اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدOn an Auxiliary Function for Log-Density Estimation
210
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
In this note we provide explicit expressions and expansions for a special function which appears in nonparametric estimation of log-densities. This function returns the integral of a log-linear function on a simplex of arbitrary dimension. In particular it is used in the R-package LogCondDEAD by Cule et al. (2007).
قيم البحث
اقرأ أيضاً
Mixture models are regularly used in density estimation applications, but the problem of estimating the mixing distribution remains a challenge. Nonparametric maximum likelihood produce estimates of the mixing distribution that are discrete, and thes
e may be hard to interpret when the true mixing distribution is believed to have a smooth density. In this paper, we investigate an algorithm that produces a sequence of smooth estimates that has been conjectured to converge to the nonparametric maximum likelihood estimator. Here we give a rigorous proof of this conjecture, and propose a new data-driven stopping rule that produces smooth near-maximum likelihood estimates of the mixing density, and simulations demonstrate the quality empirical performance of this estimator.
We consider the problem of approximating the empirical Shannon entropy of a high-frequency data stream under the relaxed strict-turnstile model, when space limitations make exact computation infeasible. An equivalent measure of entropy is the Renyi e
ntropy that depends on a constant alpha. This quantity can be estimated efficiently and unbiasedly from a low-dimensional synopsis called an alpha-stable data sketch via the method of compressed counting. An approximation to the Shannon entropy can be obtained from the Renyi entropy by taking alpha sufficiently close to 1. However, practical guidelines for parameter calibration with respect to alpha are lacking. We avoid this problem by showing that the random variables used in estimating the Renyi entropy can be transformed to have a proper distributional limit as alpha approaches 1: the maximally skewed, strictly stable distribution with alpha = 1 defined on the entire real line. We propose a family of asymptotically unbiased log-mean estimators of the Shannon entropy, indexed by a constant zeta > 0, that can be computed in a single-pass algorithm to provide an additive approximation. We recommend the log-mean estimator with zeta = 1 that has exponentially decreasing tail bounds on the error probability, asymptotic relative efficiency of 0.932, and near-optimal computational complexity.
In Statistics, log-concave density estimation is a central problem within the field of nonparametric inference under shape constraints. Despite great progress in recent years on the statistical theory of the canonical estimator, namely the log-concav
e maximum likelihood estimator, adoption of this method has been hampered by the complexities of the non-smooth convex optimization problem that underpins its computation. We provide enhanced understanding of the structural properties of this optimization problem, which motivates the proposal of new algorithms, based on both randomized and Nesterov smoothing, combined with an appropriate integral discretization of increasing accuracy. We prove that these methods enjoy, both with high probability and in expectation, a convergence rate of order $1/T$ up to logarithmic factors on the objective function scale, where $T$ denotes the number of iterations. The benefits of our new computational framework are demonstrated on both synthetic and real data, and our implementation is available in a github repository texttt{LogConcComp} (Log-Concave Computation).
Principal component analysis (PCA) is fundamental to statistical machine learning. It extracts latent principal factors that contribute to the most variation of the data. When data are stored across multiple machines, however, communication cost can
prohibit the computation of PCA in a central location and distributed algorithms for PCA are thus needed. This paper proposes and studies a distributed PCA algorithm: each node machine computes the top $K$ eigenvectors and transmits them to the central server; the central server then aggregates the information from all the node machines and conducts a PCA based on the aggregated information. We investigate the bias and variance for the resulting distributed estimator of the top $K$ eigenvectors. In particular, we show that for distributions with symmetric innovation, the empirical top eigenspaces are unbiased and hence the distributed PCA is unbiased. We derive the rate of convergence for distributed PCA estimators, which depends explicitly on the effective rank of covariance, eigen-gap, and the number of machines. We show that when the number of machines is not unreasonably large, the distributed PCA performs as well as the whole sample PCA, even without full access of whole data. The theoretical results are verified by an extensive simulation study. We also extend our analysis to the heterogeneous case where the population covariance matrices are different across local machines but share similar top eigen-structures.
Approximate Bayesian Computation (ABC) has become one of the major tools of likelihood-free statistical inference in complex mathematical models. Simultaneously, stochastic differential equations (SDEs) have developed to an established tool for model
ling time dependent, real world phenomena with underlying random effects. When applying ABC to stochastic models, two major difficulties arise. First, the derivation of effective summary statistics and proper distances is particularly challenging, since simulations from the stochastic process under the same parameter configuration result in different trajectories. Second, exact simulation schemes to generate trajectories from the stochastic model are rarely available, requiring the derivation of suitable numerical methods for the synthetic data generation. To obtain summaries that are less sensitive to the intrinsic stochasticity of the model, we propose to build up the statistical method (e.g., the choice of the summary statistics) on the underlying structural properties of the model. Here, we focus on the existence of an invariant measure and we map the data to their estimated invariant density and invariant spectral density. Then, to ensure that these model properties are kept in the synthetic data generation, we adopt measure-preserving numerical splitting schemes. The derived property-based and measure-preserving ABC method is illustrated on the broad class of partially observed Hamiltonian type SDEs, both with simulated data and with real electroencephalography (EEG) data. The proposed ingredients can be incorporated into any type of ABC algorithm and directly applied to all SDEs that are characterised by an invariant distribution and for which a measure-preserving numerical method can be derived.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
