irregular problems in hypothesis testing

The validity of many test statistics depends on various regularity conditions. For instance, Wilk’s famous theorem states that -2 times the log-likelihood converges to a ${\chi }^2$ distribution, which provides a test statistic for simple nulls and simple alternatives. But what if these regularity conditions don’t hold? Such problems are called irregular problems.

There are methods to handle such irregular problems; universal inference is one such method.

Some examples of irregular problems:

• Mixtures of Gaussians Suppose we want to test whether the samples come from a mixture of $k_0$ or $k_1$ Gaussians with $k_0<k_1$. This is an irregular problem (even for $k_1=k_0+1$).
• Shape constraints For instance, testing whether a given sample is from a log-concave distribution.
• Latent variables Problems with latent variables, such as:
  • testing if a hidden Markov model has $\le k$ states
  • testing whether a factor model has $\le k$ factors
  • conditional independence testing