DATE ~~~ 6O 300 N1. nbc OF / END FJLIEEO

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1 / AD AO6O 390 SOUTHERN METHODIST UNIV DALLAS TX DEPT OF OPERATIONS ETC F/G 12/1 APPROXIMATIONS TO THE INVERSE CUMULATIVE NORMAL FUNCTION FOR US ETCH) ) JUL 78 8 W SCHMEISER N eC Ot,25 UNCLASSIFIED OREM Nt. OF / END Pu DATE ~~~ 6O 300 N1 FJLIEEO nbc a 1

2 r ~~ v. ~~~ ~ 7 ~ echnical / ~~~~~~~ ~~~~~~~~~~~~~~~~~~ J ~~~~~~~~~~~ TO THE INVERSE CUMULATIVE NORMAL FUNCTION FOR USE ON HAND CALCULATORS I: : Bruce W/Schxneiser 1 ii Departmen t of Operations Research and Engineering Management Southern Methodist University Dallas, Texas ~ ~~~ ~~~~~~~~~~ cr ~ stributiofl is utthnuted. ~~ ~~~~~ This research was sup ~~~ ted in part ~ y Office of Naval Research Con t t ~ NOOO14_77_C Ø42 ~ j NR ) // L_. ~~~~~~~~~~~~~

3 -- Intended for use on a pocke t calculator, Page (1977) gives an approximation to the cumulative normal function 2~~ _h/2 e ~ p( u 2 /2)du and its inverse x F 1 (p). Given here are two other approximations to the inverse Iunction which are easier to apply. - (It should be noted that more accurate, but also more complicated, approximations are well known. See, for -a ~ an ~ p1e, Abramowitz and Stegun (1972), equations and ) Given a fractile p > 1/2, Page s approximation to x, denoted x, is calculated in a three step procedure: 1. Set y ~.n{p/ (l p)}/ Set u = + (y ) 1/2 ~, /3 3. Set x = u - ( u) 1. For p < 1/ 2, 1 p is used and the sign of ~~ is reversed. A simpler approximation, based on results from Ramberg and Schmeiser (1972), is x = [p (1 p) which. ~.s valid for any fractile p. Additional accuracy for large values of x may be obtained using.2 + [p ~~~ (1 p) 09 )/ where p > 1/2. For p < 1/2, 1 p is used ari d th ~ sign of ~ is reversed.1

4 - Table 1 compares the accuracy of the approximations x, x, and ~~. x and x exhibit similar behavior, both being least accurate in the tail. x is more accurate than x for most values of x, but for many applica tions the simpler x may be suitable. The most consistent accuracy is obtained with ~~~, which has maximum error of about.015 and which is also simpler than x ~ to calculate. Table 1 Comparison of the inverse cumulative normal - -function approximations x, x, and x. x P(x) x x The value.9939 shown in Page (1977) is incorrect. ~~~~~ - ~~~ - : 2

5 REFERENCES Abraniowitz, M. and I. A. Stegun (eds) (1972) Handbook of Mathematical Functions, Washington, D.C.: U.S. Government Printing Office. Page, E. (1977). Approximations to the cumulative normal function and its inverse for use on a pocket calculator, 4ppl. Statist., 26, Ramberg, J. S. and Schmeiser, B. W. (1972). An approximate method for generating symmetric random variables, Commun. Ass. Comput. Mach. 15, J 3

6 ~ ~~~~~~~ -- ~~~~ -, ; ~~~ ~~~~ SECURITY CLASSIF ICATION OF THIS PAGE 1 )V)ie ~ D.ta, EnIet.d) ~ m ~~ D( ~~~~~ ~~ r ~~~ i III ~~~~~ A READ 1 ~ ~~ STRUCT O) ~ ~ S r.lr Vr\ I L ljluml1 ~ I ~~ I ~ I ~ ~~~~~~~~ BEFORE CO ~ :PLET1 ~~ G FORM 1. REPORT NUMBER 2. GoVT AcCESS ION NO. 3. R ECIPIEN S CATA LOG ~ i ~ j M BER OR.EN TITLE (an d SubIiIle) S. TYPE OF REPORT 6 PERIOD COVEREO Approximations to the Inverse Cumulative Normal Function for Use on Hand Calculators. Technical Report 6. P E RF O RMIN G O ~~~. RE PoRT Pi IJMBER 7. AUTHOR( S) 6. CONTRACT OR GRANT NUMBER (S) Bruce W. Schmeiser N C 0425 S. PERFORMIN G ORGANIZATION NAME AND ADDRESS IC. PR OGRAM ELEMENT. PROJECT ~ TA SK AREA 6 WORK UNIT NUM BERS Southern Methodist University Dallas, Texas NR II. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DAT E August 1978 Naval Analysis Program Code 431 Office of Naval Research 3. NUMB EROF PAGES Arlington, VA t & MONITORIN G AGENCY NAME 6 ADORESS(!/ different from Controlling Office) IS. SECURITY CLASS. (ci thu repcr ~ ) ISa. OE CLA!SI FICATIC ~. Z,OWNGRADIHG SCH EDULE 16. OI STPIEUTION STATEMENT ( ~~ ( this Report) Distribution of this document is unlimited. 17. DISTRIBUTION STATEMENT (of the abstract entered in Block 20, If different from Report) Ia. SUPPL EMENTARY NOTES 9. KEY WOR DS (Cont inue on reverse aide if necessary and identify by block number) Normal distribution Inverse distribution function 2O ~ ABS ~~~ ACT (Continue on reverse side II necessary nd,dentily b ~ block number) Two approximations to the inverse distribution function of the normal distributions are given. Both approximations dominate, in terms of accuracy and case of impinmentation, a method recently proposed in the literature. I DD ~~~ 1473 EDITIoN o I NOV6S IS OBSOLETE unclassif ied SECJDI7Y C_ AS S IC ICA ION OF ~~~~~~ ~~~~~ ~~~~~~~ S ara nie,.d) - - ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~