# #StackBounty: #hypothesis-testing #t-test #p-value Estimating "population p-value" \$Pi\$ using an observed p-value

### Bounty: 100

I asked a similar question last month, but from the responses, I see how the question can be asked more precisely.

Let’s suppose a population of the form

\$\$X sim mathcal{N}(100 + t_{n-1} times sigma / sqrt{n}, sigma)\$\$

in which \$t_{n-1}\$ is the student \$t\$ quantile based on a specific value of a parameter \$Pi\$ (\$0<Pi<1)\$. For the sake of the illustration, we could suppose that \$Pi\$ is 0.025.

When performing a one-sided \$t\$ test of the null hypothesis \$H_0: mu = 100\$ on a sample taken from that population, the expected \$p\$ value is \$Pi\$, irrespective of sample size (as long as simple randomized sampling is used).

I have 4 questions:

1. Is the \$p\$ value a maximum likelihood estimator (MLE) of \$Pi\$? (Conjecture: yes, because it is based on a \$t\$ statistic which is based on a likelihood ratio test);

2. Is the \$p\$ value a biased estimator of \$Pi\$? (Conjecture: yes because (i) MLE tend to be biased, and (2) based on simulations, I noted that the median value of many \$p\$s is close to \$Pi\$ but the mean value of many \$p\$s is much larger);

3. Is the \$p\$ value a minimum variance estimate of \$Pi\$? (Conjecture: yes in the asymptotic case but no guarantee for a given sample size)

4. Can we get a confidence interval around a given \$p\$ value by using the confidence interval of the observed \$t\$ value (this is done using the non-central student \$t\$ distribution with degree of freedom \$n-1\$ and non-centrality parameter \$t\$) and computing the \$p\$ values of the lower and upper bound \$t\$ values? (Conjecture: yes because both the non-central student \$t\$ quantiles and the \$p\$ values of a one-sided test are continuous increasing functions)

Get this bounty!!!