Statistics for Environmental Engineers

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Cohen’s Maximum Likelihood Estimator Method


There are several methods to estimate the mean of a sample of censored data. Comparative studies show that none is always superior so we have chosen to present Cohen’s maximum likelihood method (Cohen, 1959, 1961; Gilliom and Helsel, 1986; Haas and Scheff, 1990). It is easy to compute for samples from a normally distributed parent population or from a distribution that can be made normal by a log-arithmic transformation.


A sample of n observations has measured values of the variable only at у > yc, where yc is a known and fixed point of censoring. In our application, yc is the MDL and it is assumed that the same MDL applies to each observation. Of the n total observations in the sample, nc observations have у < yc and are censored. The number of observations with у > yc is к = n — nc. The fraction of censored data is


Cohen’s X as a Function of h = nlnc and у = s2/yc)


yh


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 = ncln . Using the к fully measured observations, calculate crude estimates of the mean and variance of the noncensored data using:

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