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L1592_Frame_C15 Page 135 Tuesday, December 18, 2001 1:50 PM

TABLE 15.4
Cohen’s λ as a Function of h = n/n c and γ = s y – y c )
ˆ
(
2
γγ γγ \h 0.1 0.2 0.3 0.4 0.5
0.2 0.12469 0.27031 0.4422 0.6483 0.9012
0.3 0.13059 0.28193 0.4595 0.6713 0.9300
0.4 0.13595 0.29260 0.4755 0.6927 0.9570
0.5 0.14090 0.30253 0.4904 0.7129 0.9826
0.6 0.14552 0.31184 0.5045 0.7320 1.0070
Source: Cohen (1961).

h = n c /n . Using the k fully measured observations, calculate crude estimates of the mean and variance
of the noncensored data using:

(
y = ∑y i and s = ∑ y i – y) 2
2
------------------------
--------
k k
2
Note that s is calculated using a denominator of k instead of the usual n – 1.
For the case where the lower tail of the distribution is truncated, the estimated mean will be too high
and the estimated variance will be too low. These estimates are adjusted using a factor λ. The maximum
2
likelihood estimates for the mean η and the variance σ are:
(
ˆ
η ˆ = y λ y – y c )
–
(
2
2
ˆ
σ ˆ = s + λ y – y c ) 2
The hat (^) indicates that the value is an estimate (η ˆ is an estimate of the true mean η). Cohen (1961)
λ
provides tables of as a function of h and γ = s y – y c ) 2 . A portion of his table is reproduced in
(
2
ˆ
Table 15.4.
Example 15.4
Using Cohen’s method, estimate the mean and variance of the censored sample from Example
15.1. The sample of n = 27 observations includes 23 measured and 4 censored values. The
censoring limit is 6 µg/L. The 23 uncensored values are 6.9, 7.8, 8.9, 7.7, 9.6, 8.7, 6.7, 8.0, 8.5,
6.5, 9.2, 7.4, 6.3, 7.3, 8.3, 7.2, 7.5, 6.1, 9.4, 7.6, 8.1, 7.9, and 10.1. The average and variance of
the k = 23 observations with measurable values are:

y = -------- = 7.9 µg/L
∑y i
k
(
------------------------ =
s = ∑ y i – y) 2 1.1078 µg/L( ) 2
2
k
The limit of censoring is at y c = 6 and the proportion of censored data is:
h = 4 /27 = 0.1481,

which gives:

γ = s / y – y c ) = 1.1078/ 7.9 6.0) = 0.30688
(
(
2
2
2
–
© 2002 By CRC Press LLC