The heart of the bootstrap is not simply computer simulation, and bootstrapping is not perfectly synonymous with Monte Carlo. Bootstrap method relies on using an original sample or some part of it, such as residuals as an artificial population from which to randomly resampled [1]. Bootstrap resampling methods have emerged as powerful tools for constructing inferential procedures in modern statistical data analysis. The Bootstrap approach, as initiated by [2], avoids having to derive formulas via different analytical arguments by taking advantage of fast computers. The bootstrap methods have and will continue to have a profound influence throughout science, as the availability of fast, inexpensive computing has enhanced our abilities to make valid statistical inferences about the world without the need for using unrealistic or unverifiable assumptions [3].

An excellent introduction to the bootstrap maybe found in the work of [4-9]. Recently, [10,11] have independently introduced non-parametric versions of the bootstrap that are applicable to weakly dependent stationary observations. Their sampling procedure have been generalized by [12,13] and by [14] by resampling “blocks of blocks” of observations to the stationary time series process.

In this article, we introduce a new resampling method as an improvement on the stationary bootstrap of [15] and the moving block bootstrap of [16,17].

The stationary bootstrap is essentially a weighted average of the moving blocks bootstrap distributions or estimates of standard error, where the weights are determined by a geometric distribution. The difficult aspect of applying these methods is how to choose b in moving blocks scheme and how to choose p in the stationary scheme.

On this note we propose a stationary bootstrap method generated by resampling blocks of random size, where the length of each block has a “truncated geometric distribution”.

In Section 2, the actual construction of a truncated geometric bootstrap method is presented. Some theoretical properties of the method are investigated in Section 3 in the case of the mean. In Section 4, it is shown how the theory may be extended to stationary time series processes.

To overcome the difficulties of moving blocks and geometric stationary scheme in determining both b (number of blocks) and p (probabilities). We introduced a truncated form for the geometric distribution and then demonstrated that it is a suitable model for a probability problem. This truncation is of more than just theoretical interest as a number of application has been reported [18,19].

A random length L may be defined to have a truncated geometric distribution with parameter P and N terms, when it has the probability distribution or probability density function

The constant K is found, using the condition

, to be

Thus, we have

These are the probabilities of a truncated geometric distribution with parameter P and N terms. Suppose that L length, number 1 to N are to be selected randomly with replacement. Then, the process continues until it is truncated geometrically at r with an appropriate probability P attached to its random selection in form of,. We take our N to be 4, that is, the random selections could be truncated between 1 to 4 at an appropriate probabilities.

Then, a description of the resampling algorithm when r > 1 is as follows:

1) Let   be a random variables.

2) Let be determined by the r-th truncated observation X_r in the original time series.

3) Let be equal to X_r+1 with probability 1 – P and picked at random from the original N observations with probability p.

4) Let, be the block consisting of b observation starting from X_i.

5) Let be a sequence of blocks of random length determined by truncated geometric distribution.

6) The first L₁, observations in the pseudo time series are determined by the first block_, of observation the next L₂ observations in the pseudo time series are the observations in the second sampled block_.

7) The process is resampled with replacement, until the process is stopped once N observation in the pseudo time series have been generated.

8) Once has been generated, one can compute the quantities of interest for the pseudo time series.

The algorithm has two major components, the construction of a bootstrap sample and the computation of statistics on this bootstrap sample, and repeat these operation many times through some kinds of a loop.

Proposition 1. Conditional on, is stationary.

If the original observations are all distinct, then the new series is, conditional on, a stationary Markov chain. If, on the other hand, two of the original observations are identical and the remaining are distinct, then the new series is a stationary second order Markov chain. The stationary bootstrap resampling scheme proposed here is distinct from the proposed by [20] but posses the same properties with that proposed by [21].

In this section, the emphasis is on the construction of valid inferential procedures for stationary time series data, and some illustrations with real data are given. The real data are the geological data from demonstratigraphic data from Batan well at 30 m regular interval, [22]. The data is the principal oxide of sand or sandstone, which is S_iO₂ or silicon oxide. The point is that the bulk of oil reservoir rocks in Nigeria sedimentary basins is sandstone and shale, a product of sill stone [23]. In other to improve on the geological analysis and prediction of the presence of these elements, a mathematical tool which can be used to examine a wide range of data sets is developed to detect and improve new and old oil basins.

The geological data of 130 observations was subjected to our new method described in section two of this article at 500 and 1000 bootstrap replicates, for block of (1, 2, 3, and 4).

The replicates with minimum variance was selected in each case of number of bootstrap replicates.

The truncated geometric bootstrap method proposed in this article is able to determine P in (0, 1) and provides b, for stationary time series process.

We proposed an algorithm for effective truncated geometric bootstrap method, which shares the construction of resampling blocks of different sizes of observation to form pseudo-time series, so that the statistics of interest are calculated along based on the resampled data sets. We implemented this method with real geological data and generated bootstrap replications of (B = 500 and 1000) for different block sizes of (b = 1, 2, 3, 4) which represents each number of truncations.

The linear models of AR (10) are fitted to the resampled pseudo-time series data, and the choice of the order are determined by AIC and BIC. The normality test was carried out using Jargue Bera statistic on the residual variance of the fitted models. Forecast was generated for the next 10 observations based on the fitted models, which are justified by root mean square error of the models.

In conclusion, the bootstrap method gives a better and a reliable models for predictive purposes. All the models for the different block sizes are good. They preserve and maintain stationary data structure of the process and are reliable for predictive purposes, confirming the efficiency of the proposed method.