UV {mixedsde}

Computation Of The Sufficient Statistics

Description

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).

Usage

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.

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

Value

U

vector of the M statistics U(Tend)

V

list of the M matrices V(Tend)

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