799 research outputs found
Ergodicity for the -type Markov Chain
Ergodicity is a fundamental issue for a stochastic process. In this paper, we
refine results on ergodicity for a general type of Markov chain to a specific
type or the -type Markov chain, which has many interesting and
important applications in various areas. It is of interest to obtain conditions
in terms of system parameters or the given information about the process, under
which the chain has various ergodic properties. Specifically, we provide
necessary and sufficient conditions for geometric, strong and polynomial
ergodicity, respectively.Comment: 16 page
Bayesian Estimation of Inequalities with Non-Rectangular Censored Survey Data
Synthetic indices are used in Economics to measure various aspects of
monetary inequalities. These scalar indices take as input the distribution over
a finite population, for example the population of a specific country. In this
article we consider the case of the French 2004 Wealth survey. We have at hand
a partial measurement on the distribution of interest consisting of bracketed
and sometimes missing data, over a subsample of the population of interest. We
present in this article the statistical methodology used to obtain point and
interval estimates taking into account the various uncertainties. The
inequality indices being nonlinear in the input distribution, we rely on a
simulation based approach where the model for the wealth per household is
multivariate. Using the survey data as well as matched auxiliary tax
declarations data, we have at hand a quite intricate non-rectangle
multidimensional censoring. For practical issues we use a Bayesian approach.
Inference using Monte-Carlo approximations relies on a Monte-Carlo Markov chain
algorithm namely the Gibbs sampler. The quantities interesting to the decision
maker are taken to be the various inequality indices for the French population.
Their distribution conditional on the data of the subsample are assumed to be
normal centered on the design-based estimates with variance computed through
linearization and taking into account the sample design and total nonresponse.
Exogeneous selection of the subsample, in particular the nonresponse mechanism,
is assumed and we condition on the adequate covariates
Cutoff for the cyclic adjacent transposition shuffle
We study the cyclic adjacent transposition (CAT) shuffle of cards, which
is a systematic scan version of the random adjacent transposition (AT) card
shuffle. In this paper, we prove that the CAT shuffle exhibits cutoff at
, which concludes that it is twice as fast as the
AT shuffle.Comment: 26 pages, 3 figure
Mixing time of the adjacent walk on the simplex
By viewing the -simplex as the set of positions of ordered particles
on the unit interval, the adjacent walk is the continuous time Markov chain
obtained by updating independently at rate 1 the position of each particle with
a sample from the uniform distribution over the interval given by the two
particles adjacent to it. We determine its spectral gap and prove that both the
total variation distance and the separation distance to the uniform
distribution exhibit a cutoff phenomenon, with mixing times that differ by a
factor . The results are extended to the family of log-concave distributions
obtained by replacing the uniform sampling by a symmetric log-concave Beta
distribution
Convergence rates of posterior distributions for noniid observations
We consider the asymptotic behavior of posterior distributions and Bayes
estimators based on observations which are required to be neither independent
nor identically distributed. We give general results on the rate of convergence
of the posterior measure relative to distances derived from a testing
criterion. We then specialize our results to independent, nonidentically
distributed observations, Markov processes, stationary Gaussian time series and
the white noise model. We apply our general results to several examples of
infinite-dimensional statistical models including nonparametric regression with
normal errors, binary regression, Poisson regression, an interval censoring
model, Whittle estimation of the spectral density of a time series and a
nonlinear autoregressive model.Comment: Published at http://dx.doi.org/10.1214/009053606000001172 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Componentwise accurate fluid queue computations using doubling algorithms
Markov-modulated fluid queues are popular stochastic processes frequently used for modelling real-life applications. An important performance measure to evaluate in these applications is their steady-state behaviour, which is determined by the stationary density. Computing it requires solving a (nonsymmetric) M-matrix algebraic Riccati equation, and indeed computing the stationary density is the most important application of this class of equations. Xue et al. (Numer Math 120:671–700, 2012) provided a componentwise first-order perturbation analysis of this equation, proving that the solution can be computed to high relative accuracy even in the smallest entries, and suggested several algorithms for computing it. An important step in all proposed algorithms is using so-called triplet representations, which are special representations for M-matrices that allow for a high-accuracy variant of Gaussian elimination, the GTH-like algorithm. However, triplet representations for all the M-matrices needed in the algorithm were not found explicitly. This can lead to an accuracy loss that prevents the algorithms from converging in the componentwise sense. In this paper, we focus on the structured doubling algorithm, the most efficient among the proposed methods in Xue et al., and build upon their results, providing (i) explicit and cancellation-free expressions for the needed triplet representations, allowing the algorithm to be performed in a really cancellation-free fashion; (ii) an algorithm to evaluate the final part of the computation to obtain the stationary density; and (iii) a componentwise error analysis for the resulting algorithm, the first explicit one for this class of algorithms. We also present numerical results to illustrate the accuracy advantage of our method over standard (normwise-accurate) algorithms. © 2014, Springer-Verlag Berlin Heidelberg
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