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Probability, Statistics and Financial Data / Statistical Models and Data Reduction

Parametric Statistical Models and the Likelihood

Every time a desk calibrates a return-distribution model to a window of historical P&L — deciding whether daily log-returns look like N(0.02%,1.1%2)\mathcal N(0.02\%, 1.1\%^2) or N(−0.05%,1.8%2)\mathcal N(-0.05\%, 1.8\%^2) — it is running the same underlying computation: among a family of candidate distributions indexed by a parameter θ\theta, which value of θ\theta makes the observed data most plausible? That question turns the joint density, normally read as "the probability of the data given the parameter," on its head and reads it instead as a function of θ\theta for the data actually observed. This reversal is the likelihood function, and it is the single object underneath every estimation and testing procedure in the rest of this unit — maximum likelihood (Point Estimation, Topic 3), Bayesian posteriors (Bayesian Inference, Topic 4), and hypothesis tests (Neyman-Pearson Theory, Topic 6) are all, at bottom, statements about the likelihood.

This note sets up the formal machinery: what a parametric statistical model is, how the likelihood function is built from it, why it is emphatically not a probability density in θ\theta, and the identifiability condition that makes "estimating θ\theta" a well-posed question in the first place.

Learning objectives. By the end of this note you should be able to:
  • define a parametric statistical model {Pθ:θ∈Θ}\{P_\theta : \theta \in \Theta\} and identify the parameter space Θ\Theta for standard examples;
  • write down the likelihood function L(θ;x)L(\theta; x) for an i.i.d. sample and distinguish it in words and in symbols from the joint density;
  • compute and plot a likelihood (or log-likelihood) curve for a simple discrete model, and read off the value of θ\theta that maximises it;
  • define identifiability and recognise a non-identifiable parametrisation;
  • recognise a location-scale family and state why the normal family is the canonical example.

In this lesson

  1. Parametric statistical models
  2. The likelihood function
  3. Identifiability
  4. Location-scale families

Before you start

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