Problems in Ill-posed Statistical Inference

提案ID：DMS-0203942PI：Frits H.Ruymgaart标题：不适定统计推断中的问题摘要部分统计学家对求解有噪声的积分方程组重新产生了兴趣。这类问题是不适定的，因为涉及的积分算子通常有一个无界逆，而这样的模型归结为传统曲线估计的推广。这项提议的目的是追求正在发展成三个方向的一般理论。第一个问题是，尽管存在可能未知的误差密度作为无穷维扰动参数，是否可以构造足够光滑的输入信号泛函的渐近有效估计器。第二个要处理的问题是以其平均值为中心的积分平方误差的渐近分布。这种全局精度度量的精确渐近行为在模型检验中有重要的应用。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卷积算子在时间域上的逆运算，在只有输出是可观测的情况下，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，统计程序就变得中肯了。例如，医学成像中使用的计算机断层成像，以及体视学中的维克塞尔问题，其中观察到了颗粒尺寸分布的变化。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的问题。例如，当只有关于电缆形状的数据时，人们能估计出悬挂在其端点的电缆的总重量吗？构造这种输入泛函的有效估计量是本研究的目的之一。考虑的第二个问题是关于估计投入与其预期之间差异的频率分布的准确信息。这些结果可以用于检查某些先前假设的有效性，例如，可以应用于恢复银河系的光度分布。最后，利用小波变换提出了一种新的非规则情况下的输入重建方法，并将其应用于图像重建。

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.