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A portmanteau-type statistic based on sample covariance is proposed to test for the existence of the unit-root of an autoregressive model under a heavy-tailed linear noise, where the noise ɛ t = ∑ j = 0 ∞ d j η t − j and { η t } is a sequence of i.i.d. innovations belonging to the domain of attraction of an α -stable distribution for some α ∈ ( 0 , 2 ) . The proposed procedure is nonparametric and easy to implement. Under certain regular assumptions, the limit distribution of the statistics is shown to be a functional of a standard stable distribution. Further, the good finite sample studies show that the new test outperforms PP, DF and ADF tests in size and power. Applications to the daily world crude oil price and 3-month AA financial commercial paper rate are also given to illustrate the performance of the new test.

The tail index is an important parameter in the whole of extreme value theory. In this article, we consider the estimation of the tail index in the presence of a random covariate, where the conditional distribution of the variable of interest is of Pareto-type. More precisely, we use a logarithmic function to link the tail index to the nonlinear predictor induced by covariates, which forms the nonparametric tail index regression models. To estimate the unknown function, we develop an estimation procedure via a local likelihood method. Consistency and asymptotic normality of the estimated functions are established. Subsequently, these theoretical results are illustrated through simulated and real datasets.

We propose empirical likelihood-based statistics to construct confidence regions for the regression coefficient of the parametric tail index regression model. Our limited simulation study shows the method is more accurate than the normal approximation in terms of coverage probability.

This paper deals with a new class of tail index varying coefficient models with the random covariate under Pareto-type distributions. To estimate the unknown coefficient functions, we develop an es...