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分布函数检验与置信带的新方法AI 解读

We introduce new goodness-of-fit tests and corresponding confidence bands for distribution functions. They are inspired by multiscale methods of testing and based on refined laws of the iterated logarithm for the normalized uniform empirical process Un(t)/ t(1−t) and its natural limiting process, the normalized Brownian bridge process U(t)/ t(1−t). The new tests and confidence bands refine the procedures of Berk and Jones (1979) and Owen (1995). Roughly speaking, the high…

复杂抽样设计：一致极限定理及其应用AI 解读

In this paper, we develop a general approach to proving global and local uniform limit theorems for the Horvitz–Thompson empirical process arising from complex sampling designs. Global theorems such as Glivenko–Cantelli and Donsker theorems, and local theorems such as local asymptotic modulus and related ratio-type limit theorems are proved for both the Horvitz–Thompson empirical process, and its calibrated version. Limit theorems of other variants and their conditional…

对数凹密度众数的统计推断AI 解读

We study a likelihood ratio test for the location of the mode of a log-concave density. Our test is based on comparison of the log-likelihoods corresponding to the unconstrained maximum likelihood estimator of a log-concave density and the constrained maximum likelihood estimator where the constraint is that the mode of the density is fixed, say at $m$. The constrained estimation problem is studied in detail in Doss and Wellner (2018). Here, the results of that paper are…

重尾误差下最小二乘回归估计量的收敛速度AI 解读

We study the performance of the least squares estimator (LSE) in a general nonparametric regression model, when the errors are independent of the covariates but may only have a $p$th moment ($p\geq1$). In such a heavy-tailed regression setting, we show that if the model satisfies a standard “entropy condition” with exponent $\alpha\in(0,2)$, then the $L_{2}$ loss of the LSE converges at a rate \[\mathcal{O}_{\mathbf{P}}\bigl(n^{-\frac{1}{2+\alpha}}\vee n^{-\frac{1}{2}+\frac{1}{2p}}\bigr).\]…

对数凹与 s 凹密度极大似然估计的全局收敛速率AI 解读

We establish global rates of convergence for the Maximum Likelihood Estimators (MLEs) of log-concave and $s$-concave densities on $\mathbb{R}$. The main finding is that the rate of convergence of the MLE in the Hellinger metric is no worse than $n^{-2/5}$ when $-1<s<\infty$ where $s=0$ corresponds to the log-concave case. We also show that the MLE does not exist for the classes of $s$-concave densities with $s<-1$.

基于Rényi散度的s凹密度逼近与估计AI 解读

In this paper, we study the approximation and estimation of $s$-concave densities via Rényi divergence. We first show that the approximation of a probability measure $Q$ by an $s$-concave density exists and is unique via the procedure of minimizing a divergence functional proposed by [Ann. Statist. 38 (2010) 2998–3027] if and only if $Q$ admits full-dimensional support and a first moment. We also show continuity of the divergence functional in $Q$: if $Q_{n}\to Q$ in the…

两阶段抽样下的加权似然估计AI 解读

We develop asymptotic theory for weighted likelihood estimators (WLE) under two-phase stratified sampling without replacement. We also consider several variants of WLEs involving estimated weights and calibration. A set of empirical process tools are developed including a Glivenko–Cantelli theorem, a theorem for rates of convergence of $M$-estimators, and a Donsker theorem for the inverse probability weighted empirical processes under two-phase sampling and sampling without…

多元凸变换密度的非参数估计AI 解读

We study estimation of multivariate densities p of the form p(x) = h(g(x)) for x ∈ ℝ^{d} and for a fixed monotone function h and an unknown convex function g. The canonical example is h(y) = e^{−y} for y ∈ ℝ; in this case, the resulting class of densities $P ( {e}^{− y} ) = \{ p = exp ( − g ) : g \kern0.2em \mathrm{is convex} \}$ is well known as the class of log-concave densities. Other functions h allow for classes of densities with heavier tails than the log-concave…

对数凹密度最大似然估计的极限分布理论AI 解读

We find limiting distributions of the nonparametric maximum likelihood estimator (MLE) of a log-concave density, i.e. a density of the form f(0) = exp varphi(0) where varphi(0) is a concave function on R. Existence, form, characterizations and uniform rates of convergence of the MLE are given by Rufibach (2006) and Dümbgen and Rufibach (2007). The characterization of the log-concave MLE in terms of distribution functions is the same (up to sign) as the characterization of…

竞争风险下现状数据的非参数极大似然估计：极限分布AI 解读

We study nonparametric estimation for current status data with competing risks. Our main interest is in the nonparametric maximum likelihood estimator (MLE), and for comparison we also consider a simpler 'naive estimator'. Groeneboom, Maathuis and Wellner [8] proved that both types of estimators converge globally and locally at rate n(1/3). We use these results to derive the local limiting distributions of the estimators. The limiting distribution of the naive estimator is…

竞争风险当前状态数据：极大似然估计的一致性与收敛率AI 解读

We study nonparametric estimation of the sub-distribution functions for current status data with competing risks. Our main interest is in the nonparametric maximum likelihood estimator (MLE), and for comparison we also consider a simpler “naive estimator.” Both types of estimators were studied by Jewell, van der Laan and Henneman [Biometrika (2003) 90 183–197], but little was known about their large sample properties. We have started to fill this gap, by proving that the…

k 单调密度估计：极限分布理论与样条联系AI 解读

We study the asymptotic behavior of the Maximum Likelihood and Least Squares Estimators of a k-monotone density g0 at a fixed point x0 when k>2. We find that the jth derivative of the estimators at x0 converges at the rate n−(k−j)/(2k+1) for j=0, …, k−1. The limiting distribution depends on an almost surely uniquely defined stochastic process Hk that stays above (below) the k-fold integral of Brownian motion plus a deterministic drift when k is even (odd). Both the MLE and…

含协变量面板计数数据的两种基于似然的半参数估计方法AI 解读

We consider estimation in a particular semiparametric regression model for the mean of a counting process with “panel count” data. The basic model assumption is that the conditional mean function of the counting process is of the form E{ℕ(t)|Z}=exp(β0TZ)Λ0(t) where Z is a vector of covariates and Λ0 is the baseline mean function. The “panel count” observation scheme involves observation of the counting process ℕ for an individual at a random number K of random time points;…

基于 φ 散度的拟合优度检验AI 解读

A unified family of goodness-of-fit tests based on φ-divergences is introduced and studied. The new family of test statistics Sn(s) includes both the supremum version of the Anderson–Darling statistic and the test statistic of Berk and Jones [Z. Wahrsch. Verw. Gebiete 47 (1979) 47–59] as special cases (s=2 and s=1, resp.). We also introduce integral versions of the new statistics. We show that the asymptotic null distribution theory of Berk and Jones [Z. Wahrsch. Verw.…

缺失数据下 Cox 回归模型的信息界AI 解读

We derive information bounds for the regression parameters in Cox models when data are missing at random. These calculations are of interest for understanding the behavior of efficient estimation in case-cohort designs, a type of two-phase design often used in cohort studies. The derivations make use of key lemmas appearing in Robins, Rotnitzky and Zhao [J. Amer. Statist. Assoc. 89 (1994) 846–866] and Robins, Hsieh and Newey [J. Roy. Statist. Soc. Ser. B 57 (1995) 409–424],…

结果依赖两阶段抽样下半参数回归的大样本理论AI 解读

Outcome-dependent, two-phase sampling designs can dramatically reduce the costs of observational studies by judicious selection of the most informative subjects for purposes of detailed covariate measurement. Here we derive asymptotic information bounds and the form of the efficient score and influence functions for the semiparametric regression models studied by Lawless, Kalbfleisch and Wild (1999) under two-phase sampling designs. We show that the maximum likelihood…

单调函数的似然比检验AI 解读

We study the problem of testing for equality at a fixed point in the setting of nonparametric estimation of a monotone function.The likelihood ratio test for this hypothesis is derived in the particular case of interval censoring (or current status data)and its limiting distribution is obtained. The limiting distribution is that of the integral of the difference of the squared slope processes corresponding to a canonical version of the problem involving Brownian motion $+t^2$…

凸函数估计：估计量刻画与渐近理论AI 解读

We study nonparametric estimation of convexregression and density functions by methods of least squares (in the regression and density cases) and maximum likelihood (in the density estimation case).We provide characterizations of these estimators, prove that they are consistent and establish their asymptotic distributions at a fixed point of positive curvature of the functions estimated. The asymptotic distribution theory relies on the existence of an “invelope function” for…

凸函数估计的典范过程：积分布朗运动加四次漂移的“内包络”AI 解读

A process associated with integrated Brownian motion is introduced that characterizes the limit behavior of nonparametric least squares and maximum likelihood estimators of convex functions and convex densities, respectively. We call this process “the invelope” and show that it is an almost surely uniquely defined function of integrated Brownian motion. Its role is comparable to the role of the greatest convex minorant of Brownian motion plus a parabolic drift in the problem…

面板计数数据下计数过程均值的两种估计量AI 解读

We study two estimators of the mean function of a countingprocess based on “panel count data.” The setting for “panel count data” is one in which $n$ independent subjects, each with a counting process with common mean function, are observed at several possibly different times duringa study. Following a model proposed by Schick and Yu, we allow the number of observation times, and the observation times themselves, to be random variables. Our goal is to estimate the mean…

独立非同分布随机变量样本中位数弱一致性的充要条件AI 解读

Necessary and sufficient conditions for the weak consistency of the sample median of independent, but not identically distributed random variables are given and discussed.

删失数据非参数最大似然估计的混合计算算法AI 解读

We present a hybrid algorithm for nonparametric maximum likelihood estimation from censored data when the log-likelihood is concave. The hybrid algorithm uses a composite algorithmic mapping combining the expectation-maximization (EM) algorithm and the (modified) iterative convex minorant (ICM) algorithm. Global convergence of the hybrid algorithm is proven; the iterates generated by the hybrid algorithm are shown to converge to the nonparametric maximum likelihood estimator…

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已知边际分布时概率测度线性泛函的有效估计AI 解读

Suppose that $P$ is the distribution of a pair of random variables $(X, Y)$ on a product space $\mathbb{X} \times \mathbb{Y}$ with known marginal distributions $P_X$ and $P_Y$. We study efficient estimation of functions $\theta(h) = \int h dP$ for fixed $h: \mathbb{X} \times \mathbb{Y} \rightarrow R$ under iid sampling of $(X, Y)$ pairs from $P$ and a regularity condition on $P$. Our proposed estimator is based on partitions of both $\mathbb{X}$ and $\mathbb{Y}$ and the…