OPTIMAL SEMIPARAMETRIC INFERENCE FOR THE TAIL INDEX BASED ON RATIOS OF THE LARGEST EXTREMES

The k largest order statistics in a random sample from a common heavy‐tailed parent distribution with a regularly varying tail can be characterized as Fréchet extremes. This paper establishes that consecutive ratios of such Fréchet extremes are mutually independent and distributed as functions of beta random variables. The maximum likelihood estimator of the tail index based on these ratios is derived, and the exact distribution of the maximum likelihood estimator is determined for fixed k, and the asymptotic distribution as k →∞ . Inferential procedures based upon the maximum likelihood estimator are shown to be optimal. The Fréchet extremes are not directly observable, but a feasible version of the maximum likelihood estimator is equivalent to Hill's statistic. A simple diagnostic is presented that can be used to decide on the largest value of k for which an assumption of Fréchet extremes is sustainable. The results are illustrated using data on commercial insurance claims arising from fires and explosions, and from hurricanes.     

A Note on Window Length Selection in Singular Spectrum Analysis

In singular spectrum analysis (SSA) window length is a critical tuning parameter that must be assigned by the practitioner. This paper provides a theoretical analysis of signal–noise separation and time series reconstruction in SSA that can serve as a guide to optimal window choice. We establish numerical bounds on the mean squared reconstruction error and present their almost sure limits under very general regularity conditions on the underlying data generating mechanism. We also provide asymptotic bounds for the mean squared separation error. Evidence obtained using simulation experiments and real data sets indicates that the theoretical properties are reflected in observed behaviour, even in relatively small samples, and the results indicate how, in practice, an optimal assignment for the window length can be made.     

ESTIMATING COMPONENTS IN FINITE MIXTURES AND HIDDEN MARKOV MODELS

When the unobservable Markov chain in a hidden Markov model is stationary the marginal distribution of the observations is a finite mixture with the number of terms equal to the number of the states of the Markov chain. This suggests the number of states of the unobservable Markov chain can be estimated by determining the number of mixture components in the marginal distribution. This paper presents new methods for estimating the number of states in a hidden Markov model, and coincidentally the unknown number of components in a finite mixture, based on penalized quasi‐likelihood and generalized quasi‐likelihood ratio methods constructed from the marginal distribution. The procedures advocated are simple to calculate, and results obtained in empirical applications indicate that they are as effective as current available methods based on the full likelihood. Under fairly general regularity conditions, the methods proposed generate strongly consistent estimates of the unknown number of states or components.     

Two Canonical VARMA Forms: Scalar Component Models Vis-à-Vis the Echelon Form

In this article we study two methodologies which identify and specify canonical form VARMA models. The two methodologies are: (1) an extension of the scalar component methodology which specifies canonical VARMA models by identifying scalar components through canonical correlations analysis; and (2) the Echelon form methodology, which specifies canonical VARMA models through the estimation of Kronecker indices. We compare the actual forms and the methodologies on three levels. Firstly, we present a theoretical comparison. Secondly, we present a Monte Carlo simulation study that compares the performances of the two methodologies in identifying some pre-specified data generating processes. Lastly, we compare the out-of-sample forecast performance of the two forms when models are fitted to real macroeconomic data.     

Bayes Arma Model Determination: Some Empirical Evidence

In this paper the operational characteristics of a Bayesian procedure for autoregressive moving-average model determination are considered. Finite sample bias is considered and the consequences of employing a data taper investigated.     

Stable spectral factorization with applications to the estimation of time series models

A method for generating a miniphase and inveitible spectral factor from an unstable v × v full rank polynomial matrix is proposed. The zeros inside the unit circle are reflected through the boundary |z|=1 using closed form algebraic manipulations. Also included in the procedure is a technique foi determining the stability of a polynomial operator that does not require the explicit construction of the determinant al equation. Application of the technique is illustrated and the implementation of the method in the statistical context of system estimation is discussed.     

Double-blind deconvolution: the analysis of post-synaptic currents in nerve cells

This paper is concerned with the analysis of observations made on a system that is being stimulated at fixed time intervals but where the precise nature and effect of any individual stimulus is unknown. The realized values are modelled as a stochastic process consisting of a random signal embedded in noise. The aim of the analysis is to use the data to unravel the unknown structure of the system and to ascertain the probabilistic behaviour of the stimuli. A method of parameter estimation based on quasi-profile likelihood is presented and the statistical properties of the estimates are established while recognizing that there will be a discrepancy between the model and the true data-generating mechanism. A method of model validation and determination is also advanced and kernel smoothing techniques are proposed as a basis for identifying the amplitude distribution of the stimuli. The data processing techniques described have a direct application to the investigation of excitatory post-synaptic currents recorded from nerve cells in the central nervous system and their use in quantal analysis of such data is illustrated.