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.

[1]  Peter A. W. Lewis,et al.  Some Simple Models for Continuous Variate Time Series. , 1985 .

[2]  Jiří Anděl.  NON-NEGATIVE AUTOREGRESSIVE PROCESSES , 1989 .

[3]  ESTIMATION FOR NONNEGATIVE AUTOREGRESSIVE PROCESSES WITH AN UNKNOWN LOCATION PARAMETER , 1993 .

[4]  M. R. Leadbetter,et al.  Extremes and Related Properties of Random Sequences and Processes: Springer Series in Statistics , 1983 .

[5]  J. Andel.  On ar(1) processes with exponential white noise , 1988 .

[6]  Gwilym M. Jenkins,et al.  Time series analysis, forecasting and control , 1972 .

[7]  An Hongzhi.  NON‐NEGATIVE AUTOREGRESSIVE MODELS , 1992 .

[8]  P. Bougerol,et al.  Stationarity of Garch processes and of some nonnegative time series , 1992 .

[9]  D. Pollard.  Empirical Processes: Theory and Applications , 1990 .

[10]  Richard A. Davis,et al.  Time Series: Theory and Methods , 2013 .

[11]  S. Resnick.  Extreme Values, Regular Variation, and Point Processes , 1987 .

[12]  J. Andel.  Nonlinear nonnegative AR(1) processes , 1989 .

[13]  George E. P. Box,et al.  Time Series Analysis: Forecasting and Control , 1977 .

[14]  Kai Lai Chung,et al.  A Course in Probability Theory , 1949 .

[15]  B. Efron.  Jackknife‐After‐Bootstrap Standard Errors and Influence Functions , 1992 .

[16]  Sidney I. Resnick,et al.  Estimation for autoregressive processes with positive innovations , 1992 .

[17]  G. Shorack,et al.  Some results on the influence of extremes on the bootstrap , 1993 .

[18]  ESTIMATION FOR REGRESSIVE AND AUTOREGRESSIVE MODELS WITH NON‐NEGATIVE RESIDUAL ERRORS , 1993 .

[19]  Sidney I. Resnick,et al.  Limit Distributions for Linear Programming Time Series Estimators , 1994 .

[20]  B. Efron.  Bootstrap Methods: Another Look at the Jackknife , 1979 .

[21]  W. McCormick,et al.  Asymptotic Results for an Extreme Value Estimator of the Autocorrelation Coefficient for a First Order Autoregressive Sequence , 1989 .

[22]  P. Feigin.  Inference and martingale estimating equations for stochastic processes on a semigroup , 1994 .

[23]  M. A. Turkman.  BAYESian Analysis of an Autoregressiye Process with Exponential White Noise , 1990 .

[24]  C. B. Bell,et al.  Infrence for non-negative autoregressive schemes , 1986 .

[25]  C. Sim.  A mixed gamma ARMA(1, 1) Model for river flow time series , 1987 .

[26]  Richard A. Davis,et al.  Estimation for first-order autoregressive processes with positive or bounded innovations , 1989 .

[27]  Richard A. Davis,et al.  Time Series: Theory and Methods (2nd ed.). , 1992 .

[28]  P. Billingsley,et al.  Convergence of Probability Measures , 1969 .