Estimation error from estimation quantile of normal distribution

[Derk]

Hi guys,

For my (master) project I am trying to find the probability that a random variable, which is normally distributed, exceeds a quantile that is estimated by a limited number of observations. See attached for my attempt.
- Is it correct?
- How to incorporate the fact that the mean and variance of the normal distribution are unknown in reality?

Thanks in advance!

 

[mfb]

Why do you estimate it that way if you know your variable has a normal distribution? Find estimates of the mean and the standard deviation, calculate the quantile based on that. It will give you much better estimates especially for large deviations from the mean.

 

[WWGD]

I agree with mfb: If/when you don't know the standard deviation from a normal population, then the sample mean $x_s$ is t-distributed as ( for a two-sided) $( x_s- t_{ \alpha/2}SE, x_s+ t_{\alpha /2} SE )$ where SE is the standard error and $\alpha$ is the confidence level. Other statistics have different distributions. Do you have any specific one in mind? If you are computing the sampling mean and your sample is large-enough ( n>30 usually; n>= 40 for more accuracy) then you can use the CLT to assume normality.

 

[Stephen Tashi]

It may be that the purpose of @Derk 's project is not to find an optimal method of estimating a quantile, but rather to work out the distribution of a particular estimator. If he were doing a masters project in engineering, we'd expect something practical, but a mathematical project can be an "academic exercise".