UNCLASSIFIED

AD NUMBER

AD003584

NEW LIMITATION CHANGE

TOApproved for public release, distributionunlimited

FROMDistribution authorized to U.S. Gov't.agencies and their contractors;Administrative/Operational Use; FEB 1953.Other requests shall be referred toCommanding Officer, Office of NavalResearch, Arlington, VA 22217.

AUTHORITY

ONR ltr, 26 Oct 1977

THIS PAGE IS UNCLASSIFIED

THIS REPORT HAS BEEN DELIMITED

AND CLEARED FOR PUBLIC RELEASE

UNDER DOD DIRECTIVE 5200.20 AND

NO RESTRICTIONS ARE IMPOSED UPON

ITS USE A" D.SULOSURE,/

DISTRIBUTION STATEMENT A

APPROVED FOR PUBLIC RELEASE;

DISTRIBUTION UNLIMITED,

I Reproduced 6y2

Armed Services Technical Information agencyDOCUMENT SERVICE CENTER

KNOTT BUILDING, DAYTON, 2, OHIO

UCASFE

IIP

-l .... -____________

UNLAS FIE

ECHanIaL RspoR No.. 4

PREPARED UNDR COMMRCT Nonr-410(O)

O1"FICE Or, NA.L RE5tARC!!

K D~UFAM'ZUrT OF ?!ATIULATICS-5

___ IMMST

S3*M RESULTS ON T1UNCATD LIF TFSTSIN THE EXPONTIAL CA3E

by

Benjamin EpsteinDepartment of Matheatics

Wayne University

In this report we consider life tests which are truncated as follows.

n items are placed on test and it is decided in advance that the experiment

will be terminated at m (X 9 T ), where X in a random variableequal to the time at which the ro

0th failure occurs and To is a truncation

time, beyond which the experiment will not be run. Both ro and To are

assigned before experimentation starts. If the experiment is terminated

at Xron (ioe, r. failures occur before time To ) then the action in terms00

of hypothesis 'Y sting is the rejection of some specified null-hypothesis.

If the experiment is terminated at time To (i.e., the r. 'th failure occurs

atter time TO) then the action in terms of hypothesis testing is the accept-

anee of some specified null-hypothesis. While truncated procedures can be

considered for any life distribution, we limit ourselves here to the case

where the underlying life distribution is specified by a p.d, f, of the ex-

ponential form (), f(x;e) x e , > 0, 9 > 0, Two situations asr con-

edaered° The first is the non-replacement case where a failure when it occurs

during the test is not replaced by a new item. The second is the replace-

ment ca-i where failed items are replaced at once by new items drawn at

(1) 'hte practioai Justification for using this kind of distribution asa first apprnximation to a numbumr of test situations ip discuseedin a recent paper by Davis '213

F.(r), the expected number of observations to come to a decision; E (T),

•bhe expocted waiting time for reaohing a deciaion; and L(O), the probubili-y'

of aecepting the hypothesis that E - -o (90 being the value associatc %ith

the null-hypothesie) when 0 is the true value. Some procecuree are workWd

out for finding truncated tests meeting specified conditions and a practical

illustration is given. Detailed tables will appear else%ere. Truncated

tests considered as special cases of sequential procedures will also be

treated in another place.

TT. The Derivati of a Truncated Test in -the Non-ReDlasement Case

Let n itema drawn at random from a population be placed on life

test, Let the underlying p.d.f, of life be of the form f(x;9), Items

that fail are not replaced and the experiment is truncated at time (2)

min (Xr T0) , where X is the time when the r0 'th failure occurs

and ro and will be taken as preassigned. T. is a truncation time beyond

which the experiment , oee not run. If we define F(To;0) in the ustal way

as F(T o9) I f(x; E) dx, then it follows at once that the probability

of reaching a decision requiring exactly k failures is given by

. -) (n, k 2v °,p ro1l

and

(2) Pr(r-pr e) a 1 - Pr(rokle),

(2) For convenience we consider the variete to be time. It is perfectlyclear that it can be other things depending on the physical applicationone is concerned with. Generally the mt iate (e.g., it it is time) willbe non-negative.

Fu.r*her tile cxoted numbcr of observations to 'each A decialo. !.Lr t -

0

The fortaula for Ee(T), the expected waiting time for reaching a deoision is

giver, by (3)n

Suppose the t:'unc .Ao3 rule is such that a hypothesis Ho aaso~iated with( 4 is accepted i mm (X ,To) To, I.e., in the particular EsPiple

is more than T 0o Then if t(8) is defined as the probability of accepting

I e - owhen 8 is triie, it .follow that

r -1

(5 L() Pr*(r ) +?-(n'.,(X,,

In the special case where the p.d~f, of the life of items is fven1- -x/6b7 £x;G - x > 0, 9 > 0, the formulae (2.)-(5) become substantially

simpler. This is in particular true for E()o For th? exponential denaity

(3) S t should be noted that there is an essential difHernce betweenGPisake e) and Pr(rnkl O) for < <n Pr*(r-k ) is sinply the

probabiity that exartle i out o n failures will ocu in the inter-val (0, T*) while Pr(rMk ) is the probability tot a cp willbe eached after xaotly kc failures are obteha. Cleary from thedei)ton of the trunation 1 oc'ue Pr(rmk'G) - (r1 8) forO<k<r0 -1o Further fo <-ke) .for r l i< sn.

Pr(r-kBL) - 1 0 , 1, 1 .o, ro- '

andr -l

(2')1 ~ ~) f-1 Pz(r-kI e).

In the case vhere the underiying p.d.f. is exponwtial, (3) becomes

(31) . r- k b(k;n, p) + ro 1 - b(kjn, p 91

Where p. I-e aih b~jnqpq) _ (n). (I -kGk -O

It can be readily shown that (3') simplifies to

(3) E. a np [ t b(kJfPe] 0 311i l b(kjn,p ,

This is in a conveniorsi form for calalation, For any preassigned n, TO,

and ro, ES(r) can be found easily from the Binomial Tables [8 or the

Tables of the Incomplete Beta Function F6].

We now derive a simple formula for Eq(T) in the exponential case,

We first note that for any f(xee), E9(Xo.n), the expected waiting time for

the ro thf- lure, is given by

r.-1 n00

comparing (4) and (6) it is clear that

(7) E (T) -Ee(Xr ,)* Pr(rnkle)FC To - e(Xr n a)0 R

Formla (7) is perfectly general. Let us now make use of the properties of

an exponential p.d.f. to simplifY (7), This is best done through two lemma

Lorna 1: If Lo underlying p.d.f, of the life distribution ie f(x;e) = , * /Xx e O 9 > o, then the conditional expected waiting tbne for the ro th

failure in a sampie of size n given that exactly k failures, 0 < k < ro0 -l,

have occurred by time To is given by

(8) Eq(Xro,n Ir k) -T O + Eq(Xroknk), k- 1, 2, oa ro-

in (8)2 -' k) a E'inr () Oxnl k 8 ("P~n 001 <k~ To* X1,, -> o)p k 0- , i 2,9. ro-

is the conditional expected waiting tim for which we seek a f ormula

Ee(Xroknk) is the unconditional expected waiting time to get the

(r-k) 'th failure in a random sample of sise (n-k)0 The proof of Lena 1

follows directly from rem.ts in [33.

Lenma 2." e~r~nk EeEeXon (eX:) 0 <k < rEno.No XYrk,rk) aO(Xrn) - G(Xkn) ak

where ie(xo s i defned a zero for all n > 1. The proof of L=ma 2 is

immediate. In E2] it was shoi-ni that

(9 e .o1)- , 1,.0

0 , n n ~

Th',s for any integer k such that 0 < k < ro, it follows (subject to the con-

vention about k 0 0) that

(1o) e(X,.o,. e9 X,) E ( Xoo 'E -k)o

Thus Le=a 2 fo3 1,wso

U:Ling both TLenns 1 and 2, (7) becomes

(7t) *() -e(Xron) o%( O(Zkn)i

r- -1 . -S P r(r-k*)Eg(XPkn) + -kI)] E (X

- , Pr(zk~0) EXkn)9 since e(Xo n ) = o"k-1

Thus, in the exponential case, formula (4) frr B, (T) simplifies to (7').

We remark paranthetioally that for an mderl.ying life distribution

(including the exponential), the c.d.f. of the waitinvg time T, 0 8 (t), ii

given by

(11) G( - Pr(T< tl ) - r(Xr n _ f t < To

"1, ift > T0

This result am, be useful in finding out mre about the waiting time that

just its expectation.

In a practical situation one might want a truncated tot without

replacement which has the folloAng properties:

(i) T is preassigned.0

(ii) The OoC., curve should be auch that L(80 ) > z and

L(81 ) _< . e and 91 are preasined and e > el

It is quite easy to accomplish this since oonditionv (i) and (ii)

mean in effect that we are den1ing with a binomial situation in which we

are testing p - 1 eTo/% again.st P a 1" e / with L(p ) Z 1 - '.

an! L(p,) T ° 3.ated in binomial terms, we are seeking a sample size n

and a rejection number r such that we will accept the hypothesis that0

L

p Po U the number of defeatives (failures) in the sample is < r 4.

The hypothesis that p a p0 will be rejected if the number of defectives

in the sample of size n is > roo The detailed ealcalutions in any given

situation are greatly facilitated by the Binomiral. Tables C 83 or Tables

of the Ire."mplete Beta ftnetion £63. Tables of practical interest rill

appear elsewhero.

III, Form la.e in the Replacezent Cae

In bhis section we assnme througout that the underlying p.d, f

of the life of item is given by f(x;S) a...x ,>0,8>O0, 'ho teat

ia started with n ites and ary item that fails is replaced at once by a

new item drawn at random from the underlying p.d.f The experment is

truncated at time min(Xr $n9 To), where r,n is the time (measured from

the beginning of the entire experiment) when the rogth failure occurs and

To is a truncation time beyond which the experiment does not run. It is

then easy to show that the probability of terminating the experiment after

exactly k failures have occurred is given byTO/" (n TO/9) k, k ,1

(2) Pr(r-k o) I -k, 0, 1 2 , . 1

and

(13) Pr (r - r.9 - Pr(r-0kI)0ka

The expected number of observations to reach a decision is given by

k -%

*, ' /kJ and 9̂ n r/e 0

-8-.

It ,:a be readily shown that. (Vi) 9Lmpn.fies to

(E5 (r)- p(k~ j r0)j ki

Th i is ins a tonvenlrt form for computation. For any prossi d nT ard

r> E(r) can be found easily from Molina's tables on the Pois.aon diutri-.

bution . or from the tables on the Incomplbte r.-fun,.tion .

The expected waiting time E(T) is given hy a particularly uimple

rormula, .it). a_. in Lhe non-repl=ement case It .ne bo shon that

(16) E (T) B( )o 1 0 .,,nrk

For u.a (1Y.6) ,:an be nimpliflf.d )y us-iiht come, ppePe of the exponent.ii.1

p,.cifo 'Nn lemas are i.u- sSated,

L2ML : Let the underiyUiis podof,. of the lipe distribution be f{x )- . e

00- O, 0 Let n ite.,s he drawn at random .Vrvm this p, d. f, axd put. on

ltfe test. o et an item that fails be replaced at orce by a new item drawn

from the underlyi ng pd.fo then the conditional expected waiting tie for-the re 'th failure (time measured from the beginning of the experiment)

given that exactly k failures, 0 < k < ro -1 have occurred by time '.7 11

given by

(17) z n rk) - T * E gX oo , .-2

(rok)(1

00

,I.m. . E p o

- 0 .,n 0

This is a conn~quence oC t',o fact that. U (X ) * s9 E/ for any ienteger 9, 3 n :f

L ~

From Lema 3 and 4, (16) beome

r -1 F (r,-k)8

Xe=L k r.rk(e) + ro (1- P(r-ki

,TIC

Also in analogy with formula (7') in the non-replacement case, one can write

(18') ES(T) " "1 Pr(r-kl 0) Se(Xk,)

as the fozuaa which expresses the expected waiting time for the twuneated

procedure in terms of the uconditional waiting times to get the. firt ro

failures.In I _799 EX. .,. + 1_ )(non-replacemet).

8 kon ken n-I.

In (18,)9 .e(X - (replacmnt).

Also Pr(r-ke) is given respeotively by (1'), (2') or (12), (13).

Formula (18) bears a strong analogy to Wald's fundamental identity

in sequential analysis in which it is shown that umder suitable conditions

f41~ ~(~'~ * (Z)E(n),, The Zile anL identically distributed randamn

variables and n is the uallest integer for which a < _ Zi < b is notiml

satisfied and ,.'ere a and b ame preassIgned constmnts, There is theimportant differenne, however, that in the Wald case information becomes

available in disorete amounts, wherea in the life test situation informa-r tion becomes available contimuously. In the Wald jase a decision can be

made only after some integral number of observations has been taken. In

the present case it is possible to stop, hmwever, id it takes too long to

.~I0

make the k~th observation, 0 < k < r.. In & life toot, information becomes

available in auch a way that one does not actuaLly fail, a first item, if

it takes too lo g to wait for the first item to failo More generally, if

one has k fai-.ures and a decision has not yet been reached to either acc pt

or reject, then ona dues not actually fail the (kl)st item if it talkeF3 1

long to get it. Much nore %dll be said about this question v hon i treat

uequentialized life tent.

Ir - pratttcal eituation one might want to find a truncated test

with roplacement which has the following properties:

(U) T0 is preassigned,

(ii) The 04,o curve should be uch thatT(90 ) ? 1 - a arid L(eI) I '

where 0 and 01 are preassigned and % > :1 Viewed as a test of or& Pol6con

parameter 1- i/a against another , i/e (the "is are expe:ted number0

of failures per unit time), this is equivalent to finding a test procedure

which will ental. observing a Poisson process having average occurrence

= rate 1/0, for a length of time equal to nT and nkin% the action

of accepting H if the numbeir of occurrences oose-'",d in the time nTo iv0

< r o -1 and 'rejeoting H if the number of occurr;acee in the tire nTois> ro

Thc; details for carrying out the foregoing in any given situation

are facilitated by Molina's tables C 5J and the tables on the incomplete

r fmationj o Using these tables, suitable integers n and r..o can

always be found so as to make L(e) I a and L(e,) < 0. Several tables

dealing with the truncated replacement test have been peepare" ard ll].

appear olsewhere°

IV. A Test Which Is Not Truncated At A Fixed Tire To

In C-21 it was proved that the "best." r-gion of acooptace for

11o in the Neyman-Pearon sense, in the non-replacement case, for t~otin

a hypothesis Ho that 9 - 9 asainst alternatives of the form e -

(8o > 0) based on the first r out of n ordered observations from an

exponential distribution is of tho rorm > C, whem

rri

r,nr

Both r and n are preassigned integors. It is easily verified that

(20) xl,,,+O2 OXrn+(n-r)zn-nxj,n(n-l) (x2,n.-l n)%Oo+(nr~l)(xr,n=xrl,n)'

Introducing l-ew random variables definedb

(21)), ,, , , .

it is elea that 0 > C can be rewrittenl asran

(22) >i PC. *~

It is now asserted that (22) carries with It the implication that the test

is truncated. This is evident sinoe the Is are positive random variables

whioh are monotonically non-decreasing as time goes on. More precisely the

experiment wil). be truncated at time tI (with acceptance of He) if no failures

have occurred by time t1 - r C/n. More gereral!, suppose that i failures

(I1. i < r-1) have occurred, without reaching a decision; i.e., supposei

that . k < zC, then the experiment can be trumnated before the (i+l)vtk-l

failure occurs, if the time t I between the i th and (i + i)st failure is sucht~iat

(2-3) t' k, r-3.

Trw-ation wokid occur after an add.tion.,J wpjut-:r. Ame ot .*, be.ynd tho

t UIV x, (whton the i. th fnilurq. Tcr,,), f ;,.one of 1 ro :inequalties (23)

are tre2 'oo any 0 '. i < r-I (def il, l ' Or if i 0) then tho

experiment will be termina ,.d aftpr the occurrence of the first r failure

in t.his Ca!--

(24) I. < rC

The action taken on the ba~i., of (21+) is the rejection of H , ad

the total time required before taking this action would be xr n

We now proceed to derive cortain Mopertien of the test based arnTo do so we recall, [2J, that the are independent randm variabls,

each of which is dlstribu'-,ed with the same p.d1f% . x > 09 0 > 0e

Thuo the problem has been redueed to one to which the theory of section

can be applied. From the theory developed there . L fotlours at once t t).

(25) Pr( Pm k)) p'k; I r 09 1, 2, i

and

In (25) and k26) - C/O ad p(k; .; e

Thus in anaelog with (14) and (15) we have

2)) k P-r kI0)L : I:"J

Further E,(T), the expcted waiting time to reach a decision, ,in be

writter) as

(28) Es( ) P r( k~ 9) EO(Xkgn)

where Pr( *kit 9) Is given by (25) and (26) WAri Ee(Xkn) i., given by (9)

with r replaced by k,0

Finally L(e), the probability of aocepting B - 9 when 9 is true0

r-1is given by L(e) & k p(k! L),

Up to this point in the prebent section we have been treating the

non-replacement situation, It is interesting to see what happens if failed

items are replaced at once by new items drawn from the p.d.f. a • AsB

in Section 3, let xk n be the time when the k th failure occur- (whether

it be an original item or replacement itsm) measured from the beginning of

the experiment. It can be shown, in the replacement case, that if one starts

with n items, then the beit, region of acceptance for R in the Neyman-Pearson0

sense for teoting a hypothesis H that 9 = e against alternatives of the formx ise a 91 (e > 90) based on the first r failure times x, ' °° rn

of the form' > 0, ti now equal to

Thu the region of acceptance for H is of the form x > C*'0 rn

But this means , of course, that we are dealing with a truncated test,

2 > C* as a region of acceptance means in wo ds that the test is terminatedrn

at min (Xrn C) with acceptance of H if truncation occurs at C* and• 0

rejection of Ho if truncation occurs at X r.o Thus the theory of Section 3

Is completely applicable,

V. ANumerical Exaple

As an illustration of the theory we consider three test procedures

which have essenttally the same OC. curve. Specifiopaly, it i aesumed that

roll-

6hat the underlying p•dJ* is i > 0 > O o > 0ish too

testHo 0 0 -~ 1500 hoursagainstl8 1 e 1 5oohoire, with aa0- 051

ioe., we want L(80) - I - a - .95 axid L(81) - m -05o Due to the fact

that we are l imiting discuesioy, to non=randomized tests, we will be satis-

f ed with a test procedure for which L() -95 and L(81) S .05o

Usinl the results in [23nd in the earlier part of this report

it can be readily verified that the following three t eats have ilrtually the

same O.C. curvee

(A) 20 item are taken at random from the lot and plaoed on

test. Items which fail are not replaced. At each moment t, copute

-(nk+1)x .(n *! -) (t-xi n), where i is the number of failures which

have ocourred bef£ , time t. 1 this m emeeds 8150 for amy i such

that 0 < i < 9 atojp the experiment at time t and aacept o. If 10 fail-

ures occur and this 8M is less than 8350, then reject H (accept H ), As0Apointed out in Section 4 this test is equivalent to accepting H. if 61 0 > 815

and rej.cting H if '-o150o0

(B) 20 item are taken at random, fro the lot and plawd on test,

Failed item are not replaced. If min IX90 5i.01 a 51.0 truncte the

expriment at time 540 hours with the acceptance of 1o0' If min i - X1.. 1 0 92 M . 10,20

truncate the experiment at XIO 0 with the rejection a Ho.

(C) 20 items a" taken at random from the lot and placed an test.

An ita which fails in replaced at cne t7 new itea from the original loto

The tim X wbfe the i th failure occurs is measured from the beginning of

the exeimet Ifmi [O, M W,7. 5J = 407a.5, truncae the experiment '

at 07a 5hour@ with the aoeptane of H Ifin[Z 407~u 9

oP20 ,10,920truncate the experiment gt v

X1I.

-15-

It follows from Semtion 3 and 4 that the O.C4 curves and the

probability of terminating everimintation with exactly r - k observations

(0 < k < 10) and hence (in putioular) Ee(r) are exactly the same for pro-

cedures A and C. In Table 1 v give L(G), 99(r), and EO(T) for the tests

A, B, C aL selected values of e

Table I

.LRMZ.ies of Th1e Last Preue

L(e) r(T)

Mean Life G A B C A S C A B C

250 .000 0000 .000 10 10 10 167.2 167.2 125,0

500 .038 .043 .038 9.93 9.94 9.93 331.4 331.6 248. 3

750 9355 .366 .355 9,10 9,25 9o.10 444, 7 453°5 311.3

1000 .698 .702 .698 7.68 8.06 7.68 481.8 509,1 384.0

1250 S876 .877 .876 6.39 6.93 6.39 484.8 529.2 399.3

1500 .950 .950 .950 5.39 6.02 5.39 474.7 536.0 404,5

1750 .979 .979 .979 4.64 5.30 4.64 4660 538.3 4o063

2000 .991 .991 .991 4.07 4.73 4,07 458.3 539.4 407Q

2250 .996 .996 .996 3.62 4.27 3.62 452.3 539.7 407.3

2500 .998 -998 .998 3-88 3-26 5399 407.4

L It is easy to verify that for all three procedures lir E (r) - 10

and !Am Eq(r) 0 C, Further for procedures A and B, lim E(T)/V-.(Xio2) m 1.G oo e->o

Hence Ea(T): 'A668773) as BqO For procedure C, lr Eq(T)/9 1 2/2. As

GcO, liaM E (T) - 407.5 in procedures A ano C and - 540 in procedure B,

rel*ts, It should bo possible to truncate with a smaller E(r) as 4 gets

:3"1&I. and Aii' a smaller Eq(T) as I gets lauge, We shall study this question

10. a royrxro 'Iealing with sequentialized life tests In any event vw can see

(ilready that by taking advantage of the fact that failures are ordered in,

tima, IL will be *ssibl.:

(1) to come to a deciaion (rejection of H0 ) after a short waixing

time (small E T) )if the mean life is low and

(2) to come to a decision (acceptance of H ) after a small mbter0

of failures (omqll E,(r) ) if the mean life is large. If the mean life-- ,

il will be possible t.o stop (with acceptance of H ) at some time T* without0

any failures at afl ~curin °

I:I

1, D. J. Davis, "An Analwis of Some Failure Data," Journal of theAmerican Statistical Association, U, 113-150, 1952o

2, BEpstein and H. Sobel, "Life Testing I," To appear.

3. B. Epstein and M. Sobel, "Som Theorems Relevant to Life Testingfrom an Exponential Distribution," submitted for publication.

4 A. Wald, "Sequential Analysis," John Wiley and Sons, 1917.

5. E. C. Molina, "Poisson's Fjxonentil Binomial, Lit," D. Van Nostrandand Co., 149.

6o K. Pearson, editor, "Tables of the Incomplete Beta Function," Univer-sity Press, Cambridge, England, reissue, 1948.

7, K. Pearson, "Tables of the Incomplete r-Funotion," Ciabridge UniversityPress, reissued, 1951.

8. "Tables of the Binomial Probability Distribution," Nat. 3 kw. of Stds.Applied Mathematics Series 6, 1950O

DISTR TTON LIST

Wqn. University Technical Reports

Dr. ward Paulswn raHead Statistiof Branch Amy Chemical CenterOffice of Naval Research nltAnreerah

Department of the Na ality Assurance Branch

Washington 25, D, C, 5Dr. Clifford MaloneyOffie oavalesearch Chief, Statistics ranch

Office of Naval Rsearh Chemical Corps. Biological LaboratoriesDepartment of the Navy Physical Sciene Division1000 Gea7 Street Cp Detriak, Maryland 1San Francisco 9, California 2

Coending OfficerDirector, Naval Research Laboratory 9926 Technical Service UnitWashington 25, D. 0. Armed Services edical ProcurementAttns Technical Information Agency, Inspection Division

Officer 9 84 Sands Street

Chief of Naval Research Brook , Now York 1

Office of Naval Research Aasto Chief of Staff, C-4Washington 23, D. C, United Uat UW

Attn: Code 132 Procurement Division(MathematiOs Branch) 2 Standards Branch

Office o; the Assistant ahgt 25, D. C.

Attaebe for Research Chairman, Muitions BoardNaval Attach Material Inspeotion Agency

Azar1anbaTWahingto 2, 1.C. 2

Fleet Post Office Chief of OrdanneNew York, N. 7.0 2 United States Arq

Research and Developmnt DivisionPlanning Roearch Division Washington 25, D. C.Deputy Cbief of Staff Attn: Brig. General L. E, Simon 1Comptroller, U. . Force Mr,. Charles Bilking 3The PentagonWashington, D, C1 Chief, Bureau of Ordnance

Department of the NavyCommanding 0ffie uality Control DivisionSignal Corpe Procurement Agency Washington 25, D ". 22800 South 20th StreetPhiladelphia, PmfYlvania 2 Chief, Bureau of Aeronautics

Department of the NavyCommanding General Code 231New York Qa-termaster Procuwet Washington 25, D. C,, 2Agene, Inspection Division111 East 16th Street Coanding O1'ficerNew York, N. T 2 Frakfdord Arsenal

Philadelphia 37, Pennayivania

i

aos An e.es F ?Inaering Field Offt,.e Nval Ordnan!e j.abarut iryAr Research i4.d Develoent Commu Library350 Hollyvo Boulevard Silver Springs 19, Mayland 1oB Angeles 18s CaliforniaAttn: Captain Norman E. Nelson I Chairman

Research and Development BoardChief, Bureau of Shipe The PentagonAsst. Chief for Research a.r Waliton 25, D. C, 2

Development - Code 373Washington 25, D. C. 2 Asistant Chief of Staffs G_

for Research and DevelopmentCommanding General U., S. Ari yAir Hateriel Conmand Washington 25, D. C. 1Quality Control Division 0PLXPWright-Patterson Air Force Base Chief of Naval OperationsDayton, Ohio 15 Operations Evaluation Group - OP342E

The Pentagonilead*uaters, USAF Washington 2, D. C. 1Director of Research andDevelopment Ccummming General

Washington 25, D. 0. 1 Air Proving GroundEglin Air Force Base

Commanding Officer Fglin, Florida IOffice of Naval ResearchBranch Office CommanderThe John Crerar Library Building U. 3, Naval Proving GroundTenth Floor, 96 E. Randolph St. Dahigren, Virginia IChicago 1, llnos 1

CommanderComanding Officer U. S. Naval Ordnance Test StationOffice of Naval Research Inyokern, China Lake, Califo IBranch Office346 Broadway Camianding OeneralNew York 13, N. 1. 1 U. S. Army Proving Ground

Aberdeen, MrylanOfficer in Charge Attas Ballistios Research Lab. 2Office of Navwl ResearchLondon Branch Office Rand Corporationfleet Post Offi(e - NavylO0 1500 Fourth StreetNew York, N. T. 2 Santa Monica, Caliornia

Commanding Officer Office of Naval ResearchOffice of Naval Research LoMistio Branch - Code 43Branch Office T-3 Building1030 East Green Street Wahington 25, Do C. 1Pasadena 1, California 1

Logistics Research ProjectMajor M. H. Parde George Wshington UniversityDirector of Procument 707 -. -2d Street, N W.Statistical Quality Control Branch Washington 7, D. C. IAir Yteriel CommanWrigtt-Patterson Air Force Base Mr. J, L, DolbyDnyton, hiio 1 Elsetro-Heehanioal Division

Ganaral 7ngineerig Laboratory1 River RfoaSthsneotsdy % Neu Yorli

Bell Telephone '.%b,, rto4eB, Tne, Unvcraity of '!ohin ton463 West Street Sattle,, WashbngtaNew York, N, Y.

P ofe or David H. BlackwellProf. W. Allen Wallis Depatmnt of Xathematice

Committee on Statistics Howa Universi tyUniversity of Ohicago Washingto 25, D. 0,

Chicago 37, Ilnois 1Professor W, G. C*ohran

Dr. Walter Shv..tart - Depalnt of BiostatisticsBell• Telephono Laboratories, Inoo The John Hopkins UniveroityMurray H1l, New Jersey 1 Bltimore 3., Marund

Professor A. H. Bo er Professor 8. 5. WilkeDepartamnt of Statistics Departawat of HathmaticbSta ror4 University Princeton Universty

~a~iforniI P'iAnoetons Now .AX017Prof stor 14 A. Giri9ek roftesor J. walflrt5

Departmaent of Statistics Depufamn of MatheminacsStanford Unveity Comfl Univwsityt Stanford, California 1. Ithasa, New York 2

Professor. J. Heian, Depatmft at Mathmtical StatisticsStatistics Laboratory Mwwsitt of North CarolinaUniversity of California Chapel Hill, Noth Carolna-Berkeley. Calif ornia1

-~r -b N. aalperin 7Protesor RO.G. Md Natinmal Heart InstituteDept. of ateaisFederal Security AgeacyCarnegie Institute of Techology Betheda, MarylandSobevley ParkPittsbuwgh 13% ftoylvania I Dr. go Jo Oumbtl

41g1 Ocean AvuamD avAS arts Apt,. 6K

inspection Division Brockljn 26, New YorkAz'W Qdartarmat,.r Co"'p.111 East 3.6th treet Statistical Riiaeribig LaboratoryNew York, N. 1 National Bmeaa of. Standards

Wasi gta 25, D. Coul '.-.ir For", Lialn Attna r. ChOchil R'ismnhar.t

OfficeAmes -Aeronautical Laboratory Statistical Engneering Laboratory)ffett Field Baal weau of StardardsMountain Ve-, saCaliforniaa 25, D. C.A ,.: -N'ri Carl Tuach 1 AtthiI Mr. Julius Liesin

Dii.Jercme Rothstein Dr. max ScherberfCamp Evans Office of Air ResearchBeuamr, New Jersey 3. WMlt-Patterson-Al Farce Base

Professor Sebastian LitaewDatoDOPArit~mt Industrial Ears i mn DirectorCimbli' University. Evans Signal LaboratowYw Y ork # Y.O B 92ml . New Je rse

1Athte kr.J Weinstein;,IA ied:l, piae lk• .i :"I

Director of Research UsS Departmnt of the Mmtsrior

Operatioas Research Office shadWildlifeSevcUs 8. AM 1220 East Washington StreetFort Mcflair Amn Arbor# VAignTgshington 25, D). C. 1 Attu# Gec'ge F. Luimger

Aeui. for Operations Analyim 0, 8. N. InginferIng Experizent StationHeadqartere U. S. Air Force Anapolis, MarylandWIashington 25, Do C. 1 Atth: Mr. ?rancls R. DelPriori 1

Mr..illad Rsenfld r. Willia= L. KLuMr.c cur1mad Data~nch Bps"iI Weapons Department

sio~al Corps ftgineering Labs. Anorhrop Aircraft., Iwo.- - FoitMrtb New Jenq

Dr,. J, Ri Steeni Silvania Electic

S IlvaniLa Electric Prdce Inc 70FryhUm40 DroadvV -Doston# Massachusetts

New York 199 Ne York 1Dr. hgsme W. Pike

Kr. Railton Brook. Api'l1ed Phys~ics SectionCapaitorSecton ineerin& DiLvi*Loh

Weu inglouse Ele'tria -corpor'ation Ra~thebn uafactmring CompailyEast Pitt abur~, PaMITlv A I I"1. Oalifoz'pU Street

Netn assachusetts1Mr. Leonard 0. JohnsonGeneral Motors Laboratories (M%2,1) Pri~essow Andrew Schult2, Jr.Geineral Rtors haidin Dsparts" of Il4dutz'al az'Detroit, ich~igan 1 - Adinistrat ion

Professor R. A*fltup5 Jr. Itbuat Now Yzo*k'Graduate -6ohoo~of fingnerijasCambridge, ftmaui ttfl 1 Dr. 06 West, Churchman

Case Institut of Tecbnolftg

fthy, CoporaionProfessor T, J. Dolan

Deptof EletrZ.rningr 1xbaa ... r .u-n

Stanford UniversityPtaford, COalifo'ri Professor RowBsoif

Department of Industrial al in

Professor W. G. Shei Co1lubla UniversityDept. of flecrical *MgIZ151ifl Nowv York, N. Y.

£ University of NXimeuataMlrneapolies HMS60e~ta It.akf ofNva ateriel

DrM. M. Flood Dwp*tmnt of-the NMDepartmenrt of S06i10 Vetig c 25, D, 0.

* ~Columbia UniverstYM VArk Su.a JaonesNew rori, New York A-M. ak. oe

M-1tl Contract Div±ionISO Jim uspefr Awa. 9 X W.Ws-shz~gbn 6D.O,4