Estimation - Note 2

Likelihood based estimation for models on trees

Stefka Asenova

2023-02-11


For application of this estimator, see Vignette “Code - Note 2”.


\[\begin{equation} \{\mu_{W_u,u}(\theta)\}_v = -\frac{1}{2}\sum_{e \in p(u,v)} \theta_{e}^2, \quad v\in W_u \setminus u \end{equation}\]

\[\begin{equation} \big(\Lambda(\theta)\big)_{ij} = \lambda^2_{ij}(\theta) = \frac{1}{4}\sum_{e \in p(i,j)} \theta_e^2\, , \qquad i,j\in V, \ i \ne j, e\in E. \end{equation}\]

\[\begin{equation} \label{eq:hrdist} \big(\Sigma_{W_u,u}(\Lambda)\big)_{ij} = 2(\lambda_{iu}^2 + \lambda_{ju}^2 - \lambda^2_{ij}), \qquad i,j\in W_u\setminus u. \end{equation}\]

Maximum likelihood method - Version 1

The estimator of \((\theta_e, e\in E)\) is obtained in a two-step procedure:

Maximum likelihood method - Version 2

Consider the likelihood function of a random sample \(y_{i}, i=1, \ldots, k\) of multivariate normal distribution with mean vector \(\mu\) and covariance matrix \(\Sigma\), where \(y_i\) is of dimension \(d\).

\[\begin{align*} L(\mu,\Sigma;\, &y_1,\ldots,y_k) = \prod_{i=1}^k\phi_d(y_i-\mu;\Sigma) \\&= (2\pi)^{-kd/2}(\det \Sigma^{-1})^{k/2} %\\& %\times \exp\Big( -\frac{1}{2}\sum_{i=1}^k(y_i-\mu)^T\Sigma^{-1}(y_i-\mu) \Big)\, . \end{align*}\]

The method of composite likelihoods consists of optimizing a function that collects the likelihood functions across all the sets \(W_u, u\in U\). So let for all \(u\in U\) the subsets \(W_u\) be given.

Consider the composite likelihood function \[\begin{equation} \begin{split} L\big(\theta; \, & \{\Delta_{uv,i}: v\in W_u\setminus u,\, i\in I_u, u\in U\}\big) \\&= \prod_{u\in U}L\big(\theta_{W_u}; \{\Delta_{uv,i}: v\in W_u\setminus u, i\in I_u\}\big) \\&= \prod_{u\in U}\prod_{i\in I_u} \phi\Big(\{\Delta_{uv,i}: v\in W_u\setminus u, i\in I_u\} - \mu_{W_u, u}(\theta); \Sigma_{W_u, u}(\theta) \Big)\, . \end{split} \end{equation}\]

The estimator is given by

\[\begin{equation} \hat{\theta}^{MLE2}_{k,n} = \arg\max_{\theta\in(0,\infty)^{|E|}} L\big(\theta; \{\Delta_{uv,i}: v\in W_u\setminus u, i\in I_u, u\in U\}\big) \end{equation}\]

The assumption under this definition is that for any \(u,v \in U\) we have \(\Delta_{W_u\setminus u}\perp \Delta_{W_v\setminus v}\), which is clearly not true for overlapping vertex sets \(W_u\) and \(W_v\). However this simplifies the joint likelihood function and simulation results show that the estimator has comparable qualities to the moment estimator or the one based on extremal coefficients.

References