STOR Colloquium: Changryong Baek (The University of North Carolina at Chapel Hill)

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

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 analyzing 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. To reduce bias, several least squares estimators, generalizing rank-based and QQ-estimators, and conditional maximum likelihood estimators, based on the exact form of second order regular variation, are introduced and the their basic asymptotics are established. 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. This is a joint work with Patrice Abry, Vladas Pipiras and Herwig Wendt.

Refreshments will be served at 3:30pm in the 3rd floor lobby of Hanes Hall