yuima (version 1.15.34)

qmleDegenerate: Quasi-likelihood Estimation for Degenerate Diffusion Processes

Description

Calculate a quasi maximum likelihood type estimator for the parameters of a degenerate diffusion process based on discrete observation data.

Usage

qmleDegenerate(yuima, start, method = "L-BFGS-B", 
                fixed = list(), print = FALSE, 
                envir = globalenv(), lower, upper, 
                joint = FALSE, ...)

Value

An object of class yuima.qmle-class.

When joint=FALSE, the slots min, details and minuslogl are for estimation of the drift parameter of the degenerate components, but the details slot has an additional component initial containing the initial estimator.

Arguments

yuima

a yuima object.

start

initial values to be passed to the optimizer.

method

passed to optim.

fixed

a list of fixed parameters.

print

you can see a progress of the estimation when print is TRUE.

envir

an environment where the model coefficients are evaluated.

lower

a named list for specifying lower bounds of parameters.

upper

a named list for specifying upper bounds of parameters.

joint

perform joint estimation or adaptive estimation with one-step correction? by default joint=FALSE.

...

passed to optim.

Author

Yuta Koike with YUIMA Project Team

Details

When joint=TRUE, the function calculates the non-adaptive estimator presented in Gloter and Yoshida (2024).

When joint=FALSE, the function calculates the adaptive one-step estimator presented in Section 4 of Gloter and Yoshida (2021) with the initial estimator constructed as in Section 3 of Gloter and Yoshida (2021).

References

Gloter, A. and Yoshida, N. (2020) Adaptive and non-adaptive estimation for degenerate diffusion processes, working paper, available at tools:::Rd_expr_doi("10.48550/arXiv.2002.10164").

Gloter, A. and Yoshida, N. (2021) Adaptive estimation for degenerate diffusion processes, Electronic Journal of Statistics, 15, 1424--1472.

Gloter, A. and Yoshida, N. (2024) Non-adaptive estimation for degenerate diffusion processes, Theory of Probability and Mathematical Statistics, 110, 75--99.

See Also

qmle

Examples

## dXt = theta3*Yt*dt
## dYt = -theta2*(Xt+Yt)dt + theta1*dWt

## Setting of the model
drift <- c("theta3*y", "-theta2*(x+y)")
diff.mat <- matrix(c(0, 0, 0, "theta1"), 2, 2)
ymod <- setModel(drift = drift, diffusion = diff.mat,
                 state.variable=c("x", "y"),solve.variable=c("x", "y"))
param <- list(theta1=1, theta2=2, theta3=0.7)

## Setting of the sampling scheme 
n <- 900
ysamp <- setSampling(Terminal = n/100, n = 10*n)
ysubsamp <- setSampling(Terminal = n/100, n = n)
yuima1 <- setYuima(model = ymod, sampling = ysamp)

## Simulation
set.seed(111)
yuima <- simulate(yuima1, xinit=c(1, 1), true.parameter = param)
yuima <- subsampling(yuima, ysubsamp)
plot(yuima)

## Adaptive one-step estimation 
start <- list(theta1=0.5, theta2=0.5, theta3=0.5)
lower <- list(theta1=0.1, theta2=0.1, theta3=0.1)
upper <- list(theta1=5, theta2=5, theta3=5)
res <- qmleDegenerate(yuima, start = start, lower = lower,upper = upper)
print(res)

if (FALSE) {
## Joint estimation
set.seed(123)
yuima <- simulate(yuima1, xinit=c(1, 1), true.parameter = param)
yuima <- subsampling(yuima, ysubsamp)
res <- qmleDegenerate(yuima, start = start, lower = lower,upper = upper, joint = TRUE)
print(res)

## Fix theta1 to the true value
res <- qmleDegenerate(yuima, start = start, lower = lower,upper = upper, 
                       fixed = list(theta1 =1))
print(res)

## Multi-dimensional case
drift <- c("a1*z", "a2*(z+w)", "-b1*x-z", "-b2*(x+y)-w")
diff.mat <- matrix(c(0,0,0,0,0,0,0,0,0,0,0, "sigma1", 0, 0,"sigma2", "sigma3"), 4, 4,byrow=TRUE)
ymod <- setModel(drift = drift, diffusion = diff.mat,
                 state.variable=c("x", "y", "z", "w"),solve.variable=c("x", "y", "z", "w"))
param <- list(a1=1,a2=0.5,b1=1,b2=0.5,sigma1=1,sigma2=0.5,sigma3=0.5)

n <- 100
ysamp <- setSampling(Terminal = n/10, n = 10*n)
ysubsamp <- setSampling(Terminal = n/10, n = n)
yuima1 <- setYuima(model = ymod, sampling = ysamp)

set.seed(111)
yuima <- simulate(yuima1, xinit=c(1, 1, 1, 1), true.parameter = param)
yuima <- subsampling(yuima, ysubsamp)
plot(yuima)

start <- list(a1=2,a2=2,b1=2,b2=2,sigma1=2,sigma2=2,sigma3=2)
lower <- list(a1=0.1,a2=0.1,b1=0.1,b2=0.1,sigma1=0.1,sigma2=0.1,sigma3=0.1)
upper <- list(a1=5,a2=5,b1=5,b2=5,sigma1=5,sigma2=5,sigma3=5)
res <- qmleDegenerate(yuima, start = start, lower = lower,upper = upper)
print(res)
}