Truncated and Censored Samples

### Theory and Applications

Truncated and Censored Samples

### Theory and Applications

ByA. Clifford Cohen

Edition 1st Edition

First Published 1991

eBook Published 19 April 2016

Pub. location Boca Raton

Imprint CRC Press

Pages 328 pages

eBook ISBN 9781482277036

SubjectsMathematics & Statistics

KeywordsEstimating Equations, Truncated Samples, Maximum Likelihood, Loglikelihood Function, Maximum Likelihood Estimators

Get Citation

#### Get Citation

Clifford Cohen, A. (1991). Truncated and Censored Samples. Boca Raton: CRC Press, https://doi.org/10.1201/b16946

ABOUT THIS BOOK

TABLE OF CONTENTS

1.1 PRELIMINARY CONSIDERATIONS of the sample space are, depending on the nature of the restriction, It is perhaps more accurate

of modem of American trotting horses. Sample data were extracted from Wallace's

of truncation. of n observations, x T, where T is of truncation. of a total of N observations of which n are fully measured while c < T, whereas for each of Tis a fixed (known) of censoring. In Type II samples, T that is, the (c+ l)st of size N. of a total of N ob- > of I

of an unrestricted (i.e., complete) distribution with parameters

2.1 PRELIMINARY REMARKS of estimators for

of the standard normal distribution. 2.3 MOMENT ESTIMATORS FOR SINGLY TRUNCATED SAMPLES of the truncated normal population.

2.4.2 An Illustrative Example Example 2.4.1. A complete sample of 40 observations was selected from of random observations from a normal population

of expected values of the second-order partial derivatives of the

of 12.00 of 50 units was selected from the screened pro- of Figure 2.1). fl and &are calculated from (2.3.9) and (2.3.12) as = + 0.00916(9.35- = 1.1907, rl = 9.35 - = 9.37. It follows that = 1.091 = (12.00 -= 2.41.

3.1 INTRODUCTION of the normal distribution are derived for of the complete distribution, the truncation points are = Tz -

of (3.3.1), differentiate with respect to these parameters, and = 0.

-52.9767 1791.2545 3.4 PROGRESSIVELY CENSORED SAMPLES 2a i=I + 2: +

of which are considered later +A.) -1.0 -0.083 -0.205 -0.5

of products of the complete (uncensored) ob- of right censored observations. It follows that N + n,

ByN is the total sample size, n is the number of complete (uncensored)

(W'V- It then follows that E'fX (lnV- -i and < J). = c= c, and 1V-E = E'V- It follows that

Standard errors follow as

Bya .... 1.69\10.13613 = 0.624 and

of prominence in the field of reliability and life testing where samples

of the first two of these equations are identical to the first two of (6.4.4). The third of the preceding equations, which in this case of (6.4.4), can be written in an expanded form as + + aE(Z

of an estimate of the threshold parameter of total size N should approximately equal the of a corresponding estimate from a complete sample. This result was

7.1 THE INVERSE GAUSSIAN DISTRIBUTION

2: ln(x; - > I, maximum likelihood estimating equations may be obtained by 1 ~ ~ cÂ· aF

ByJ=d-F J=li-F

Of course, Tis a lower bound on all and we

ByofT for first approximations is convenient, and it usually

8.1 THE EXPONENTIAL DISTRIBUTION 8.1.1 = 0, and in these cases the

[F(nJ '. of size n from a truncated dis- t 7) = Thus [expk

ByMaximum Likelihood Estimators

of and in small samples, the of the preceding equations is identical with the second equation

= 100 ( , of the W eibull distribution is of both sides of the first equation of (8.1.31), we obtain of ln (x -of + = H- Ink Â·

of rainfall, flood flow, earthquake, and other of material, cor- of extreme value is of limiting distributions, which approximate = exp [ - = exp (>0), and (>0) are parameters. of "extreme value" distributions. Many authors consider it to be "the"

of the two-parameter We ibull distribution is o < < > o. > o. (8.2.4) of the Type I distribution of greatest extreme values is (8.2.5)

of a sample as thus described from a distribution that is of least extreme values is * = +

(5.3.6) for calculating the Weibull estimate 8. of censoring cj items are removed (censored) from further -nina+ -'-a-

ByN from a Type I dis- a uaa =

9.1 INTRODUCTION of acoustical of which is normally distributed (0, u of the Rayleigh distribution (i.e., the pdf of X) follows as

of the of the estimators presented in Chapter 5 for Weibull of Weibull parameters might of Rayleigh

10.1 of economics who formulated it ( 1897)

of the pdf and the cdf with a of this of a are included in Table 10.1 for selected values of this argument. More complete

ByaiX) and aiX) as functions

143.2632- of as It follows that = 20.66, and the approximate 95% CI is It is noted that differences between the MLE and the MMLE calculated from of its smaller bias. However, readers are again reminded that the only

11.1 of Cohen ( 1941,

= 0; of the same sign, D f.l;, D = and = a of Figure of Craig (1936).

Byrimaginary, bÂ¥ 2bf.l;, D

of (11.4.1) plus (11.4.2) to obtain h* = 38.600670, of the sample data based on these Shook's Graduation For Right Singly For Complete Sample Truncated Sample 159.95 79.9 0 0.2 89.9 12.8

Bybf 5.247727, and J*

of accepted specimens, and c = N - of rejected Nand care unknown. Only n, the number of acceptances, is known. In selected samples, full measurement of the screening ---------===----

of the associated variates. Accordingly, of Chapters 2 and 3 are applicable here just as they of size N. of maximum likelihood estimators for parameters of the multivariate normal

Sample Sample 67.6664 67.7033 67.6794 1.7857 1.6927 1. 5235 1. 5172 1. 5318 0.5239 0.5265 0.5318 0.5446 0. 4872 0.4924 0.7339 0.7053 0.7037 N â€¢ 119 n "' 108 Asymptotic Variances* 3.377 1.836 0.695 1. 959 1.041

Byv 1.389

13.1 of zero are not observed. As an example, consider the distribution of the of children per family in developing nations, where records are maintained

... '

ByX= 0, 1, 2, pt-x, 0,

l, 2, = -nA + = = 0, - = A [ + f(a 1)].

Byni!nA- n!n[P(a)]-

xis the mean of the n uncensored observations. 13.2.6 Doubly Censored Samples-Total Number of Censored Observations Known, But Not the Number in Each Tail Separately + IW. + P(d + + (1 f(a 1))].

Byf(a 1) ) ] . -!\a)+ P f(a - x =A

= _ ni _ n [f(a - + [f(a + n [ !_(d) ] + I) 1 -+ (! ))<flnL ni

By-!_(a - f(d - f(a - ni + n [f(d - of censored observation [f(a- -J(a- (f(a-

of all two-parameter /(0) It follows that = = =

ByBinomial with Zero Class Missing

= 0.2113 and = = = 13.4 THE BINOMIAL DISTRIBUTION = 0, 1,2, = I - + n(n -

Byf(x; n, p) ,n, (13.4.1)

... ,K-

Byk<K. j(ynA;p)= y=K,K+ j(ynR;p); y k,k j(ynR;p)+j(ynA;p); y=K,K+

of defectives found in the ith accepted lot (z; of defectives found in each rejected lot. This sample could be described of the paired values (z;, y;), i 1, 2, . . . , m

0.0000 7 0.0000 0.10 0.0128 7.74 0.10 0.0432 17.57 0.15 0.0450 8.09 0.15 0.1702 17.86 0.20 0.1209 8.35 0.20 0.3704 17.49 0.0000 0.01 0.0466 78.99 0.02 0.2156 74.70 0.0000 60 0.03 0.4319 68.02 0.01 0.0224 59.65 0.04 0.6252 60.54

100, k 120, 0.0000 0 0.0000 0.01 0.0794 97.76 0.01 0.0330 119.1 0.02 0.3233 89.37 0.02 0.2200 112.9 0.03 0.5802 77.94 0.03 0.4867 101.4

of x + 1 were reported as x = k with probability of defects per item, becomes + 2) + + 1)], x + 1)!, x k + 1, = 1, 2,

ByX= 0, 1,

+ 60 = 2.1 and

Byx 3. Following is a tabulation of the reported inspection results.

14.2.6 An Illustrative Example-Misclassified Binomial Data 14.3 An example generated by Cohen ( 1960a) consisted of N of defectives in samples of n 40 from a = of + = = 0.00000075, V(e) = 0.0017, Cov({J, e) = 0.0000025, and = 0.07.

of(l4.3.15), we calculate = 23/56 = 0.4107, and from the = = = = of Neyman's contageous distributions and they calculated expected fre-

of censored observations in of observations censored at of variation = of of the W eibull shape pa- of the gamma function.

Bynis sometimes

of Poisson, binomial and negative

ByThe Lognormal Distribution. Cam- Proc. Edinburgh Math. Soc., 4, 106-110.

of fitting the truncated negative binomial of the parameters of the distribution-a reconsideration. Austral. J. Statist., 3, 185-190.

of Michigan, Ann Arbor. of truncated

ByProc. Res. Forum, IBM Corporation, 40-43.

7. Kotz, S., Johnson, N. L., and Read, C. B., eds. Wiley, New York, of estimating the mean and standard

TABLE OF CONTENTS

1.1 PRELIMINARY CONSIDERATIONS of the sample space are, depending on the nature of the restriction, It is perhaps more accurate

of modem of American trotting horses. Sample data were extracted from Wallace's

of truncation. of n observations, x T, where T is of truncation. of a total of N observations of which n are fully measured while c < T, whereas for each of Tis a fixed (known) of censoring. In Type II samples, T that is, the (c+ l)st of size N. of a total of N ob- > of I

of an unrestricted (i.e., complete) distribution with parameters

2.1 PRELIMINARY REMARKS of estimators for

of the standard normal distribution. 2.3 MOMENT ESTIMATORS FOR SINGLY TRUNCATED SAMPLES of the truncated normal population.

2.4.2 An Illustrative Example Example 2.4.1. A complete sample of 40 observations was selected from of random observations from a normal population

of expected values of the second-order partial derivatives of the

of 12.00 of 50 units was selected from the screened pro- of Figure 2.1). fl and &are calculated from (2.3.9) and (2.3.12) as = + 0.00916(9.35- = 1.1907, rl = 9.35 - = 9.37. It follows that = 1.091 = (12.00 -= 2.41.

3.1 INTRODUCTION of the normal distribution are derived for of the complete distribution, the truncation points are = Tz -

of (3.3.1), differentiate with respect to these parameters, and = 0.

-52.9767 1791.2545 3.4 PROGRESSIVELY CENSORED SAMPLES 2a i=I + 2: +

of which are considered later +A.) -1.0 -0.083 -0.205 -0.5

of products of the complete (uncensored) ob- of right censored observations. It follows that N + n,

ByN is the total sample size, n is the number of complete (uncensored)

(W'V- It then follows that E'fX (lnV- -i and < J). = c= c, and 1V-E = E'V- It follows that

Standard errors follow as

Bya .... 1.69\10.13613 = 0.624 and

of prominence in the field of reliability and life testing where samples

of the first two of these equations are identical to the first two of (6.4.4). The third of the preceding equations, which in this case of (6.4.4), can be written in an expanded form as + + aE(Z

of an estimate of the threshold parameter of total size N should approximately equal the of a corresponding estimate from a complete sample. This result was

7.1 THE INVERSE GAUSSIAN DISTRIBUTION

2: ln(x; - > I, maximum likelihood estimating equations may be obtained by 1 ~ ~ cÂ· aF

ByJ=d-F J=li-F

Of course, Tis a lower bound on all and we

ByofT for first approximations is convenient, and it usually

8.1 THE EXPONENTIAL DISTRIBUTION 8.1.1 = 0, and in these cases the

[F(nJ '. of size n from a truncated dis- t 7) = Thus [expk

ByMaximum Likelihood Estimators

of and in small samples, the of the preceding equations is identical with the second equation

= 100 ( , of the W eibull distribution is of both sides of the first equation of (8.1.31), we obtain of ln (x -of + = H- Ink Â·

of rainfall, flood flow, earthquake, and other of material, cor- of extreme value is of limiting distributions, which approximate = exp [ - = exp (>0), and (>0) are parameters. of "extreme value" distributions. Many authors consider it to be "the"

of the two-parameter We ibull distribution is o < < > o. > o. (8.2.4) of the Type I distribution of greatest extreme values is (8.2.5)

of a sample as thus described from a distribution that is of least extreme values is * = +

(5.3.6) for calculating the Weibull estimate 8. of censoring cj items are removed (censored) from further -nina+ -'-a-

ByN from a Type I dis- a uaa =

9.1 INTRODUCTION of acoustical of which is normally distributed (0, u of the Rayleigh distribution (i.e., the pdf of X) follows as

of the of the estimators presented in Chapter 5 for Weibull of Weibull parameters might of Rayleigh

10.1 of economics who formulated it ( 1897)

of the pdf and the cdf with a of this of a are included in Table 10.1 for selected values of this argument. More complete

ByaiX) and aiX) as functions

143.2632- of as It follows that = 20.66, and the approximate 95% CI is It is noted that differences between the MLE and the MMLE calculated from of its smaller bias. However, readers are again reminded that the only

11.1 of Cohen ( 1941,

= 0; of the same sign, D f.l;, D = and = a of Figure of Craig (1936).

Byrimaginary, bÂ¥ 2bf.l;, D

of (11.4.1) plus (11.4.2) to obtain h* = 38.600670, of the sample data based on these Shook's Graduation For Right Singly For Complete Sample Truncated Sample 159.95 79.9 0 0.2 89.9 12.8

Bybf 5.247727, and J*

of accepted specimens, and c = N - of rejected Nand care unknown. Only n, the number of acceptances, is known. In selected samples, full measurement of the screening ---------===----

of the associated variates. Accordingly, of Chapters 2 and 3 are applicable here just as they of size N. of maximum likelihood estimators for parameters of the multivariate normal

Sample Sample 67.6664 67.7033 67.6794 1.7857 1.6927 1. 5235 1. 5172 1. 5318 0.5239 0.5265 0.5318 0.5446 0. 4872 0.4924 0.7339 0.7053 0.7037 N â€¢ 119 n "' 108 Asymptotic Variances* 3.377 1.836 0.695 1. 959 1.041

Byv 1.389

13.1 of zero are not observed. As an example, consider the distribution of the of children per family in developing nations, where records are maintained

... '

ByX= 0, 1, 2, pt-x, 0,

l, 2, = -nA + = = 0, - = A [ + f(a 1)].

Byni!nA- n!n[P(a)]-

xis the mean of the n uncensored observations. 13.2.6 Doubly Censored Samples-Total Number of Censored Observations Known, But Not the Number in Each Tail Separately + IW. + P(d + + (1 f(a 1))].

Byf(a 1) ) ] . -!\a)+ P f(a - x =A

= _ ni _ n [f(a - + [f(a + n [ !_(d) ] + I) 1 -+ (! ))<flnL ni

By-!_(a - f(d - f(a - ni + n [f(d - of censored observation [f(a- -J(a- (f(a-

of all two-parameter /(0) It follows that = = =

ByBinomial with Zero Class Missing

= 0.2113 and = = = 13.4 THE BINOMIAL DISTRIBUTION = 0, 1,2, = I - + n(n -

Byf(x; n, p) ,n, (13.4.1)

... ,K-

Byk<K. j(ynA;p)= y=K,K+ j(ynR;p); y k,k j(ynR;p)+j(ynA;p); y=K,K+

of defectives found in the ith accepted lot (z; of defectives found in each rejected lot. This sample could be described of the paired values (z;, y;), i 1, 2, . . . , m

0.0000 7 0.0000 0.10 0.0128 7.74 0.10 0.0432 17.57 0.15 0.0450 8.09 0.15 0.1702 17.86 0.20 0.1209 8.35 0.20 0.3704 17.49 0.0000 0.01 0.0466 78.99 0.02 0.2156 74.70 0.0000 60 0.03 0.4319 68.02 0.01 0.0224 59.65 0.04 0.6252 60.54

100, k 120, 0.0000 0 0.0000 0.01 0.0794 97.76 0.01 0.0330 119.1 0.02 0.3233 89.37 0.02 0.2200 112.9 0.03 0.5802 77.94 0.03 0.4867 101.4

of x + 1 were reported as x = k with probability of defects per item, becomes + 2) + + 1)], x + 1)!, x k + 1, = 1, 2,

ByX= 0, 1,

+ 60 = 2.1 and

Byx 3. Following is a tabulation of the reported inspection results.

14.2.6 An Illustrative Example-Misclassified Binomial Data 14.3 An example generated by Cohen ( 1960a) consisted of N of defectives in samples of n 40 from a = of + = = 0.00000075, V(e) = 0.0017, Cov({J, e) = 0.0000025, and = 0.07.

of(l4.3.15), we calculate = 23/56 = 0.4107, and from the = = = = of Neyman's contageous distributions and they calculated expected fre-

of censored observations in of observations censored at of variation = of of the W eibull shape pa- of the gamma function.

Bynis sometimes

of Poisson, binomial and negative

ByThe Lognormal Distribution. Cam- Proc. Edinburgh Math. Soc., 4, 106-110.

of fitting the truncated negative binomial of the parameters of the distribution-a reconsideration. Austral. J. Statist., 3, 185-190.

of Michigan, Ann Arbor. of truncated

ByProc. Res. Forum, IBM Corporation, 40-43.

7. Kotz, S., Johnson, N. L., and Read, C. B., eds. Wiley, New York, of estimating the mean and standard

CONTENTS

ABOUT THIS BOOK

TABLE OF CONTENTS

1.1 PRELIMINARY CONSIDERATIONS of the sample space are, depending on the nature of the restriction, It is perhaps more accurate

of modem of American trotting horses. Sample data were extracted from Wallace's

of truncation. of n observations, x T, where T is of truncation. of a total of N observations of which n are fully measured while c < T, whereas for each of Tis a fixed (known) of censoring. In Type II samples, T that is, the (c+ l)st of size N. of a total of N ob- > of I

of an unrestricted (i.e., complete) distribution with parameters

2.1 PRELIMINARY REMARKS of estimators for

of the standard normal distribution. 2.3 MOMENT ESTIMATORS FOR SINGLY TRUNCATED SAMPLES of the truncated normal population.

2.4.2 An Illustrative Example Example 2.4.1. A complete sample of 40 observations was selected from of random observations from a normal population

of expected values of the second-order partial derivatives of the

of 12.00 of 50 units was selected from the screened pro- of Figure 2.1). fl and &are calculated from (2.3.9) and (2.3.12) as = + 0.00916(9.35- = 1.1907, rl = 9.35 - = 9.37. It follows that = 1.091 = (12.00 -= 2.41.

3.1 INTRODUCTION of the normal distribution are derived for of the complete distribution, the truncation points are = Tz -

of (3.3.1), differentiate with respect to these parameters, and = 0.

-52.9767 1791.2545 3.4 PROGRESSIVELY CENSORED SAMPLES 2a i=I + 2: +

of which are considered later +A.) -1.0 -0.083 -0.205 -0.5

of products of the complete (uncensored) ob- of right censored observations. It follows that N + n,

ByN is the total sample size, n is the number of complete (uncensored)

(W'V- It then follows that E'fX (lnV- -i and < J). = c= c, and 1V-E = E'V- It follows that

Standard errors follow as

Bya .... 1.69\10.13613 = 0.624 and

of prominence in the field of reliability and life testing where samples

of the first two of these equations are identical to the first two of (6.4.4). The third of the preceding equations, which in this case of (6.4.4), can be written in an expanded form as + + aE(Z

of an estimate of the threshold parameter of total size N should approximately equal the of a corresponding estimate from a complete sample. This result was

7.1 THE INVERSE GAUSSIAN DISTRIBUTION

2: ln(x; - > I, maximum likelihood estimating equations may be obtained by 1 ~ ~ cÂ· aF

ByJ=d-F J=li-F

Of course, Tis a lower bound on all and we

ByofT for first approximations is convenient, and it usually

8.1 THE EXPONENTIAL DISTRIBUTION 8.1.1 = 0, and in these cases the

[F(nJ '. of size n from a truncated dis- t 7) = Thus [expk

ByMaximum Likelihood Estimators

of and in small samples, the of the preceding equations is identical with the second equation

= 100 ( , of the W eibull distribution is of both sides of the first equation of (8.1.31), we obtain of ln (x -of + = H- Ink Â·

of rainfall, flood flow, earthquake, and other of material, cor- of extreme value is of limiting distributions, which approximate = exp [ - = exp (>0), and (>0) are parameters. of "extreme value" distributions. Many authors consider it to be "the"

of the two-parameter We ibull distribution is o < < > o. > o. (8.2.4) of the Type I distribution of greatest extreme values is (8.2.5)

of a sample as thus described from a distribution that is of least extreme values is * = +

(5.3.6) for calculating the Weibull estimate 8. of censoring cj items are removed (censored) from further -nina+ -'-a-

ByN from a Type I dis- a uaa =

9.1 INTRODUCTION of acoustical of which is normally distributed (0, u of the Rayleigh distribution (i.e., the pdf of X) follows as

of the of the estimators presented in Chapter 5 for Weibull of Weibull parameters might of Rayleigh

10.1 of economics who formulated it ( 1897)

of the pdf and the cdf with a of this of a are included in Table 10.1 for selected values of this argument. More complete

ByaiX) and aiX) as functions

143.2632- of as It follows that = 20.66, and the approximate 95% CI is It is noted that differences between the MLE and the MMLE calculated from of its smaller bias. However, readers are again reminded that the only

11.1 of Cohen ( 1941,

= 0; of the same sign, D f.l;, D = and = a of Figure of Craig (1936).

Byrimaginary, bÂ¥ 2bf.l;, D

of (11.4.1) plus (11.4.2) to obtain h* = 38.600670, of the sample data based on these Shook's Graduation For Right Singly For Complete Sample Truncated Sample 159.95 79.9 0 0.2 89.9 12.8

Bybf 5.247727, and J*

of accepted specimens, and c = N - of rejected Nand care unknown. Only n, the number of acceptances, is known. In selected samples, full measurement of the screening ---------===----

of the associated variates. Accordingly, of Chapters 2 and 3 are applicable here just as they of size N. of maximum likelihood estimators for parameters of the multivariate normal

Sample Sample 67.6664 67.7033 67.6794 1.7857 1.6927 1. 5235 1. 5172 1. 5318 0.5239 0.5265 0.5318 0.5446 0. 4872 0.4924 0.7339 0.7053 0.7037 N â€¢ 119 n "' 108 Asymptotic Variances* 3.377 1.836 0.695 1. 959 1.041

Byv 1.389

13.1 of zero are not observed. As an example, consider the distribution of the of children per family in developing nations, where records are maintained

... '

ByX= 0, 1, 2, pt-x, 0,

l, 2, = -nA + = = 0, - = A [ + f(a 1)].

Byni!nA- n!n[P(a)]-

xis the mean of the n uncensored observations. 13.2.6 Doubly Censored Samples-Total Number of Censored Observations Known, But Not the Number in Each Tail Separately + IW. + P(d + + (1 f(a 1))].

Byf(a 1) ) ] . -!\a)+ P f(a - x =A

= _ ni _ n [f(a - + [f(a + n [ !_(d) ] + I) 1 -+ (! ))<flnL ni

By-!_(a - f(d - f(a - ni + n [f(d - of censored observation [f(a- -J(a- (f(a-

of all two-parameter /(0) It follows that = = =

ByBinomial with Zero Class Missing

= 0.2113 and = = = 13.4 THE BINOMIAL DISTRIBUTION = 0, 1,2, = I - + n(n -

Byf(x; n, p) ,n, (13.4.1)

... ,K-

Byk<K. j(ynA;p)= y=K,K+ j(ynR;p); y k,k j(ynR;p)+j(ynA;p); y=K,K+

of defectives found in the ith accepted lot (z; of defectives found in each rejected lot. This sample could be described of the paired values (z;, y;), i 1, 2, . . . , m

0.0000 7 0.0000 0.10 0.0128 7.74 0.10 0.0432 17.57 0.15 0.0450 8.09 0.15 0.1702 17.86 0.20 0.1209 8.35 0.20 0.3704 17.49 0.0000 0.01 0.0466 78.99 0.02 0.2156 74.70 0.0000 60 0.03 0.4319 68.02 0.01 0.0224 59.65 0.04 0.6252 60.54

100, k 120, 0.0000 0 0.0000 0.01 0.0794 97.76 0.01 0.0330 119.1 0.02 0.3233 89.37 0.02 0.2200 112.9 0.03 0.5802 77.94 0.03 0.4867 101.4

of x + 1 were reported as x = k with probability of defects per item, becomes + 2) + + 1)], x + 1)!, x k + 1, = 1, 2,

ByX= 0, 1,

+ 60 = 2.1 and

Byx 3. Following is a tabulation of the reported inspection results.

14.2.6 An Illustrative Example-Misclassified Binomial Data 14.3 An example generated by Cohen ( 1960a) consisted of N of defectives in samples of n 40 from a = of + = = 0.00000075, V(e) = 0.0017, Cov({J, e) = 0.0000025, and = 0.07.

of(l4.3.15), we calculate = 23/56 = 0.4107, and from the = = = = of Neyman's contageous distributions and they calculated expected fre-

of censored observations in of observations censored at of variation = of of the W eibull shape pa- of the gamma function.

Bynis sometimes

of Poisson, binomial and negative

ByThe Lognormal Distribution. Cam- Proc. Edinburgh Math. Soc., 4, 106-110.

of fitting the truncated negative binomial of the parameters of the distribution-a reconsideration. Austral. J. Statist., 3, 185-190.

of Michigan, Ann Arbor. of truncated

ByProc. Res. Forum, IBM Corporation, 40-43.

7. Kotz, S., Johnson, N. L., and Read, C. B., eds. Wiley, New York, of estimating the mean and standard

TABLE OF CONTENTS

of modem of American trotting horses. Sample data were extracted from Wallace's

of an unrestricted (i.e., complete) distribution with parameters

2.1 PRELIMINARY REMARKS of estimators for

of expected values of the second-order partial derivatives of the

of (3.3.1), differentiate with respect to these parameters, and = 0.

-52.9767 1791.2545 3.4 PROGRESSIVELY CENSORED SAMPLES 2a i=I + 2: +

of which are considered later +A.) -1.0 -0.083 -0.205 -0.5

of products of the complete (uncensored) ob- of right censored observations. It follows that N + n,

ByN is the total sample size, n is the number of complete (uncensored)

(W'V- It then follows that E'fX (lnV- -i and < J). = c= c, and 1V-E = E'V- It follows that

Standard errors follow as

Bya .... 1.69\10.13613 = 0.624 and

of prominence in the field of reliability and life testing where samples

7.1 THE INVERSE GAUSSIAN DISTRIBUTION

2: ln(x; - > I, maximum likelihood estimating equations may be obtained by 1 ~ ~ cÂ· aF

ByJ=d-F J=li-F

Of course, Tis a lower bound on all and we

ByofT for first approximations is convenient, and it usually

8.1 THE EXPONENTIAL DISTRIBUTION 8.1.1 = 0, and in these cases the

[F(nJ '. of size n from a truncated dis- t 7) = Thus [expk

ByMaximum Likelihood Estimators

of and in small samples, the of the preceding equations is identical with the second equation

of a sample as thus described from a distribution that is of least extreme values is * = +

ByN from a Type I dis- a uaa =

of the of the estimators presented in Chapter 5 for Weibull of Weibull parameters might of Rayleigh

10.1 of economics who formulated it ( 1897)

ByaiX) and aiX) as functions

11.1 of Cohen ( 1941,

= 0; of the same sign, D f.l;, D = and = a of Figure of Craig (1936).

Byrimaginary, bÂ¥ 2bf.l;, D

Bybf 5.247727, and J*

Byv 1.389

... '

ByX= 0, 1, 2, pt-x, 0,

l, 2, = -nA + = = 0, - = A [ + f(a 1)].

Byni!nA- n!n[P(a)]-

Byf(a 1) ) ] . -!\a)+ P f(a - x =A

= _ ni _ n [f(a - + [f(a + n [ !_(d) ] + I) 1 -+ (! ))<flnL ni

By-!_(a - f(d - f(a - ni + n [f(d - of censored observation [f(a- -J(a- (f(a-

of all two-parameter /(0) It follows that = = =

ByBinomial with Zero Class Missing

= 0.2113 and = = = 13.4 THE BINOMIAL DISTRIBUTION = 0, 1,2, = I - + n(n -

Byf(x; n, p) ,n, (13.4.1)

... ,K-

Byk<K. j(ynA;p)= y=K,K+ j(ynR;p); y k,k j(ynR;p)+j(ynA;p); y=K,K+

ByX= 0, 1,

+ 60 = 2.1 and

Byx 3. Following is a tabulation of the reported inspection results.

Bynis sometimes

of Poisson, binomial and negative

ByThe Lognormal Distribution. Cam- Proc. Edinburgh Math. Soc., 4, 106-110.

of Michigan, Ann Arbor. of truncated

ByProc. Res. Forum, IBM Corporation, 40-43.