linguistic:translation, structural:expansion
Likelihood equations

If the log-likelihood function is smooth, its gradient with respect to the parameter, known as the score and written s n (θ) ≡ ∇ θ ℓ n (θ) {\textstyle s_{n}(\theta)\equiv \nabla _{\theta }\ell _{n}(\theta)}, exists and allows for the application of differential calculus. The basic way to maximize a differentiable function is to find the stationary points (the points where the derivative is zero); since the derivative of a sum is just the sum of the derivatives, but the derivative of a product requires the product rule, it is easier to compute the stationary points of the log-likelihood of independent events than for the likelihood of independent events.

The equations defined by the stationary point of the score function serve as estimating equations for the maximum likelihood estimator. s n (θ) = 0 {\displaystyle s_{n}(\theta)=\mathbf {0} } In that sense, the maximum likelihood estimator is implicitly defined by the value at 0 {\textstyle \mathbf {0} } of the inverse function s n − 1: E d → Θ {\textstyle s_{n}^{-1}:\mathbb {E} ^{d}\to \Theta }, where E d {\textstyle \mathbb {E} ^{d}} is the d -dimensional Euclidean space, and Θ {\textstyle \Theta } is the parameter space. Using the inverse function theorem, it can be shown that s n − 1 {\textstyle s_{n}^{-1}} is well-defined in an open neighborhood about 0 {\textstyle \mathbf {0} } with probability going to one, and θ ^ n = s n − 1 (0) {\textstyle {\hat {\theta }}_{n}=s_{n}^{-1}(\mathbf {0})} is a consistent estimate of θ {\textstyle \theta }. As a consequence there exists a sequence { θ ^ n } {\textstyle \left\{{\hat {\theta }}_{n}\right\}} such that s n (θ ^ n) = 0 {\textstyle s_{n}({\hat {\theta }}_{n})=\mathbf {0} } asymptotically almost surely, and θ ^ n → p θ 0 {\textstyle {\hat {\theta }}_{n}\xrightarrow {\text{p}} \theta _{0}}. A similar result can be established using Rolle's theorem.

The second derivative evaluated at θ ^ {\textstyle {\hat {\theta }}}, known as Fisher information, determines the curvature of the likelihood surface, and thus indicates the precision of the estimate.