1.
On the distribution of the largest eigenvalue in principal
components analysis - Johnstone, Iain M.
Let x(1) denote the square of the largest
singular value of an n × p matrix X, all of whose
entries are independent standard Gaussian variates. Equivalently,
x(1) is the largest principal component variance of the
covariance matrix $X'X$, or the largest eigenvalue of a pvariate
Wishart distribution on n degrees of freedom with identity covariance.
¶ Consider the limit of large p and n with $n/p =
\gamma \ge 1$. When centered by $\mu_p = (\sqrt{n-1} + \sqrt{p})^2$ and scaled
by $\sigma_p = (\sqrt{n-1} + \sqrt{p})(1/\sqrt{n-1} + 1/\sqrt{p}^{1/3}$, the
distribution of x(1) approaches the Tracey-Widom law of order
1, which is defined in terms of the Painlevé II differential...
2.
Tail probabilities of the maxima of multilinear forms and their
applications - Kuriki, Satoshi; Takemura, Akimichi
Let Z be a kway array consisting of
independent standard normal variables. For column vectors h1,
, hk, define a multilinear form of degree
k by $(h_1 \otimes \cdots \otimes h_k)' \vec(Z)$. We derive formulas for
upper tail probabilities of the maximum of a multilinear form with respect to
the his under the condition that the
his are unit vectors, and of its standardized
statistic obtained by dividing by the norm of Z. We also give formulas
for the maximum of a symmetric multilinear form $(h_1 \otimes \cdots \otimes
h_k)' \sym(Z)$, where sym (Z) denotes the symmetrization of Z
with respect to indices. These classes of statistics are used for...
3.
Nonparametric estimation in null recurrent time series - Karlsen, Hans Arnfinn; Tjøstheim, Dag
We develop a nonparametric estimation theory in a nonstationary
environment, more precisely in the framework of null recurrent Markov chains.
An essential tool is the split chain, which makes it possible to decompose the
times series under consideration into independent and identical parts. A tail
condition on the distribution of the recurrence time is introduced. This
condition makes it possible to prove weak convergence results for sums of
functions of the process depending on a smoothing parameter. These limit
results are subsequently used to obtain consistency and asymptotic normality
for local density estimators and for estimators of the conditional mean and the
conditional variance. In contradistinction to the parametric...
4.
Nonasymptotic bounds for autoregressive time series
modeling - Goldenshluger, Alexander; Zeevi, Assaf
The subject of this paper is autoregressive (AR) modeling of a
stationary, Gaussian discrete time process, based on a finite sequence of
observations. The process is assumed to admit an AR($\infty$) representation
with exponentially decaying coefficients. We adopt the nonparametric minimax
framework and study how well the process can be approximated by a
finiteorder AR model. A lower bound on the accuracy of AR
approximations is derived, and a nonasymptotic upper bound on the accuracy of
the regularized least squares estimator is established. It is shown that with a
proper choice of the model order, this estimator is minimax
optimal in order. These considerations lead also to a nonasymptotic...
5.
Empirical process of the squared residuals of an arch
sequence - Horváth, Lajos; Teyssière, Gilles
We derive the asymptotic distribution of the sequential empirical
process of the squared residuals of an ARCH(p) sequence. Unlike the
residuals of an ARMA process, these residuals do not behave in this context
like asymptotically independent random variables, and the asymptotic
distribution involves a term depending on the parameters of the model. We show
that in certain applications, including the detection of changes in the
distribution of the unobservable innovations, our result leads to
asymptotically distribution free statistics.
6.
Selection criteria for scatterplot smoothers - Efron, Bradley
Scatterplot smoothers estimate a regression function y =
f(x) by local averaging of the observed data points
(xi, yi). In using a
smoother, the statistician must choose a window width, a
crucial smoothing parameter that says just how locally the averaging is done.
This paper concerns the databased choice of a smoothing parameter for
splinelike smoothers, focusing on the comparison of two popular methods,
Cp and generalized maximum likelihood. The latter is
the MLE within a normaltheory empirical Bayes model. We show that
Cp is also maximum likelihood within a closely related
nonnormal family, both methods being examples of a class of selection criteria.
Each member of the class is the...
7.
Stratified exponential families: Graphical models and model
selection - Geiger, Dan; Heckerman, David; King, Henry; Meek, Christopher
We describe a hierarchy of exponential families which is useful
for distinguishing types of graphical models. Undirected graphical models with
no hidden variables are linear exponential families (LEFs). Directed acyclic
graphical (DAG) models and chain graphs with no hidden variables,
includ ing DAG models with several families of local distributions, are
curved exponential families (CEFs). Graphical models with hidden variables are
what we term stratified exponential families (SEFs). A SEF is a finite union of
CEFs of various dimensions satisfying some regularity conditions. We also show
that this hierarchy of exponential families is noncollapsing with respect to
graphical models by providing a graphical model which is a CEF but...
8.
Blocked regular fractional factorial designs with maximum
estimation capacity - Cheng, Ching-Shui; Mukerjee, Rahul
In this paper, the problem of constructing optimal blocked regular
fractional factorial designs is considered. The concept of minimum aberration
due to Fries and Hunter is a wellaccepted criterion for selecting good
unblocked fractional factorial designs. Cheng, Steinberg and Sun showed that a
minimum aberration design of resolution three or higher maximizes the number of
twofactor interactions which are not aliases of main effects and also
tends to distribute these interactions over the alias sets very uniformly. We
extend this to construct block designs in which (i) no main effect is aliased
with any other main effect not confounded with blocks, (ii) the number of
twofactor interactions that are...
9.
Generalized minimum aberration for asymmetrical fractional
factorial designs - Xu, Hongquan; Wu, C.F.J.
By studying treatment contrasts and ANOVA models, we propose a
generalized minimum aberration criterion for comparing asymmetrical fractional
factorial designs. The criterion is independent of the choice of treatment
contrasts and thus modelfree. It works for symmetrical and asymmetrical
designs, regular and nonregular designs. In particular, it reduces to the
minimum aberration criterion for regular designs and the minimum
G2 aberration criterion for twolevel
nonregular designs. In addition, by exploring the connection between factorial
design theory and coding theory, we develop a complementary design theory for
general symmetrical designs, which covers many existing results as special
cases.
10.
Maximin designs for exponential growth models and
heteroscedastic polynomial models - Imhof, Lorens A.
This paper is concerned with nonsequential optimal designs for a
class of nonlinear growth models, which includes the asymptotic regression
model. This design problem is intimately related to the problem of finding
optimal designs for polynomial regression models with only partially known
heteroscedastic structure. In each case, a straightforward application of the
usual Doptimality criterion would lead to designs which depend
on the unknown underlying parameters. To overcome this undesirable dependence a
maximin approach is adopted. The theorem of Perron and Frobenius on primitive
matrices plays a crucial role in the analysis.
11.
Optimality of partial geometric designs - Bagchi, Bhaskar; Bagchi, Sunanda
We find a sufficient condition on the spectrum of a partial
geometric design d* such that, when d* satisfies this condition,
it is better (with respect to all convex decreasing optimality criteria) than
all unequally replicated designs (binary or not) with the same parameters
b, v, k as d*.
¶ Combining this with existing results, we obtain the following
results:
¶ (i) For any q \ge 3, a linked block design with parameters
b = q2, v = q2 + q,
k = q2 -1 is optimal with respect to all convex
decreasing optimality criteria in the unrestricted class of all connected
designs with the same parameters.
¶ (ii) A large class...
12.
Direct estimation of the index coefficient in a single-index
model - Hristache, Marian; Juditsky, Anatoli; Spokoiny, Vladimir
Single-index modeling is widely applied in,for example,econometric
studies as a compromise between too restrictive parametric models and flexible
but hardly estimable purely nonparametric models. By such modeling the
statistical analysis usually focuses on estimating the index coefficients. The
average derivative estimator (ADE) of the index vector is based on the fact
that the average gradient of a single index function $f(x^{\top}\beta)$ is
proportional to the index vector $\beta$. Unfortunately,a straightforward
application of this idea meets the so-called curse of
dimensionality problem if the dimensionality $d$ of the model is larger
than 2. However, prior information about the vector $\beta$ can be used for
improving the quality of gradient estimation by...
13.
Nonparametric kernel regression subject to monotonicity
constraints - Hall, Peter; Huang, Li-Shan
We suggest a method for monotonizing general kernel-type
estimators, for example local linear estimators and Nadaraya .Watson
estimators. Attributes of our approach include the fact that it produces smooth
estimates, indeed with the same smoothness as the unconstrained estimate. The
method is applicable to a particularly wide range of estimator types, it can be
trivially modified to render an estimator strictly monotone and it can be
employed after the smoothing step has been implemented. Therefore,an
experimenter may use his or her favorite kernel estimator, and their favorite
bandwidth selector, to construct the basic nonparametric smoother and then use
our technique to render it monotone in a smooth way. Implementation...
14.
Least squares estimators of the mode of a unimodal regression
function - Shoung, Jyh-Ming; Zhang, Cun-Hui
In this paper, we consider nonparametric least squares estimators
of the mode of an unknown unimodal regression function. We establish almost
sure convergence of these estimators with nearly optimal convergence rates,
under the assumption of the exponential tail for the error distributions.
15.
On posterior consistency of survival models - Kim, Yongdai; Lee, Jaeyong
Ghosh and Ramamoorthi studied posterior consistency for survival
models and showed that the posterior was consistent when the prior on the
distribution of survival times was the Dirichlet process prior. In this
paper,we study posterior consistency of survival models with neutral to the
right process priors which include Dirichlet process priors. A set of
sufficient conditions for posterior consistency with neutral to the right
process priors are given. Interestingly, not all the neutral to the right
process priors have consistent posteriors, but most of the popular priors such
as Dirichlet processes, beta processes and gamma processes have consistent
posteriors. With a class of priors which includes beta processes, a...
16.
Rates of convergence of posterior distributions - Shen, Xiaotong; Wasserman, Larry
We compute the rate at which the posterior distribution
concentrates around the true parameter value. The spaces we work in are quite
general and include in finite dimensional cases. The rates are driven by two
quantities: the size of the space, as measured by bracketing entropy, and the
degree to which the prior concentrates in a small ball around the true
parameter. We consider two examples.
17.
Prior induction in log-linear models for general contingency
table analysis. - King, R.; Brooks, S.P.
Log-linear modelling plays an important role in many statistical
applications, particularly in the analysis of contingency table data.With the
advent of powerful new computational techniques such as reversible jump MCMC,
Bayesian analyses of these models, and in particular model selection and
averaging, have become feasible. Coupled with this is the desire to construct
and use suitably flexible prior structures which allow efficient computation
while facilitating prior elicitation. The latter is greatly improved in the
case where priors can be specified on interpretable parameters about which
relevant experts can express their beliefs.
¶ In this paper, we show how the specification of a general
multivariate normal prior on the log-linear parameters...
18.
Weak convergence of the empirical process of residuals in linear
models with many parameters - Chen, Gemai and; Lockhart, Richard A.
When fitting, by least squares, a linear model (with an intercept
term) with $p$ parameters to $n$ data points, the asymptotic behavior of the
residual empirical process is shown to be the same as in the single sample
problem provided $p^3 \log^2 (p) /n \to 0$ for any error density having finite
variance and a bounded first derivative. No further conditions are imposed on
the sequence of design matrices. The result is extended to more general
estimates with the property that the average error and average squared error in
the fitted values are on the same order as for least squares.
19.
E-optimal designs for rational models - Imhof, Lorens; Studden, William J.
E-optimal and standardized-E-optimal designs for
various types of rational regression models are determined. In most cases,
optimal designs are found for every parameter subsystem. The design points and
weights are given explicitlyin terms of Bernstein-Szeg? polynomials.The
analysis is based on a general theorem on E-optimal designs for
Chebyshev systems.
20.
M-estimation for location and regression parameters in
group models: A case study using Stiefel manifolds - Chang, Ted; Rivest, Louis-Paul
We discuss here a general approach to the calculation of the
asymptotic covariance of $M$-estimates for location parameters in statistical
group models when an invariant objective function is used. The calculation
reduces to standard tools in group representation theory and the calculation of
some constants. Only the constants depend upon the precise forms of the density
or of the objective function. If the group is sufficiently large this
represents a major simplification in the computation of the asymptotic
covariance.
¶ Following the approach of Chang and Tsai we define a regression
model for group models and derive the asymptotic distribution of estimates in
the regression model from the corresponding distribution...
1.
