# UV: Computation Of The Sufficient Statistics In mixedsde: Estimation Methods for Stochastic Differential Mixed Effects Models

## Description

Computation of U and V, the two sufficient statistics of the likelihood of the mixed SDE dX_j(t)= (α_j- β_j X_j(t))dt + σ a(X_j(t)) dW_j(t).

## Usage

 1 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)= (α_j- β_j X_j(t))dt + σ a(X_j(t)) dW_j(t) = (α_j, β_j)b(X_j(t))dt + σ 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

mixedsde documentation built on May 1, 2019, 7:33 p.m.