functions
≡ : identity; when variables or functions are equivalent for all defined values of the parameters and the arguments.
= : equality; when two expressions are not equivalent, but have equal principal values or are equal for a certain range of parameter or argument values.
∼ : distributed as.
C , N , R, Z : the sets of complex, natural, real, and integer numbers, respectively.
pdf: probability density function.
cdf: cumulative distribution function.
: the imaginary unit.
|z| : modulus (or absolute value) of z.
B(x, y) = Γ(x)Γ(y)/Γ(x + y) : Beta function.
Γ(ν) : gamma function.
Γ 1 Γ 1 j j : B i n o m i a l C o e f f i c i e n t s .
1 j 1 Γ
j Γ : Pochhammer’s symbol.
γ ( ν , z ) , Γ ( ν, z ) : i n c o m p l e t e g a m m a f u n c t i o n s .
: g e n e r a l i z e d h y p e r g e o m e t r i c s e r i e s .
: G a u s s h y p e r g e o m e t r i c s e r i e s ( t h e h y p e r g e o m e t r i c f u n c t i o n ) .
: Ku m m er ’ s f u n c ti o n ( c o n f l u e n t / d e g e n e r a t e h y p e r g e o m e t r i c f u n c t i o n ) .
φ ( z ) , Φ ( z ) : s t a n d a r d N o r m a l p d f a n d c d f r e s p e c t i v e l y .
i n t ( . ) : i n t e g e r p a r t o f t h e a r g u m e n t .
: m o d i f i e d B e s s e l f u n c t i o n o f t h e f i r s t k in d o f o rd er .
