Math 2.0

Statistics Definition

Parametric model

Builds on Statistical model

Definition

Let 𝒫={Pθ:θ∈Θ}\mathcal{P} = \set{P_\theta : \theta \in \Theta} be a statistical model on a measurable space (S,𝒮)(S, \mathcal{S}). The model is a parametric model if its parameter space Θ\Theta is a subset of ℝk\R^k for some finite k≥1k \ge 1. The number kk is the dimension of the parameter.

Remarks

Example: the normal model on S=ℝS = \R with P(μ,σ2)=𝒩(μ,σ2)P_{(\mu, \sigma^2)} = \mathcal{N}(\mu, \sigma^2) and Θ=ℝ×(0,∞)⊆ℝ2\Theta = \R \times (0, \infty) \subseteq \R^2 is parametric with k=2k = 2. Non-example: the model of all probability measures on ℝ\R with finite variance is not given by a natural finite-dimensional parameter, and it is called nonparametric. In practice one also asks that the map θ↦Pθ\theta \mapsto P_\theta is regular, for example that the densities depend smoothly on θ\theta. Without such a condition the definition is weak, since any set of at most continuum many laws can be indexed by a subset of ℝ\R.

Reading path

To understand this page from the ground up, read these 29 pages in order. The foundations come first.

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Construction of the reals from Cauchy sequences, concept, no evidence levelConstruction of the r…conceptElement, definition, no evidence levelElementdefinitionIntegers, object, no evidence levelIntegersobjectOrdered field, definition, no evidence levelOrdered fielddefinitionProposition, concept, no evidence levelPropositionconceptRandom experiment, concept, no evidence levelRandom experimentconceptRational numbers, object, no evidence levelRational numbersobjectSet, definition, no evidence levelSetdefinitionRing of integers, statement, no evidence levelRing of integersstatementDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionUnion, definition, no evidence levelUniondefinitionSample space, definition, no evidence levelSample spacedefinitionThe rationals form a field, statement, no evidence levelThe rationals form a …statementUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionSubset, definition, no evidence levelSubsetdefinitionThe rationals form an ordered field, statement, no evidence levelThe rationals form an…statementPower set, definition, no evidence levelPower setdefinitionUniversal set, definition, no evidence levelUniversal setdefinitionOrder on the reals, definition, no evidence levelOrder on the realsdefinitionComplement, definition, no evidence levelComplementdefinitionInterval, definition, no evidence levelIntervaldefinitionSigma-algebra, definition, no evidence levelSigma-algebradefinitionEvent, definition, no evidence levelEventdefinitionProbability axioms, concept, no evidence levelProbability axiomsconceptProbability measure, definition, no evidence levelProbability measuredefinitionStatistical model, definition, no evidence levelStatistical modeldefinitionParametric model, definition, no evidence levelParametric modeldefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Construction of the reals from Cauchy sequences concept
  2. Element definition
  3. Integers object
  4. Ordered field definition
  5. Proposition concept
  6. Random experiment concept
  7. Rational numbers object
  8. Set definition
  9. Ring of integers statement
  10. Disjunction definition
  11. Implication definition
  12. Predicate definition
  13. Union definition
  14. Sample space definition
  15. The rationals form a field statement
  16. Universal quantifier definition
  17. Subset definition
  18. The rationals form an ordered field statement
  19. Power set definition
  20. Universal set definition
  21. Order on the reals definition
  22. Complement definition
  23. Interval definition
  24. Sigma-algebra definition
  25. Event definition
  26. Probability axioms concept
  27. Probability measure definition
  28. Statistical model definition
  29. Parametric model definition

Where this leads

5 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Parametric_model, Definition
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