Second order properties of distribution tails and estimation of tail exponents in random difference equations

Extremes, 12 (2009), 361-400
http://dx.doi.org/10.1007/s10687-009-0082-x

According to a celebrated result of Kesten (Acta Math 131:207–248, 1973), random difference equations have a power-law distribution tail in the asymptotic sense. Empirical evidence shows that classical estimators of tail exponent of random difference equations, such as Hill estimator, are extremely biased for larger values of tail exponents. It is argued in this work that the bias occurs because the power-tail region is too far in the tail from a practical perspective. This is supported by analysis of a few examples where a stationary distribution of random difference equation is known explicitly, and by proving a weaker form of the so-called second order regular variation of distribution tails of random difference equations, which measures deviations from the asymptotic power tail. The latter, in particular, suggests a specific second order term for a distribution tail. Estimation of tail exponents can be adapted by taking this second order term into account. One such method available in the literature is examined, and a new, simple, regression type estimator is proposed. Simulation study shows that the proposed estimator works very well. ARCH models of interest in Finance and multiplicative cascades used in Physics are considered as motivating examples throughout the work. Extension to multidimensional random difference equations with nonnegative entries is also considered.