BSL (version 3.2.5)

semiparaKernelEstimate: Estimate the semi-parametric synthetic (log) likelihood

Description

This function computes the semi-parametric synthetic likelihood estimator of An2018BSL. The advantage of this semi-parametric estimator over the standard synthetic likelihood estimator is that the semi-parametric one is more robust to non-normal summary statistics. Kernel density estimation is used for modelling each univariate marginal distribution, and the dependence structure between summaries are captured using a Gaussian copula. Shrinkage on the correlation matrix parameter of the Gaussian copula is helpful in decreasing the number of simulations.

Usage

semiparaKernelEstimate(
  ssy,
  ssx,
  kernel = "gaussian",
  shrinkage = NULL,
  penalty = NULL,
  log = TRUE
)

Value

The estimated synthetic (log) likelihood value.

Arguments

ssy

The observed summary statisic.

ssx

A matrix of the simulated summary statistics. The number of rows is the same as the number of simulations per iteration.

kernel

A string argument indicating the smoothing kernel to pass into density for estimating the marginal distribution of each summary statistic. Only ``gaussian" and ``epanechnikov" are available. The default is ``gaussian".

shrinkage

A string argument indicating which shrinkage method to be used. The default is NULL, which means no shrinkage is used. Current options are ``glasso'' for the graphical lasso method of Friedman2008;textualBSL and ``Warton'' for the ridge regularisation method of Warton2008;textualBSL.

penalty

The penalty value to be used for the specified shrinkage method. Must be between zero and one if the shrinkage method is ``Warton''.

log

A logical argument indicating if the log of likelihood is given as the result. The default is TRUE.

References

Friedman2008BSL

Warton2008BSL

Boudt2012BSL

See Also

Other available synthetic likelihood estimators: gaussianSynLike for the standard synthetic likelihood estimator, gaussianSynLikeGhuryeOlkin for the unbiased synthetic likelihood estimator, synLikeMisspec for the Gaussian synthetic likelihood estimator for model misspecification.

Examples

data(ma2)
ssy <- ma2_sum(ma2$data)
m <- newModel(fnSim = ma2_sim, fnSum = ma2_sum, simArgs = ma2$sim_args,
              theta0 = ma2$start, sumArgs = list(delta = 0.5))
ssx <- simulation(m, n = 300, theta = c(0.6, 0.2), seed = 10)$ssx

# check the distribution of the first summary statistic: highly non-normal
plot(density(ssx[, 1]))

# the standard synthetic likelihood estimator over-estimates the likelihood here
gaussianSynLike(ssy, ssx)
# the semi-parametric synthetic likelihood estimator is more robust to non-normality
semiparaKernelEstimate(ssy, ssx)
# using shrinkage on the correlation matrix of the Gaussian copula is also possible
semiparaKernelEstimate(ssy, ssx, shrinkage = "Warton", penalty = 0.8)