On the use of the bootstrap methods in uncovering the sampling distribution of threshold value estimates.

This study aims at investigating the performance of bootstrap methods in un covering the sampling distribution of parameter estimates of threshold mod els, particularly threshold estimates, which are known for mathematical in tractability. It is impossible to establish theoretical results regarding true distributions of the threshold value. Hence, in this thesis, Efron’s (1992a) bootstrap method is used to study the sampling distributions of parameter es timates of threshold models, particularly, the threshold value estimates in the unknown threshold case. Monte Carlo estimation of the bootstrap distribution is applied. The consistency of bootstrap parameter estimates – i.e. the effect of increasing sample size of bootstrap estimates – and their standard errors are studied. Moreover, to assess the performance of the bootstrap method in threshold models, data are simulated through Gaussian white noise using the statistical software R for different sample sizes (small and large). Then, an investigation into the behaviour of the sampling distributions of parameter es timates of threshold models and the effect of sample size is done for both known and unknown threshold values allows for judging the performance of the boot strap method. All bootstrap parameter estimates are checked for normality by calculating the coefficients of skewness, kurtosis, the plotting of histograms, and box plots. It is worth mentioning that the fitting is done for a fixed number of thresholds, delays, and orders. The findings are interesting and promising. The percentile method, based on bootstrap distribution, is used to construct 100(1−2α)% confidence intervals for model parameters, and compares them with the classical confidence interval, based on large sample theory (approxi mate normality). Finally, to assess the performance of bootstrap methods, all results from the simulated normal errors, or, “the true sampling distribution” case, are compared with results from the bootstrap sampling distribution.
A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy, May 2019