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,
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 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 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
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
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-
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
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).
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
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
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 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))].
chapter |3 pages
ni _ n [f(a - + f(d)]
= _ ni _ n [f(a - + [f(a + n [ !_(d) ] + I) 1 -+ (! ))<flnL ni
chapter 13|8 pages
3.2 An Illustrative Example
= 0.2113 and = = = 13.4 THE BINOMIAL DISTRIBUTION = 0, 1,2, = I - + n(n -
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,
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.
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
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