6,908 research outputs found
Multivariate goodness-of-fit tests based on kernel density estimators
The paper is devoted to multivariate goodness-of-fit ests based on kernel density estimators. Both simple and composite null hypotheses are investigated. The test statistic is considered in the form of maximum of the normalized deviation of the estimate from its expected value. The produced comparative Monte Carlo power study shows that the proposed test is a powerful competitor to the existing classical criteria for testing goodness of fit against a specific type of an alternative hypothesis. An analytical way to establish the asymptotic distribution of the test statistic is discussed, using the approximation results for the probabilities of high excursions of differentiable Gaussian random fields
New Entropy Estimator with an Application to Test of Normality
In the present paper we propose a new estimator of entropy based on smooth
estimators of quantile density. The consistency and asymptotic distribution of
the proposed estimates are obtained. As a consequence, a new test of normality
is proposed. A small power comparison is provided. A simulation study for the
comparison, in terms of mean squared error, of all estimators under study is
performed
Improved kernel estimation of copulas: Weak convergence and goodness-of-fit testing
We reconsider the existing kernel estimators for a copula function, as
proposed in Gijbels and Mielniczuk [Comm. Statist. Theory Methods 19 (1990)
445--464], Fermanian, Radulovi\v{c} and Wegkamp [Bernoulli 10 (2004) 847--860]
and Chen and Huang [Canad. J. Statist. 35 (2007) 265--282]. All of these
estimators have as a drawback that they can suffer from a corner bias problem.
A way to deal with this is to impose rather stringent conditions on the copula,
outruling as such many classical families of copulas. In this paper, we propose
improved estimators that take care of the typical corner bias problem. For
Gijbels and Mielniczuk [Comm. Statist. Theory Methods 19 (1990) 445--464] and
Chen and Huang [Canad. J. Statist. 35 (2007) 265--282], the improvement
involves shrinking the bandwidth with an appropriate functional factor; for
Fermanian, Radulovi\v{c} and Wegkamp [Bernoulli 10 (2004) 847--860], this is
done by using a transformation. The theoretical contribution of the paper is a
weak convergence result for the three improved estimators under conditions that
are met for most copula families. We also discuss the choice of bandwidth
parameters, theoretically and practically, and illustrate the finite-sample
behaviour of the estimators in a simulation study. The improved estimators are
applied to goodness-of-fit testing for copulas.Comment: Published in at http://dx.doi.org/10.1214/08-AOS666 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Parametric versus nonparametric: the fitness coefficient
The fitness coefficient, introduced in this paper, results from a competition
between parametric and nonparametric density estimators within the likelihood
of the data. As illustrated on several real datasets, the fitness coefficient
generally agrees with p-values but is easier to compute and interpret. Namely,
the fitness coefficient can be interpreted as the proportion of data coming
from the parametric model. Moreover, the fitness coefficient can be used to
build a semiparamteric compromise which improves inference over the parametric
and nonparametric approaches. From a theoretical perspective, the fitness
coefficient is shown to converge in probability to one if the model is true and
to zero if the model is false. From a practical perspective, the utility of the
fitness coefficient is illustrated on real and simulated datasets
Goodness-of-fit testing and quadratic functional estimation from indirect observations
We consider the convolution model where i.i.d. random variables having
unknown density are observed with additive i.i.d. noise, independent of the
's. We assume that the density belongs to either a Sobolev class or a
class of supersmooth functions. The noise distribution is known and its
characteristic function decays either polynomially or exponentially
asymptotically. We consider the problem of goodness-of-fit testing in the
convolution model. We prove upper bounds for the risk of a test statistic
derived from a kernel estimator of the quadratic functional based on
indirect observations. When the unknown density is smoother enough than the
noise density, we prove that this estimator is consistent,
asymptotically normal and efficient (for the variance we compute). Otherwise,
we give nonparametric upper bounds for the risk of the same estimator. We give
an approach unifying the proof of nonparametric minimax lower bounds for both
problems. We establish them for Sobolev densities and for supersmooth densities
less smooth than exponential noise. In the two setups we obtain exact testing
constants associated with the asymptotic minimax rates.Comment: Published in at http://dx.doi.org/10.1214/009053607000000118 the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Recent advances in directional statistics
Mainstream statistical methodology is generally applicable to data observed
in Euclidean space. There are, however, numerous contexts of considerable
scientific interest in which the natural supports for the data under
consideration are Riemannian manifolds like the unit circle, torus, sphere and
their extensions. Typically, such data can be represented using one or more
directions, and directional statistics is the branch of statistics that deals
with their analysis. In this paper we provide a review of the many recent
developments in the field since the publication of Mardia and Jupp (1999),
still the most comprehensive text on directional statistics. Many of those
developments have been stimulated by interesting applications in fields as
diverse as astronomy, medicine, genetics, neurology, aeronautics, acoustics,
image analysis, text mining, environmetrics, and machine learning. We begin by
considering developments for the exploratory analysis of directional data
before progressing to distributional models, general approaches to inference,
hypothesis testing, regression, nonparametric curve estimation, methods for
dimension reduction, classification and clustering, and the modelling of time
series, spatial and spatio-temporal data. An overview of currently available
software for analysing directional data is also provided, and potential future
developments discussed.Comment: 61 page
Goodness-of-Fit Tests for Symmetric Stable Distributions -- Empirical Characteristic Function Approach
We consider goodness-of-fit tests of symmetric stable distributions based on
weighted integrals of the squared distance between the empirical characteristic
function of the standardized data and the characteristic function of the
standard symmetric stable distribution with the characteristic exponent
estimated from the data. We treat as an unknown parameter,
but for theoretical simplicity we also consider the case that is
fixed. For estimation of parameters and the standardization of data we use
maximum likelihood estimator (MLE) and an equivariant integrated squared error
estimator (EISE) which minimizes the weighted integral. We derive the
asymptotic covariance function of the characteristic function process with
parameters estimated by MLE and EISE. For the case of MLE, the eigenvalues of
the covariance function are numerically evaluated and asymptotic distribution
of the test statistic is obtained using complex integration. Simulation studies
show that the asymptotic distribution of the test statistics is very accurate.
We also present a formula of the asymptotic covariance function of the
characteristic function process with parameters estimated by an efficient
estimator for general distributions
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