Math 2.0

Statistics Definition

Risk function

Builds on Statistical modelLoss functionExpected valueMean squared errorNormal distribution

Definition

Let {Pθ:θ∈Θ}\set{P_\theta : \theta \in \Theta} be a statistical model for an observation XX with values in a measurable space 𝒳\mathcal{X}. Let 𝒜\mathcal{A} be a measurable space of actions and let L ⁣:Θ×𝒜→[0,∞)L \colon \Theta \times \mathcal{A} \to [0, \infty) be a loss function, measurable in aa for each θ\theta. A (non-randomized) decision rule is a measurable map δ ⁣:𝒳→𝒜\delta \colon \mathcal{X} \to \mathcal{A}. The risk function of δ\delta is the map R(⋅,δ) ⁣:Θ→[0,∞]R(\cdot, \delta) \colon \Theta \to [0, \infty] given by R(θ,δ)=Eθ[L(θ,δ(X))]=∫𝒳L(θ,δ(x)) Pθ(dx).R(\theta, \delta) = E_\theta\big[ L(\theta, \delta(X)) \big] = \int_{\mathcal{X}} L(\theta, \delta(x)) \, P_\theta(dx).

Remarks

The risk is an expected value over the data, for a fixed θ\theta. Under the squared error loss L(θ,a)=(a−g(θ))2L(\theta, a) = (a - g(\theta))^2, the risk of an estimator is its mean squared error. For randomized rules, one also averages over the randomization.

Example: let X1,…,XnX_1, \dots, X_n be independent normal with mean θ\theta and variance 11, with squared error loss. For δ=Xˉ\delta = \bar X, the risk is R(θ,δ)=Eθ[(Xˉ−θ)2]=Var⁡θ(Xˉ)=1/nR(\theta, \delta) = E_\theta[(\bar X - \theta)^2] = \operatorname{Var}_\theta(\bar X) = 1/n for every θ\theta.

Non-example: the realized loss L(θ,δ(x))L(\theta, \delta(x)) at the observed data xx is not the risk. It depends on the data, while the risk is an average over all possible data.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Expected value, concept, no evidence levelExpected valueconceptLoss function, concept, no evidence levelLoss functionconceptMean squared error, concept, no evidence levelMean squared errorconceptNormal distribution, concept, no evidence levelNormal distributionconceptStatistical model, concept, no evidence levelStatistical modelconceptRisk function, definition, no evidence levelRisk functiondefinition
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. Expected value concept
  2. Loss function concept
  3. Mean squared error concept
  4. Normal distribution concept
  5. Statistical model concept
  6. Risk function definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Loss_function
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