Statistical Papers 32, 233-241 (1991) stallBue Mrs 9 Springer-Verlag 1991

Exact two-sided Lieberman-Resnikoff sampling plans M. Bruhn-Suhr and W. Krumbholz

Received: Oct. 10, 1989; revised version: April 25, 1991

We deal with sampling by ~ria51es with two-way-protection in the case of a N(p, a ~) distributed characteristic with unknown ~t. For the sampling plan by Lieber- man and R~nikoff (1955), which is based on the MVU estimator of the percent defec- tive, we prove a formula for the OC. If the sampling parameters Px (AQL), Pz (LQ) and o, fl (type I, II errors) are given, we are sble to compute the true type I and II errors of the usual (one-sided) approximation plans. Furthermore it is possible to compute exact two-sided Lieberman-Resnikoff sampling plans.

I I n t r o d u c t i o n

We deal with sampling by variables in case of a normally distributed quality characteristic X with mean p and u.known variance c# > O. Ass,,m|tlg that there are both upper and lower specification limits U, L (L < U) the percent defective p is given by

(1.1)

with

P = PL + PU

(I.2) pL:= P(X<L)=r (1.3) pu := P ( x > u) = 9 ( V - ~ ) ,

and ~ denoting the standard normal distribution function. Let Xx, . . . . X . be a sample on X (n > 3), and

~ = I ~ x,, $2 1 ~<. ~<. - ,- i. (x, - X) 2.

Let (1.4)

(I .5)

1 1 X - L ~ } V : = m a x 0,~ 2 S ~ = i '

W := max - - S--n- I '

234

(I.6) b(z) := r c n - 2) z ~ - 2 r l - z ) ~ -2 (0 < z < 1), rc~-~)r(~) '

r denoting the gamma function,

(I.7) B(z) : = / : b(t) dt. B and b are the distribution and the density function of a symmetrical beta distribution of the first kind.

It is well-known ( cL Kolmogorov (1953), Lieberman/Resnikoff (1955)) that

(L8) ~L := B(V), (L9) ~ := B(W), (I.10) ~ := ~L + ~ u

are the minimum variance unbiased (MVU) estimators of PL, p~r and p respectively. Lieberman/ttesnikoff (1955) proposed the sampling plan (n, k) based on ~, for which

the lot is accepted if 1~ _< k. The MIL-STD 414 sampling scheme is based on plans of this kind.

For given px (AQL), P2 (LQ) and a, fl (probabilities of the errors of the first and second kind) the Lieberman-Resnikoff plan (n, k) (n >_ 3) may be computed approximately in the same way as the corresponding plan in the case of a one-sided specification by

(i) F,_l,v~.-,(p,)(l ) >_. l - a (I.11) (ii) F,_, ,v~- , (~)( l ) <_ fl

(ili) n " rain (iv) k = B( 89 + ~ ) ,

with F,.a denoting the distribution function of the noncentral t distribution with r degrees of freedom and the noncentrality parameter/L

Remark 1: In the case of a one-sided specification limit which without loss of generality may be assumed to be the upper one U, consider the sampling plans (n, l) and (n, k) for which the lot is accepted if v/'nX-~ _< I and l~br _< k respectively. These plans are equivalent iff (I.11)(iv) holds. Given l~,/~, a and fl, the plan (n,l) is determined by (I.11) (i)-(iii) (cf. Brulm-Suhr/Krumbholz (1987)).

Numerical investigations by Resnikoff (1952) and also Baillie (1987) seem to confu'm that the plan computed by (I.11) (i)-(iv) represents a reasonable approximation of the corre- sponding exact plan. This conjecture is also supported by comparisons between the exact two-sided plans of Bruhn-Suhr/KrumbhoLz (1987), using instead of ~ the estimator

of p (which is essentially the ML estimator), with the corresponding one-sided plans. In the present paper we prove a formula for the operating characteristic (OC) of (n, k).

This enables us to compute exact two-sided Lieberman-Resnikoff plans.

235

II T h e OC of (n,k)

Before we derive the formula for the OC

(II.1) L(/~,a) := P.,o@ < k)

of (n, k), we have to prove some lemmata. Define

(II.2) F(z, %t) := P ( V < z, W < ~I),

and denote the density of the X 2 distribution with r degrees of freedom by gr.

L e m m a II.1 I f z >_ O, ~/ >__ O, then

(II.3) F(z,~t) = [ , /(='~){~ ( - 6 u q - ( 2 ~ 1 - 1 ) ~ >

with

(II.4) A :~ n ( V - L) ~

n - 1 4a 2

(II.5) A(z,y) := { ~oo

(II.6) 6u := V r ,

(n.T) ~. := V~ " - L O"

, i f z + ~ < 1

, else.

Proof : ~a-~1S2 is X 2 distributed with n - 1 degrees of .freedom. Applying the total probability decomposition rule to (I.4), (I.5) and using the indepondeno~ of X and S 2 we get:

FCz,y ) = _-

= /re -~+(~-2~) ,(.~-~)_<V~ x-~ _<-~+(2y-I) g._,C,)d,

236

In the last step we used that

is equivalent to

v -L j n >_ (1-,-v),/i (II.8) "2~ V n - 1 "

I f z + V > 1, (/1.8) always holds because the left hand side is positive. then (11.8)/s equivalent to

A t _< ( 1 - z - ~,)2 -

Thus we have proved Lemma 11.1.

L e m m a 11.2 Let z _> 0, V > 0. Then

If z + y < 1,

c,,.~) o~c,,~) : ~. r " " o ' (-,,.,-,-c~.,,-1)~) ~ 1)~,,-,co,,,. OV Jo

Proof: (i) If z + V >- 1, we have in (11.5) A(z, V) = co. We obtain (11.9) by partial differentiation of the inteamnd of (II.3) ~th respect to y.

(6) I f m + V < 1, define

(II.10) c(x,v,O := ~ ( - 6 0 ' + ( 2 y - 1 ) t ( n ~ ' ~ - l ) ) -

- O ( - 6 r . + ( 1 - 2 z ) ~ ) .

We get (cf. Mangoldt/Knopp (1963) p.387) .from (I1.3)

(ii.n) 0F(z,V) Ov

F r o m (11.4)- (II.V) we get

oA(~'y) 0

aAC,,v) +C(.,v, AC.,v))g.-1CAC.,v)) ov

-,~§247 cl-2.)~c=,~)c.-1)= ~ c ~ - L ) - 2 ~ . = which yields (II.12) C(=, V, A(=, V)) = 0.

B,om (/1.11) and (II.12) we get (11.9). Thus we have proved Lemma II.2.

237

R e m a r k 2: We wish to thank Dipl. Math. A. ZSUer, TH Darmstadt, for pointing to the fact that C(z, y, A(z, y)) = O.

L e m m a II.3 For 0 ~_ k < 1 we have

P(f_< k) [a-'(k) OF(z,y)

(ILl3) = F(B-I (k ) 'O) + ao+ Oy "I =ffi-,b(v) F(o,o)

with (II.14)

Proof :

dy .for O < k < l f o r k = O

r := B - 1 ( ~ - B(=) ) .

(i) i f k = 0 we get from (I.4) - (I.10) and (II.3)

P@ <_ k) = P(~ = o) = eCV = o , w = o) = F ( 0 , 0 ) .

(ii) I f 0 < k < 1 we define for z, y > 0

(II.15) GI(y) := P ( V = 0,0 < W <_ y) = F(O,y) - F(O,O),

(II.16) G2(z) := e(0 < V < z ,W = 0) = F ( z , 0 ) - F(0,0) ,

(II.17) G3(z,y) := P(0 < V < z,0 < W < y)

= F ( z , y) - F ( 0 , y) - F ( z , 0) + F ( 0 , 0).

(1.4) - (I.lO) and (II.14) - (ILl7) yield

_!,@ _< k) = ~'0~ = o) + P@L = o,o < ~v _< t=) + P(O < ~. _< k , ~ = o) + P(~ _< k,~. > o,~v > o)

= F(0,0) + GI(B-I(~)) + G2(B-I(k)) +

+ //~BC,)+B(,)<_k~>O,,>O) O ~ y)d=dy (I1.18)

= F(o ,B-I (~) ) + F(B-I (~ ) ,0 ) - F(0 ,0) +

9 ~ az ! ay. + Jo+ \Jo+ ~xOy ]

0F(z,y) Prom (11.15) and (II.17) and lira = G'z(y) we get

=--,0 Oy z > o

+ OzOy az) dy = (II.19) [a-*(k) OF(z, y) = .,o+ ~ ==,m) dY - G ~ C S - ~ C k ) ) .

Combining (11.15), (11.18) and (11.19) we finally obtain (11.13). Thus toe have proved /,emma II.3.

238

R e m a r k 3: We wish to thank Prof. Dr. H. ScheUhaas, TH Darmstadt, for attracting our attention to an error in an earlier version of Lemma II.3.

The following theorem gives a formula for the OC of the Lieberman-Resnikoff plan (n, k).

T h e o r e m II.1 For 0 ~_ k < 1 toe have

(II.20) LCp, a) = o, {~ ) _

+2/o 9 , c , F ; z 1) ~.-~c,) d,} d~,.

Proof : (II.20) is established immediately by the Lemmata ILI-IL3. Thus toe have proved the theorem.

R e m a r k 4: Analogous to the proof of (H.20) the following equivalent formula for the OC can be shown:

(II.21) L(~,~)

Using (H.20) and (II.21) it is not difficult to show that L has the symmetry property

(1L22)

with p0 defined by (III.1).

I I I Computation of exact Lieberman-Resnikoff sampling plans

Formula (II.20) or (II.21) enables the computation of exact plans (n, k) when the sampling parameters pl, p~, a, fl are given. For this, we make use of the same procedure that

239

(Ill.l)

(III.2)

and define for given p (0 < p < 1) and cr (0 < ~ _< %(p))

was employed by Bruhn-Suhr/Krumbholz (1987) to compute exact plans based on the estimator p* in (I.12). Put

L+U #o := 2 '

L - U ,,o(p) .- 2~_,(D,

~=.(~,p) by

(III.3)

We set (III.4) LC~;p) := LC~C~,p), ~). In exactly the same way as in Bruhn-Suhr/Krumbholz (1987) we obtain that the exact plan (n, k) is given by the conditions

(re.s)

(i) rain. L(a;pl) > t - a o<,<~o(p)

(ii) m ~ LC~,;p,) < o<~_<Oo(~)

(iii) n " rain

( 0 < p x <P2 < 1; 0 < / ~ < l - a < 1).

The practical computation of(n, k) can be done in the same way as in Br, hn-Stthr]Krumbholz (1987, p.13-14). Starting with a ~ = a, ~* = fl, we successively compute for/>I,/o2, a*,/~* the approximation plan (h, X:) given by (I.11)(i)-(iv), check (III.5)(i), (ii), and vary a*,/~* until (III.5) (i),(ii) are true.

Example 1

L = 1.0, U : 9.0, P l = 0.01, P2 : 0.06, /~ = ~ - - 0.10.

The approzimation plan according to (I.11) (i)-(iv) is given by (h, Ir = (33; 0.02272).

The steps of computing an ezact plan (n, k) are given in the following table :

0.10 0.099 0.098 0.098 0.098 0.098 0.098

(n, k) = (34, 0.02262)

0.10 0.099 0.098 0.097 0.096 0.095 0.094

(a,~) (33,0.02272) (33,0.02271) (34,0.02285) (34,0.02279) (34,0.02274) (34,0.02268) (34,0.02262)

m2nLCo';p,) 0.8990 0.8989 0.9025 0.9019 0.9013 0.9007 0.9001

satisfies (III.5) (i), (ii).

m~/,(<,; p2) 0.1051 0.1050 0.1018 0.1013 0.1008 0.1003 0.0998

240

E x a m p l e 2

L = 1.0, U = 9.0, Pl = 0.01,/o2 = 0.03, a =/3 = 0.10. The approximation plan according to (I.11) (i)-(iv) is given by

(~., k) = (108;0.01679).

~S

0.10 0.099 0.098 0.097 0.096 0.096 0.096

0.I0 0.099 0.098 0.097 0.096 0.095 0.094

(108,0.01679) (109,0.01680) (110,0.01680) (111,0.01681) (112,0.01682) (112,0.01680) (113,0.01679)

0.8968 0.8978 0.8988 0.8999 0.9009 0.9003 0.9008

(n,k) = (113,0.01679) satisfies (1"]I.5) (i), (ii).

0.1061 0.1051 0.1040 0.1030 0.1020 0.1014 0.i000

R e m a r k 5: An unreasonable amount of computing time would be required to check whether the plans (n, k) in Example 1 and 2 also satisfy (III.5) (ili). This makes little sense if the sample size could at most be reduced to n - 1.

R e m a r k 6: Numerical investigations showed that there is no need for L(o;p) to have either its maximum or its mjnlmum in the region near 0 or in ~0(P). Furthermore, Z(*;p) may have several local maxima or m~n~ma. Therefore the computation of (n, k), which is carried out by a FORTRAN 77 program, needs a large amount of time. For each of the plans (n, k) in Example 1 and 2 we need about lh CPU time. The corresponding plans of Brnhn-Suhr/KrumbhoLz (1987) in Example 1 and 2 have the sample size n -- 36 and n -- 115 respectively. The reason for this little loss of sample size is that the MVU estimator ~ is superior to the estimator p* in (I.12). The little loss of sample size is compensated by a saving of about 80 % of the computing time, which is caused by the considerable simpler formula of the OC in Br~lhn-Suhr/Krtunbholz (1987).

References

D.H. B~i]He (1987) : Multivariate Acceptance Sampling. Frontiers in Statistical Quality Control 3, H.-3. Lenz et al. (eds.), 83 - 115.

M. Brutm-Suhr, W. Krumbholz (1987) : Ein neuer Variablenpr,]fplan f~r den Fall der Normalverteilung mit unbekannter Varianz und zweiseitigen Toleranzgrenzen. Disknssionsbeitr~ge zur Statistik und Quantitafiven ()konomik Nr. 27, Universit~t der Bundeswehr Hamburg

241

A.N. Kolmogorov (1953) : Unbias~ Estimates. American Mathematical Society Translations 98.

G. J. Lieberman, G. J. Resnikotf (1955) : Sampling Plans for Inspection by Variables. Journal of the American Statistical Association 50, 457-516.

H.v. Mangoldt and K. Knopp (1963) : Einhihrung in die h6here Mathematik. Bd.31 12.Auflage, Stuttgart

G. J. Resnikotf (1952) : A New Two Sided Acceptance Re@ion .for Sampling by Vari- ables. Technical Report Nr. 8, Applied Mathematics and Statistics Laboratory, Stanford University

Dr. Marlon Br~lh--Suhr Universit~t Hamburg Fernstudienzentrum Averhoffstr.38 D 2000 Hamburg 76

Prof. Dr. Wolf Krumbholz Institut Far Statistik und Quantitative 6konomlk, Universit~t der Bundeswehr Hamburg Holstenhofweg 85 D 2000 Hamburg 70