calculate The maximum likelihood estimator of parameter $\mu$ according to $T$

Question

suppose $X_1,X_2,\ldots,X_n$ be a random sample of $N(\mu,1)$. if $T=\sum_{i=1}^n I_{(X_i<0)}$ how can I calculate The maximum likelihood estimator of parameter $\mu$ according to $T$. ($\Phi$ is Standard normal distribution function)

Answer 1

First we can write $P(X < 0) = P(X - \mu < - \mu) = \Phi(-\mu)$, and so $T \sim$ binomial$(n, p = \Phi(-\mu))$. The likelihood function on observing $T = t$ is then just the binomial mass function at the point $t$,

$$ L(\mu) = \binom{n}{t} \Phi(-\mu)^t [1 - \Phi(-\mu)]^{n - t} . $$

But the maximum likelihood estimate of the success probability for a sample of Bernoulli trials is just the sample mean, so we know this function is maximized when $\Phi(- \mu) = t / n$, which means $\hat{\mu}_\text{MLE} = - \Phi^{-1}(t / n)$.