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# Truncated and Censored Samples

DOI link for Truncated and Censored Samples

Truncated and Censored Samples book

Theory and Applications

# Truncated and Censored Samples

DOI link for Truncated and Censored Samples

Truncated and Censored Samples book

Theory and Applications

ByA. Clifford Cohen

Edition 1st Edition

First Published 1991

eBook Published 16 May 2014

Pub. Location Boca Raton

Imprint CRC Press

Pages 328

eBook ISBN 9780429081743

Subjects Mathematics & Statistics

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#### Get Citation

Clifford Cohen, A. (1991). Truncated and Censored Samples: Theory and Applications (1st ed.). CRC Press. https://doi.org/10.1201/b16946

## ABSTRACT

This book deals with the development of methodology for the analysis of truncated and censored sample data. It is primarily intended as a handbook for practitioners who need simple and efficient methods for the analysis of incomplete sample data.

## TABLE OF CONTENTS

chapter 1|1 pages

#### INTRODUCTION Preliminary Considerations

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

chapter 1|2 pages

#### 2A HISTORICAL ACCOUNT

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

chapter |2 pages

#### of truncation

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

chapter 1|1 pages

#### 5 LIKELIHOOD FUNCTIONS

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

chapter 2|1 pages

#### Singly Truncated and Singly Censored Samples from the Normal Distribution

2.1 PRELIMINARY REMARKS of estimators for

chapter |1 pages

#### ) and ) are defined by (2.2.3), and of course = F(T).

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

chapter |9 pages

#### =.X-

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

chapter 2|1 pages

#### 6 SAMPLING ERRORS OF ESTIMATES

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

chapter |1 pages

#### of miles of service we have

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.

chapter 3|13 pages

#### Multirestricted Samples from the Normal Distribution

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

chapter 3|4 pages

#### 3 DOUBL V CENSORED SAMPLES

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

chapter |2 pages

#### 52.97671-l--0.000 026

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

chapter |5 pages

#### of iterations required for a specified degree of accuracy will to

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

chapter |2 pages

#### of significant proportions, and the elimination of bias more than compensates

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)

chapter |10 pages

#### of 9* is = = (v;). (4.3.11)

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

chapter |1 pages

#### From Table 4.2, read 0.13613 a 0.23712 a a*)= 0.02671 a

Standard errors follow as

Bya .... 1.69\10.13613 = 0.624 and

chapter 5|2 pages

#### Truncated and Censored Samples from the Weibull Distribution

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

chapter |4 pages

#### It is, of course, necessary that we solve the three equations of (6.4.4) si-jl, d-). A straightforward trial-and-error iterative

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

chapter 6|2 pages

#### 6 ERRORS OF ESTIMATES

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

chapter 7|3 pages

#### Truncated and Censored Samples from the Inverse Gaussian and the Gamma Distributions

7.1 THE INVERSE GAUSSIAN DISTRIBUTION

chapter |1 pages

#### (x-"{) (x-

-exp- + const. (7.l.II) -,-,

By,M":: .x>"{, X-"{ L = - n In 0' X;-"{ a a =

chapter 7|2 pages

#### 2.2 Maximum Likelihood Estimators for Censored Samples of a progressively censored sample from a gamma

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

ByJ=d-F J=li-F

chapter |6 pages

#### = i,

Of course, Tis a lower bound on all and we

ByofT for first approximations is convenient, and it usually

chapter 8|3 pages

#### Truncated and Censored Samples from the Exponential and the Extreme Value Distributions

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

chapter |1 pages

#### = 0 for all j, = n and ST

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

ByMaximum Likelihood Estimators

chapter |2 pages

#### of (8 .1.18) become

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

chapter |1 pages

#### of survivors immediately of a corresponding sample item. Estimates of the hazard or

= 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 ·

chapter 8|1 pages

#### 2 THE EXTREME VALUE DISTRIBUTION

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"

chapter |3 pages

#### (1943). It was Gumbel who pioneered application

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)

chapter |1 pages

#### > Let c (N -

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

chapter |2 pages

#### of these equations. With & determined from equation (8.2.18), we

(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 =

chapter 9|2 pages

#### Truncated and Censored Samples from the Rayleigh Distribution

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

chapter 9|1 pages

#### 6 SOME CONCLUDING REMARKS

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

chapter 10|2 pages

#### Truncated and Censored Samples from the Pareto Distribution

10.1 of economics who formulated it ( 1897)

chapter |4 pages

#### of the Pareto distribution as ~ is a degenerate form of the two-parameter exponential distribution (8.1.1) in which

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

chapter |1 pages

#### Maximum Likelihood Estimates of (10.3.6), we have 1010, and from

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

chapter 11|4 pages

#### Higher-Moment Estimators of Pearson Distribution Parameters from Truncated Samples

11.1 of Cohen ( 1941,

chapter 11|3 pages

#### 5 DETERMINING THE DISTRIBUTION TYPE of distribution can be established from the original Pearson criteria, or from

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

Byrimaginary, b¥ 2bf.l;, D

chapter |2 pages

#### of H* suggests

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*

chapter 12|1 pages

#### Truncated and Censored Samples from Bivariate and Multivariate Normal Distributions

12.1

chapter |2 pages

#### of the quadratic form in the exponent is the of the variance-covariance matrix llaijll and has the positive determinant

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 ---------===----

chapter |2 pages

#### of the multivariate

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

chapter |3 pages

#### Complete Truncated Censored Parameters Sample Estimates Estimates Estimates -1.379 -1.342 138.2353 138.4883 138.2376

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 • 119 n "' 108 Asymptotic Variances* 3.377 1.836 0.695 1. 959 1.041

Byv 1.389

chapter 13|1 pages

#### Truncated and Censored Samples from Discrete Distributions

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

chapter |1 pages

#### of scientific endeavor. It was first derived by Poisson ( 1837)

... '

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

chapter 13|2 pages

#### 2.2 Singly Left Truncated Samples

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

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

chapter x|1 pages

#### + S (

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

chapter |3 pages

#### ni _ n [f(a - + f(d)]

= _ 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-

chapter |2 pages

#### + k.

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

ByBinomial with Zero Class Missing

chapter 13|8 pages

#### 3.2 An Illustrative Example

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

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

chapter |2 pages

#### of K nondefectives are found, in which case inspection

... ,K-

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

chapter |2 pages

#### of defectives found and the number of items inspected be recorded of the paired values (zy

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

chapter 10|1 pages

#### , k 4, = 20, 5, 16

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

chapter |1 pages

#### = = 2, K= =

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

chapter 14|2 pages

#### 2.1 Misclassification in the Poisson Distribution

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,

chapter 14|1 pages

#### 2.3 Illustrative Examples-Misclassified Poisson Data

+ 60 = 2.1 and

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

chapter |1 pages

#### PiJ.e--

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.

chapter 5|3 pages

#### + Total

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

chapter |1 pages

#### of a

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

Bynis sometimes

chapter |1 pages

#### Bibliography

of Poisson, binomial and negative

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

chapter |1 pages

#### of contageous distributions when ap-

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

chapter C|2 pages

#### , and Whitten, B. (1983) The standardized inverse

of Michigan, Ann Arbor. of truncated

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

chapter L|4 pages

#### and Moore, A. H. (1967) Asymptotic variances and covariances

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