Journal of Mathematics and Statistics 6 (2): 84-88, 2010 ISSN 1549-3644 © 2010 Science Publications Corresponding Author: T.O. Olatayo, Department of Mathematical Sciences, Olabisio Onabanjo University, Ago-Iwoye, Ogun State, Nigeria 84 Bootstrap Method for Dependent Data Structure and Measure of Statistical Precision 1 T.O. Olatayo, 2 G.N. Amahia and 3 T.O. Obilade 1 Department of Mathematical Sciences, Olabisio Onabanjo University, Ago-Iwoye, Ogun State, Nigeria 2 Department of Statistics University of Ibadan, Ibadane, Nigeria 3 Department of Mathematics, Obafemi Awolowo University, Ile-Ife, Nigeria Abstract: Problem statement: This article emphasized on the construction of valid inferential procedures for an estimator ˆ θ as a measure of its statistical precision for dependent data structure. Approach: The truncated geometric bootstrap estimates of standard error and other measures of statistical precision such as bias, coefficient of variation, ratio and root mean square error are considered. Results: We extend it to other measures of statistical precision such as bootstrap confidence interval for an estimator ˆ θ and illustrate with real geological data. Conclusion/Recommendations: The bootstrap estimates of standard error and other measures of statistical accuracy such as bias, ratio, coefficient of variation and root mean square error reveals the suitability of the method for dependent data structure. Key words: Truncated geometric bootstrap, standard error, bias, coefficient of variation, ratio, root mean square error and bootstrap-t confidence interval INTRODUCTION Ever since its introduction by Efron (1979), considerable attention has been given to bootstrap methods as an application of theoretical and methodological problems for statistics. The bootstrap method for estimating the distribution of an estimator or test statistic by resampling one’s data or a model estimated from the data, are available for implementing the bootstrap and the accuracy of bootstrap estimates depend on whether the data are a random sample from a distribution or a time series process. A typical problem in applied statistics involves the estimation of an unknown parameter θ. The two main questions asked are (i) what estimator ˆ θ should be used? (ii) Having chosen to use a particular ˆ θ , how accurate is it as an estimator of θ? (Efron and Tibshiran, 1993). The bootstrap is a general methodology for answering the second question. It is a computer based method, which substitutes considerable amounts of computation in place of theoretical analysis. This study is concerned with application of bootstrap method to stochastic time series process and we proposed a non-parametric bootstrap method called a truncated geometric bootstrap method for stationary time series data. The procedure attempts to mimic the original model by retaining the stationarity property of the original series in the resample pseudo-time series. The pseudo time series is generated by resampling blocks of random size at each truncation, where the length L of each blocks has a truncated geometric distribution with appropriate probability attached to it. This method shares the construction of resampling blocks of observation with replacement to form pseudo- time series of equal or less, with the original series, so that the statistics of interest may be recalculated base on the resampled data set. The method has two major components, the construction of a bootstrap samples and the computation of statistics on the bootstrap samples, through some kind of a loop. The procedure provides and estimates different measures of statistical accuracy for an estimator ˆ θ , such as standard error, bias, coefficient of variation and root mean square error. We extended it to other measure of statistical accuracy by application of bootstrap-t confidence interval with a goal to improve by an order of magnitude upon the accuracy of the standard intervals ( ) ˆ ˆ Z α σ θ± , in a way that allows routine application even to a complicated problems and it produced good approximate confidence interval. Most