Chi-squared asymptotic approximation

A Chi-squared asymptotic approximation uses a Chi-squared distribution as the limiting null law of a statistic. In a regular identifiable fixed-dimensional model, Wilks theorem gives $2(\ell (\hat{\theta })-\ell ({\hat{\theta }}_0))⟹{\chi }_q^2$ for $q$ smooth independent null restrictions, provided the true parameter is an interior point and the usual differentiability and information conditions hold. This is a limit as Sample size grows, not an exact finite-sample identity.

For a fully specified multinomial model with a fixed number $r$ of categories and probabilities $p_j>0$, Pearson's statistic isThe constraint ${\sum }_j{N}_j=n$ removes one Gaussian degree of freedom. Its usual fixed-category assumptions make all expected counts $n{p}_j$ grow. Growing dimension or persistent rare counts require separate approximation analysis; merely choosing chi-squared critical values does not supply it.

Some Gaussian models instead yield an exact chi-squared law. If $X_i$ are iid $N({\mu }_0,{\sigma }^2)$ with known $\sigma >0$, then $n(\overline{X}-{\mu }_0{)}^2/{\sigma }^2\sim {\chi }_1^2$ for every $n$. Parameter boundaries can lead to a Chi-bar-squared distribution or another limit rather than the regular Wilks law. The model and null geometry determine which calibration is justified.