Estimation in a random censoring model with incomplete information: exponential lifetime distribution

1 1

IEEE TRANSACTIONS ON RELIABILITY, VOL. 37, NO. 2,1988 JUNE 223

Estimation in a Random Censoring Model with Incomplete Information: Exponential Lifetime Distribution

T. Elperin

I. Gertsbakh Ben-Gurion University of the Negev, Beer-Sheva

Ben-Gurion University of the Negev, Beer-Sheva

Key Words - Maximum likelihood method, Point estimation, Interval estimation, Exponential distribution, Random censoring

Rwdcr Ai& - Purpose: Widen and advance state of the art Special math needed for explanations: Theory of statistical in-

Special math needed to use results: Same Results useful to: Reliability analysts and theoreticians

ference

SUmmmy & C~nchuions - Analysss Of methods and simaln- don results for estimating the exponential mean lifetime in a random censoring model with incomplete information are prerrented. The instnnt of an item’s failure is observed if it occus before a ran- domly chosen inspection time and the failore was signalled. Other- wise, the experiment is terminated at the instant of inspection dur- ing which the true state of the item is disesvered. The maximum likelihood method 0 is ased to obtain point and interval &imam for item mean l i fehe, for the exponential model. It is demonstrated, using Monte Carlo simulation, that MLM provides positively biased estimates for the mean Wethe and that the large sample approximation to the log-likelibood ratio produces rather accurate confidence intervals. Tbe quality of the estimates is very little influenced by the value of the probability of fdure-to4gnd. Properties of the Fisher information in the censored sample are in- vestigated theoretically a d numerically.

1. INTRODUCTION

This paper presents methods and simulation results for estimating the exponential mean lifetime in the following random censoring model with incomplete information. Let 7 be the item lifetime, and let 7 be the random inspection time, 7 and g are assumed to be s-independent. If the item fails before the inspection, ie, if 7 = x < g = y, then either with probability p, 0 < p 4 1, the failure is immediately signalled and then the true lifetime 7 = x is determined, or with probability 1 - p the failure is not signalled and the item’s true state will be discovered only at the instant of the inspection g = y. In the latter case, the information received will be that 7 < y. If the lifetime 7 > g, the experiment terminates at the instant of the inspection and the information received is that 7 > 71 = y .

If p = 0, ie, a failure is never signalled, we arrive at the well-known model termed by Nelson [4] as a quantal response model. If p = 1, ie, a failure is immediately

displayed, we arrive at the random censoring model, see eg, Lawless [2].

We consider in this paper the case of exponentially distributed lifetime and concentrate on point estimation and s-confidence interval for the mean lifetime. The rest of this section presents background material on the maximum likelihood methodology for obtaining point estimates and sconfidence intervals.

Section 2 presents the results of a simulation study for the above model, for various sample sizes n = 10, 20, 40, 100; for various random censoring patterns (uniform and exponential); and for various values of a probability of failure-to-signal, p. The maximum likelihood estimate (MLE) e of the mean lifetime 9, has on the average a positive bias, which for n 2 20 does not exceed 5%. Two methods of constructing s-confidence intervals are com- pared and it is shown that the method based on the large- sample approximation to the log-likelihood ratio provides rather accurate estimation, for small sample size n 2 20. It is demonstrated that the bias in the point estimator and the empirical s-confidence levels are very little influenced by the value of a probability of failure-to-signal.

Section 3 is devoted to the study of the Fisher information IF in the sample. It is proved that IF is a linear combination, with the weightsp and 1 - p, of the Fisher information in the quantal response model and in the random censoring model, respectively. It is demonstrated numerically that both extreme casesp = 0 a n d p = 1 differ relatively little in the values of the Fisher information.

I

Notation

7 random time to item failure; random_ time to item inspection (censoring); rl

A t ; e), et; e), F (t; 8) pdf, Cdf, Sf of 7; with parameter 8; g 0 , G69, GQ pdf, Cdf, Sf of rl; - implies “is distributed as”; - 42 probability of failure-to-signal; 6 i 9 null set; D , D2 - D i IDi! C set of censored observations; z,,, (1 - a/2)-quantile of the standard normal

X;-Jl) (1 - or)-quantile of the chi-square distribution with

as implies “is asymptotically distributed as”;

maximum likelihood estimate of 8; item number, i = 1, ..., n;

set of noncensored and nonsignalled observations; set of noncensored and signalled observations; complement to the set Di; i = 1, 2,; number of elements in the set Di;

distribution;

one degree of freedom;

0018-9529/88/o600-0223$01 .oOO 1988 IEEE

224 IEEE TRANSACTIONS ON RELIABILITY, VOL. 37, NO. 2,1988 JUNE

Ex(@ V(a, b) implies a uniform distribution with support [a, bl, N(p, v) implies a s-normal (Gaussian) distribution with

L(zi; e) likelihood associated with item i in the sample, L(z; e) likelihood function associated with the sample,

implies an exponential distribution with mean 8, We deal with a noninformative censoring, ie, the Cdf G(y) does not depend on 8, the parameter of interest.

After a little algebra we obtain from (1) that: mean, p , and variance, v,

log L ( ~ ~ ; e) = logflzi; e) + a ~ l - pyOg fiZi; e)

E[ - L(zi; e) 1 = the Fisher information in

ae2 one experiment associated with item i,

- L(z; e) [ e= e the observed informa- ae2 10

tion in the sample, Fisher information in one experiment for p = 1, ie, for random censoring scheme, Fisher information in one experiment for p = 0, ie, for the quantal response model,

rF"

1:

ML maximum likelihood MLE ML estimator

Other, standard notation is given in "Information for Readers & Authors" at rear of each issue.

2. BACKGROUND ON MAXIMUM LIKELIHOOD

Consider item i; it has lifetime T~ with Cdf F(t; e). After time y i since the beginning of the item's operation it will be inspected. Let yi be a realization of a r.v. 11 with Cdf G(y) . R.v.3, T~ and 11 are s-independent. Three possibilities can occur in the experiment with item i:

i. The item fails in the interval (xi, xi + dx), xi < yi, and the failure is imgediately signalled. The probability of this event is f(xi; 8)G(xi) - p dx.

ii. The item fails at some instant xi, xi < yi; the failure is not signalled and remains undetected until it will be discovered at the inspection. The inspection will take place in the interval (yi, yi + dy). The probability of this event is g(yi)F(yi; e)(l - p)dy;

iii. The inspection takes place in the interval (yi, yi + dy) and it reveals that the item did not fail o~ the interval [0, yi]. The probability of this event is g(yi)F (yi; 8)dy.

The likelihood associated with testing item i is:

where zi is the observed lifetime or inspection time, and a, are zero-one indicator variables:

1, if the lifetime is not censored;

1, if the failure is signalled; 0, otherwise.

(3)

$(zi, p) does not depend on 8 and is thus dropped from the likelihood function.

For the exponential lifetime, F(t; e) = 1 - exp( - t/O):

The equation for finding the MLE e is alog L(z; t9)/ae = 0. After simple algebra we obtain from (5) that e is the root of -

+ ID21 - 8 = zi. (6) Zi i E D 1

C iEDl exp(- zi/O) - 1

This equation always has one and only one solution, e > 0, except for the following cases: D1 = Dz = 0, or El = 0. Obvious changes must be made in (1) for two extreme cases p = 1 andp = 0. The first one corresponds to the random censoring scheme with complete information, for which D1 = 0. The corresponding equation for 8 is obtained from

(a) by dropping the first sum, and then e = i=l zi/lDzl, [2, p

1051. The case p = 0 corresponds to the quantal response model [4, chapter 91 in which D2 = 0. Then the equation for e is obtained from (6) by setting lDzl = 0; it is a well known ML equation for grouped data with one group of random length [l , chapter 21.

The following two large-sample approximations are used for obtaining s-confidence intervals for 8, [3, chapter 61, [5, chapter 131:

"

As asymptotically equivalent version of (7) is based on the observed information Io:

' I '

ELPERIN/GERTSBAKH: ESTIMATION IN A RANDOM CENSORING MODEL WITH INCOMPLETE INFORMATION

~

225

The latter expression is very convenient because the com- putation of I, is straightforward. It follows from (8) that an approximate (1 - a)-confidence interval for 0 is:

(9)

Another large-sample result useful for s-confidence intervals is the following. Let 0 = Bo be the true parameter value. Then [5 , chapter 131 - - L(Z- e A(&) = - 2 1 0 g A XZ(1).

L(2; ij)

Then it follows from the theory of statistical inference that the set:

is an approximate (1 - +confidence set for 8. In our model, (11) takes the form:

log 1 -- exp(- zih) e + (Dz(l0g 7

ED^ 1 - exp(- zJ0) e J2(0) = is:

The value of J2(6) can be found numerically. Several sources, eg, [2, p 1081, indicate that (11) might be more accurate than (8) and thus might produce more accurate s-confidence intervals. This is verified by our simulation study, the results of which are presented in the next section.

3. POINT AND INTERVAL ESTIMATION: A SIMULATION STUDY

3.1 Goals of the Simulation

i. To investigate the influence of the sample size n, failure signalling probability p, and the censoring (inspection time) pattern on the bias and variance of e^;

ii. To assess the performance of the two methods of s-confidence estimation for 8; to determine which of them produces more accurate s-confidence intervals; and to verify the effect of the sample size, failure signalling probability, and the censoring pattern on such s-confidence intervals.

3.2 Scope of Simulation

Without loss of generality, let 8 = 1. We simulated samples of size n = 10,20,40, 100 with

p = 0.0 (0.25) 1 .O, and several statistical patterns of the inspection (censoring) time, namely 11 - Ex(0.5), Ex(l), Ex(1.5), U(0.5, 1.5). For each particular choice of n, p, and the pdf of 7, 1 OOO replicas of a random sample were

generated. The MLE e ^ c i ) was found for replica i, i = 1, . . . , 1OOO, by solving (6). The replicas for which (6) did not have a solution were excluded from consideration. _The sample average e, the sample standard deviation of 0, $6, and the t-statistic t = ,KiXlJ(e - l)& were computed from the sample of loo0 replicas. Besides, for each generated random sample, s-confidence intervals from (9) and (12) were computed, for 1 - a = 0.90, 0.95, 0.975. The sample average of the ratio of the lengths of these intervals, p E P2/P1, were computed, where Pl and P2 are the lengths of the s-confidence intervals obtained by the log- likelihood ratio method and s-normal approximation respectively. The s-confidence levels were computed as N/lOOO, where Ni s the number of replicas out of lOOO, for which the s-confidence interval did cover the true value.

3.3 Results of the Simulation

The results regarding the point estimation are presented in table 1. The main conclusions drawn from these results are the following.

i. e^is always positively biased, and the magnitude of the bias is between 1% and 22%. The bias decreases considerably with increase in sample size, and for n = 20 it does not exceed 5Vo. The largest bias corresponds to small sample n = 10 and strong censoring (7 - Ex(0.5)). The values of the t-statistic reveal that the bias is strongly s-significant for sample sizes n = 10, 20;

ii. For any fixed sample size and fmed censoring pattern, the properties of the point estimator are relatively little affected by the magnitude of p. Table 1 shows that the 6i and the t-statistic have small variations when p is changed;

iii. Eq @) does not have a solution when either D1 = D2 = 0 or D 1 = 0. The highest probability of this event (about 0.008) corresponds to n = 10 and the uniform censoring pattern. The results in table 1 are presented only for samples for which (6) has a solution. For n exceeding 10, the probability that the ML equation does not have a solution becomes negligibly small.

Typical results for the s-confidence intervals are presented in table 2. It shows the values of the s-confidence for various censoring patterns, sample sizes, and p-values, for the nominal sconfidence value 1 - a = 0.95. The figures in table 2 are obtained by using the log-likelihood ratio method, ( l l ) , (12). The figures in parentheses in table 2 represent the s-confidence for the s-normal approximation in (9). The standard deviation for the s-confidence is 0.007 and thus the values in the range 0.94-0.96 might be considered as agreeing with the results.

The main conclusions are:

i. The log-likelihood ratio method is adequate for sample sizes n = 10, 20,40, 100;

ii. The s-normal approximation provides strongly negatively biased s-confidence levels for n = 10, 20; for n = 40, 100 this method is still inferior to the log-likelihood


TABLE 1 Properties of e

loo0 - 1 (The Empirical Average 8 = - e ^ ( i J ; Ci; and the t-statistic)

i=l

p = o p = 0.5 p = 1.0 - - - -

t e 35 t 8 ‘75 t

Ex(0.5) 1.216 0.91 7.5 1.217 0.91 7.6 1.220 0.89 7.9

V(O.5;l.S) 1.063 0.40 4.9 1.055 0.40 4.3 1.062 0.40 5.0

Ex(0.5) 1.050 0.36 4.1 1.051 0.38 4.2 1.053 0.38 4.4

Censoring Sample Pattern 8 ‘7i

1.120 0.72 5.4 1.035 0.44 2.5 1.041 0.42 3.1 1.071 0.59 3.8 1.026 0.39 2.2

Ex(U Ex(1.5) 1.088 0.63 4.4

1.022 0.34 2.0 1.024 0.33 2.3 1.025 0.32 2.4 W l )

U(0.5;l.S) 1.036 0.31 3.7 1.036 0.30 3.7 1.039 0.30 4.1

Ex(O.5) 1.034 0.3! 3.2 1.023 0.28 2.6 1.027 0.28 2.9

Ex(1.5) 1.017 0.24 2.2 1.013 0.23 1.9 1.009 0.21 1.4

2o Ex(1.5) 1.039 0.38 3.2 1.023 0.31 2.3 1.020 0.29 2.1

1.019 0.26 2.3 1.016 0.24 2.0 1.015 0.24 2.0

U(0.5;1.5) 1.023 0.22 3.3 1.022 0.22 3.2 1.024 0.21 3.7

Ex(1)

E W

Ex(O.5) 1.016 0.20 2.6 1.016 0.19 2.6 1.016 0.19 2.6 1.009 0.16 1.7 1.008 0.16 1.5 1.009 0.15 1.8

Ex(l.5) 1.008 0.17 1.6 1.007 0.15 1.5 1.008 0.14 1.7 V(0.5;1.5) 1.014 0.14 3.2 1.012 0.14 2.7 1.012 0.14 2.8

ratio method and underestimates the s-confidence level by n P 1 “70-2.5 VO ;

iii. The data in table 2 do not provide evidence that the p-value influences the s-confidence level;

iv. For n = 10 and nominal sconfidence level 1 - a = 0.975 the log-likelihood ratio method produced often enormously long sconfidence intervals whose lengths ex- ceeded 10. This phenomenon happened with rather high probability 0.30-0.15, depending on the censoring pattern, and can be explained by the long plateau on the log- likelihood ratio curve. These “pathological” samples were excluded from consideration, and the resulting sconfidence levels were still superior to those provided by the large-sample s-normal approximation. Therefore the log-likelihood ratio is a reliable method for generating sconfidence intervals even in small samples (n = 10) provided that the nominal s-confidence level is not too high. The alternative method based on the s-normal approximation is negatively biased by 5%-9%;

v. The s-confidence interval generated by the log- likelihood ratio is very little influenced by the increase of the sample size (for n = 20, 40, 100), and it remains quite accurate for all these n-values. The seonfidence levels provided by the s-normal approximation improve considerably their properties with the increase of the sample size;

vi. There is, however, a price for the better performance of the log-likelihood ratio method, namely the s-confidence intervals are, on the average, longer than those obtained by the s-normal approximation. Denote by p E{Pz/P1} where PI, Pz are the lengths of the sconfidence intervals obtained by log-likelihood and s-normal approximation, respectively. Our computations based on results of simulation show that:

10 20 40

100

0.7-0.9 0.83-0.86 0.92-0.94 0.97-0.98

4. SOME PROPERTIES OF THE FISHER INFORMATION IN THE SAMPLE

These properties are summarized in theorem 1. State- ment i of this theorem asserts that the Fisher information is a linear combination of the Fisher information of two extreme cases, p = 0 (quantal response data) and p = 1 (random censoring with complete information). The statement ii of the theorem asserts that the random censoring scheme is not less informative than the quantal response scheme while statements iii and iv provide formulas for computing the information which is used for a numerical comparison between IF and IS.

Theorem I

i. I~ = p r , ~ + (I - p ) ~ :

iv. For 7 - Ex@),

F = V2 1,” (1 - exp( - v/fl))g(v)dv, (17)

: I

ELPERIN/GERTSBAKH: ESTIMATION IN A RANDOM CENSORING MODEL WITH INCOMPLETE INFORMATION 227

TABLE 2

(Using the Log-Likelihood Ratio Method) s-Confidence for a nominal s-confidence level of 1 - a = 0.95

Sample Censoring Size Pattern p = 0.0 p = 0.5 p = l

Ex(0.5) Ex(1) Ex(1.5) U(0.5;1.5)

Ex(0.5)

Ex(1.5) U(0.5;1.5)

Ex(0.5) E x W Ex(1.5) U(0.5; 1.5)

Ex(0.5)

Ex(l.5) U(0.5;1.5)

n=lO

Ex(1) n=20

n =40

n=lOO Ex(1)

0.950 0.952 0.956 0.967

0.956 0.963 0.956 0.951

0.948 0.950 0.956 0.957

0.939 0.940 0.939 0.945

(0.865) (0.898) (0.902) (0.922)

(0.903) (0.914) (0.902) (0.924)

(0.931) (0.937) (0.943) (0.944)

(0.937) (0.938) (0.925) (0.948)

0.940 0.953 0.946 0.954

0.955 0.956 0.953 0.952

0.957 0.984 0.951 0.958

0.937 0.943 0.925 0.944

(0.865) (0.881) (0.895) (0.901)

(0.910) (0.905) (0.917) (0.914)

(0.928) (0.927) (0.934) (0.942)

(0.935) (0.938) (0.924) (0.946)

~~

0.952 0.957 0.958 0.963

0.960 0.954 0.945 0.948

0.956 0.948 0.945 0.958

0.941 0.943 0.940 0.950

(0.875) (0.884) (0.896) (0.900)

(0.907) (0.903) (0.912) (0.908)

(0.926) (0.931) (0.944) (0.946)

(0.932) (0.942) (0.928) (0.944)

The value in parentheses corresponds to the large-sample snormal approximation.

TABLE 4

Censoring Pattern 1 - Z$ZF R

The proof is presented in the appendix. It is instructive to compare numerically I: with e. ~ ~ ( 0 . 5 ) -- 0.308 - 0.924 1.52

Without loss of generality, let 8 = 1. Consider first the

values of +(q) for various 1. 0.500

0.333

-- oA04 - 0.808 1 .08 case of a constant censoring time 7 . Table 3 shows the

ExU)

Ex(1.5) -- 0‘424 - 0.707 1.17 TABLE 3 0.600

The ratio $(q) e/& as a function of TJ

U(0.5; 1 3) -- OS60 - 0.908 1.09 TJ 0.4 0.8 1.2 1.6 2.0 2.4 2.6 0.617

$(TJ) 1.013 1.054 1.126 1.23 1.38 1.58 1.85 R is the average sample value of Ziy’ divided by b Z 2 + (1 - p)Z:] - ’ I , for n = 0, p = 0.5.

Table 3 shows that +($ grows surprisingly slowly. For ex- ample, for 7 = 2, ie, when almost 86% of all observations are not cencored at all, the relative increase in the variance for the “blind” quantal response scheme is only about 38%. Slow increase of $(q) explains why the relative increase of r,“ with respect to I: is rather moderate when the random censoring is applied. Table 4 compares the values of I:, r , ~ for various censoring patterns.

We have already pointed out that the properties of point estimators are very little affected by the value of p, within fixed censoring pattern and sample size. We explain this robustness by a relatively small change in the information in the sample when the value of p changes, as was demonstrated above.

APPENDIX

Proof of Theorem I

ment is: By definition, the Fisher information in one experi-

where zi, a, 0 must be treated as random variables. For the sake of brevity we will use the following notation:


f .f" - f f 2

f From (3), the Fisher information in one experiment con- { . } = Pr{a = 1) i', ( - sisting of testing one item is:

f .f" - f ' 2 IF=E[B( - CY + (1 - a) f F 2

F2

Note that:

Now note that Pr{a = 1) = Pr{7 d y } = F(y; e), and that for all 8,

s', f (v ; 8)dv + F ( y ; e) = 1.

Assume that this expression can be differentiated twice

with respect to 0 by exchanging - (-) and the integration.

(A'1)

d dB

Then

s',f"dv - F;(y; e) = 0.

In view of that, we obtain (after simple algebra) that { - } in (A.7) is exactly the expression in the outside braces in the integral (15). Averaging with respect to the values of r ] ,

gives the final expression for E. In a similar way, one obtains (16) for 1:. A straightforward substitution of F(v; e) = 1 - exp( - v/O) into r,C & I: yields (17), (18).

In order to prove statement ii it is sufficient to prove it for a fixed censoring time r] = y . Thus one has to prove:

1 E[- CY + (1 - a) -

F . F " + F ' 2 1 E[- a

f .f" - f f 2 F . F" + F'2

f F 2

(A*2) = E

F 2 F . F" - F'2 + (1 - a)

F2

= I: (A.3)

Averaging with respect to p, one obtains from (A.l)-(A.3) that:

t 7 d v + f f 2 (F%- 2 ' e))2 - ( m y ; e))z IF = P ( p = 1 ) E + P(/3 = 0)I: = pr,E + (1 - p)I:, (A.4)

To prove statements iii & iv, compute E, I$ by condi-

F ( y ; e) RY; e)

which proves statement i. + F(Y; e)

Y2 tioning first on the value r] = y of the censoring time. Then: s', d v 2

f .f" - f f z

f The last expression can be proved with the aid of the

ACKNOWLEDGMENT We are most grateful to Professor A. M. Kagan for

For a given r] = y , zi is a mixed-type random variable which has pdff(v; 6) for v < y and a mass of F(y; e) con- centrated at the censoring point y. Given CY = 1 (failure before censoring), the pdf of zi isf(v; B)/F(v; e), 0 d v d y , and given a = 0 (censoring before failure), zi is a 111 G. Kulldorff, Estimation from Grouped and Partially Grouped degenerate r.v. with mass 1 placed at the censoring point r]

J. F. Lawless, Statistical Models and Methods for Lifetime Data, John Wiley & Sons, 1982. = y. Then the sexpectation in (A.5) is:

valuable suggestions.

REFERENCES

samPles~ John Wiley 8~ Sons, lg60. [2]

/I

ELPERINICERTSBAKH: ESTIMATION IN A RANDOM CENSORING MODEL WITH INCOMPLETE INFORMATION 229

E. L. Lehmann, Theory of Point Estimation, John Wiley & Sons, 1983. W. Nelson, Applied Life Data Analysis, John Wiley & Sons, 1982. S. Wilks, Mathematical Statistics, John Wiley & Sons, 1962.

AUTHORS Dr. Tov I. Elperin; The Pearlstone Center for Aeronautical Engineering Studies; Department of Mechanical Engineering; Ben-Gurion University of the Negev; POBox 653; Beer-Sheva 84105 ISRAEL.

Tov I. Elperin is a lecturer in the Department of Mechanical Engineering at Ben-Gurion University of the Negev. He received his BS and MS in Theoretical Physics from the Byelorussian State University, Minsk, USSR, and PhD in Nuclear Engineering from the Ben-Gurion University of the Negev. His research interests include applications of

stochastic models in applied physics and engineering, Monte Carlo methods and reliability analysis.

Dr. Ilya B. Gertsbakh; Department of Mathematics and Computer Sciences; Ben-Curion University of the Negev; POBox 653; Beer-Sheva 84105 ISRAEL.

nyn B. Gertsbnkb is an Associate Professor in the Department of Mathematics and Computer Sciences at the Ben-Curion University of the Negev. He received his MS in Mechanical Engineering and Mathematics from the Latvian State University, Riga, USSR, and a PhD in Applied Probability and Statistics from the Latvian Academy of Sciences, Riga, USSR. His research interests include operations research and mathematical reliability theory.

Manuscript TR86-159 received 1986 December 1; revised 1987 June 24.

IEEE Log Number 17332 4 TR b

.

Journal of Quality Technology - Tables of Contents A Quarterly Journal of Methods, Applications, and Related Topics

1987 July 1987 October

Variability Reduction Through Subvessel CUSUM Comparison of Poisson Means: The General Case.. ........ Control.. ................ K. K. Hockman & J. M. Lucas 113 ......................................... L. S. Nelson 173

Use of Boxplots for Process Evaluation. .................. .......................... B. Iglewicz and D. C. Hoaglin 180 Plans for Very Low Fraction Nonconforming. ............. ...................................... L. Pesotchinsky 191 An Evaluation of the Laboratory Ranking Test. ............

Multivariate Tolerance Regions and F-Tests . . . . . . . . . . . . . . ............................... C. Fuchs & R. S. Kenett 122 Tools for Computer-Aided Design of Experiments. ........ .................................... C. J. Nachtsheim 132

.......................................... R. S. Elder 197 ANSVASQC Z1.4 Performance Without Limit Numbers . . . . . . . . . R. B. Grinde, E. D. McDowell& S. U. Randhawa 204 Allocating Observations in the Random Effects Balanced One-

Computer Programs Average Run Lengths of Exponentially Weighted Moving Average Control Charts.. .............. S. V. Crowder 161

Technical Aids Way ANOVA.. ..................... J. J. Pignatiello, Jr. 216 Upper lo%, 5% and 1 Vo Points of the Maximum F-Ratio . ...................................... L. S. Nelson 165 Computer Programs

A Computer Program for Allocating Observations in the Book Reviews Random Effects Balanced One-way ANOVA. ...........

................................ J. J. Pignatiello, Jr. 221 Understanding Statistical Process Control. . . . . . . . . . . . . . ..................... 11. J. Wheeler & D. S. Chambers 168 Truncated and Censored Samples from Normal Popula- tions.. ................................ H . Schneider 169 The Elements of Graphing Data. ....... W. S. Cleveland 169

Technical Aids A Chi-square Control Chart for Several Proportions. .... ...................................... L. S. Nelson 229

Experimental Measurements: Precision, Error and Truth. . Book Reviews ..................................... N. C. Barford 170 Out of the Crisis.. ..................... W. E. Deming 232

Introduction to Statistical Quality Control. ............. ................................. D. C. Montgomery 233 Introduction to Statistical Quality Control .............. ................................. D. C. Montgomery 238 Probability Distributions. . V. Rothschild & N. Logothetis 238

Index.. .............................................. 240 4 T R b

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Estimation in a random censoring model with incomplete information: exponential lifetime distribution