Let's assume that we have a statistical model that is given by the family of $N(μ, σ^2)$ distributions where $θ = (μ, σ^2) ∈ R^1 × R^+$ is unknown. If our interest is in making inferences about the first quartile of the true distribution, then determine $ψ(μ, σ^2)$.
$\textbf{SOLUTION}$: We have that $ψ(μ, σ^2) = x_{0.25} = μ + σz_{0.25}$ where $\Phi(z_{0.25}) = 0.25$.
How? And What does that solution mean? What is $z_{0.25}$ and what is $\Phi(z_{0.25})$?
The first quartile ($x_{0.25}$) of the true distribution satisfies: $$ P(X \le x_{0.25}) = 0.25. $$
Rescaling gives $$ P \left ( \frac{X-\mu}{\sigma} \le \frac{x_{0.25} - \mu}{\sigma} \right) = 0.25 $$ Note that $\frac{X-\mu}{\sigma} $ is equal in distribution to a standard normal $Z\sim N(0,1)$, with CDF $\Phi$, so we can equivalently write $$ \Phi \left ( \frac{x_{0.25} - \mu}{\sigma}\right) = 0.25 $$ Rearranging yields $$ x_{0.25} = \sigma \Phi^{-1}(0.25) + \mu = \sigma z_{0.25} + \mu $$ where $\Phi^{-1}(0.25) := z_{0.25}$ is the quartile of the standard normal.
If we want to do inference, we should come up with an estimator. The natural thing to do is MLE estimation. Note that the thing we are trying to estimate is a function of the parameters $\theta$, and it is a fact that the MLE of a function of parameters is the function of the MLEs of those parameters, so an estimate here would be $$ \hat{x}_{0.25} = \hat{\sigma} z_{0.25} + \hat{\mu} $$ where $\hat{\sigma} = \frac{1}{n} \sum_{i=1}^n (X_i - \bar{X})^2$ is the MLE estimate of the variance, and $\hat{\mu} = \bar{X} = \frac{1}{n} \sum_{i=1}^n X_i$ is the MLE estimate of the mean