1,118 research outputs found
Nonparametric estimates of low bias
We consider the problem of estimating an arbitrary smooth functional of distribution functions (d.f.s.) in terms of random samples from them.
The natural estimate replaces the d.f.s by their empirical d.f.s. Its bias is
generally , where is the minimum sample size, with a {\it
th order} iterative estimate of bias for any . For , we give an explicit estimate in terms of the first von Mises
derivatives of the functional evaluated at the empirical d.f.s. These may be
used to obtain {\it unbiased} estimates, where these exist and are of known
form in terms of the sample sizes; our form for such unbiased estimates is much
simpler than that obtained using polykays and tables of the symmetric
functions. Examples include functions of a mean vector (such as the ratio of
two means and the inverse of a mean), standard deviation, correlation, return
times and exceedances. These th order estimates require only
calculations. This is in sharp contrast with computationally intensive bias
reduction methods such as the th order bootstrap and jackknife, which
require calculations
Expansions about the gamma for the distribution and quantiles of a standard estimate
We give expansions for the distribution, density, and quantiles of an
estimate, building on results of Cornish, Fisher, Hill, Davis and the authors.
The estimate is assumed to be non-lattice with the standard expansions for its
cumulants. By expanding about a skew variable with matched skewness, one can
drastically reduce the number of terms needed for a given level of accuracy.
The building blocks generalize the Hermite polynomials. We demonstrate with
expansions about the gamma
The distribution of the maximum of a second order autoregressive process: the continuous case
We give the distribution function of , the maximum of a sequence of
observations from an autoregressive process of order 2. Solutions are first
given in terms of repeated integrals and then for the case, where the
underlying random variables are absolutely continuous. When the correlations
are positive, P(M_n \leq x) =a_{n,x}, where a_{n,x}= \sum_{j=1}^\infty
\beta_{jx} \nu_{jx}^{n} = O (\nu_{1x}^{n}), where are the
eigenvalues of a non-symmetric Fredholm kernel, and is the
eigenvalue of maximum magnitude. The weights depend on the th
left and right eigenfunctions of the kernel.
These results are large deviations expansions for estimates, since the
maximum need not be standardized to have a limit. In fact such a limit need not
exist.Comment: 8 pages This version removes an inappropriate not
The distribution of the maximum of an ARMA(1, 1) process
We give the cumulative distribution function of , the maximum of a
sequence of observations from an ARMA(1, 1) process. Solutions are first
given in terms of repeated integrals and then for the case, where the
underlying random variables are absolutely continuous. The distribution of
is then given as a weighted sum of the th powers of the eigenvalues of
a non-symmetric Fredholm kernel. The weights are given in terms of the left and
right eigenfunctions of the kernel.
These results are large deviations expansions for estimates, since the
maximum need not be standardized to have a limit. In fact, such a limit need
not exist.Comment: arXiv admin note: text overlap with arXiv:1001.526
Accurate inference for a one parameter distribution based on the mean of a transformed sample
A great deal of inference in statistics is based on making the approximation
that a statistic is normally distributed. The error in doing so is generally
and can be very considerable when the distribution is heavily
biased or skew. This note shows how one may reduce this error to
, where is a given integer. The case considered is when
the statistic is the mean of the sample values from a continuous one-parameter
distribution, after the sample has undergone an initial transformation
The chain rule for functionals with applications to functions of moments
The chain rule for derivatives of a function of a function is extended to a
function of a statistical functional, and applied to obtain approximations to
the cumulants, distribution and quantiles of functions of sample moments, and
so to obtain third order confidence intervals and estimates of reduced bias for
functions of moments. As an example we give the distribution of the
standardized skewness for a normal sample to magnitude , where
is the sample size
The distribution and quantiles of functionals of weighted empirical distributions when observations have different distributions
This paper extends Edgeworth-Cornish-Fisher expansions for the distribution
and quantiles of nonparametric estimates in two ways. Firstly it allows
observations to have different distributions. Secondly it allows the
observations to be weighted in a predetermined way. The use of weighted
estimates has a long history including applications to regression, rank
statistics and Bayes theory. However, asymptotic results have generally been
only first order (the CLT and weak convergence). We give third order
asymptotics for the distribution and percentiles of any smooth functional of a
weighted empirical distribution, thus allowing a considerable increase in
accuracy over earlier CLT results.
Consider independent non-identically distributed ({\it non-iid}) observations
in . Let be their {\it weighted
empirical distribution} with weights . We obtain cumulant
expansions and hence Edgeworth-Cornish-Fisher expansions for for
any smooth functional by extending the concepts of von Mises
derivatives to signed measures of total measure 1. As an example we give the
cumulant coefficients needed for Edgeworth-Cornish-Fisher expansions to
for the sample variance when observations are non-iid
Expansions for Quantiles and Multivariate Moments of Extremes for Distributions of Pareto Type
Let be the th largest of a random sample of size from a
distribution for
and . An inversion theorem is proved and used to derive
an expansion for the quantile and powers of it. From this an
expansion in powers of is given for the
multivariate moments of the extremes for fixed , where .
Examples include the Cauchy, Student , , second extreme distributions and
stable laws of index
Fredholm equations for non-symmetric kernels, with applications to iterated integral operators
We give the Jordan form and the Singular Value Decomposition for an integral
operator with a non-symmetric kernel . This is used to give
solutions of Fredholm equations for non-symmetric kernels, and to determine the
behaviour of and for large .Comment: 12 A4 page
The distribution of the maximum of a first order moving average: the continuous case
We give the distribution of , the maximum of a sequence of
observations from a moving average of order 1. Solutions are first given in
terms of repeated integrals and then for the case where the underlying
independent random variables have an absolutely continuous density. When the
correlation is positive, where
% is a moving average of order 1 with positive correlation, and
are the eigenvalues (singular values) of a Fredholm kernel and
is the eigenvalue of maximum magnitude. A similar result is given
when the correlation is negative. The result is analogous to large deviations
expansions for estimates, since the maximum need not be standardized to have a
limit. % there are more terms, and
For the continuous case the integral equations for the left and right
eigenfunctions are converted to first order linear differential equations. The
eigenvalues satisfy an equation of the form for certain known weights
and eigenvalues of a given matrix. This can be solved
by truncating the sum to an increasing number of terms.Comment: 15 A4 pages. Version 4 corrects (3.8). Version 3 expands Section 2.
Version 2 corrected recurrence relation (2.5
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