William P. McCormick

Bootstrap Inference for a First-Order Autoregression with Positive Innovations

Abstract In this article we consider statistical inference for the autoregressive parameter of a first-order autoregressive sequence with positive innovations via an extreme value estimator ϕ. We show that a bootstrap procedure correctly estimates the sampling distribution of an asymptotically pivotal quantity (whose distribution depends only on the exponent of regular variation of the innovation distribution) based on ϕ, provided that the ratio of the bootstrap sample size m and the original sample size n converges to zero. This result enables us to construct a totally nonparametric confidence interval for the autoregressive parameter. We also consider bootstrapping a normalized version of ϕ with an application toward bias correction. To obtain the bootstrap validity results, we develop a continuous convergence result for certain associated point processes. We also present results of simulation studies and a numerical example.

Estimation for first-order autoregressive processes with positive or bounded innovations

We consider estimates motivated by extreme value theory for the correlation parameter of a first-order autoregressive process whose innovation distribution F is either positive or supported on a finite interval. In the positive support case, F is assumed to be regularly varying at zero, whereas in the finite support case, F is assumed to be regularly varying at the two endpoints of the support. Examples include the exponential distribution and the uniform distribution on [-1, 1 ]. The limit distribution of the proposed estimators is derived using point process techniques. These estimators can be vastly superior to the classical least squares estimator especially when the exponent of regular variation is small.

Asymptotic expansions for infinite weighted convolutions of heavy tail distributions and applications

We establish some asymptotic expansions for infinite weighted convolution of distributions having regular varying tails. Various applications to statistics and probability are developed.

Inference for the Tail Parameters of a Linear Process with Heavy Tail Innovations

AbstractConsider a linear process

Extremes for shot noise processes with heavy tailed amplitudes

X_t = \sum\nolimits_{i = 0}^\infty {c_i Z_{t - 1} }

Extreme value theory for processes with periodic variances

where the innovations Zs are i.i.d. satisfying a standard tail regularity and balance condition, vis., P(Z > z) ∼ rz-αL1(z), P(Z < -z) ∼ sz-αL1(z), as z →∞, where r + s = 1, r, s ≥ 0, α > 0 and L1 is a slowly varying function. It turns out that in this setup, P(X > x) ∼ px-αL(x), P(X < -x) ∼ qx-αL(x), as x →∞, where α is the same as above, p is a convex combination of r and s, p + q = 1, p, q ≥ 0 and L =

Nonlinear Autoregression with Positive Innovations

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