An Efficient Polynomial Approximation of the Normal Distribution Function & Its Inverse Function

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1 A Efficiet Polyomial Approximatio of the Normal Distributio Fuctio & Its Iverse Fuctio Wisto A. Richards, 1 Robi Atoie, * 1 Asho Sahai, ad 3 M. Raghuadh Acharya 1 Departmet of Mathematics & Computer Sciece; Faculty of Sciece & Agriculture St. Augustie Campus; The Uiversity of The West Idies. Triidad & Tobago; West Idies. Departmet of Mathematics ad Statistics The Pesylvaia State Uiversity, Harrisburg, USA 3 Departmet of Statistics ad Computer Sciece, Aurora s Post Graduate College; Osmaia Uiversity, Hyderabad (Adhra Pradesh); Idia. Abstract I this paper, we propose a sigle-polyomial approximatio to the ormal cumulative distributio fuctio, as also that of the iverse of the ormal cumulative distributio fuctio too, usig this polyomial. Our approximatio has sigificatly less absolute error of approximatio relative to other popular approximatios available i the literature, icludig the recet improved approximatio achieved by Aludaat K. M. & M. T. Alodat (8). This paper is motivated by the powerful polyomial approximatio operator of Sahai (4), which uses a probabilistic approach. We compare all the competig approximatios empirically, relative to the relevat exact values, via calculatig their respective Percetage Absolute Relative Errors. Keywords: Feller fuctio, ormal distributio, probabilistic polyomial-approximatio operator. MSC1: 3E3. *Correspodig Author s sahai.asho@gmail.com 1 P a g e

2 1. Itroductio The problem of approximatio arises i may areas of sciece ad egieerig i which umerical aalysis ad computig are ivolved. I 1885, Weierstrass proved his celebrated approximatio theorem: if f ε C [a, b]; for every δ > ; there is a polyomial p such that f - p < δ. This result mared the begiig of mathematicias iterest i polyomial approximatio of a uow fuctio usig its values geerated experimetally or ow otherwise at certai chose Kots of iterest i the domai of relevat variable. Later, Russia mathematicia S. N. Berstei proved the Weierstrass approximatio theorem i a maer which was very stimulatig ad iterestig i may ways. He first oted a simple but importat fact that if the Weierstrass theorem holds for C [, 1], it also holds for C [a, b] ad coversely. Essetially C [, 1] ad C [a, b] are idetical, for all practical purposes, as they are liearly isometric as ormed spaces ad order isomorphic as algebras (rigs). Perhaps the most importat cotributio i Berstei s proof of this theorem cosisted i the fact that Berstei actually displayed a sequece of polyomials that approximate a give fuctio f(x) ε C [, 1]. If f (x) is ay bouded fuctio o C [, 1], the sequece of Berstei s Polyomials for f (x) is defied by: ( ( f ))( x) B w f, ( / ) * (1.1) P a g e

3 Where, = are the respective weights for the values f (/) of the fuctio at the ots (/) [ = (1) ]. The most sigificat fact to be oted, at this stage, is that ay polyomial is such a ice cotiuous fuctio. The polyomials are ifiitely itegrable ad may be differetiated arbitrarily may times till they cease to exist. Thus, our proposed sigle-polyomial approximatio of the ormal distributio fuctio will ot oly be more efficiet tha existig approximatios but will also be very easy to calculate, eve with a pocet calculator. We proceed to itroduce this i what follows. Let X be the stadard ormal radom variable, i.e., a radom variable with the followig probability desity fuctio. f (x) = (1/ (π)) exp (-x /); - < x <. (1.) Hece, the distributio fuctio [F(x) = P (X < x)] for the stadard ormal radom variable X is: F( x) - x (1/ ) exp (- t / ) dt (1.3) The aim of this paper is to achieve a sigle-polyomial approximatio for approximatig F (x) i (1.3) above, ad the iverse distributio fuctio F -1 (x)= IF(x) (Say), for the stadard ormal distributio. 3 P a g e

4 . The Efficiet Probabilistic Polyomial Approximatio Operator of Sahai (4). We ow that for the stadard ormal distributio, the value of the distributio fuctio F (x) for x=3. happes to be: F (3.) =.9987 = (1/ ).exp( x / ) dx (.1) Hece, for our proposed polyomial-approximatio the target is effectively: 3. (1/ ).exp( x / ) dx 1 (3/ ).exp( (4.5) * x ) dx (.) Now, this coforms to the ambit-iterval C [, 1] of Sahai (4) s computerizable quadrature-polyomial formula usig the probabilistic approach. It is desirable, for ease of referece, to detail here the geesis of this simple probabilistic polyomial approximator to be used for our target as i (.1), above. As such, the iterval of itegratio happes to be [, 1], while x = ad x = 1. We cosider the equidistat odes: x i = (i/); i=, 1,.... (.3) Now, cosiderig the lie [, 1], let us visualize a radomly sittig poit x o it. It is obvious that the probability of a poit o this lie beig less tha x (o its left, o the lie) is x, whereas the probability of a poit o this lie beig more tha x (o its right, o the lie) is 1 - x, i.e., P(X < x) = x ad P(X > x) = 1 x. (.4) 4 P a g e

5 Hece, the expected umber of poits out of equidistat-poits o the lie which are o the left of the poit x (or smaller tha x) will be x, ad the expected umber of poits out of equidistat poits o the lie which are o the right of the poit x (or greater tha x) will be (.x) or equivaletly (1 x). Now, to devise the weight fuctio A (x) associated with the ode x, we simply place it i the shoes of x. However, we ow that accordig to our choice of the + 1 odes i (.3), for ay ode x there are odes o the left of the ode x, ad that there are ( ) odes o the right of the ode x. Cosequetly, i this probabilistic setup, the probability of our choice of the ode x is x.(1 x) * / x.(1 x) * = A (x) [As 1] (.5) The equatio i (.5) might well be expressed i terms of the well-ow Gamma fuctios for computatioal purposes to accommodate ay real value of x i [, 1]. Therefore, the probabilistic polyomial approximator for the distributio fuctio F (x), as i (.) (resultig from usig the aforesaid probabilistic perspective of polyomial approximatio) is simply: F (x).5 + x ( x) * f ( x ). dx; wherei f (x ) = 3.(1/ ) exp( (4.5)*( x ) (.6) A The last itegral i (.) has o closed form. Most basic statistical boos give the values of this itegral for differet values of x i a table called the stadard ormal table. 5 P a g e

6 From this table we ca also fid the value of x whe Φ(x) is ow. Several authors gave approximatios usig polyomials (Chori, 3; Johso, 1994; Bailey, 1981; Polya, 1945). These approximatios give quite high accuracy, but computer programs are eeded to obtai their values ad they have a maximum absolute error of more tha.3. Oly the Polya's approximatio F (x) =.5*[1+ (1- exp ((-/π)*x )] (.7) has oe-term to calculate, while the others eed more tha oe term. They are reviewed i Johso et al. (1994) as follows: 1. Let F1 (x) 1.5*( a 1 + a x + a 3 x + a 4 x 3 + a 5 x 4 + a 6 x 5 ) -16 ; wherei, (.8) a 1 = , a = , a 3 = , a 4 = , a 5 = , ad a 6 = Let F (x) 1- (. π) -1/ * exp (-.5 x -.94 x - ), [x 5.5/ Thus. excluded!] (.9) 3. Let F (x) exp (*y) / (1 + exp (*y)), y =.7988.x ( x ). (.1) 4. Let F3 (x) 1.5*exp [- (83*x +351)*x + 56) / (73/x + 165)]. (.11) 5. Let F4 (x).5*[1 + (1 - exp (- (π/8)*x ))] (.1) Whereas the approximatio i (.1) was proposed by Aludaat ad Alodat (8) as a improvemet of that by Polya s i (.7), all the other approximatios eed computer programs to calculate, sice their iverse fuctios are quite itricate ad implicit. 6 P a g e

7 Now, usig our probabilistic polyomial approximatio as i (.6) with =8 [i.e. 9 ots], we get the 8 th.-degree polyomial: A ( x) * f ( x ) = x x x x x x x x 8. (.13) Ad hece, usig (.6), we get the followig sigle-polyomial approximatio to the distributio fuctio of the stadard ormal, a ith-degree polyomial: 6. Let F5 (x).5 + x A ( x) * f ( x ). dx= x x x x x x x x x 9. (.14) Now, as metioed earlier i the itroductio, we will tae up the approximatio of the iverse distributio fuctio F -1 (p) [F (x) = p F -1 (p) =x (where p 1)]. This will have may applicatios i practical situatios. Oe such applicatio will be i geeratig radom x-values for stadard Normal variate. The probability p ( p 1) may be geerated usig a radom-umber geerator from the Uiform Distributio U [, 1]. Suppose we would have geerated p 1, p, p 3,, p that could be used to geerate the stadard Normal Variates: { x α ; α = 1(1) }, usig the iverse distributio Fuctio for the Stadard Normal distributio, amely F -1 (p α ) [α = 1 (1) ]. Therefore we ow cosider the approximatios to F -1 (p) i what follows. As F1 (x) i (.8) would have ifiite terms, it could ot be expressed i a closed form via a fiite degree polyomial. 7 P a g e

8 I a Closed form, it would very implicit ad tedious to geerate a good approximatio to the iverse fuctio, F -1 [1] (p). Hece we cosider oly the approximatios to the iverse fuctios, say F -1 [I] (p); I = (1) 5 i (.1), (.11), (.1) ad (.14), as follows. F -1 [] (p) = Real Root [Betwee to ] of the equatio: ~ x. ( x ) = {log (p) log (1-p)}/. (.15) F -1 [3] (p) = Real Root [Betwee to ] of the equatio: ~ (83.x + 351).x ((73/x) + 165). (log (-.p)) =. (.16) F -1 [4] (p) = [- log (1- (.p-1) )/ (π/8)] (.17) Ad; F -1 [5] (p) = Real Root [Betwee to ] of the equatio: ~ F5 (x) -.5 = p. [F5 (x), as per (.14)] (.18) 3. A Numerical Compariso of the Approximatios to F (x) ad F -1 (p). I this sectio, we compare the exact value of F (x) with its approximated oes, amely F1 (x), F (x), F3 (x), F4 (x), ad F5 (x) [As per their expressios i equatios (.8), (.1), (. 11), (.1), ad (.14), i the precedig sectio. The compariso is afforded per their umerical values so calculated vis-à-vis the exact value of F (x), for each of the illustrative example-values of x (=.1,.3,.6, 1., 1.5, ad.), respectively. These values are tabulated i the Table A.1 give i the APPENDIX. The followig Table A. displays the values of Abs. Per. Rel. Error [APRE] For Various Approximatig Fuctios F ( ) (x). 8 P a g e

9 Wherei; APRE [F (J) (x)] = F( J )( x) F( x) *1% F( x) ; J = 1 (1) 5. The most favourable approximatio F ( ) (x) s value/ APRE value has bee highlighted. It is quite evidet that our proposed approximatio F5 (x) is doig rather well, ad is cosistetly better tha that by Aludaat ad Alodat (8) approximatio F4 (x)! Similarly, we compare the exact value of F -1 (p) with its approximated oes, amely F -1 [] (p), F -1 [3] (p), F -1 [4] (p), ad F -1 [5] (p) [As per their expressios i equatios (.15), (. 16), (.17), ad (.18), i the precedig sectio. The compariso is afforded per their umerical values so calculated vis-à-vis the exact value of F -1 (p), for each of the illustrative example-values of p (=.53988, ,.75747, , , ad.9775), respectively, tabulated i Table A.3 i the APPENDIX. The followig Table A.4 displays the values of Abs. Per. Rel. Error [APRE] for various approximatig fuctios F -1 ( ) (p). Wherei; APRE [F -1 (J) (p)] = = F 1 ( J )( p) F F 1 1 ( p) ( p) *1% ; J = 1 (1) 5. The most favourable approximatio F -1 ( ) (p) s value/ APRE value has bee highlighted; maig it evidet that our proposed approximatio F -1 [5] (p) is doig rather well, ad is cosistetly better tha that by Aludaat ad Alodat (8) approximatio F4 (x)! 9 P a g e

10 Refereces [1] Aludaat, K.M. ad Alodat, M.T. (8). A ote o approximatig the ormal distributio fuctios. Applied Mathematical Scieces;, 9,45-9. [] Bailey, B. J. R. (1981). Alteratives to Hastig's approximatio to the iverse of the Normal cumulative distributio Fuctio. Applied Statistics, 3, (3) [3] Chori (3). A short ote o the umerical approximatio of the stadard ormal cumulative distributio ad its Iverse. Real 3-T-7, olie mauscript. [4] Johso, N. I., Kotz, S. ad Balarisha, N. (1994); Cotiuous uivariate distributio s. Joh Wiley & Sos. [5] LeBoeuf, C., Guegad, J., Roque, J. L. ad Ladry, P. (1987) Probabilites Ellipses. [6] Polya, G. (1945) Remars o computig the probability itegral i oe ad two dimesios. Proceedig of the first Berele Symposium o mathematical statistics ad probability, [7] Reyi, A. (197). Probability theory. North Hollad. [8] Sahai, A., Jaju, R.P., ad Mashwama, P.M. (4) A ew computerizable quadrature formula usig probabilistic Approach. Applied Mathematics ad Computatio. 158; P a g e

11 APPENDIX. Table A.1: Values of Various Approximatig Fuctios F ( ) (x) & Actual Value of Normal Feller Fuctio F (x). x-values Apxg. Fs. F (1) (x) F () (x) F (3) (x) F (4) (x) F (5) (x) F (x)-values: Table A.: Values of Abs. Per. Rel. Error [APRE] For Various Approximatig Fuctios F ( ) (x). x-values Apxg. Fs. APREF (1) (x) APREF () (x) APREF (3) (x) APREF (4) (x) APREF (5) (x) Table A.3: Values of Approximatig Iverse Fuctios F -1 ( ) (p) & Actual Value of The Iverse Fuctio F -1 (p). p-values Apxg. Fs. F -1 () (p) F -1 (3) (p) F -1 (4) (p) F -1 (5) (p) F -1 (p)-values: P a g e

12 Table A.4: Values of Abs. Per. Rel. Error [APRE] For Various Approximatig Fuctios F ( ) (x) p-values Apxg. Fs. APREF -1 () (p) APREF -1 (3) (p) APREF -1 (4) (p) APREF -1 (5) (p) F -1 (p)-values: P a g e