Truncated maximum likelihood, goodness of fit tests and tail analysis

We propose a new method of tail analysis for data featuring a high degree of leptokurtosis. Heavy tails can typically be found in financial series, like for example, the stock returns or durations between arrivals of trades. In our framework, the shape of tails can be assessed by fitting some selected pseudo-models to extremely valued observations in the sample. A global examination of the density tails is performed using a sequence of truncated pseudo-maximum likelihood estimators, called the tail parameter function (t.p.f.). In practice, data often exhibit local irregularities in the tail behaviour. To detect and approximate these patterns we introduce the local parameter function (l.p.f.), a method of tail analysis involving a selected interval of extreme observations. An immediate extension of the pseudo-value based approach to density analysis yields a new procedure for testing the goodness of fit. We also develop a new nonparametric estimator of the density function. The method is applied to unequally spaced high frequency data. We study an intradaily series of returns on the Alcatel stock, one among the most frequently traded stocks on the Paris Stock Exchange.