UV function

Computation Of The Sufficient Statistics

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)

Arguments

• X: matrix of the M trajectories.
• model: name of the SDE: 'OU' (Ornstein-Uhlenbeck) or 'CIR' (Cox-Ingersoll-Ross).
• random: 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: fixed effects in the drift: value of the fixed effect when there is only one random effect, 0 otherwise.
• times: times vector of observation times.

Returns

• U: vector of the M statistics U(Tend)

• V: list of the M matrices V(Tend)

Details

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 : $V(Tend) = \int_0^{Tend} b(X(s))^2/a^2(X(s))ds$

References

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