Problems in Ill-posed Statistical Inference

题目：病态统计推理中的问题摘要统计学家对求解有噪声积分方程重新产生了兴趣。这类问题是不适定的，因为所涉及的积分算子通常具有无界逆，这类模型可以归结为传统曲线估计的推广。这一建议的目的是追求正在向三个方向发展的一般理论。第一个问题是，尽管存在可能未知的误差密度作为无限大维度的干扰参数，是否可以构造输入信号的足够光滑函数的渐近有效估计。要处理的第二个问题是以其均值为中心的积分平方误差的渐近分布。这种精度全局测度的精确渐近行为对模型检验具有重要的应用。最后提出了一种适用于具有数学不规则性的卷积方程的新方法。该方法基于小波基展开和时域卷积算子的反演。当只有输出可观察时，实验者经常面临恢复系统输入的问题。通常情况下，观测结果会因测量误差而变得模糊，而统计程序则变得相关。例子包括医学成像中使用的计算机断层扫描，以及观察到粒度分布变换的立体学中的Wicksell问题。逆热传导要求在给定当前热分布（输出）时恢复初始热分布（输入）。还有更多的例子和相关问题。例如，当只有关于其形状的数据可用时，人们可以估计悬挂在其端点的电缆的总重量吗？构造输入函数的有效估计是本研究的目的之一。考虑的第二个问题是关于估计输入与其期望之间差异的频率分布的精确信息。结果可以用于检查某些先前假设的有效性，例如，可以用于恢复银河系的光度分布。最后，提出了一种基于小波的不规则输入重建方法，该方法在图像重建中具有潜在的应用前景。

Proposal ID: DMS-0203942PI: Frits H. RuymgaartTitle: Problems in ill-posed statistical inferenceAbstractThere is a renewed interest from the part of statisticians in solving noisy integral equations. This kind of problem is ill-posed because the integral operator involved typically has an unbounded inverse and such models boil down to a generalization of traditional curve estimation. The purpose of this proposal is to pursue the general theory that is being developed into three directions. The first question is whether asymptotically efficient estimators of sufficiently smooth functionals of the input signal can be constructed despite the presence of the possibly unknown error density as an infinite dimensional nuisance parameter. The second question to be dealt with is the asymptotic distribution of the integrated squared error centered at its mean. Such precise asymptotic behavior of a global measure of accuracy has important applications to model checks. Finally a new method expedient for convolution equations with mathematical irregularities will be developed. The method is based on expansion in a wavelet basis coupled with inversion of the convolution operator in the time domain.Experimenters are often faced with the problem of recovering the input of a system when only the output is observable. Typically the observations will be blurred by measurement error and statistical procedures become pertinent. Examples include computer tomography employed in medical imaging, and Wicksell's problem in stereology where a transform of the particle size distribution is observed. Inverse heat conduction requires the recovery of the initial heat distribution (input) when the present one (output) is given. There are many more examples and related questions. For instance, can one estimate the total weight of a cable suspended at its endpoints, when only data regarding its shape are available? The construction of efficient estimators of such functionals of the input is one purpose of this research. A second question considered is precise information about the frequency distribution of the discrepancy between the estimated input and its expectation. Results can be useful in checking the validity of certain prior assumptions and could, for instance, be applied in recovering the luminosity distribution of the Milky Way. Finally a new method of input reconstruction in irregular cases will be developed using wavelets with potential application to image reconstruction.