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Given the random sample $X_1,...,X_n \sim N(\mu, \sigma^2)$, I want to perform a Wald test for:

$\mathrm{H}_\mathrm{0}: \mu = \mathrm{\mu}_\mathrm{0}$

$\mathrm{H}_\mathrm{1}: \mu \neq \mathrm{\mu}_\mathrm{0}$

What would be the test statistic here?

Generally, it is $T = |W| = (\hat{\theta} - \mathrm{\theta}_\mathrm{0})/ \hat{se}$

Is it simply $\hat{\mu}$ in this case? And in order to find what is it equal to, I need to find the MLE for $\mu$, correct? and that is by finding Fisher Information $I(\mu)$.

Any help would be much appreciated.

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  • $\begingroup$ the shortest answer I can think of is the $t$-test. For some results on the use of MLE see here and for the $t$-test see here. $\endgroup$
    – utobi
    Commented Nov 22, 2022 at 10:44
  • $\begingroup$ What do you mean by "and that is by finding Fisher Information $I(\mu)$." ? $\endgroup$
    – utobi
    Commented Nov 22, 2022 at 10:50
  • $\begingroup$ Is $\sigma^2$ assumed to be known? $\endgroup$
    – statmerkur
    Commented Jan 4, 2023 at 8:50

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