96JCGS04\P0315------------------------------------------------
Estimating and Visualizing Conditional Densities
Rob J. Hyndman David M. Bashtannyk and Gary K. Grunwald
We consider the kernel estimator of conditional density and derive
its asymptotic bias, variance, and mean-square error. Optimal
bandwidth (with respect to integrated mean-square error) are found
and it is shown that the convergence rate of the density estimator
is order $n^{-2/3}$. We also note that the conditional mean
function obtained from the estimator is equivalent to a kernel
smoother. Given the undesirable bias properties of kernel
smoothers, we seek a modified conditional density estimator that
has mean equivalent to some other nonparametric regression smoother
with better bias properties. It is also shown that our modified
estimator has smaller mean square error than the standard estimator
in some commonly occurring situations. Finally, three graphical
methods for visualizing conditional density estimators are
discussed and applied to a data set consisting of maximum daily
temperatures in Melbourne, Australia.
Key Words: Bandwidth; Conditional Density; Data visualization;
density estimation; Kernel smoothing; Nonparametric
regression.
96JCGS04\P0337-------------------------------------------------
Fast Computation of Auxiliary Quantities in Local
Polynomial Regression
B. A. Turlach and M. P. Wand
We investigate the extension of binning methodology to fast
computation of several auxiliary quantities that arise in
local polynomial smoothing. Examples include degrees of
freedom measures, cross-validation functions, variance estimates,
and exact measures of error. It is shown that the computational
effort required for such approximations is of the same order
of magnitude as that required for a binned local polynomial
smooth.
Key Words: Binning; Cross-validation; Error degrees of freedom; Kernel
estimator; Linear smoother; Mean average squared error;
Smoother matrix; Standard error.
96JCGS04\P0351---------------------------------------------------
Algorithms for Analyzing Nonstationary Time Series
with Fractal Noise
Peter Hall, David Matthews, and Eckhard Platen
Arguably he best-known applications of fractal methods are in
relatively homogeneous, stationary settings, where the environment
is controllable by scientists or engineers. For example, in
applications to surface science, an unblemished portion of a
surface is selected for analysis; and in environmental science,
an artificial soil bed of controlled homogeneity is subjected to
uniformly distributed water droplets, to model the effect of
actual rain on a real soil surface. In some applications, however,
the environment is uncontrollable, with the result that
measurements are subject to irregular fluctuations that are not
so plausibly modeled by fractal processes. The fluctuations may
include discontinuities and nonlinear drift in the mean. Some
approaches to analysis do not distinguish between this
nonstationary contamination and the ``background,'' with the
result that a jump process may provide a significantly better
explanation of the data than a fractal process. In this
article we suggest decomposing an irregular time series into
at least three components --- a jump process, a nonlinear drift,
and a fractal background. We identify the jump process using
threshold methods, and subtract it out. Then we estimate the
nonlinear drift using local regression. After the jumps and
drift have been removed, the fractal background is relatively
homogeneous. It may be analyzed using techniques based on
the variogram, and its dimension used to quantify the
``erraticism'' of the environment that produced the data.
Key Words: Brownian motion; Exchange rate; Fractal dimension;
Gaussian process; Jump process; Local linear regression;
Nonparametric regression; Ornstein-Ublenbeck process;
Point process; Surface science; Wavelets; Wiener process.
96JCGS04\P0365---------------------------------------------------
Sequential Linearlization of Empirical Likelihood
Constraints with Application to U-Statistics
A.T.A. Wood, K.-A. Do, and B. M. Broom
Empirical likelihood for a mean is straightforward to compute, but
for nonlinear statistics significant computational difficulties
arise because of the presence of nonlinear constraints in the
underlying optimization problem. It is certainly the case that
these difficulties can be overcome with sufficient time, care, and
programming effort. However, they do make it difficult to write
general software for implementing empirical likelihood, and
therefore these difficulties are likely to hinder the widespread use
of empirical likelihood in applied work. The purpose of this
article is to suggest an approximate approach that sidesteps the
difficult computational issues. The basic idea, which may be
described as ``sequential linearization of constraints,'' is a very
simple one, but we believe it could have significant ramifications
for the implementation and practical use of empirical likelihood
methodology. One application of the linearization approach, which
we consider in this article, is to the problem of constructing
empirical likelihood for U-statistics. However, the sequential
linearization idea can be applied in the empirical likelihood
context to a broad range of smooth statistical functionals.
Key Words: Bootstrap calibration; Nondegenerate; Nonlinear constraint;
Nonlinear functional; Wilk's theorem.
96JCGS04\P0365------------------------------------------------------
Computer-Assisted Statistics Education at Delft University
of Technology
Piet Groeneboom, Peter de Jong, Dimitri B. Tischenko, & Bert C. van Zomeren
At Delft University of Technology many students experience
difficulties in mastering basic concepts of probability and
statistics. In the past few years the lectures have undergone
a radical change --- the lecture notes now contain modern data
analysis techniques, like kernel density estimation, simulation,
and bootstrapping. In the TWI-Stat project, a computer-aided
instruction course was developed to help students become more
familiar with modern statistical analysis. The course presents
itself as a dynamic, interactive, personal book. Highly
interactive analysis tools are available. The software will be
available for MS-Windows.
Key Words: Computer-aided instruction; Data analysis; Statistical education.
96JCGS04\P0400-------------------------------------------------------
Constraint-Based Representations of Statistical Graphs
Allan R. Wilks
Pictor is an environment for statistical graphics that promotes
simple commands for common uses and offers the ability to
experiment with whole new paradigms. Pictor describes graphs as
graphical objects whose component pieces are related by several
sorts of constraints. This article describes in detail the
constraint system that Pictor uses.
Key Words: Constraint; Layout; Statistical graphics
96JCGS04\P0415------------------------------------------------------
Correction to
Genshiro Kitagawa, ``Monte Carlo Filter and Smoother for
Non-Gaussian Non-linear State Space Models,'' 5, No. 1, 1--25}
The filtering and smoothing methods presented in the article were
originally shown by Kitagawa (1993). However, the same filtering
method, named ``bootstrap filter,'' was developed independently
and published earlier by Gordon, Salmond, and Smith (1993).
Similar fixed-lag smoothing method was noted by Avitzour (1995).
The references for the applications of their method to target
tracking problems were listed by Gordon (1995).
I wish to thank Neil J. Gordon for giving me these references.
Key Words: State Space Models; Nonlinear; Smoothing.
96JCGS04\P0418-------------------------------------------------------
INDEX to Volume 5