scorevektori

The scorevektori, or score vector, is a central object in likelihood-based statistics. It is defined as the gradient of the log-likelihood with respect to the parameter vector. For data x1, …, xn with a density f(x; θ) and log-likelihood l(θ) = ∑ log f(xi; θ), the scorevektori is U(θ) = ∂l(θ)/∂θ, a p-dimensional vector where p is the number of parameters.

Key properties of the scorevektori include its expectation and variability under the true parameter value θ0, assuming regularity conditions hold. The expectation satisfies Eθ0[U(θ0)] = 0. The Fisher information matrix I(θ) is defined as Varθ[U(θ)] = -Eθ[∂^2 l(θ)/∂θ∂θ^T]. The observed information is the negative Hessian J(θ) = -∂^2 l(θ)/∂θ∂θ^T evaluated at a specific θ.

In large samples, the scorevektori plays a key role in inference. If θ̂ is the maximum likelihood estimator, then under standard regularity conditions sqrt(n)(θ̂ − θ) converges in distribution to N(0, I(θ)^{-1}). At the MLE, the scorevektori is approximately zero: U(θ̂) ≈ 0.

The score test, or Lagrange multiplier test, uses U(θ0) to assess H0: θ = θ0. The test statistic often takes the form U(θ0)^T I(θ0)^{-1} U(θ0) and has an asymptotic χ^2 distribution with p degrees of freedom under the null. The scorevektori thus underpins several inference approaches, including tests and asymptotic theory, and extends to multivariate and latent-variable contexts.