Jesus, J; (2012) Contributions to inference without likelihoods. Doctoral thesis , UCL (University College London).

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Abstract

This thesis is concerned with statistical inference in situations where one is unwilling or unable to formulate a likelihood function. The theory of estimating functions (EFs) provides an alternative inference framework in such settings. The research was motivated by problems arising in the application of a class of stochastic models for rainfall based on point processes. These models are often used by hydrologists to produce synthetic rainfall sequences for risk assessment purposes, notably in the UKCP09 climate change projections for the UK. In the absence of a likelihood function, the models are usually fitted by minimizing some measure of disagreement between theoretical properties and the observed counterparts. In general situations of this type, two ”subjective” decisions are required: what properties to use, and how to weight their contribution to the objective function. The choice of weights can be formalised by defining a minimum variance criterion for the estimator. This is equivalent to the Generalized Method of Moments estimator which is widely used in econometrics. The first contribution of this thesis is to translate the problem to an EF framework which is much more familiar to statisticians. Simulations show that the theory has poor finite sample performance for point process rainfall models. This is associated with inaccurate estimation of the covariance matrix of observed properties. A two-stage approach is developed to overcome this problem. The second main contribution is to apply EF theory to the Whittle likelihood, which is based on the periodogram of the data. A problem here is that the covariance matrix of the estimators depends on fourth-order properties which are often intractable. An EF approach provides a feasible alternative in practical applications. After establishing the conditions under which EF theory can be applied to Whittle estimation, simulations are once again used to explore the finite sample performance.

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