This function tests the null hypothesis that mu(F) = mu versus not equal.
We assume the data given is current status censored data.
The definition of the mean, mu(F) is
$$
\mu(F) = \int_0^M [1- F(t)] d \Psi(t)
$$
and its estimator based on \( (\delta_i, t_i) \)
or \( \hat F_n \) is (assume \( \min(t_i) =0\) or \(t_{(1)} =0\))
$$
{\mu(\hat F_n )} = \sum_{i=1}^n [1-\hat F_n(t_{(i)})] \Delta \Psi(t_{(i)})~,
$$
where \( \Psi (t) \) is a given function and
\( \Delta \Psi(t_{(i)})= \Psi (t_{(i+1)}) - \Psi(t_{(i)}) \).
If \( \Psi(t) =t \) in the above, then this
is the ordinary mean (assuming F(t) has support (0, M) ).
The NPMLE \( \hat F_n(t)\) is convergent at cubic root n speed, but the mean estimator is
convergent at ordinary root n speed. The -2LLR has chi square DF=1 null distribution.
It goes without saying that we assume the NPMLE mu(hat F)
has finite asymptotic variance (when normalized by root n).