Uniform Asymptotic Expansion for the Incomplete Beta Function
SSymmetry, Integrability and Geometry: Methods and Applications SIGMA (2016), 101, 5 pages Uniform Asymptotic Expansionfor the Incomplete Beta Function
Gerg˝o NEMES and Adri B. OLDE DAALHUISMaxwell Institute and School of Mathematics, The University of Edinburgh,Peter Guthrie Tait Road, Edinburgh EH9 3FD, UK
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Received September 12, 2016, in final form October 21, 2016; Published online October 25, 2016http://dx.doi.org/10.3842/SIGMA.2016.101
Abstract.
In [Temme N.M., Special functions. An introduction to the classical functionsof mathematical physics,
A Wiley-Interscience Publication , John Wiley & Sons, Inc., NewYork, 1996, Section 11.3.3.1] a uniform asymptotic expansion for the incomplete beta func-tion was derived. It was not obvious from those results that the expansion is actually anasymptotic expansion. We derive a remainder estimate that clearly shows that the resultindeed has an asymptotic property, and we also give a recurrence relation for the coefficients.
Key words: incomplete beta function; uniform asymptotic expansion
For positive real numbers a , b and x ∈ [0 , I x ( a, b )is defined by I x ( a, b ) = 1 B ( a, b ) (cid:90) x t a − (1 − t ) b − d t, where B ( a, b ) denotes the ordinary beta function: B ( a, b ) = (cid:90) t a − (1 − t ) b − d t = Γ( a )Γ( b )Γ( a + b )(see, e.g., [2, Section 8.17(i)]). In this paper, we will use the notation of [2, Section 8.18(ii)].The incomplete beta function plays an important role in statistics in connection with the betadistribution (see, for instance, [1, pp. 210–275]). Large parameter asymptotic approximationsare useful in these applications. For fixed x and b , one could use the asymptotic expansion I x ( a, b ) = x a (1 − x ) b − aB ( a, b ) F (cid:18) , − ba + 1 ; xx − (cid:19) ∼ x a (1 − x ) b − aB ( a, b ) ∞ (cid:88) n =0 (1 − b ) n ( a + 1) n (cid:18) xx − (cid:19) n , (1)as a → + ∞ . The right-hand side of (1) converges only for x ∈ [0 , ), but for any fixed x ∈ [0 , a → + ∞ . For more details, see [3,Section 11.3.3]. However, it is readily seen that (1) breaks down as x →
1. Since this limithas significant importance in applications, Temme derived in [3, Section 11.3.3.1] an asymptoticexpansion as a → + ∞ that holds uniformly for x ∈ (0 , a r X i v : . [ m a t h . C A ] O c t G. Nemes and A.B. Olde Daalhuis
Theorem 1.
Let ξ = − ln x . Then for any f ixed positive integer N and fixed positive real b , I x ( a, b ) = Γ( a + b )Γ( a ) (cid:32) N − (cid:88) n =0 d n F n + O (cid:0) a − N (cid:1) F (cid:33) , (2) as a → + ∞ , uniformly for x ∈ (0 , . The functions F n = F n ( ξ, a, b ) are defined by the recurrencerelation aF n +1 = ( n + b − aξ ) F n + nξF n − , (3) with F = a − b Q ( b, aξ ) , F = b − aξa F + ξ b e − aξ a Γ( b ) , and Q ( a, z ) = Γ( a, z ) / Γ( a ) is the normalised incomplete gamma function ( see [2, Section 8.2(i)]) .The coefficients d n = d n ( ξ, b ) are defined by the generating function (cid:18) − e − t t (cid:19) b − = ∞ (cid:88) n =0 d n ( t − ξ ) n . (4) In particular, d = (cid:18) − xξ (cid:19) b − , d = xξ + x − − x ) ξ ( b − d . They satisfy the recurrence relation ξ ( n + 1)( n + 2) d d n +2 = ξ n (cid:88) m =0 ( m + 1) (cid:18) n − m + 1 + m − n − b − (cid:19) d m +1 d n − m +1 + n (cid:88) m =0 ( m + 1) (cid:18) n − m − − ξ + m − nb − (cid:19) d m +1 d n − m + n (cid:88) m =0 (1 − m − b ) d m d n − m . (5) In the case that b = 1 , we have d = 1 and d n = 0 for n ≥ . Our contribution is the remainder estimate in (2) and the recurrence relation (5). In fact, it isnot at all obvious from (3) that the sequence { F n } ∞ n =0 has an asymptotic property as a → + ∞ .We will show that for any non-negative integer n ,0 < F n +1 ≤ n + βa F n , (6)where β = max(1 , b ).In [4, Section 38.2.8] the function F n is identified as a Kummer U -function: F n = ξ n + b e − aξ n !Γ( b ) U ( n + 1 , n + b + 1 , aξ ) . niform Asymptotic Expansion for the Incomplete Beta Function 3 We proceed similarly as in [3, Section 11.3.3.1] and start with the integral representation I x ( a, b ) = 1 B ( a, b ) (cid:90) + ∞ ξ t b − e − at (cid:18) − e − t t (cid:19) b − d t. (7)We substitute the truncated Taylor series expansion (cid:18) − e − t t (cid:19) b − = N − (cid:88) n =0 d n ( t − ξ ) n + r N ( t )into (7) and obtain I x ( a, b ) = Γ( a + b )Γ( a ) (cid:32) N − (cid:88) n =0 d n F n + R N ( a, b, x ) (cid:33) , where F n is given by the integral representation F n = 1Γ( b ) (cid:90) + ∞ ξ t b − e − at ( t − ξ ) n d t = e − aξ Γ( b ) (cid:90) + ∞ ( τ + ξ ) b − τ n e − aτ d τ, (8)and the remainder term R N ( a, b, x ) is defined by R N ( a, b, x ) = 1Γ( b ) (cid:90) + ∞ ξ t b − e − at r N ( t ) d t. (9)The recurrence relation (3) can be obtained from (8) via a simple integration by parts.Let, for a moment, c n ( a, b ) = (cid:90) + ∞ ( τ + ξ ) b − τ n e − aτ d τ. Then via integration by parts we find ac n +1 ( a, b ) = ( n + b ) c n ( a, b ) + ξ (1 − b ) c n ( a, b − . (10)We make the observation that0 ≤ ξc n ( a, b −
1) = ξ (cid:90) + ∞ ( τ + ξ ) b − τ n e − aτ d τ ≤ c n ( a, b ) . (11)It follows from (10) and (11) that ac n +1 ( a, b ) ≤ (cid:40) ( n + 1) c n ( a, b ) if 0 < b ≤ , ( n + b ) c n ( a, b ) if b ≥ . Since F n = e − aξ c n ( a, b ) / Γ( b ), this inequality implies (6).To obtain the remainder estimate in (2), we use the Cauchy integral representation r N ( t ) = ( t − ξ ) N π i (cid:73) { ξ,t } (cid:16) − e − τ τ (cid:17) b − ( τ − t )( τ − ξ ) N d τ, (12) G. Nemes and A.B. Olde Daalhuiswhere the contour encircles the points ξ and t once in the positive sense. From the integralrepresentation (9), we have that 0 ≤ ξ ≤ t . Thus, in the case that N ≥
1, we can deform thecontour in (12) to the path[1 + ∞ i , π i] ∪ [1 + π i , − π i] ∪ [ − π i , − − π i] ∪ [ − − π i , − π i] ∪ [1 − π i , − ∞ i] . For the integrals along the final three portions of the path, we have the estimates (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) π i (cid:90) − − π i − π i (cid:16) − e − τ τ (cid:17) b − ( τ − t )( τ − ξ ) N d τ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ max (cid:18) (e − b − , (cid:16) e+1 √ π +1 (cid:17) b − (cid:19) (1 + ξ ) N +1 , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) π i (cid:90) − π i − − π i (cid:16) − e − τ τ (cid:17) b − ( τ − t )( τ − ξ ) N d τ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ max (cid:18)(cid:16) e ± +1 √ π +1 (cid:17) b − (cid:19) π N +2 , (13)and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) π i (cid:90) −∞ i1 − π i (cid:16) − e − τ τ (cid:17) b − ( τ − t )( τ − ξ ) N d τ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ π (cid:90) + ∞ π max (cid:0)(cid:0) ± e − (cid:1) b − (cid:1)(cid:0) s + 1 (cid:1) (1 − b ) / (cid:112) s + (1 − t ) (cid:0) s + (1 − ξ ) (cid:1) N/ d s ≤ max (cid:0)(cid:0) ± e − (cid:1) b − (cid:1) π (cid:90) + ∞ π (cid:0) s + 1 (cid:1) (1 − b ) / s N +1 d s, (14)respectively. The integrals along the first two portions can be estimated similarly to (13)and (14). Hence, for 0 ≤ ξ ≤ t and N ≥
1, we have | r N ( t ) | ≤ C N ( b )( t − ξ ) N , where the constant C N ( b ) does not depend on ξ . Using this result in the integral representa-tion (9), we can infer that | R N ( a, b, x ) | ≤ C N ( b ) F N . Finally, combining this result with the inequalities (6), we obtain the required remainder estimatein (2).The reader can check that the function f ( t ) = (cid:16) − e − t t (cid:17) b − is a solution of the nonlineardifferential equation tf ( t ) f (cid:48)(cid:48) ( t ) − b − b − tf (cid:48) ( t ) + ( t + 2) f ( t ) f (cid:48) ( t ) + ( b − f ( t ) = 0 . If we substitute the Taylor series (4) into this differential equation and rearrange the result, weobtain the recurrence relation (5).
Acknowledgements
This research was supported by a research grant (GRANT11863412/70NANB15H221) from theNational Institute of Standards and Technology. The authors thank the anonymous referees fortheir helpful comments and suggestions on the manuscript.niform Asymptotic Expansion for the Incomplete Beta Function 5
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