# What is this parameter estimation strategy called?

Let $$X_1, X_2, \ldots, X_n$$ be a random sample from a normal distribution with mean $$\mu$$ and variance $$\sigma^2$$. Consider the problem of estimating $$P(X > 100)$$.

One way to accomplish this is to calculate $$n^{-1}\sum_{i=1}^n \mathbb{1}(X_i > 100)$$. This "plug-in" estimator is consistent, and its bias and MSE are easy to compute.

A smaller group of my students came up with another way to approach the problem: calculate $$1 - \Phi\left(\frac{100 - \bar{x}}{s} \right).$$ This can be motivated by the fact that $$P(X > 100) = 1 - \Phi[(100 - \mu)/\sigma].$$ This estimator is also consistent, but its bias and MSE are more difficult to compute.

My question is this: does this kind of strategy have a name? I ask because we are still plugging things in, but this is not a so-called plug-in estimator.

• Isn't this just the Gaussian MLE? – shadowtalker Oct 3 '18 at 3:40
• @shadowtalker yeah it’d be the invariance principle if you were using the MLE for the variance, but we are not as long I define this sample variance as the one where you divide by $n-1$. – Taylor Oct 3 '18 at 3:45
• I would call the second one the "plug-in" estimator, while the first one is the moment estimator. – Xi'an Oct 4 '18 at 20:35
• It's Rao-Blackwellized...but that isn't very specific – Taylor Oct 11 '18 at 13:08
• The second method is much more powerful because it uses the distributional assumption of normality. However $\frac{100-\bar x}{s}$ is not normally distributed due to the uncertainty in $s$. Look up the $t$ distribution with $n$ degrees of freedom. – dave fournier Jan 27 '19 at 21:20

• thanks (+1) I'm not sure if you saw the discussion above, but I believe $s^2 = \sum_i(x_i - \bar{x})^2/(n-1)$ is not the MLE for iid normal data. – Taylor Jan 27 '19 at 20:29
• Yes, so replace with $n$ as divisor. Rao-Blackwellization at least can be done approximately by simulation, exact calculation I am not so sure ... – kjetil b halvorsen Jan 28 '19 at 1:54