UNIV1"'RSITY OF NORTH CAROLINA Department f Statistics Chapel Hill, N. C. CUMULATIVE SUM CONTROL CHARTS FOR THE FOLDED NORMAL DISTRIBUTION by N. L. Jlmsn December 1962 Grant N. AFOSR -62..148 Methds f cnstructin f cumulative sum cntrl charts fr flded nrmal variates are described. These charts are likely t be useful \-Then the sign f an apprximately nrmally distributed quantity is lst in measurement. Sme assessment is given f the infrmatin lst by missin f the sign. This research was supprted by the Air Frce Office f Scientific Research. Institute f Statistics Mime Series N. 346
CUMUIATIVE SUM CONTROL CHARTS FOR TEE FOlDED NORMAL DISTRIBUTION by I N. L. Jhnsn University f Nrth Carlina 1. Intrductin. The distributin f the mdulus f a nrmal variable is knwn as the 'flded' nrmal distributin. It is reasnable t apply this distributin when tbe sign f a variable, which can be justifiably represented by a nrmally distributed randm variable, is irretrievably lst in measurement. Sme examples are given by Lene et al ~7_7. This paper als describes prperties f the flded nrmal distributin. It als includes a discussin f methds f estimating tbe parameters, which are further discussed by Elandt,['1_7 and Jhnsn ~3_7. Many standard cntrl chart prcedures are based n the assumptin that tbe bservatins used (either individually r as sample arithmetic means) in pltting the chart can be represented by nrmally distributed randm variables. It is evident that, n ccasin, it may be desirable t cnstruct cntrl chart prcedures when the apprpriate distributin is that f a flded nrmal variable. In this paper the cnstructin f cumulative sum cntrl charts fr such variables will be described. The methds used will be based n the ideas described by 2. Statement f Prblem. If the riginal variable (x', say) is nrmally distributed with expected value and standard deviatin g, then the prbability density functin f the flded nrmal variable x = lx' lis 1. This research was supprted by the Mathematics Divisin f the Air Frce Office f Scientific Research.
2 1[j2 2J - '2 G1 + (xl (J) e csh [~1C[ ]-, where 9 = F;,I(J We will cnsider tw situatins (a) where we are trying t keep F;, = 0, s that the riginal variable has zer mean (b) where we are trying t cntrl (J at a specified value (J In case (a) we suppse (J is knwn; in case (b) we suppse F;, is knwn. 3. Cntrl f Mean. If we have a sequence f m independent randm variables xl' x 2 ', x, m each having a flded nrmal distributin, then the likelihd rati f the hypthesis H l : (IF;, I = ~l) against the hypthesis H : (r;, = 0) is Hence the sequential prbability rati test discriminating between these tw hyptheses has its cntinuatin regin defined by the inequalities Ct l 12m l-a II ( ) Q <... II h [Qlx~/"" J < dn(~) + -2 1 2-1"'1 "n 1 _ a + '2 m 1. L" "n cs... v "\.h!lj.'el 1 ~. ~=l 0 where a. = Prraccept H l. IH. ] (i = 0, 1). 1 - -1. 1 Nw vie apply the methd described in ["2_7, regarding the cumulative sum cntrl chart (cscd as a reversed sequential test, with a very small value fr a l, and using nly the right-hand inequality f (3) which nw becmes, effectively (4)
3 T apply this methd we must (i) transfrm the bserved values xi t 'scres' y. = in csh rlx./u] and (ii) plt the cumulative sum ~.y. Then 1-1 ]. 1 change in the value f Is I frm zer t 0lu is indicated if any pltted pints 1\ fall belw the line PQ in Figu~e 1. Here 0 is the last pltted pint, tan OPQ = ~ Q~ and OQ = -in a ' s that OP = (-2 Kn a )Oi 2 -----"'It------:/'p Fig. 1 With this system f pltting Yi is never negative, s the value f ~Yi is nn-decreasing. T reduce the amunt f paper needed fr the chart it may smetimes be preferred t use, instead f y., the mdified scre 1 Then ~ Yi' = y. - 1: 0 2 ]. 2 1 y! is pltted and we simply use a hrizntal line, (- in a ) belw the 1 0 last pltted pint, as bundary. The need t transfrm bserved vlaues xi t scres, y. r y!, makes the ]. 1 cnstructin f the CSCC a little trublesme. Hwever a t~le f values f in csh X as a functin f X makes the task quite easy, while if a cnsiderable amunt f wrk is t be dne fr well-established values f 0l( = sl/u) and u, a special table f y = in COSh(Olx/u) can be cnstructed nce fr all. (It may be nted that when x is large, y - 0lx/u - fn2.)
4 4. Cntrl f Mean-Average Sample Number. We will restrict urselves t cnsideratin f the situatin when the true mean has in fact shifted t ~l = 0la. The expected number f bservatins befre this will be detected is apprximately (-in a )E- l where 1 2-1 =--0 +a 2 1!(2!n) 1 2 = - - 0 + 2 1!(2!n) 11"\2 CD e - '2...1 r v dx The integral can be evaluated as an infinite series using the expansin Hence c = I;) Y - "n 2 + E 2jO Y fi (_l)j+l J.-l e- -1, 1. 1 J= and
5 ( 6) ;:; ~l [J{2/rr.) { 2F( Q1) - l} ] - in 2 + I ( ~1) (Xl -1 + I: (_l)j+l {j(j+l)} I«2j+l)01) j=l where _ 1:02 (]) I(rO ) l =J(2/rr.) e 2 1. ~ J Finally we btain - 1:,l ( 7) E ;:; 01 [J{2/rr.} e 2 1 + 01 {2F(Ol) - ~ f- ] - in 2 + 1 - F(~l) Mills' rati An apprximate frmula fr E can be btained by using the apprximatin t The resulting frmula ( $) - 1:rl E ;" <_1_ e 21). J21c gives useful results fr &1 larger than 3/4.
6 Using further terms in the asympttic expressin (a = 1; fr i ~ 1) fr MillIs rati we get the frmal expressin where E. is the j-th Euler number. J Evaluating the cefficients in the asympttic expansin we btain (8)'... ] Sme values f E- l as a functin f 01 are shwn in Table 1. Values given by the apprximate frmulas (8) and (8)' (excluding the term 0.006030i5) are als shwn. TABLE 1. Values f E- l = (-in a )-1 x (Expected number f bservatins) 1 E -1 Values frm -1 Value frm E 1 (8) (8)' (8) r (8)' 0.25 (980) 1.25 2.95 2.93 0.50 73 5 (26.4) (1.12) 1.50 1.67 1.67 0.75 16.6 14.6 i'7'~ 1. 75 1.06 1.06 1.00 6.13 6.01 61'7 2.00 0.73 0.73 {~ J~ I
7 1 2 d When 9 1 is large, E... '2 B l - ~n 2 (fr ~l = 2, E = 1.36735 and ( ~ 9~ - in 2 = 1.30685). The average number f bservatins is then apprximately (-2 in a ) [~~ - in 4 J-l, as cmpared with (-2 in a)~i2 (= OP) fr the CSCC fr means f nrmal variables (see ["2_7). The average sample number fr the flded nrmal distributin is thus greater by a factr f apprximately. This may be regarded as reflecting the lss in infrmatin abut the signs f the variables in the flded nrmal. As wuld be expected, the rati tends t 1 as 01 increases. 5. Cntrl f Standard Deviatin. We nw suppse that ~ is knwn and that we wish t cntrl the value f rr at (J' If we desire t detect a-'change in the value f rr frm rr t rr (fr definiteness we will suppse rr l l > rr ) then the apprpriate likelihd rati is m 2 l( ms 2 + 2)( -2-2) l.: x. Cl -CTl CT m. 1 k 0 m csh(x.s/rri) ] = (~) e k= IT, 1 2 Cl l i=l csh(x.s/ct ) - 1 0 We will infer that there has been a change in Cl if the inequality ( 10) < - in ex ] is nt satisfied (the XIS being numbered backwards, starting frm the last bservatin). Examinatin f (10) is mre trublesme than applying the cnditin (4) fr the cntrl f mean value. Hwever, (10) can be written in the frm
8 (11) S if tables f 'scres' (j == 0, 1) are available we can plt E(y(l) - y(»against number f bservatins t frm a CSCC. T check whether there is evidence f a change in the value f cr we see whether any pints fall belw the line PQ (see Figure 1) where, as befre, 0 is 1\ the last pltted pint and OQ == (- in a). In the present case tan OPQ == {/ ( / ) 1 2( - 2-2) xn (J cr l - "2 ~ cr - cr l ( It shuld be nted that tan O~ can be negative. If s2 > @Kn(crO/cr l )] -2-2)-1 cr - cr l then the pint P will be t the left f 0, as in Figure 2, which exhibits a situatin where a change in the value f cr wuld be indicated at T """=-------jl{o Figure 2. Q b se,-v..ti..y\. yu)..w\.6 e -{ In the special case where we can assume that the mean (s) is cntrlled at s == 0, the prblem reduces t that f cnstructing a CSCC fr the variance f a nrmal ppulatin with knwn mean (in this case zer). This case can be cvered by the methds similar t thse described by Jhnsn and Lene ~5_7. It shuld be nted, hwever, that the methds described in~5_7 which use sample range cannt be emplyed when the sign f each bservatin is lst. There
9 is a tecbnically crrect way f intrducing range, even in tbis case, by assigning psitive and negative signs at randm (witb prbability 1/2 eacb) t tbe bservatins. Hwever, apart frm tbe labr invlved, tbe element f arbitrariness militates against acceptance f tbis metbd as a practical prcedure. 6. Average Sample Number under Mre General Cnditins. Tbe average number f bservatins needed fr tbe cntrl f mean prcedure (described in sectin 3) indicates a cbange in ~ wben tbe true value f ~ is g I (a general value) is (- in a )E, - l wbere E' :::: _1 ci + 2 1 _1012J ~(2rIr.) e 2 1 2 - -y 2 e csb ~ly in csb ~ly dy and "I ::: ~ 1/(J Tbe integral in tbis expressin can be evaluated in a similar way t tbat in (5), tbugb tbe result is a little mre cmplicated. Numerical evaluatin f tbis quantity, and f similar quantities needed t assess tbe perfrmance f tbe prcedure described in sectin 5, will be discussed elsewbere. Tbe present paper is intended t sbw bw cumulative sum cntrl cbarts can be cnstructed fr certain types f cntrl based n signless bservatins.
10 REFERENCES Elandt, Regina C. The flded nrmal distributin: ~lo methds f estimating parameters frm mments", Technmetrics, Vl. 3 (1961), pp. 551-562 Jhnsn, N. L. "A simple theretical apprach t cumulative sum cntrl charts", J. Amer. Statist. Ass., Vl. 56 (1961), pp. 835-840. Jhnsn, N. L. "The flded nrmal distributin: Accuracy f estimatin by maximum likelihd", Technmetrics, Vl. 4 (1962), pp. 249-256. Jhnsn, N. L. and Lene, F. C. "Cumulative SlJlU cntrl charts: Mathematical principles applied t their cnstructin and use. I", Indust. Qual. Cntrl, Vl. l8(12} (1962}~ pp. l5-2l.,, " d II li, d, Vl. 19(1) (1962).. pp. 29-36. " d III", do, Vl. 19(2) (1962), 'pp. 22-28. Lene, F. C., Nelsn and Nttingham ( 1961) "The flded nrmal distributin", Technmetrics, Vl. 3 (1961), pp. 543~550.