Since the probabilities are not actually known, their values must be estimated
from the observed numbers in the R random samples. When the null hypothesisH0 is
true, the R random samples are actually drawn from the same distribution. Therefore,
the M.L.E. of the probability that an observation in each of these samples will be of
type j is simply the proportion of all the observations in the R samples that are of
type j . In other words, the M.L.E. of is the same for all values of i (i = 1, . . . , R),
and this estimator is ˆ = /n. When this M.L.E. is substituted into (9.4.2), we obtain the statistic
Since the probabilities are not actually known, their values must be estimated
from the observed numbers in the R random samples. When the null hypothesisH0 is
true, the R random samples are actually drawn from the same distribution. Therefore,
the M.L.E. of the probability that an observation in each of these samples will be of
type j is simply the proportion of all the observations in the R samples that are of
type j . In other words, the M.L.E. of is the same for all values of i (i = 1, . . . , R),
and this estimator is ˆ = /n. When this M.L.E. is substituted into (9.4.2), we obtain the statistic
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