Goodness-of-Fit Tests

Every test built so far in this topic asks a parametric question — is $\theta$ equal to some value, does a restriction on $\theta$ hold — always within a family the analyst has already committed to. Goodness-of-fit testing asks the more basic question underneath all of that: does the data plausibly come from the assumed family at all? This is the closing question of the unit for a good reason: every likelihood-based test built in Topics 6–7 is only as trustworthy as the parametric model it is computed inside, and a regulator, model-validation committee, or backtesting framework needs a direct, model-checking answer before any of that earlier machinery can be trusted. This note builds the two standard tools. For binned/categorical data, the generalized likelihood ratio test applied to the multinomial distribution reduces, via the same Taylor-expansion argument as Wilks' theorem, to Pearson's chi-squared statistic — the single most widely used goodness-of-fit test in applied statistics. For continuous data, the empirical CDF supplies the Kolmogorov-Smirnov test, comparing the entire fitted curve against the entire data-driven curve at once, with no arbitrary binning decision at all. The finance stakes are concrete: a bank's internal-models approach to regulatory capital lives or dies on whether its assumed P&L distribution — normal, Student-$t$, or otherwise — actually survives one of these tests against realised trading outcomes.

Learning objectives. By the end of this note you should be able to: