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二元结果概率预测的诚实校准评估AI 解读

Summary Probability predictions from binary regressions or machine learning methods ought to be calibrated: if an event is predicted to occur with probability $x$, it should materialize with approximately that frequency, which means that the so-called calibration curve $p(\cdot)$ should equal the identity, i.e., $p(x) = x$ for all $x$ in the unit interval. We propose honest calibration assessment based on novel confidence bands for the calibration curve, which are valid…

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…

反卷积中的形状约束多尺度方法：定性特征的置信陈述AI 解读

We derive multiscale statistics for deconvolution in order to detect qualitative features of the unknown density. An important example covered within this framework is to test for local monotonicity on all scales simultaneously. We investigate the moderately ill-posed setting, where the Fourier transform of the error density in the deconvolution model is of polynomial decay. For multiscale testing, we consider a calibration, motivated by the modulus of continuity of Brownian…

对数凹分布逼近及其回归应用AI 解读

We study the approximation of arbitrary distributions P on d-dimensional space by distributions with log-concave density. Approximation means minimizing a Kullback–Leibler-type functional. We show that such an approximation exists if and only if P has finite first moments and is not supported by some hyperplane. Furthermore we show that this approximation depends continuously on P with respect to Mallows distance D1(⋅, ⋅). This result implies consistency of the maximum…

密度的多尺度推断AI 解读

We introduce a multiscale test statistic based on local order statistics and spacings that provides simultaneous confidence statements for the existence and location of local increases and decreases of a density or a failure rate. The procedure provides guaranteed finite-sample significance levels, is easy to implement and possesses certain asymptotic optimality and adaptivity properties.

反卷积密度估计中的非参数置信带AI 解读

Summary Uniform confidence bands for densities f via non-parametric kernel estimates were first constructed by Bickel and Rosenblatt. In this paper this is extended to confidence bands in the deconvolution problem g=f*ψ for an ordinary smooth error density ψ. Under certain regularity conditions, we obtain asymptotic uniform confidence bands based on the asymptotic distribution of the maximal deviation (L∞-distance) between a deconvolution kernel estimator f^ and f. Further…

从区间删失数据估计单峰分布AI 解读

In this article we consider three nonparametric maximum likelihood estimators based on mixed-case interval-censored data. Apart from the unrestricted estimator, we consider estimators under the assumption that the underlying distribution function of event times is concave or unimodal. Characterizations of the estimates are derived, and algorithms are proposed for their computation. The estimators are shown to be asymptotically consistent, and the benefits of additional…

定性假设的多尺度检验AI 解读

Suppose that one observes a process Y on the unit interval, where dY(t) =n1/2 f(t)dt +dW (t) with an unknown function parameter f, given scale parameter n <=1 ("sample size") and standard Brownian motion W. We propose two classes of tests of qualitative nonparametric hypotheses about f such as monotonicity or concavity. These tests are asymptotically optimal and adaptive in a certain sense. They are constructed via a new class of multiscale statistics and an extension of…

估计量与置信集的自适应调节AI 解读

An unknown signal plus white noise is observed at $n$ discrete time points. Within a large convex class of linear estimators of $\xi$, we choose the estimator $\hat{\xi}$ that minimizes estimated quadratic risk. By construction, $\hat{\xi}$ is nonlinear. This estimation is done after orthogonal transformation of the data to a reasonable coordinate system. The procedure adaptively tapers the coefficients of the transformed data. If the class of candidate estimators satisfies…

新拟合优度检验及其在非参数置信集中的应用AI 解读

Suppose one observes a process V on the unit interval, where $dV = f_o + dW$ with an unknown parameter $f_o \epsilon L_1[0, 1]$ and standard Brownian motion W. We propose a particular test of one-point hypotheses about $f_o$ which is based on suitably standardized increments of V. This test is shown to have desirable consistency properties if, for instance, $f_o$ is restricted to various Hölder classes of functions. The test is mimicked in the context of nonparametric…