problem¶

The cppad_mixed Maximum Likelihood Problem¶

Notation¶

The densities below are known functions of \(y\), \(z\), \(\theta\), and \(u\):

Fixed Effects, theta¶

For each cppad_mixed object, there is a vector of fixed effects theta . The number of the fixed effects, and their order is constant (for each cppad_mixed object). We sometimes use \(\theta\) to denote the vector of fixed effects.

Random Effects, u¶

For each cppad_mixed object, there is a vector of random effects u . The number of the random effects, and their order is constant (for each cppad_mixed object).

Data, y, z¶

We use \(y\), (\(z\)) to denote the set of data points that are dependent on (independent of) the random effects.

Fixed Prior Density, p(theta)¶

The fixed effects prior density is denoted by \(\B{p}( \theta )\).

Fixed Data Density, p(z|theta)¶

The data density, that does not depend on the random effects, is denoted by \(\B{p}( z | \theta )\).

Random Prior Density, p(u|theta)¶

The random effects prior density, given the fixed effects, is denoted by \(\B{p}( u | \theta)\).

Random Data Density, p(y|theta,u)¶

The density for the data, given both the fixed and random effects, is denoted by \(\B{p}( y | \theta, u)\).

Fixed Constraint Function, c(theta)¶

The fixed effects constraint function, denoted by \(c( \theta )\), is a smooth function of the fixed effects,

Optimal Random Effects, u^(theta)¶

Given a value for the fixed effects \(\theta\), the corresponding optimal random effects are defined by

\[\hat{u} ( \theta ) = \R{argmax} \; \B{p} ( y | \theta , u) \; \B{p} ( u | \theta )\]

Note that this definition agrees with the other definition for u^(theta) .

Random Constraint Matrix, A¶

The random constraint matrix is denoted by \(A\). It has row dimension equal to the number of constraints and column dimension equal to the number of random effects.

Random Constraint Function, A*u^(theta)¶

We refer to \(A \; \hat{u} ( \theta )\) as the random constraint function.

Maximum Likelihood¶

We are given the problem of estimating the fixed effects by maximizing the likelihood with respect to \(\theta\); i.e.,

\[\B{p}( y , z , \theta ) = \B{p} ( \theta ) \B{p} ( z | \theta ) \int_{-\infty}^{+\infty} \B{p} ( y | \theta, u ) \B{p} ( u | \theta ) \; \B{d} u\]

No Random Effects¶

In the case where there are not random effects, the vector \(u\) is empty and the optimal fixed effects maximize

\[\B{p} ( \theta ) \B{p} ( z | \theta )\]

Fixed Constraints, c¶

The fixed effects have constraints of the form

\[c_L \leq c(\theta) \leq c_U\]

where and \(c_L\), \(c_U\) are vectors in the range of \(c(x)\). In the case where there are no constraints on the fixed effects, the range space is the empty vector.

Random Constraints¶

The random constraints are defined by the equation

\[0 = A \; \hat{u} ( \theta )\]

Negative Log-Density Vector¶

If vec is a density vector corresponding to \(\B{p}(x)\), the corresponding negative log-density is given by

\(- \log [ \B{p} (x) ] =\)

vec [0] + fabs ( vec [1]) + … fabs ( vec [ s -1 ])

where s = vec . size () .