اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدA law of the iterated logarithm for Grenanders estimator
138
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
In this note we prove the following law of the iterated logarithm for the Grenander estimator of a monotone decreasing density: If $f(t_0) > 0$, $f(t_0) < 0$, and $f$ is continuous in a neighborhood of $t_0$, then begin{eqnarray*} limsup_{nrightarrow infty} left ( frac{n}{2log log n} right )^{1/3} ( widehat{f}_n (t_0 ) - f(t_0) ) = left| f(t_0) f(t_0)/2 right|^{1/3} 2M end{eqnarray*} almost surely where $ M equiv sup_{g in {cal G}} T_g = (3/4)^{1/3}$ and $ T_g equiv mbox{argmax}_u { g(u) - u^2 } $; here ${cal G}$ is the two-sided Strassen limit set on $R$. The proof relies on laws of the iterated logarithm for local empirical processes, Groenebooms switching relation, and properties of Strassens limit set analogous to distributional properties of Brownian motion.
قيم البحث
اقرأ أيضاً
We present a general law of the iterated logarithm for stochastic processes on the open unit interval having subexponential tails in a locally uniform fashion. It applies to standard Brownian bridge but also to suitably standardized empirical distrib
ution functions. This leads to new goodness-of-fit tests and confidence bands which refine the procedures of Berk and Jones (1979) and Owen (1995). Roughly speaking, the high power and accuracy of the latter procedures in the tail regions of distributions are essentially preserved while gaining considerably in the central region.
We study the law of the iterated logarithm (LIL) for the maximum likelihood estimation of the parameters (as a convex optimization problem) in the generalized linear models with independent or weakly dependent ($rho$-mixing, $m$-dependent) responses
under mild conditions. The LIL is useful to derive the asymptotic bounds for the discrepancy between the empirical process of the log-likelihood function and the true log-likelihood. As the application of the LIL, the strong consistency of some penalized likelihood based model selection criteria can be shown. Under some regularity conditions, the model selection criterion will be helpful to select the simplest correct model almost surely when the penalty term increases with model dimension and the penalty term has an order higher than $O({rm{loglog}}n)$ but lower than $O(n)$. Simulation studies are implemented to verify the selection consistency of BIC.
We propose a censored quantile regression estimator motivated by unbiased estimating equations. Under the usual conditional independence assumption of the survival time and the censoring time given the covariates, we show that the proposed estimator
is consistent and asymptotically normal. We develop an efficient computational algorithm which uses existing quantile regression code. As a result, bootstrap-type inference can be efficiently implemented. We illustrate the finite-sample performance of the proposed method by simulation studies and analysis of a survival data set.
We prove the consistency of the Power-Law Fit PLFit method proposed by Clauset et al.(2009) to estimate the power-law exponent in data coming from a distribution function with regularly-varying tail. In the complex systems community, PLFit has emerge
d as the method of choice to estimate the power-law exponent. Yet, its mathematical properties are still poorly understood. The difficulty in PLFit is that it is a minimum-distance estimator. It first chooses a threshold that minimizes the Kolmogorov-Smirnov distance between the data points larger than the threshold and the Pareto tail, and then applies the Hill estimator to this restricted data. Since the number of order statistics used is random, the general theory of consistency of power-law exponents from extreme value theory does not apply. Our proof consists in first showing that the Hill estimator is consistent for general intermediate sequences for the number of order statistics used, even when that number is random. Here, we call a sequence intermediate when it grows to infinity, while remaining much smaller than the sample size. The second, and most involved, step is to prove that the optimizer in PLFit is with high probability an intermediate sequence, unless the distribution has a Pareto tail above a certain value. For the latter special case, we give a separate proof.
We present theoretical properties of the log-concave maximum likelihood estimator of a density based on an independent and identically distributed sample in $mathbb{R}^d$. Our study covers both the case where the true underlying density is log-concav
e, and where this model is misspecified. We begin by showing that for a sequence of log-concave densities, convergence in distribution implies much stronger types of convergence -- in particular, it implies convergence in Hellinger distance and even in certain exponentially weighted total variation norms. In our main result, we prove the existence and uniqueness of a log-concave density that minimises the Kullback--Leibler divergence from the true density over the class all log-concave densities, and also show that the log-concave maximum likelihood estimator converges almost surely in these exponentially weighted total variation norms to this minimiser. In the case of a correctly specified model, this demonstrates a strong type of consistency for the estimator; in a misspecified model, it shows that the estimator converges to the log-concave density that is closest in the Kullback--Leibler sense to the true density.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
