Editing Likelihood function (section)

===Likelihood equations===
If the log-likelihood function is [[Smoothness|smooth]], its [[gradient]] with respect to the parameter, known as the [[Score (statistics)|score]] and written <math display="inline">s_{n}(\theta) \equiv \nabla_{\theta} \ell_{n}(\theta)</math>, exists and allows for the application of [[differential calculus]]. The basic way to maximize a differentiable function is to find the [[stationary point]]s (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.
<math display="block">s_{n}(\theta) = \mathbf{0}</math>
In that sense, the maximum likelihood estimator is implicitly defined by the value at <math display="inline">\mathbf{0}</math> of the [[inverse function]] <math display="inline">s_{n}^{-1}: \mathbb{E}^{d} \to \Theta</math>, where <math display="inline">\mathbb{E}^{d}</math> is the <var>d</var>-dimensional [[Euclidean space]], and <math display="inline">\Theta</math> is the parameter space. Using the [[inverse function theorem]], it can be shown that <math display="inline">s_{n}^{-1}</math> is [[well-defined]] in an [[open neighborhood]] about <math display="inline">\mathbf{0}</math> with probability going to one, and <math display="inline">\hat{\theta}_{n} = s_{n}^{-1}(\mathbf{0})</math> is a consistent estimate of <math display="inline">\theta</math>. As a consequence there exists a sequence <math display="inline">\left\{ \hat{\theta}_{n} \right\}</math> such that <math display="inline">s_{n}(\hat{\theta}_{n}) = \mathbf{0}</math> asymptotically [[almost surely]], and <math display="inline">\hat{\theta}_{n} \xrightarrow{\text{p}} \theta_{0}</math>.<ref>{{cite journal |first=Robert V. |last=Foutz |title=On the Unique Consistent Solution to the Likelihood Equations |journal=[[Journal of the American Statistical Association]] |volume=72 |year=1977 |issue=357 |pages=147–148 |doi=10.1080/01621459.1977.10479926 }}</ref> A similar result can be established using [[Rolle's theorem]].<ref>{{cite journal |first1=Robert E. |last1=Tarone |first2=Gary |last2=Gruenhage |title=A Note on the Uniqueness of Roots of the Likelihood Equations for Vector-Valued Parameters |journal=Journal of the American Statistical Association |volume=70 |year=1975 |issue=352 |pages=903–904 |doi=10.1080/01621459.1975.10480321 }}</ref><ref>{{cite journal |first1=Kamta |last1=Rai |first2=John |last2=Van Ryzin |title=A Note on a Multivariate Version of Rolle's Theorem and Uniqueness of Maximum Likelihood Roots |journal=Communications in Statistics |series=Theory and Methods |volume=11 |year=1982 |issue=13 |pages=1505–1510 |doi=10.1080/03610928208828325 }}</ref>

The second derivative evaluated at <math display="inline">\hat{\theta}</math>, known as [[Fisher information]], determines the curvature of the likelihood surface,<ref>{{citation |first=B. Raja |last=Rao |title=A formula for the curvature of the likelihood surface of a sample drawn from a distribution admitting sufficient statistics |journal=[[Biometrika]] |volume=47 |issue=1–2 |year=1960 |pages=203–207 |doi=10.1093/biomet/47.1-2.203 |mode=cs1 }}</ref> and thus indicates the [[Precision (statistics)|precision]] of the estimate.<ref>{{citation |first1=Michael D. |last1=Ward |first2=John S. |last2=Ahlquist |title=Maximum Likelihood for Social Science : Strategies for Analysis |publisher= [[Cambridge University Press]] |year=2018 |pages=25–27 |mode=cs1 }}</ref>