Daren B. H. Cline

Subexponentiality of the product of independent random variables

Suppose X and Y are independent nonnegative random variables. We study the behavior of P(XY>t), as t --> [infinity], when X has a subexponential distribution. Particular attention is given to obtaining sufficient conditions on P(Y>t) for XY to have a subexponential distribution. The relationship between P(X>t) and P(XY>t) is further studied for the special cases where the former satisfies one of the extensions of regular variation.

Convolution tails, product tails and domains of attraction

SummaryA distribution function is said to have an exponential tail F(t) = F(t, ∞) if eαuF(t+u) is asymptotically equivalent to F(t), t→∞, t→∞, for all u. In this case F(lnt) is regularly varying. For two such distributions, F and G, the convolution H=F*G also has an exponential tail. We investigate the relationship between H and its components F and G, providing conditions for lim H/F to exist. In addition, we are able to describe the asymptotic nature of H when the limit is infinite, for many cases. This corresponds to determining both the domain of attraction and the norming constants for the product of independent variables whose distributions have regularly varying tails.In addition, we compare the tails of H=F*G with H1=F1*G1when F is asymptotically equivalent to F and G is equivalent to G1. Such a comparison corresponds to the “balancing” consideration for the product of independent variables in stable domains of attraction. We discover that there are several distinct comparisons possible.

Convolutions of Distributions With Exponential and Subexponential Tails

Distribution tails F(t) = F(t, ∞) are considered for which and as t → ∞ . A real analytic proof is obtained of a theorem by Chover, Wainger and Ney, namely that . In doing so, a technique is introduced which provides many other results with a minimum of analysis. One such result strengthens and generalizes the various known results on distribution tails of random sums. Additionally, the closure and factorization properties for subexponential distributions are investigated further and extended to distributions with exponential tails.

Kernel Estimation of Densities with Discontinuities or Discontinuous Derivatives

In kernel density estimation, one usually assumes the density has two continous derivatives. In this paper we give precise experssions for the asymptotic mean integrated squared error in case the density has m-1 continuous derivatives and two more derivatives with simple discontinuities. We show that the convergence rate for the mean integrated squared error, with appropriate kernel, is proportional to n-v,ν =2m+1/2m+2. Furthermore the ratio of the (random) integrated squared error to the mean integrated squared error converges to 1 almost surely, if the bandwidth is chosen by cross-validation. In particular, it is possible to have n-5/6 with only one continuous derivative, if the kernel and bandwidth are appropriately chosen. When the density has known points of discontinuity, a symmetrization device suggested by SCHUSTER (1985) can be used to improve the convergence rate. The mean integrated squared error of the symmetrization estimator will converge as if the density had no discontinuities at those poi...

Linear prediction of ARMA processes with infinite variance

In order to predict unobserved values of a linear process with infinite variance, we introduce a linear predictor which minimizes the dispersion (suitably defined) of the error distribution. When the linear process is driven by symmetric stable white noise this predictor minimizes the scale parameter of the error distribution. In the more general case when the driving white noise process has regularly varying tails with index [alpha], the predictor minimizes the size of the error tail probabilities. The procedure can be interpreted also as minimizing an appropriately defined l[alpha]-distance between the predictor and the random variable to be predicted. We derive explicitly the best linear predictor of Xn+1 in terms of X1,..., Xn for the process ARMA(1, 1) and for the process AR(p). For higher order processes general analytic expressions are cumbersome, but we indicate how predictors can be determined numerically.

Mechanical Properties of High-Strength Concrete for Prestressed Members

High-strength concrete (HSC) is widely used in prestressed concrete bridges (PCBs). Current design guidelines for PCB structures such as the AASHTO LRFD specifications, however, were developed based on mechanical properties of normal-strength concrete (NSC). As a first step toward evaluating applicability of current AASHTO design provisions for HSC prestressed bridge members, statistical parameters for the mechanical properties of plant-produced HSC were determined. In addition, prediction equations relating mechanical properties with the compressive strength were evaluated. HSC samples were collected in the field from precasters in Texas and tested in the lab at different ages for compressive strength, modulus of rupture, splitting tensile strength, and modulus of elasticity. Statistical analyses were conducted to determine the probability distribution, bias factors (actual mean-to-specified design ratios), and coefficients of variation for each mechanical property. It was found that for each mechanical property, the mean values are not significantly different among the considered factors (precaster, age, specified strength class) or combination of these factors, regardless of specified design compressive strength. Overall, 28-day bias factors (mean-to-nominal ratios) decrease with increases in specified design compressive strength due to relative uniformity of mixture proportions provided for the specified strength range. Still, the 28-day bias factors for compressive strength are higher than those used to calibrate AASHTO LRFD specifications. With few exceptions, the coefficients of variation were uniform for each mechanical property. In addition, the coefficients of variation for the compressive strength and splitting tensile strength of HSC in this study are lower than those for NSC used in the development of the AASHTO LRFD specifications.

Verifying irreducibility and continuity of a nonlinear time series

When considering the stability of a nonlinear time series, verifying aperiodicity, irreducibility and smoothness of the transitions for the corresponding Markov chain is often the first step. Here, we provide reasonably general conditions applicable to nonlinear autoregressive time series, including many with nonadditive errors.

Stability and the Lyapounov exponent of threshold AR-ARCH Models

The Lyapounov exponent and sharp conditions for geometric ergodicity are determined of a time series model with both a threshold autoregression term and threshold autoregressive conditional heteroscedastic (ARCH) errors. The conditions require studying or simulating the behavior of a bounded, ergodic Markov chain. The method of proof is based on a new approach, called the piggyback method, that exploits the relationship between the time series and the bounded chain. The piggyback method also provides a means for evaluating the Lyapounov exponent by simulation and provides a new perspective on moments, illuminating recent results for the distribution tails of GARCH models.

Stability of nonlinear AR(1) time series with delay

The stability of generally defined nonlinear time series is of interest as nonparametric and other nonlinear methods are used more and more to fit time series. We provide sufficient conditions for stability or nonstability of general nonlinear AR(1) models having delay d[greater-or-equal, slanted]1. Our results include conditions for each of the following modes of the associated Markov chain: geometric ergodicity, ergodicity, null recurrence, transience and geometric transience. The conditions are sharp for threshold-like models and they characterize parametric threshold AR(1) models with delay.

Stability of nonlinear stochastic recursions with application to nonlinear AR-GARCH models

We characterize the Lyapunov exponent and ergodicity of nonlinear stochastic recursion models, including nonlinear AR-GARCH models, in terms of an easily defined, uniformly ergodic process. Properties of this latter process, known as the collapsed process, also determine the existence of moments for the stochastic recursion when it is stationary. As a result, both the stability of a given model and the existence of its moments may be evaluated with relative ease. The method of proof involves piggybacking a Foster-Lyapunov drift condition on certain characteristic behavior of the collapsed process.