7RD-Ai33 487 MOMENT ETTIMATORS OF DEFECT PATTERN FREQUENCIES i/i

ALLOWING FOR INSPECTION E. .(U) MARYLAND UNIV COLLEGEPARK DEPT OF MANAGEMENT SCIENCES AND ST.-

UNCLASSIFIED N L JOHNSON ET AL. SEP 83 UMD-DKSS-83/5 F/G 12/1 NL

. 1-1 302

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ILNIL &4

MICROCOPY RESOLUTION TEST CHARTNATIONAL BUREAU Of STANDARDS-1963-A

SESMTRS OF • FREUENCIES,...

-4. ..... _.

. ALLDWING F( DGI CKE Q EUMO

X3 by:

University of North Carolina Univetsity of MarylandChapel Hill, North Carolina College Park, Maryland

ABSrRACr

Unbiased estimators of the nmrber of individuals in a lot

possessing various patterns of types of defects are constructed.

Explicit formulas are given for cases of two and three types of

defect. Application of the formlas requires knowledge of the

probabilities of various kinds of errors in the inspection process.

Key Words 4 Phrases: Inspection errors; Estimation; Quality

-' control; Sampling inspection. DTCSOST

121983

A

L.J This document has been approved.- j for public release and sale; its

?1 .distribution is unlimited.

PLA

83 10 11 026iL .... ... . .4

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1. Introduction

In [1] [2], we introduced the term defect pattern, to reflect the incidence

of defects of different types in the same individual. If there are k types of

defect, in all, under consideration, then an individual's defect pattern is

represented by a binary (0,1) sequence of k digits, with the j-th digit equal

to 0, or 1 according as the j-th type of defect is not, or is, present in the

individual. There are 2k possible different defect patterns.

We will denote the defect pattern (9 1 g2 9...,gk) by (g), and the number

of individuals, in a lot of size N, possessing defect pattern (g), by D g

Clearly

~ D Nwhee 1 g) 1 1 1 2

whee g) s I ... I denotes summation over all 2 patterns.- g1 00 g2 0 gk=0

If a random sample of size n is chosen (without replacement) from the lot,

and Z I of these are found, on inspection, to have defect pattern (1), then, if

inspection is perfect a natural, (and unbiased) estimator of D . obtained-by

equating observed and expected values of Z,, is NZn. If inspection is

imperfect, the situation is not so simple, but the same method ,can be used,

provided the probabilities of correct and incorrect assignments of defect

patterns are koown. Since the resultant estimators - , say - are obtained'

by equating first mments of saple and population values of the Z I's we

will call them moment estitors.

We will suppose that b:

i) if defect of type is present, the probabili C,, -." -z.-..--

detected is Pj; 1[3~EM F-4 0 %

-2-

(ii) if a defect of type j is not present, the probability that it will be

(erroneously) detected is P!;

and

(iii) for any defect pattern, doisuiog in regard to presence or absence of

each type (or disjoint sets of types) of defect are mutually independent.

This last assumption may not always be justified, but it serves as a useful

point of reference.

With assumptions (i)-(iii), the probability that an individual with defect.

pattern (f) is classified, as a result of inspection, as having pattern (g) is

pgjp gf(1- (1-g_1: (1-g()(1-f)P sig" -1 j J(1-pi) 3 J(1-p!)}(1

We will further assume that pj > p! for all j. This does not affect many

of our formulas, and it is a very reasonable assumption. It merely means that

an individual is more likely to be classified as possessing defect type j if

indeed it does possess this type of defect, than if it does not.

2. ,bunt Estimators

. Our problem is to estimate the D 's from the observed values 'of the Z 's.

The expected value of D is

ERZ 1mn (2)

wnd so the mument estimators, D£, satisfy the 2k equations

Z P(3)

or, in matrix fom ,,"

k vectr with elements (Zred.{,,,(3)

" .. " Z ' b ,e-.ectivy, .

mi f isa zk otrix witht -th ow andf -th

column. Prom (3)',

D - Pin- Lp"lz (4)

where Z-1 is the left inverse of ?. Since

EZ - (P D

it follows that E[] n D, so D is an unbiased estimator of (E[D9 - D for

all (9).)

However it is possible for some of the D 's to be negative (though not

all of them, because X() - N). Similarly some (but not all) of the D 's

' . might exceed N. This latter possibility occurs much less frequently.

For k-l (a single type of defect) we have, of course

D1 =N- Do , and E[Z1l - N-n{D p1 +N-D)p t

- 9-n{Nl' D(pl-pj'))

whence-l-

D1 - N(pl-p'l 1 I(n'IZl-pj) (5)

_jSince pl > Pl' , D, < 0 if h; < "P. (6.1)

> D1 N if Z, > Pi (6.2)

In practice, one would use a iodified estimator such as

I m if 0.:l IN (7)

N if N <

This estimator is wot. in geneal, unbiased.Table 1 shows values of

(iI) Pr[(b, > NJ

fra(is) Diss s(oau ) o the Dret-rsinfo a few net of valuss Of the Paramters.

' . -4-

3. Distribution Theory

Conditional on the numbers Y - (Yf) of individuals with actual defect

W patterns {(f)) in the sample, the joint distribution of the 2k numers

{W If) of the Yf individuals with pattern (f) which are assigned patterns

{ ()} as a result of inspection is multinomial with parameters (Yf;{p I£f). For

different (f) and (f*) and any (1), (f) (whether or not (g) =-g*)) "1.f

and Wg. £* are ntually independent binomial variables with parameters

(Yf-2 Pg f) (Yf*pg*jf*) respectively. Since

zgW.~ (8)z . '.±, ".g'

the conditional joint distribution of Z, given Y, can be represented symboli-

cally. z[Yo ,* Muitinomiai ( f }( f) (9)

where * stands for "convolution" and " " means "is distributed as".

To obtain the unconditional distribution of Z, this conditional distri-

bution has to be cmpounded with respect to Y, which has a multivariate

hypergeometric distribution with parameters n; D; N, so that

Pr(Y-] -N-1 fDf. ,(01h"I 1 )-l (0 1_ yf 1 Df;I (f)yf n) (10),

[!'Z]" - Cg[I ...

Fra (8) and (9), mments and product-oments of the Z 's can'be

evaluated (see [1], equations (17)).

4. Variances of the Estimators

From (4) the variance-covariance matrix of D is

Var(P) " nl)ZplVar(Z) (p)T (11)

(where the supefix CM denotes 'transpose').

• . ., ,, ... . . ..., . . .. ... ,- - . ... .. ., . . , ... . . . .

Praii [1] (equation (17))

var(Z .-n ((-~ + Nn)p )(N-1) (12.1)

cov(Z .Z -n f((fl-)-y** (N-n)F~g*)CN-1f' (12.2)

where p N 1f)Ift!;p 1jfn2

and N fgIfg

S. &Seial Case: Uniform Accuracy of Inspection

If pj p and pj - p' for all j-l,...,.k - that is, the accuracy of inspec-

tion is the same for all types of defect -then

Pgff a p lp (1-P) (1-PI) 0

*where s ab 'number of types (J) of defect for which f. -bpg. a.

For k-2, we have

(1) cQ- 00o 01 10 11 200 (1-p')2 (1;-p)(1-p I) (1-P) (1-p') (1-p)2

01 P, (1-p') P(1-p') (1-p)p' p(1-p)P.

10 p(-) (1ppp(-) p(p) (13.1)

11 P0 P1 p , ~ 2

and2 -p(l-p) -p(1-p) (1-p)2

(P -ppo p(-p') (1-p)p' - (1-P) (1-p') (32

.-Ypp (1-p)p' p(1-P') -(1-p)(1-p') (32

p' 2 -P, (1-p') -p' (1-p') (1-p') 2

-6-

D - (RN1 ) (p-p')2{ppZ 00p(1-p)(ZZ.(-P)pZ 1 1-)(1p) 1) 14

B1 _ (o ~) P_')-2(_-pp'Z.o(1-pl)pZol+(-P)Z-('P(-,Zbl N-1) -l-(pio+'-)lo~(-l o(1-P) (1-p')Z 11) 4)

D - (Nn1l)(P-P) 2 {p 2Z00 -p,' C-P')(Z01.Z10)4+(1-p') 2Z11 )

* For exauple, if p -0.90 and pt -0.95 then

bo0 - W71{1.121Z00 - 0.12S(Zo01 +Z10) + 0.014Z11)

-1D a Nui (-0.062Z + 1.183Z0 +. 0.007Z1 0.3Z)

10 - ?hf1(-0.O62ZO+ 0.007Z10 + 1.183Z10 :-O.131Z11)

NI- No 1(0.003Z00 - 0.066(Z01.Z10) + 1.249Z11)

As is to be expected the ujor coefficient in the exression, for Dbis that

of Zabut the Contributions of the other toxins is not negligible.

M- r.6 ~.

4 V-

I B. ?. B.

m. me 0. B. 6

0. m.B 4. N_4

a 1 V4 -a 4 ~ . B . B .

B. B B a a4

aV- -4- -

-~~C 1 - ,4 14 -. a S a a N-

1 4 B. e4 B. B 1 4.4 % #

B4 C6 v 4

r 4 V-4 4 ''-

'4~~- %..#B B B

9.4.

%- -4 f- 4 1 ,a -0S I -- - 4 ~ ' . - B

B. as as as la . .B. '' . - -

-r.6

0..-..

For moxmple

,a5011 -(Nn "1) (p-pI)'-3{(p 22000-PP' (1-p,) (2001 20101- (1-p)p,22100

+p (-p') 2 Z0 1 1 +(1-p)p' Cl-p') (Z10i +Z1 10 ) - (l-p) (1-p') 2Z111} (16)

In these circumstances, if also D: - 2"kN for all C), then

p = 2 -k cpp')m(2-p-p,)k 'm, (17.1)

where m(- m(g)) is the number of l's in (g)

= 2-k (p24,2)m(-p)2 (l-p')2 k-m (17.2)

_k 2+1 n)o~l((_)+,P) (17.3)and pg.g (P "kpp'm1{p(1-p'+(1-Ppl"m1m0 1p2(-'2 m o (75

where

mab mab(g,g*)) is the number of types () of defect with gj = a, g! b.

(M-l + M=ol +"Do , k).

The further specialization p' - 1-p, corresponding to a constant probabil-

ity (1-p) of error, leads to further simplification. In particular,

2 k - 2"(p (l-p)2}k for all (g)

- . 2 kp2 (1-p) 2 'O (18.2)'

Note that (moo +.l equals the number of j's for which gj g;end N

p2 (1-p) 2 - 1-2p(l-p). We will use the notation "

Moo +O I - 0; P2+(1-p) 2 -A; 2p(l-p) - 1-A- B (19)

From (12) and (18)

0a(Z ) = .2 kc l- Wn . 2 - nl Ak) (20.1) ".

kovZ,,-n N - k n - k-0 (20.2). . . . . .. -n - 2 + A

" .- •-9-

As an exuile consider

Nil1 a (Nn1) (2p-l) 6(p(1-p) 2 (Z00 0+Z1 0 1 +Z1 10 )-p 2 (1-p) (Z1 1 I Z0 01 +Z0 1 0 )

+ p3 Z0 1 1 - (l-p) 3Z100) (21)

obtained frsI (16) by putting p' - 1-p.

We obtain from (20) and (21).2

* 1N 2 A3 - N-9n.8var1 ,ll a 1 6(2(2p-1) (22)

(Details are given in the Appendix. Note that

A3(2p-1)'6 _ 1 {1*(1_2p)'2}3.)

The proportion D011/N (=2"k in this case)"is estimated unbiasedly by

50111N and this estimator has variance

1 [l1-223 N-9n+8 -1 -2 3Vn [{1*(1- 2p) - -- ]- I - n[{l+(1-2p) '}-1] if n -cN.

REFEREM

[1] Kotz, S. and Johnson, N.L. (1983) Some distributions arising from faultyinspection with mltitype defectives, and an application to grading,cun.m, Statist. Theory Meth., 12 CTo appear).

[2] Jdhnsn, N.L. and Notz, S. (1983). Mltivariate distributions arisingin faulty saling inspecti. Pr. th Internat. SY p. Multiv.Analysis, Pittsburh (sd. P.R. n-) iterdii NorthHolland (To appear).

ACKNOWLEDGEMENT

Samuel Kotz's work was supported by the U.S. Office ofNaval Research under Contract N00014-81-K-0301.

.5_, . , , ' . -. . - . . . .. . . . . . .

APPENDIX: Derivation of the Variance of Dl

From (20) and (21)

(k)(p1vr('0 011) V5 - S 1L-1 C42A S2 +

2{p 2 (1-p)4 (3AB 2)+p4(1-p)2 (3AB2)

-p 3 (1-p) 3 (B2+ZAB2+4A 2 +2B 3) -p 3 (1 -p) 3B 3

-pci p)S5(3AZB)-p 5 (1 p)(3A2B) 1]

1 n-n) _Z Ln-1)[ r 3

PSf2 -63M-) 1 S! 9-0 ") 2-6(p(1-p) S +2p3 (1-p) 3+p S(1-p) IA2B

+12{p 2(1-P) 4.04 (1-p) 2 )AB2 -8p3 ( -p) 3B3]

_ n(-n) 2_ nn-l3A4B + B 6)'22 P cW-s S 4L 3ABB)

where S2 sum of squares of coefficients of Z's in (Nnl -2plti

a p2(1-p) 4*304(_P)2 + p6+(1-p)6 - A3

and S, - sun of coefficients of Z's i n ) 1(Zp-1) 3 501

- 3p(1-p) 2-3p 2(1-p).p3-(1-p)3 - (2p -1) 3

..jting that since A.B -1,

AS- 3A4B2 3A2B4-B 6 (A2-B2) 3 -(A-B) 3 -(2p-1) 62

w obtain

N2 A 3_ N-9n+8~ N2 - n1Var(D01 1 n 1 N2 A3 a *n.3T [{l.(2p-1) )2 1 + _-1 L (2).ii T (2p-1)0

The prportion D,/N is estimted unbiasedly by bo/Nl; this estimator has

Variance-1+()-

1 (+2pl- 1 C. 1+ ( - -. (12) - -

SECURITY CLASUICATION OF TH41S PACE (WhiN 1104 9 ___________________

REPORT DOCUMENTATION PAGE BRE INSTRUCIhONSo~

REPORT~2 RU 0ZVT ACCESSION NO: 3. RECIPIENT's CATALOG NURSER

4. TILC (Hd #"as) . TYPE OF REPORT I PE1RIOD COVERED

?.*xnet Estimators of Defect Pattern Frequenciest7Allowing for Inspection Error Technical

6. PERFORMING OkG. REPORT uUmeaR

UMD,DMSS-83Z51. Aurnonfe) 6.CONTRACT ON GRANT HUM99R(a)

Norman L. johnson & Samuel lotz ORCnrcN00014-81-K-0301

S. PERFORMING ^RC 10;Z ATION N4AME AND ADDRESS SO- PROGRAM ELEMENT. PROJECT. TASK

Dep~riiient of Management Sciences and* Statistics, University of Maryland '

.College-Park, Md-. 20742 ___________

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Approved for Pulic Release - Distribution Unlimited

17. DISTRI1UUTION STATEMENT (at the abatraer sneered n Ri.f.ti 20. ot.I.f .-111-m on R~potf)

III. SUPPLEMENTARY NOTES

IS. KEY WORDS (Ctinus a" reveirse side It necessman~ md identity 61- 611ch ~MIae)

Inspection errors; Estimation; Quality control; Samling inspection.

20. AtSTRACT (Centie an roersnaide it necesare d idw.itify 6y back MaM16")

tkhbiased estimators of the amber of individu4s in a lot possessingvarious patterns of types of defects are constructed. Explicit fomilas aregiven for cases of two mnd three types of defect. j Application of thefoimias requires knowledge of the probabilities of various kinds of errorsin the inspection process.

FOR 1473 EITION of I Nov GsIS, OBSOLETE

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