# Recommended significance test for coefficients in regression mixture with bootstrap Standard errors?

I have fit a model that comprises of a mixture of regressions. Within each regression I have an estimate of the coefficient of each covariate within that component regression. The model was fit using the EM algorithm.

Eg I have estimates $$\hat{\beta_{kj}}$$ for $$k=1,2,\dots,K$$ the number of regressions in the mixture, and $$j=1,2,\dots,J$$ the number of coefficients.

Along side these estimates I have bootstrap estimates of their standard errors (the function used to fit the model did not include standard error estimates).

I'm unsure what the appropriate/formal way of determining which coefficients are significant is?

Informally, I can compare the estimates to their standard errors. And I know in a normal regression can use the Wald test by computing the ratio

$$\frac{\hat{\beta_{kj}} - 0}{SE(\hat{\beta_{kj}})}$$

I am unsure if this test is appropriate here. Since the EM algorithm was used, the estimates obtained asymptotically converge to the true ML estimates, so that assumption is forfilled.

But what about thee standard errors? And the fact the model is a mixture? The wikipedia page for the Wald test do not seem to list assumptions other than them being ML estimates. So I'm not sure if there are additional assumptions that might not hold?