Mean and variance of implicitly defined biased estimators (such as penalized maximum likelihood): Applications to tomography
Mean and variance of implicitly defined biased estimators (such as penalized maximum likelihood): Applications to tomography
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DOI：
10.1109/83.491322
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发表时间：
1996-03-01
影响因子：
10.6
通讯作者：
Fessler, JA
中科院分区：
文献类型：
--
作者：
Fessler, JA
信号处理问题中的许多估计量都被隐式定义为某个目标函数的最大值，隐式定义的估计量的例子包括最大似然估计、惩罚似然估计、最大后验估计和非线性最小二乘估计。对于这类估计量，均值和方差的精确解析表达式通常是得不到的，因此，研究人员通常求助于数值模拟来检验这类估计量的均值和方差的性质。本文描述了无约束连续参数隐式定义的估计量的均值和方差的近似表达式，我们利用隐函数定理、泰勒展开和链规则推导了近似表达式，这些表达式仅根据用于估计的目标函数的偏导数来定义，作为例证，我们证明了这种近似在两种泊松统计层析成像应用中是有效的。我们还描述了一种‘’插件‘’近似，即使从单个噪声的泊松正弦图测量，它也提供了非常准确的变异性估计，该近似在广泛的估计问题中应该是有用的。
Many estimators in signal processing problems are defined implicitly as the maximum of some objective function, Examples of implicitly defined estimators include maximum likelihood, penalized likelihood, maximum a posteriori, and nonlinear least squares estimation. For such estimators, exact analytical expressions for the mean and variance are usually unavailable, Therefore, investigators usually resort to numerical simulations to examine properties of the mean and variance of such estimators, This paper describes approximate expressions for the mean and variance of implicitly defined estimators of unconstrained continuous parameters, We derive the approximations using the implicit function theorem, the Taylor expansion, and the chain rule, The expressions are defined solely in terms of the partial derivatives of whatever objective function one uses for estimation, As illustrations, we demonstrate that the approximations work well in two tomographic imaging applications with Poisson statistics, We also describe a ''plug-in'' approximation that provides a remarkably accurate estimate of variability even from a single noisy Poisson sinogram measurement, The approximations should be useful in a wide range of estimation problems.