# Wald test for gamma distributed data?

One of the assumptions of Wald test is that the data is normally distributed. If my data follows Gamma distribution. And I have estimated the two parameters of the gamma distributions. Is there a test like Wald test I can use?

## 1 Answer

The Wald test assumes that the estimator is asymptotically normally distributed. This is the case for the mean from most distributions, as well as maximum likelihood estimators. Therefore you can use the Wald test for estimating the parameters of a gamma distribution by the MLEs and their standard errors.

If $\hat{\alpha}$ is your MLE for $\alpha$ and your null hypothesis has $\alpha=\alpha_0$, the Wald test statistic is $$Z=\frac{\hat{\alpha}-\alpha_0}{SE(\hat{\alpha})}.$$ A reasonable (asymptotically correct) estimate for $SE(\hat{\alpha})$ is $1/\sqrt{I_n(\hat{\alpha})}$, where $I_n(\hat{\alpha})$ is the Fisher information. You should be able to get either the standard errors or Fisher information from whatever software you used to find the MLE.

• Sir, can an explanation be given in detail specifically with respect to the shape and scale parameters?
– Vani
May 14 '14 at 10:03