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{-# LANGUAGE BangPatterns, ScopedTypeVariables #-}
-- |
-- Module : Numeric.SpecFunctions
-- Copyright : (c) 2009, 2011, 2012 Bryan O'Sullivan
-- License : BSD3
--
-- Maintainer : bos@serpentine.com
-- Stability : experimental
-- Portability : portable
--
-- Special functions and factorials.
module Numeric.SpecFunctions (
-- * Error function
erf
, erfc
, invErf
, invErfc
-- * Gamma function
, logGamma
, logGammaL
, incompleteGamma
, invIncompleteGamma
-- * Beta function
, logBeta
, incompleteBeta
, incompleteBeta_
, invIncompleteBeta
-- * Logarithm
, log1p
, log2
-- * Factorial
, factorial
, logFactorial
, stirlingError
-- * Combinatorics
, choose
-- * References
-- $references
) where
import Data.Bits ((.&.), (.|.), shiftR)
import Data.Int (Int64)
import Data.Word (Word64)
import qualified Data.Number.Erf as Erf (erfc,erf)
import qualified Data.Vector.Unboxed as U
import Numeric.Polynomial.Chebyshev (chebyshevBroucke)
import Numeric.MathFunctions.Constants (m_epsilon, m_sqrt_2_pi, m_ln_sqrt_2_pi,
m_NaN, m_neg_inf, m_pos_inf, m_sqrt_2)
----------------------------------------------------------------
-- Error function
----------------------------------------------------------------
-- | Error function.
--
-- > erf -∞ = -1
-- > erf 0 = 0
-- > erf +∞ = 1
erf :: Double -> Double
{-# INLINE erf #-}
erf = Erf.erf
-- | Complementary error function.
--
-- > erfc -∞ = 2
-- > erfc 0 = 1
-- > errc +∞ = 0
erfc :: Double -> Double
{-# INLINE erfc #-}
erfc = Erf.erfc
-- | Inverse of 'erf'.
invErf :: Double -- ^ /p/ ∈ [-1,1]
-> Double
invErf p = invErfc (1 - p)
-- | Inverse of 'erfc'.
invErfc :: Double -- ^ /p/ ∈ [0,2]
-> Double
invErfc p
| p == 2 = m_neg_inf
| p == 0 = m_pos_inf
| p > 2 = error "x>2"
| p < 0 = error "x<0"
| p <= 1 = r
| otherwise = -r
where
pp = if p <= 1 then p else 2 - p
t = sqrt $ -2 * log( 0.5 * pp)
-- Initial guess
x0 = -0.70711 * ((2.30753 + t * 0.27061) / (1 + t * (0.99229 + t * 0.04481)) - t)
r = loop 0 x0
--
loop !j !x
| j >= 2 = x
| otherwise = let err = erfc x - pp
x' = x + err / (1.12837916709551257 * exp(-sqr x) - x * err) -- // Halley
in loop (j+1) x'
sqr x = x * x
----------------------------------------------------------------
-- Gamma function
----------------------------------------------------------------
-- Adapted from http://people.sc.fsu.edu/~burkardt/f_src/asa245/asa245.html
-- | Compute the logarithm of the gamma function Γ(/x/). Uses
-- Algorithm AS 245 by Macleod.
--
-- Gives an accuracy of 10-12 significant decimal digits, except
-- for small regions around /x/ = 1 and /x/ = 2, where the function
-- goes to zero. For greater accuracy, use 'logGammaL'.
--
-- Returns ∞ if the input is outside of the range (0 < /x/ ≤ 1e305).
logGamma :: Double -> Double
logGamma x
| x <= 0 = m_pos_inf
| x < 1.5 = a + c *
((((r1_4 * b + r1_3) * b + r1_2) * b + r1_1) * b + r1_0) /
((((b + r1_8) * b + r1_7) * b + r1_6) * b + r1_5)
| x < 4 = (x - 2) *
((((r2_4 * x + r2_3) * x + r2_2) * x + r2_1) * x + r2_0) /
((((x + r2_8) * x + r2_7) * x + r2_6) * x + r2_5)
| x < 12 = ((((r3_4 * x + r3_3) * x + r3_2) * x + r3_1) * x + r3_0) /
((((x + r3_8) * x + r3_7) * x + r3_6) * x + r3_5)
| x > 3e6 = k
| otherwise = k + x1 *
((r4_2 * x2 + r4_1) * x2 + r4_0) /
((x2 + r4_4) * x2 + r4_3)
where
(a , b , c)
| x < 0.5 = (-y , x + 1 , x)
| otherwise = (0 , x , x - 1)
y = log x
k = x * (y-1) - 0.5 * y + alr2pi
alr2pi = 0.918938533204673
x1 = 1 / x
x2 = x1 * x1
r1_0 = -2.66685511495; r1_1 = -24.4387534237; r1_2 = -21.9698958928
r1_3 = 11.1667541262; r1_4 = 3.13060547623; r1_5 = 0.607771387771
r1_6 = 11.9400905721; r1_7 = 31.4690115749; r1_8 = 15.2346874070
r2_0 = -78.3359299449; r2_1 = -142.046296688; r2_2 = 137.519416416
r2_3 = 78.6994924154; r2_4 = 4.16438922228; r2_5 = 47.0668766060
r2_6 = 313.399215894; r2_7 = 263.505074721; r2_8 = 43.3400022514
r3_0 = -2.12159572323e5; r3_1 = 2.30661510616e5; r3_2 = 2.74647644705e4
r3_3 = -4.02621119975e4; r3_4 = -2.29660729780e3; r3_5 = -1.16328495004e5
r3_6 = -1.46025937511e5; r3_7 = -2.42357409629e4; r3_8 = -5.70691009324e2
r4_0 = 0.279195317918525; r4_1 = 0.4917317610505968;
r4_2 = 0.0692910599291889; r4_3 = 3.350343815022304
r4_4 = 6.012459259764103
data L = L {-# UNPACK #-} !Double {-# UNPACK #-} !Double
-- | Compute the logarithm of the gamma function, &#915;(/x/). Uses a
-- Lanczos approximation.
--
-- This function is slower than 'logGamma', but gives 14 or more
-- significant decimal digits of accuracy, except around /x/ = 1 and
-- /x/ = 2, where the function goes to zero.
--
-- Returns &#8734; if the input is outside of the range (0 < /x/
-- &#8804; 1e305).
logGammaL :: Double -> Double
logGammaL x
| x <= 0 = m_pos_inf
| otherwise = fini . U.foldl' go (L 0 (x+7)) $ a
where fini (L l _) = log (l+a0) + log m_sqrt_2_pi - x65 + (x-0.5) * log x65
go (L l t) k = L (l + k / t) (t-1)
x65 = x + 6.5
a0 = 0.9999999999995183
a = U.fromList [ 0.1659470187408462e-06
, 0.9934937113930748e-05
, -0.1385710331296526
, 12.50734324009056
, -176.6150291498386
, 771.3234287757674
, -1259.139216722289
, 676.5203681218835
]
-- | Compute the log gamma correction factor for @x@ &#8805; 10. This
-- correction factor is suitable for an alternate (but less
-- numerically accurate) definition of 'logGamma':
--
-- >lgg x = 0.5 * log(2*pi) + (x-0.5) * log x - x + logGammaCorrection x
logGammaCorrection :: Double -> Double
logGammaCorrection x
| x < 10 = m_NaN
| x < big = chebyshevBroucke (t * t * 2 - 1) coeffs / x
| otherwise = 1 / (x * 12)
where
big = 94906265.62425156
t = 10 / x
coeffs = U.fromList [
0.1666389480451863247205729650822e+0,
-0.1384948176067563840732986059135e-4,
0.9810825646924729426157171547487e-8,
-0.1809129475572494194263306266719e-10,
0.6221098041892605227126015543416e-13,
-0.3399615005417721944303330599666e-15,
0.2683181998482698748957538846666e-17
]
-- | Compute the normalized lower incomplete gamma function
-- γ(/s/,/x/). Normalization means that
-- γ(/s/,∞)=1. Uses Algorithm AS 239 by Shea.
incompleteGamma :: Double -- ^ /s/ ∈ (0,∞)
-> Double -- ^ /x/ ∈ (0,∞)
-> Double
incompleteGamma p x
| isNaN p || isNaN x = m_NaN
| x < 0 || p <= 0 = m_pos_inf
| x == 0 = 0
| p >= 1000 = norm (3 * sqrt p * ((x/p) ** (1/3) + 1/(9*p) - 1))
| x >= 1e8 = 1
| x <= 1 || x < p = let a = p * log x - x - logGamma (p + 1)
g = a + log (pearson p 1 1)
in if g > limit then exp g else 0
| otherwise = let g = p * log x - x - logGamma p + log cf
in if g > limit then 1 - exp g else 1
where
norm a = 0.5 * erfc (- a / m_sqrt_2)
pearson !a !c !g
| c' <= tolerance = g'
| otherwise = pearson a' c' g'
where a' = a + 1
c' = c * x / a'
g' = g + c'
cf = let a = 1 - p
b = a + x + 1
p3 = x + 1
p4 = x * b
in contFrac a b 0 1 x p3 p4 (p3/p4)
contFrac !a !b !c !p1 !p2 !p3 !p4 !g
| abs (g - rn) <= min tolerance (tolerance * rn) = g
| otherwise = contFrac a' b' c' (f p3) (f p4) (f p5) (f p6) rn
where a' = a + 1
b' = b + 2
c' = c + 1
an = a' * c'
p5 = b' * p3 - an * p1
p6 = b' * p4 - an * p2
rn = p5 / p6
f n | abs p5 > overflow = n / overflow
| otherwise = n
limit = -88
tolerance = 1e-14
overflow = 1e37
-- Adapted from Numerical Recipes §6.2.1
-- | Inverse incomplete gamma function. It's approximately inverse of
-- 'incompleteGamma' for the same /s/. So following equality
-- approximately holds:
--
-- > invIncompleteGamma s . incompleteGamma s = id
invIncompleteGamma :: Double -- ^ /s/ ∈ (0,∞)
-> Double -- ^ /p/ ∈ [0,1]
-> Double
invIncompleteGamma a p
| a <= 0 =
error $ "Statistics.Math.invIncompleteGamma: a must be positive. Got: " ++ show a
| p < 0 || p > 1 =
error $ "Statistics.Math.invIncompleteGamma: p must be in [0,1] range. Got: " ++ show p
| p == 0 = 0
| p == 1 = 1 / 0
| otherwise = loop 0 guess
where
-- Solve equation γ(a,x) = p using Halley method
loop :: Int -> Double -> Double
loop i x
| i >= 12 = x'
-- For small s derivative becomes approximately 1/x*exp(-x) and
-- skyrockets for small x. If it happens correct answer is 0.
| isInfinite f' = 0
| abs dx < eps * x' = x'
| otherwise = loop (i + 1) x'
where
-- Value of γ(a,x) - p
f = incompleteGamma a x - p
-- dγ(a,x)/dx
f' | a > 1 = afac * exp( -(x - a1) + a1 * (log x - lna1))
| otherwise = exp( -x + a1 * log x - gln)
u = f / f'
-- Halley correction to Newton-Rapson step
corr = u * (a1 / x - 1)
dx = u / (1 - 0.5 * min 1.0 corr)
-- New approximation to x
x' | x < dx = 0.5 * x -- Do not go below 0
| otherwise = x - dx
-- Calculate inital guess for root
guess
--
| a > 1 =
let t = sqrt $ -2 * log(if p < 0.5 then p else 1 - p)
x1 = (2.30753 + t * 0.27061) / (1 + t * (0.99229 + t * 0.04481)) - t
x2 = if p < 0.5 then -x1 else x1
in max 1e-3 (a * (1 - 1/(9*a) - x2 / (3 * sqrt a)) ** 3)
-- For a <= 1 use following approximations:
-- γ(a,1) ≈ 0.253a + 0.12a²
--
-- γ(a,x) ≈ γ(a,1)·x^a x < 1
-- γ(a,x) ≈ γ(a,1) + (1 - γ(a,1))(1 - exp(1 - x)) x >= 1
| otherwise =
let t = 1 - a * (0.253 + a*0.12)
in if p < t
then (p / t) ** (1 / a)
else 1 - log( 1 - (p-t) / (1-t))
-- Constants
a1 = a - 1
lna1 = log a1
afac = exp( a1 * (lna1 - 1) - gln )
gln = logGamma a
eps = 1e-8
----------------------------------------------------------------
-- Beta function
----------------------------------------------------------------
-- | Compute the natural logarithm of the beta function.
logBeta :: Double -> Double -> Double
logBeta a b
| p < 0 = m_NaN
| p == 0 = m_pos_inf
| p >= 10 = log q * (-0.5) + m_ln_sqrt_2_pi + logGammaCorrection p + c +
(p - 0.5) * log ppq + q * log1p(-ppq)
| q >= 10 = logGamma p + c + p - p * log pq + (q - 0.5) * log1p(-ppq)
| otherwise = logGamma p + logGamma q - logGamma pq
where
p = min a b
q = max a b
ppq = p / pq
pq = p + q
c = logGammaCorrection q - logGammaCorrection pq
-- | Regularized incomplete beta function. Uses algorithm AS63 by
-- Majumder and Bhattachrjee.
incompleteBeta :: Double -- ^ /p/ > 0
-> Double -- ^ /q/ > 0
-> Double -- ^ /x/, must lie in [0,1] range
-> Double
incompleteBeta p q = incompleteBeta_ (logBeta p q) p q
-- | Regularized incomplete beta function. Same as 'incompleteBeta'
-- but also takes logarithm of beta function as parameter.
incompleteBeta_ :: Double -- ^ logarithm of beta function for given /p/ and /q/
-> Double -- ^ /p/ > 0
-> Double -- ^ /q/ > 0
-> Double -- ^ /x/, must lie in [0,1] range
-> Double
incompleteBeta_ beta p q x
| p <= 0 || q <= 0 = error "p <= 0 || q <= 0"
| x < 0 || x > 1 || isNaN x = error "x out of [0,1] range"
| x == 0 || x == 1 = x
| p >= (p+q) * x = incompleteBetaWorker beta p q x
| otherwise = 1 - incompleteBetaWorker beta q p (1 - x)
-- Worker for incomplete beta function. It is separate function to
-- avoid confusion with parameter during parameter swapping
incompleteBetaWorker :: Double -> Double -> Double -> Double -> Double
incompleteBetaWorker beta p q x = loop (p+q) (truncate $ q + cx * (p+q)) 1 1 1
where
-- Constants
eps = 1e-15
cx = 1 - x
-- Loop
loop !psq (ns :: Int) ai term betain
| done = betain' * exp( p * log x + (q - 1) * log cx - beta) / p
| otherwise = loop psq' (ns - 1) (ai + 1) term' betain'
where
-- New values
term' = term * fact / (p + ai)
betain' = betain + term'
fact | ns > 0 = (q - ai) * x/cx
| ns == 0 = (q - ai) * x
| otherwise = psq * x
-- Iterations are complete
done = db <= eps && db <= eps*betain' where db = abs term'
psq' = if ns < 0 then psq + 1 else psq
-- | Compute inverse of regularized incomplete beta function. Uses
-- initial approximation from AS109 and Halley method to solve equation.
invIncompleteBeta :: Double -- ^ /p/ > 0
-> Double -- ^ /q/ > 0
-> Double -- ^ /a/ ∈ [0,1]
-> Double
invIncompleteBeta p q a
| p <= 0 || q <= 0 = error "p <= 0 || q <= 0"
| a < 0 || a > 1 = error "bad a"
| a == 0 || a == 1 = a
| a > 0.5 = 1 - invIncompleteBetaWorker (logBeta p q) q p (1 - a)
| otherwise = invIncompleteBetaWorker (logBeta p q) p q a
invIncompleteBetaWorker :: Double -> Double -> Double -> Double -> Double
invIncompleteBetaWorker beta p q a = loop (0::Int) guess
where
p1 = p - 1
q1 = q - 1
-- Solve equation using Halley method
loop !i !x
| x == 0 || x == 1 = x
| i >= 10 = x
| abs dx <= 16 * m_epsilon * x = x
| otherwise = loop (i+1) x'
where
f = incompleteBeta_ beta p q x - a
f' = exp $ p1 * log x + q1 * log (1 - x) - beta
u = f / f'
dx = u / (1 - 0.5 * min 1 (u * (p1 / x - q1 / (1 - x))))
x' | z < 0 = x / 2
| z > 1 = (x + 1) / 2
| otherwise = z
where z = x - dx
-- Calculate initial guess
guess
| p > 1 && q > 1 =
let rr = (y*y - 3) / 6
ss = 1 / (2*p - 1)
tt = 1 / (2*q - 1)
hh = 2 / (ss + tt)
ww = y * sqrt(hh + rr) / hh - (tt - ss) * (rr + 5/6 - 2 / (3 * hh))
in p / (p + q * exp(2 * ww))
| t' <= 0 = 1 - exp( (log((1 - a) * q) + beta) / q )
| t'' <= 1 = exp( (log(a * p) + beta) / p )
| otherwise = 1 - 2 / (t'' + 1)
where
r = sqrt $ - 2 * log a
y = r - ( 2.30753 + 0.27061 * r )
/ ( 1.0 + ( 0.99229 + 0.04481 * r ) * r )
t = 1 / (9 * q)
t' = 2 * q * (1 - t + y * sqrt t) ** 3
t'' = (4*p + 2*q - 2) / t'
----------------------------------------------------------------
-- Logarithm
----------------------------------------------------------------
-- | Compute the natural logarithm of 1 + @x@. This is accurate even
-- for values of @x@ near zero, where use of @log(1+x)@ would lose
-- precision.
log1p :: Double -> Double
log1p x
| x == 0 = 0
| x == -1 = m_neg_inf
| x < -1 = m_NaN
| x' < m_epsilon * 0.5 = x
| (x >= 0 && x < 1e-8) || (x >= -1e-9 && x < 0)
= x * (1 - x * 0.5)
| x' < 0.375 = x * (1 - x * chebyshevBroucke (x / 0.375) coeffs)
| otherwise = log (1 + x)
where
x' = abs x
coeffs = U.fromList [
0.10378693562743769800686267719098e+1,
-0.13364301504908918098766041553133e+0,
0.19408249135520563357926199374750e-1,
-0.30107551127535777690376537776592e-2,
0.48694614797154850090456366509137e-3,
-0.81054881893175356066809943008622e-4,
0.13778847799559524782938251496059e-4,
-0.23802210894358970251369992914935e-5,
0.41640416213865183476391859901989e-6,
-0.73595828378075994984266837031998e-7,
0.13117611876241674949152294345011e-7,
-0.23546709317742425136696092330175e-8,
0.42522773276034997775638052962567e-9,
-0.77190894134840796826108107493300e-10,
0.14075746481359069909215356472191e-10,
-0.25769072058024680627537078627584e-11,
0.47342406666294421849154395005938e-12,
-0.87249012674742641745301263292675e-13,
0.16124614902740551465739833119115e-13,
-0.29875652015665773006710792416815e-14,
0.55480701209082887983041321697279e-15,
-0.10324619158271569595141333961932e-15
]
-- | /O(log n)/ Compute the logarithm in base 2 of the given value.
log2 :: Int -> Int
log2 v0
| v0 <= 0 = error "Statistics.Math.log2: invalid input"
| otherwise = go 5 0 v0
where
go !i !r !v | i == -1 = r
| v .&. b i /= 0 = let si = U.unsafeIndex sv i
in go (i-1) (r .|. si) (v `shiftR` si)
| otherwise = go (i-1) r v
b = U.unsafeIndex bv
!bv = U.fromList [0x2, 0xc, 0xf0, 0xff00, 0xffff0000, 0xffffffff00000000]
!sv = U.fromList [1,2,4,8,16,32]
----------------------------------------------------------------
-- Factorial
----------------------------------------------------------------
-- | Compute the factorial function /n/!. Returns +∞ if the
-- input is above 170 (above which the result cannot be represented by
-- a 64-bit 'Double').
factorial :: Int -> Double
factorial n
| n < 0 = error "Statistics.Math.factorial: negative input"
| n <= 1 = 1
| n <= 170 = U.product $ U.map fromIntegral $ U.enumFromTo 2 n
| otherwise = m_pos_inf
-- | Compute the natural logarithm of the factorial function. Gives
-- 16 decimal digits of precision.
logFactorial :: Int -> Double
logFactorial n
| n <= 14 = log (factorial n)
| otherwise = (x - 0.5) * log x - x + 9.1893853320467e-1 + z / x
where x = fromIntegral (n + 1)
y = 1 / (x * x)
z = ((-(5.95238095238e-4 * y) + 7.936500793651e-4) * y -
2.7777777777778e-3) * y + 8.3333333333333e-2
-- | Calculate the error term of the Stirling approximation. This is
-- only defined for non-negative values.
--
-- > stirlingError @n@ = @log(n!) - log(sqrt(2*pi*n)*(n/e)^n)
stirlingError :: Double -> Double
stirlingError n
| n <= 15.0 = case properFraction (n+n) of
(i,0) -> sfe `U.unsafeIndex` i
_ -> logGamma (n+1.0) - (n+0.5) * log n + n -
m_ln_sqrt_2_pi
| n > 500 = (s0-s1/nn)/n
| n > 80 = (s0-(s1-s2/nn)/nn)/n
| n > 35 = (s0-(s1-(s2-s3/nn)/nn)/nn)/n
| otherwise = (s0-(s1-(s2-(s3-s4/nn)/nn)/nn)/nn)/n
where
nn = n*n
s0 = 0.083333333333333333333 -- 1/12
s1 = 0.00277777777777777777778 -- 1/360
s2 = 0.00079365079365079365079365 -- 1/1260
s3 = 0.000595238095238095238095238 -- 1/1680
s4 = 0.0008417508417508417508417508 -- 1/1188
sfe = U.fromList [ 0.0,
0.1534264097200273452913848, 0.0810614667953272582196702,
0.0548141210519176538961390, 0.0413406959554092940938221,
0.03316287351993628748511048, 0.02767792568499833914878929,
0.02374616365629749597132920, 0.02079067210376509311152277,
0.01848845053267318523077934, 0.01664469118982119216319487,
0.01513497322191737887351255, 0.01387612882307074799874573,
0.01281046524292022692424986, 0.01189670994589177009505572,
0.01110455975820691732662991, 0.010411265261972096497478567,
0.009799416126158803298389475, 0.009255462182712732917728637,
0.008768700134139385462952823, 0.008330563433362871256469318,
0.007934114564314020547248100, 0.007573675487951840794972024,
0.007244554301320383179543912, 0.006942840107209529865664152,
0.006665247032707682442354394, 0.006408994188004207068439631,
0.006171712263039457647532867, 0.005951370112758847735624416,
0.005746216513010115682023589, 0.005554733551962801371038690 ]
----------------------------------------------------------------
-- Combinatorics
----------------------------------------------------------------
-- | Quickly compute the natural logarithm of /n/ @`choose`@ /k/, with
-- no checking.
logChooseFast :: Double -> Double -> Double
logChooseFast n k = -log (n + 1) - logBeta (n - k + 1) (k + 1)
-- | Compute the binomial coefficient /n/ @\``choose`\`@ /k/. For
-- values of /k/ > 30, this uses an approximation for performance
-- reasons. The approximation is accurate to 12 decimal places in the
-- worst case
--
-- Example:
--
-- > 7 `choose` 3 == 35
choose :: Int -> Int -> Double
n `choose` k
| k > n = 0
| k' < 50 = U.foldl' go 1 . U.enumFromTo 1 $ k'
| approx < max64 = fromIntegral . round64 $ approx
| otherwise = approx
where
k' = min k (n-k)
approx = exp $ logChooseFast (fromIntegral n) (fromIntegral k')
-- Less numerically stable:
-- exp $ lg (n+1) - lg (k+1) - lg (n-k+1)
-- where lg = logGamma . fromIntegral
go a i = a * (nk + j) / j
where j = fromIntegral i :: Double
nk = fromIntegral (n - k')
max64 = fromIntegral (maxBound :: Int64)
round64 x = round x :: Int64
-- $references
--
-- * Lanczos, C. (1964) A precision approximation of the gamma
-- function. /SIAM Journal on Numerical Analysis B/
-- 1:86&#8211;96. <http://www.jstor.org/stable/2949767>
--
-- * Loader, C. (2000) Fast and Accurate Computation of Binomial
-- Probabilities. <http://projects.scipy.org/scipy/raw-attachment/ticket/620/loader2000Fast.pdf>
--
-- * Macleod, A.J. (1989) Algorithm AS 245: A robust and reliable
-- algorithm for the logarithm of the gamma function.
-- /Journal of the Royal Statistical Society, Series C (Applied Statistics)/
-- 38(2):397&#8211;402. <http://www.jstor.org/stable/2348078>
--
-- * Shea, B. (1988) Algorithm AS 239: Chi-squared and incomplete
-- gamma integral. /Applied Statistics/
-- 37(3):466&#8211;473. <http://www.jstor.org/stable/2347328>
--
-- * Majumder, K.L., Bhattacharjee, G.P. (1973) Algorithm AS 63: The
-- Incomplete Beta Integral. /Journal of the Royal Statistical
-- Society. Series C (Applied Statistics)/ Vol. 22, No. 3 (1973),
-- pp. 409-411. <http://www.jstor.org/pss/2346797>
--
-- * Majumder, K.L., Bhattacharjee, G.P. (1973) Algorithm AS 64:
-- Inverse of the Incomplete Beta Function Ratio. /Journal of the
-- Royal Statistical Society. Series C (Applied Statistics)/
-- Vol. 22, No. 3 (1973), pp. 411-414
-- <http://www.jstor.org/pss/2346798>
--
-- * Cran, G.W., Martin, K.J., Thomas, G.E. (1977) Remark AS R19
-- and Algorithm AS 109: A Remark on Algorithms: AS 63: The
-- Incomplete Beta Integral AS 64: Inverse of the Incomplete Beta
-- Function Ratio. /Journal of the Royal Statistical Society. Series
-- C (Applied Statistics)/ Vol. 26, No. 1 (1977), pp. 111-114
-- <http://www.jstor.org/pss/2346887>
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