Computation of U and V, the two sufficient statistics of the likelihood of the mixed SDE \(dX_j(t)= (\alpha_j- \beta_j X_j(t))dt + \sigma a(X_j(t)) dW_j(t)\).
UV(X, model, random, fixed, times)matrix of the M trajectories.
name of the SDE: 'OU' (Ornstein-Uhlenbeck) or 'CIR' (Cox-Ingersoll-Ross).
random effects in the drift: 1 if one additive random effect, 2 if one multiplicative random effect or c(1,2) if 2 random effects.
fixed effects in the drift: value of the fixed effect when there is only one random effect, 0 otherwise.
times vector of observation times.
vector of the M statistics U(Tend)
list of the M matrices V(Tend)
Computation of U and V, the two sufficient statistics of the likelihood of the mixed SDE \(dX_j(t)= (\alpha_j- \beta_j X_j(t))dt + \sigma a(X_j(t)) dW_j(t) = (\alpha_j, \beta_j)b(X_j(t))dt + \sigma a(X_j(t)) dW_j(t)\) with \(b(x)=(1,-x)^t\):
U : \(U(Tend) = \int_0^{Tend} b(X(s))/a^2(X(s))dX(s) \)
V : \(V(Tend) = \int_0^{Tend} b(X(s))^2/a^2(X(s))ds \)
See Bidimensional random effect estimation in mixed stochastic differential model, C. Dion and V. Genon-Catalot, Stochastic Inference for Stochastic Processes 2015, Springer Netherlands 1--28