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Quantitative Finance · शब्दावली

Likelihood function, maximum likelihood estimator, score, Fisher information क्या है?

अन्य नाम: likelihood function · maximum likelihood estimator · score function · Fisher information

Definition 11.2 Quantitative Methods · अध्याय 11 — Estimation

For independent observations with density f(x;θ)f(x; \theta), the likelihood function is θ↦∏if(Xi;θ)\theta \mapsto \prod_if(X_i; \theta) and the log-likelihood ℓn(θ)=∑iln⁡f(Xi;θ)\ell_n(\theta) = \sum_i\ln f(X_i; \theta). The maximum likelihood estimator (MLE) maximises it. The score function is ∇θln⁡f(X;θ)\nabla_\theta\ln f(X; \theta), and the Fisher information is I(θ)=E[∇ln⁡f ∇ln⁡f⊤]=−E[∇2ln⁡f]\mathcal I(\theta) = \E[\nabla\ln f\,\nabla\ln f^\top] = -\E[\nabla^2\ln f] per observation.

उदाहरण

Example 11.4 (The rate of arrivals)

For nn exponential waiting times with rate λ\lambda, ℓn(λ)=nln⁡λ−λ∑ti\ell_n(\lambda) = n\ln\lambda - \lambda\sum t_i, so λ^=n/∑ti\hat\lambda = n/\sum t_i, I(λ)=1/λ2\mathcal I(\lambda) = 1/\lambda^2, and se(λ^)=λ^/n\mathrm{se}(\hat\lambda) = \hat\lambda/\sqrt n. A hundred gaps summing to 50 seconds give 2±0.22 \pm 0.2 arrivals per second. The Hawkes fits of chapter 7 are the same computation with a harder likelihood.

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