Chapter 5: Conclusions

Quantile estimation and hypothesis testing have become increasingly important in applications where the mean-based inference fails to capture important distributional characteristics. In this dissertation, we have developed accurate and computationally efficient methods to help practitioners and researchers in performing inference on normal quantiles and their functions.

We have observed and proved that [chakraborti2007confidence] CI for a normal quantile and the classical one based on the noncentral $t$ distribution are the same. We also proposed a simple alternative normal approximate CI as an alternative to the classical NCT CI. The normal approximate CIs perform as good as the exact classical CIs. The normal approximation is also used to find approximate fiducial distributions for normal quantiles, which are useful to find CIs for the ratio or difference of two normal quantiles. The fiducial approach based on the normal approximation is not only conceptually simple, but also produces accurate CIs even for small samples. Our simulation studies and examples showed that the normal approximate fiducial CIs for a ratio and difference of normal quantiles are very satisfactory with better precision than the NCT fiducial CIs.

Having developed reliable interval estimation procedures, we next addressed formal hypothesis testing for quantiles in multi-group comparison settings. In many clinical and scientific studies, groups may have similar means but differ substantially in other distributional features, making quantile comparisons more informative. We addressed clinical and other comparison studies, several groups are often compared in terms of means. The results of those studies may indicate that the mean effects of the treatments are similar, still does not mean that the treatments have similar effect. Recent articles have shown that a thorough comparison among the groups is to compare them in terms of quantiles. In Chapter LABEL:ch3, we have proposed PB test and a modified LRT. We have compared the proposed tests for quantiles in terms of type I error rates and powers. The proposed MLRT controls the type I error rates very satisfactorily with better power properties. We have also noted that the MLRTs and other tests can also be used to compare the means of normal populations. The MLRTs are better alternative tests for comparing means than the Welch test. The proposed tests are also applicable to compare quantiles of several log-normal distributions because of the one-to-one relations between the normal and lognormal quantiles. However, it should be noted that the MLRTs are not applicable for testing the equality of the means of several lognormal distributions.

We also noted that the pairwise CIs based on the classical noncentral pivotal quantities and the one based on the unbiased estimators of the normal quantiles are essentially the same. As an estimator of a quantile, the minimum variance unbiased estimator ${\widehat{\xi }}_u=\overline{X}+z_{p}aS$ maybe preferred to the usual point estimator $\overline{X}+z_{p}S$ of $\mu +z_p\sigma$, but the CIs based on ${\xi }_u$ is the same as the one based on the classical pivotal quantity. Although we have theoretical evidence from the one-sample problem and strong numerical evidence indicating that these two simultaneous CIs are similar, we are unable to prove it analytically.

Building on the testing framework, we then developed confidence intervals for a common quantile across several normal populations. A common approach to constructing confidence intervals for a parameter of interest is to invert a corresponding hypothesis test. When the test is exact—in the sense that the null distribution of the test statistic does not depend on unknown parameters—the resulting confidence interval is also exact. This framework has been used for combining information from independent samples to construct confidence intervals for a common parameter of several populations. In Chapter LABEL:ch4, we have investigated some numerical methods of obtaining CIs for a common quantile of several normal populations by inverting a combination of independent tests. Inference on a common percentile has direct relevance in medical, pharmaceutical, and industrial applications. The proposed CIs based on combination of NCT tests are exact, but their calculations are numerically involved. To facilitate practical implementation, we provide R code for computing confidence intervals for a common quantile based on the Fisher, inverse chi-square, and inverse normal methods. This R code also computes the fiducial CIs.

To help the practitioners and future researchers, we provide R code in the appendix which can be used to compute CIs for a ratio and difference of two normal quantiles, the p-value of the DK-MLRT, and CIs for the common quantile of several normal populations.