Quantile Estimation in Dependent Sequences

Standard nonparametric estimators of quantiles based on order statistics can be used not only when the data are i.i.d., but also when the data are drawn from a stationary, I•-mixing process of continuous random variables. However, when the random variables are highly positively correlated, the sample sizes needed for estimating extreme quantiles become computationally unmanageable. This paper gives a practical scheme, based on a maximum transformation in a two-way layout of the data, that reduces the sample size sufficiently to allow an experimenter to obtain a point estimate of an extreme quantile. The paper gives three schemes that lead to confidence interval estimates for the quantile. One uses a spectral analysis of the reduced sample. The other two, averaged group quantiles and nested group quantiles, are extensions of the method of batched means to quantile estimation. These two schemes give even greater data compaction than the first scheme. None of the schemes requires that the process being simulated is regenerative.

[1]  Philip Heidelberger,et al.  Simulation Run Length Control in the Presence of an Initial Transient , 1983, Oper. Res..

[2]  Leonard Kleinrock,et al.  Theory, Volume 1, Queueing Systems , 1975 .

[3]  David Walter Robinson.  Non-parametric quantile estimation through stochastic approximation. , 1975 .

[4]  E. S. Pearson.  Biometrika tables for statisticians , 1967 .

[5]  E. Parzen.  On Estimation of a Probability Density Function and Mode , 1962 .

[6]  P. Heidelberger,et al.  Adaptive spectral methods for simulation output analysis , 1981 .

[7]  A. J. Lawrance,et al.  A new autoregressive time series model in exponential variables (NEAR(1)) , 1981, Advances in Applied Probability.

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

[9]  Ralph L. Disney,et al.  Applied Probability— Computer Science: The Interface , 1982, Progress in Computer Science.

[10]  Donald L. Iglehart,et al.  Simulating Stable Stochastic Systems, VI: Quantile Estimation , 1976, JACM.

[11]  M. Rosenblatt.  Remarks on Some Nonparametric Estimates of a Density Function , 1956 .

[12]  L. Schruben.  A Coverage Function for Interval Estimators of Simulation Response , 1980 .

[13]  Gutti Jogesh Babu,et al.  On deviations between empirical and quantile processes for mixing random variables , 1978 .

[14]  A. J. Lawrance,et al.  Generation of Some First-Order Autoregressive Markovian Sequences of Positive Random Variables with Given Marginal Distributions, , 1981 .

[15]  Leonard Kleinrock,et al.  Queueing Systems: Volume I-Theory , 1975 .

[16]  Averill M. Law,et al.  A Sequential Procedure for Determining the Length of a Steady-State Simulation , 1979, Oper. Res..

[17]  W. J. Conover,et al.  Practical Nonparametric Statistics , 1972 .

[18]  A. F. Seila,et al.  A Batching Approach to Quantile Estimation in Regenerative Simulations , 1982 .

[19]  G. S. Fishman.  Principles of Discrete Event Simulation , 1978 .

[20]  Philip Heidelberger,et al.  A spectral method for confidence interval generation and run length control in simulations , 1981, CACM.

[21]  P. Sen.  On the Bahadur representation of sample quantiles for sequences of φ-mixing random variables , 1972 .