Maximum likelihood style estimators possesses a number of ideal characteristics, but require prior identification of the distribution of errors to ensure exact unbiasedness. Independent of the focus of the primary statistical analysis, the estimation of a covariance matrix \(S^{P \times P}\approx Σ^{P \times P}\) must possess a specific structure and regularity constraints. The need to estimate a linear Gaussian covariance models appear in various applications as a formal precondition for scientific investigation and predictive analytics. In this work, we construct an \(\ell_{2}\)-norm based quasi-likelihood framework, identified by binomial comparisons between all pairs \(X_{n},Y_{n}, \forall {n}\). Our work here focuses upon the quasi-likelihood basis for estimation of an exactly unbiased linear regression Hájek projection, within which the Kemeny metric space is operationalised via Whitney embedding to obtain exact unbiased minimum variance multivariate covariance estimators upon both discrete and continuous random variables (i.e., exact unbiased identification in the presence of ties upon finite samples). While the covariance estimator is inherently useful, expansion of the Wilcoxon rank-sum testing framework to handle multiple covariates with exact unbiasedness upon finite samples is a currently unresolved research problem, as it maintains identification in the presence of linear surjective mappings onto common points: this model space, by definition, expands our likelihood framework into a consistent non-parametric form of the standard general linear model, which we extend to address both unknown heterogeneity and the problem of weak inferential instruments.

翻译：最大似然类估计器具备若干理想特性，但需要预先识别误差分布以确保精确无偏性。独立于主要统计分析的重点，协方差矩阵 \(S^{P \times P}\approx Σ^{P \times P}\) 的估计必须具有特定结构和正则性约束。估计线性高斯协方差模型的需求作为科学研究和预测分析的形式前提出现在各种应用中。本文构建了一个基于 \(\ell_{2}\) 范数的拟似然框架，通过对所有配对 \(X_{n},Y_{n}, \forall {n}\) 进行二项比较来识别。本研究聚焦于估计精确无偏线性回归Hájek投影的拟似然基础，其中通过Whitney嵌入实现Kemeny度量空间的可操作性，从而获得在离散和连续随机变量上均具有精确无偏最小方差的多元协方差估计器（即在有限样本存在结值情况下的精确无偏识别）。虽然协方差估计器本身具有实用性，但将Wilcoxon秩和检验框架扩展至处理具有多个协变量且在有限样本上保持精确无偏性的问题，目前仍是一个尚未解决的研究难题，因为该框架能在存在线性满射映射至公共点的情况下保持识别性：该模型空间通过定义将我们的似然框架扩展为标准一般线性模型的一致非参数形式，我们进一步扩展该框架以同时处理未知异质性和弱推断工具问题。