Semi-parametric density estimation

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Abstract

The local likelihood method of Copas (1995a) allows for the incorporation into our parametric model of influence from data local to the point t at which we are estimating the true density function g(t). This is achieved through an analogy with censored data; we define the probability of a data point being considered observed, given that it has taken value xi, as

where K is a scaled kernel function with smoothing parameter h. This leads to a likelihood function which gives more weight to observations close to t, hence the term ‘local likelihood’.

After constructing this local likelihood function and maximising it at t, the resulting density estimate f(tiOt) can be described as semi-parametric in terms of its limits with respect to h. As h--oo, it approximates a standard parametric' fit f(I.O) whereas when h decreases towards 0, it approximates

the non - parametric kernel density estimate.

My thesis develops this idea, initially proving its asymptotic superiority over the standard parametric estimate under certain conditions.

We then consider the improvements possible by making smoothing parameter h a function of /, enabling our semi parametric estimate to vary from approximating y(l) in regions of high density to f(t,0) in regions where we believe the true density to be low. Our improvement in accuracy is demonstrated in both simulated and real data examples, and the limits with respect

to h and the new adaption parameter oo are examined. Methods for choosing h and oo are given and evaluated, along with a procedure for incorporating prior belief about the true form of the density into these choices.
Further practical examples illustrate the effectiveness of I these ideas when applied to a wide range of data sets.

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