# MsdeParEst-package: Parametric Estimation in Mixed-Effects Stochastic... In MsdeParEst: Parametric Estimation in Mixed-Effects Stochastic Differential Equations

## Description

Parametric estimation in mixed-effects stochastic differential equations

## Details

This package is dedicated to parametric estimation in the following mixed-effects stochastic differential equations:

dX_j(t)= (α_j- β_j X_j(t))dt + σ_j \ a(X_j(t)) dW_j(t),

j=1,…,M, where the (W_j(t)) are independant Wiener processes and the (X_j(t)) are observed without noise. The volatility function a(x) is known and can be either a(x)=1 (Ornstein-Uhlenbeck process) or a(x)=√{x} (Cox-Ingersoll-Ross process).

Different estimation methods are implemented depending on whether there are random effects in the drift and/or in the diffusion coefficient:

1. The diffusion coefficient is fixed σ_j \equiv σ and the parameters in the drift are Gaussian random variables:

1. either α_j \equiv α and β_j \sim N(μ,Ω), j=1,…,M,

2. or β_j \equiv β and α_j \sim N(μ,Ω), j=1,…,M,

3. or (α_j,β_j) \sim N(μ,Ω), j=1,…,M.

μ, Ω and potentially the fixed effects σ, α, β are estimated as proposed in [1] and [4]. The extension to mixtures of Gaussian distributions is also implemented by following [3].

2. The coefficients in the drift are fixed α_j \equiv α and β_j \equiv β and the diffusion coefficient 1/σ_j^2 follows a Gamma distribution 1/σ_j^2 \sim Γ(a,λ), j=1,…,M. a, λ, and potentially the fixed effects α and β are estimated by the method published in [2].

3. There are random effects in the drift and in the diffusion, such that 1/σ_j^2 \sim Γ(a,λ) and

1. either α_j \equiv α and β_j|σ_j \sim N(μ,σ_j^2 Ω),

2. or β_j \equiv β and α_j|σ_j \sim N(μ,σ_j^2 Ω),

3. or (α_j,β_j)|σ_j \sim N(μ,σ_j^2 Ω).

a, λ, μ, Ω and potentially the fixed effects α and β are estimated by following [5].

## References

[1] Maximum Likelihood Estimation for Stochastic Differential Equations with Random Effects, Delattre, M., Genon-Catalot, V. and Samson, A. Scandinavian Journal of Statistics 40(2) 2012 322-343.

[2] Estimation of population parameters in stochastic differential equations with random effects in the diffusion coefficient, Delattre, M., Genon-Catalot, V. and Samson, A. ESAIM:PS 19 2015 671-688.

[3] Mixtures of stochastic differential equations with random effects: application to data clustering, Delattre, M., Genon-Catalot, V. and Samson, A. Journal of Statistical Planning and Inference 173 2016 109-124.

[4] Parametric inference for discrete observations of diffusion processes with mixed effects, Delattre, M., Genon-Catalot, V. and Laredo, C. Stochastic Processes and their Applications. To appear. (Available pre-publication hal-0133263, 2016).

[5] Estimation of the joint distribution of random effects for a discretely observed diffusion with random effects, Delattre, M., Genon-Catalot, V. and Laredo, C. hal-01446063 2017.

MsdeParEst documentation built on May 2, 2019, 3:01 a.m.