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It is known under regularity conditions, Maximum Likelihood Estimator is asymptotically normal. But why do we use score test, Wald test, Likelihood Ratio Test instead of $Z$ test?

If for an MLE $\hat{\theta}$ we can use z-tests of $\theta=\theta_0$ based on $\hat{\theta}$ having approximately a standard normal distribution, why do we use chi-square tests based on Wald, score, or maximum-likelihood ratio statistics?

ZHU
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    What situation(s) are you talking about (e.g. what research question, kind of data, kind of model, etc.)? IMO the question as put forward now is too broad/vague. Please edit the question with additional information. – IWS Apr 11 '17 at 09:53
  • Not sure what the Z-test is that you're contrasting these with. Note that a Z-test for a single parameter can be constructed based on the directional Wald, score, or likelihood ratio test statistic. Please do edit your question as @IWS suggests if the link doesn't give you what you need. – Scortchi - Reinstate Monica Apr 11 '17 at 10:23
  • @Scortchi test under normality? – ZHU Apr 11 '17 at 10:41
  • Well suppose you're estimating the mean of a normal distribution (known variance) from a simple random sample. What's the maximum-likelihood estimator? What's its distribution? How do the likelihood-ratio, score, & Wald tests differ - between themselves & from a z-test you might perform? – Scortchi - Reinstate Monica Apr 11 '17 at 11:06
  • Well, what's the distribution of a squared standard normal random variable? – Scortchi - Reinstate Monica Apr 11 '17 at 11:41
  • Anyway, I think your question's something like "If for an MLE $\hat \theta$ we can use z-tests of $\theta=\theta_0$ vs $\theta = \hat \theta$ based on $\frac{\hat \theta - \theta_0}{\sqrt{\operatorname{Var}\hat \theta}}$'s having approximately a standard normal distribution, why do we use chi-square tests based on Wald, score, or maximum-likelihood ratio statistics?" if you want to edit it. Or it might be answered [here](http://stats.stackexchange.com/q/60074/17230). – Scortchi - Reinstate Monica Apr 11 '17 at 12:20
  • @Scortchi Hi, I have put your comment into the question – ZHU Apr 11 '17 at 19:40
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    I don't see how this isn't covered by the duplicate. – gung - Reinstate Monica Apr 11 '17 at 20:09
  • @gung the duplicate discusses how three tests are different, but my question is why another test is different from the three. – ZHU Apr 11 '17 at 22:50
  • @ZHU, the z-test you refer to is a Wald test. It isn't different from the three. – gung - Reinstate Monica Apr 12 '17 at 00:31

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