math_notation

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Mathematical Notation

A

The notation \(A\) is used for the random constraint matrix .

B

The notation B(beta, theta, u) ( \(B( \beta , \theta , u )\) ) is used for the second order random constraint approximation.

c

The notation c(theta) ( \(c( \theta )\) ) is used for the fixed effects constraints as a function.

c_L

The notation c_L ( \(c_L\) ) is used for the lower limit of the constraints on the fixed effects.

c_U

The notation c_U ( \(c_U\) ) is used for the upper limit of the constraints on the fixed effects.

f

The notation f(theta, u) ( \(f( \theta , u )\) ) is used for the random likelihood function.

g

The notation g(theta) ( \(g( \theta )\) ) is used for the fixed likelihood function.

Lower h

The notation h(theta, u) ( \(h( \theta , u )\) ) is used for the Laplace approximation function.

Capital H

The notation H(beta, theta, u) ( \(H( \beta , \theta , u )\) ) is used for the second order Laplace objective.

p

The notations \(\B{p} ( \cdot )\) and \(\B{p} ( \cdot | \cdot )\) are use for the probability density (conditional probability density) functions; see p(theta) , p(z | theta) , p(u | theta) , p(y | theta, u) .

r

The notation r(theta) ( \(r( \theta )\) ) is used for the Laplace objective function.

L

The notation L(theta) ( \(L( \theta )\) ) is used for the fixed effects objective function.

Lower u

The notation u ( \(\theta\) ) is used for the vector of fixed effects.

u^(theta)

The notation u^(theta) ( \(\hat{u}( \theta )\) ) is used for the optimal random effects.

Capital U

The notation U(beta, theta, u) ( \(U( \beta , \theta , u )\) ) is used for the first order optimal random effects.

W

The notation W(beta, theta, u) ( \(W( \beta , \theta , u )\) ) is used for the second order approximation of optimal random effects.

theta

The notation theta ( \(\theta\) ) is used for the vector of fixed effects.

y

The notation y is used for data that depends on the random effects.

z

The notation z is used for data that does not depend on the random effects.