# Efficiency of the pseudolikelihood for multivariate normal and von Mises distributions

Kanti V. Mardia, Gareth Hughes & Charles C. Taylor

In certain circumstances inference based on the likelihood function can be hindered by, for example, computational complexity; new applications of directional statistics to bioinformatics problems give many obvious examples. In such cases it is necessary to seek an alternative method of estimation. Two pseudolikelihoods, each based on conditional distributions, are assessed in terms of their efficiency for the multivariate normal distribution and a bivariate von Mises distribution. It is shown that both the pseudolikelihoods are fully efficient for the multivariate normal distribution with all variances equal to $\sigma^2$ and all correlations equal to $\rho$. Loss of efficiency is shown for the estimator of equal $\rho$ in the case of known $\sigma^2$. We also prove a remarkable result that both pseudolikelihoods are fully efficient for the general multivariate normal distribution.

This work helps us to understand the behaviour of various von Mises distributions, since for large concentrations, these tend to a normal distribution. We consider here a particular bivariate von Mises distribution, but the results will extend to the multivariate case. Specifically, for the bivariate von Mises distribution, we calculate the efficiency numerically for the two key parameters and study its behaviour. The distribution is approximately normal for highly concentrated data, and the efficiency is shown to tend to unity for increasing concentration. Simulations support the numerical calculations obtained. With the exception of the bimodal case studied, it is seen that the bias of the pseudolikelihood estimator is very similar to that of the maximum likelihood estimator, and for one parameter configuration, the bias is smaller.

Keywords:
Bias, Composite likelihood, Full pseudolikelihood, Fisher information, Pairwise pseudolikelihood.

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