semiparaKernelEstimate {BSL} | R Documentation |

## Estimate the semi-parametric synthetic (log) likelihood

### Description

This function computes the semi-parametric synthetic likelihood estimator of (An et al. 2019). 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
)
```

### 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 |

`shrinkage` |
A string argument indicating which shrinkage method to
be used. The default is |

`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 |

### Value

The estimated synthetic (log) likelihood value.

### References

An Z, Nott DJ, Drovandi C (2019).
“Robust Bayesian Synthetic Likelihood via a Semi-Parametric Approach.”
*Statistics and Computing (In Press)*.

Friedman J, Hastie T, Tibshirani R (2008).
“Sparse Inverse Covariance Estimation with the Graphical Lasso.”
*Biostatistics*, **9**(3), 432–441.

Warton DI (2008).
“Penalized Normal Likelihood and Ridge Regularization of Correlation and Covariance Matrices.”
*Journal of the American Statistical Association*, **103**(481), 340–349.
doi: 10.1198/016214508000000021.

Friedman J, Hastie T, Tibshirani R (2008).
“Sparse Inverse Covariance Estimation with the Graphical Lasso.”
*Biostatistics*, **9**(3), 432–441.

Warton DI (2008).
“Penalized Normal Likelihood and Ridge Regularization of Correlation and Covariance Matrices.”
*Journal of the American Statistical Association*, **103**(481), 340–349.
doi: 10.1198/016214508000000021.

Boudt K, Cornelissen J, Croux C (2012).
“The Gaussian Rank Correlation Estimator: Robustness Properties.”
*Statistics and Computing*, **22**(2), 471–483.
doi: 10.1007/s11222-011-9237-0.

### 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)
```

*BSL*version 3.2.5 Index]