491 research outputs found
Rearranging Edgeworth-Cornish-Fisher Expansions
This paper applies a regularization procedure called increasing rearrangement
to monotonize Edgeworth and Cornish-Fisher expansions and any other related
approximations of distribution and quantile functions of sample statistics.
Besides satisfying the logical monotonicity, required of distribution and
quantile functions, the procedure often delivers strikingly better
approximations to the distribution and quantile functions of the sample mean
than the original Edgeworth-Cornish-Fisher expansions.Comment: 17 pages, 3 figure
Objective Bayes and Conditional Frequentist Inference
Objective Bayesian methods have garnered considerable interest and support among statisticians,
particularly over the past two decades. It has often been ignored, however, that in
some cases the appropriate frequentist inference to match is a conditional one. We present
various methods for extending the probability matching prior (PMP) methods to conditional
settings. A method based on saddlepoint approximations is found to be the most
tractable and we demonstrate its use in the most common exact ancillary statistic models.
As part of this analysis, we give a proof of an exactness property of a particular PMP in
location-scale models. We use the proposed matching methods to investigate the relationships
between conditional and unconditional PMPs. A key component of our analysis is a
numerical study of the performance of probability matching priors from both a conditional
and unconditional perspective in exact ancillary models. In concluding remarks we propose
many routes for future research
On the Effect of Bias Estimation on Coverage Accuracy in Nonparametric Inference
Nonparametric methods play a central role in modern empirical work. While
they provide inference procedures that are more robust to parametric
misspecification bias, they may be quite sensitive to tuning parameter choices.
We study the effects of bias correction on confidence interval coverage in the
context of kernel density and local polynomial regression estimation, and prove
that bias correction can be preferred to undersmoothing for minimizing coverage
error and increasing robustness to tuning parameter choice. This is achieved
using a novel, yet simple, Studentization, which leads to a new way of
constructing kernel-based bias-corrected confidence intervals. In addition, for
practical cases, we derive coverage error optimal bandwidths and discuss
easy-to-implement bandwidth selectors. For interior points, we show that the
MSE-optimal bandwidth for the original point estimator (before bias correction)
delivers the fastest coverage error decay rate after bias correction when
second-order (equivalent) kernels are employed, but is otherwise suboptimal
because it is too "large". Finally, for odd-degree local polynomial regression,
we show that, as with point estimation, coverage error adapts to boundary
points automatically when appropriate Studentization is used; however, the
MSE-optimal bandwidth for the original point estimator is suboptimal. All the
results are established using valid Edgeworth expansions and illustrated with
simulated data. Our findings have important consequences for empirical work as
they indicate that bias-corrected confidence intervals, coupled with
appropriate standard errors, have smaller coverage error and are less sensitive
to tuning parameter choices in practically relevant cases where additional
smoothness is available
Higher-Order Improvements of the Sieve Bootstrap for Fractionally Integrated Processes
This paper investigates the accuracy of bootstrap-based inference in the case
of long memory fractionally integrated processes. The re-sampling method is
based on the semi-parametric sieve approach, whereby the dynamics in the
process used to produce the bootstrap draws are captured by an autoregressive
approximation. Application of the sieve method to data pre-filtered by a
semi-parametric estimate of the long memory parameter is also explored.
Higher-order improvements yielded by both forms of re-sampling are demonstrated
using Edgeworth expansions for a broad class of statistics that includes first-
and second-order moments, the discrete Fourier transform and regression
coefficients. The methods are then applied to the problem of estimating the
sampling distributions of the sample mean and of selected sample
autocorrelation coefficients, in experimental settings. In the case of the
sample mean, the pre-filtered version of the bootstrap is shown to avoid the
distinct underestimation of the sampling variance of the mean which the raw
sieve method demonstrates in finite samples, higher order accuracy of the
latter notwithstanding. Pre-filtering also produces gains in terms of the
accuracy with which the sampling distributions of the sample autocorrelations
are reproduced, most notably in the part of the parameter space in which
asymptotic normality does not obtain. Most importantly, the sieve bootstrap is
shown to reproduce the (empirically infeasible) Edgeworth expansion of the
sampling distribution of the autocorrelation coefficients, in the part of the
parameter space in which the expansion is valid
Number Counts and Non-Gaussianity
We describe a general procedure for using number counts of any object to
constrain the probability distribution of the primordial fluctuations, allowing
for generic weak non-Gaussianity. We apply this procedure to use limits on the
abundance of primordial black holes and dark matter ultracompact minihalos
(UCMHs) to characterize the allowed statistics of primordial fluctuations on
very small scales. We present constraints on the power spectrum and the
amplitude of the skewness for two different families of non-Gaussian
distributions, distinguished by the relative importance of higher moments.
Although primordial black holes probe the smallest scales, ultracompact
minihalos provide significantly stronger constraints on the power spectrum and
so are more likely to eventually provide small-scale constraints on
non-Gaussianity.Comment: 19 pages; v2 is published PRD versio
The bootstrap -A review
The bootstrap, extensively studied during the last decade, has become a powerful tool in different areas of Statistical Inference. In this work, we present the main ideas of bootstrap methodology in several contexts, citing the most relevant contributions and illustrating with examples and simulation studies some interesting aspects
Adjusted empirical likelihood with high-order precision
Empirical likelihood is a popular nonparametric or semi-parametric
statistical method with many nice statistical properties. Yet when the sample
size is small, or the dimension of the accompanying estimating function is
high, the application of the empirical likelihood method can be hindered by low
precision of the chi-square approximation and by nonexistence of solutions to
the estimating equations. In this paper, we show that the adjusted empirical
likelihood is effective at addressing both problems. With a specific level of
adjustment, the adjusted empirical likelihood achieves the high-order precision
of the Bartlett correction, in addition to the advantage of a guaranteed
solution to the estimating equations. Simulation results indicate that the
confidence regions constructed by the adjusted empirical likelihood have
coverage probabilities comparable to or substantially more accurate than the
original empirical likelihood enhanced by the Bartlett correction.Comment: Published in at http://dx.doi.org/10.1214/09-AOS750 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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