The scope of this dissertation is three-dimensional. First, following
Hodges and Lehmann's (1967) quasimedians, we study weighted
averages of two order statistics as the estimates of the median,
quartiles and quantiles, by obtaining the large sample expressions
for bias and variance, up to order $O(n^{-3})$. These are used to
suggest certain reduced-bias and reduced-variance modifications.
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The second and the main purpose of this thesis is to derive and
study a new monotone nonparametric estimator of the quantile
function based upon cubic monotone Hermite interpolation and
hence called the Hermitian estimator. It combines the simplicity of
linear interpolation with some desirable properties of the kernel
quantile estimators. The asymptotic expansions up to
order $O(n^{-3})$ for the expectation and variance of the Hermitian
estimator are provided. Some special cases of practical importance,
namely Hermitian medians and quartiles, are further examined. The
approach to construction is carried further by using numerical
methods involving rational functions.
The third component of this
thesis is to develop some applications of the
Hermitian quantile function estimator. One application consists of
new continuous graphical tools such as Hermitian Q-Q curves and
Hermitian TTT-curves, designed for exploratory data analysis.
Another application of the Hermitian estimator, and a major thrust
of the future work, is the estimation of the LQ-moments which are
developed in this thesis as extensions of the L-moments. The
LQ-moments are constructed by using functionals defining the quick
estimators, such as the median and trimean, in place of the
expectations in L-moments. They have extensive uses in applications
such as hydrology, meteorology and engineering. Advantages of
LQ-moments are that they always exist and are often easier to
evaluate,
and their estimates based upon the Hermitian estimator are
considerably simpler to implement and analyze. The theory of
LQ-moments parallels that of L-moments, and they have the same
applied potential as L-moments. Also, because of their pivotal role
in defining the LQ-moments, and their wide use as measures of
location and scale, we examine the conditions for the optimality and
unacceptability of the quick estimators. The dissertation is
organized in eleven chapters, the last given to several problems
stimulated by the work for the dissertation are outlined.