M-statistics: Optimal Statistical Inference for a Small Sample 1st Edition

A comprehensive resource providing new statistical methodologies and demonstrating how new approaches work for applications

M-statistics introduces a new approach to statistical inference, redesigning the fundamentals of statistics and improving on the classical methods we already use. This book targets exact optimal statistical inference for a small sample under one methodological umbrella. Two competing approaches are offered: maximum concentration (MC) and mode (MO) statistics combined under one methodological umbrella, which is why the symbolic equation M=MC+MO. M-statistics defines an estimator as the limit point of the MC or MO exact optimal confidence interval when the confidence level approaches zero, the MC and MO estimator, respectively. Neither mean nor variance plays a role in M-statistics theory.

Novel statistical methodologies in the form of double-sided unbiased and short confidence intervals and tests apply to major statistical parameters:

• Exact statistical inference for small sample sizes is illustrated with effect size and coefficient of variation, the rate parameter of the Pareto distribution, two-sample statistical inference for normal variance, and the rate of exponential distributions.
• M-statistics is illustrated with discrete, binomial and Poisson distributions. Novel estimators eliminate paradoxes with the classic unbiased estimators when the outcome is zero.
• Exact optimal statistical inference applies to correlation analysis including Pearson correlation, squared correlation coefficient, and coefficient of determination. New MC and MO estimators along with optimal statistical tests, accompanied by respective power functions, are developed.
• M-statistics is extended to the multidimensional parameter and illustrated with the simultaneous statistical inference for the mean and standard deviation, shape parameters of the beta distribution, the two-sample binomial distribution, and finally, nonlinear regression.

Our new developments are accompanied by respective algorithms and R codes, available at GitHub, and as such readily available for applications.

M-statistics is suitable for professionals and students alike. It is highly useful for theoretical statisticians and teachers, researchers, and data science analysts as an alternative to classical and approximate statistical inference.

《M-統計學》是一本全面的資源，提供了新的統計方法論，並展示了這些新方法在應用中的運作方式。該書介紹了一種新的統計推論方法，重新設計了統計學的基礎，並改進了我們已經使用的傳統方法。本書的目標是在一個方法論框架下，為小樣本提供精確的最佳統計推論。提供了兩種競爭的方法：最大集中（MC）和模態（MO）統計，結合在一個方法論框架下，這就是為什麼符號方程式為 M=MC+MO。M-統計將估計量定義為當置信水平趨近於零時，MC或MO精確最佳置信區間的極限點，分別是MC和MO估計量。在M-統計理論中，均值和變異數都不起作用。

該書提供了新穎的統計方法，包括雙側無偏和短置信區間以及測試，適用於主要的統計參數，例如：小樣本的精確統計推論，包括效應大小和變異係數、帕雷托分佈的速率參數、正態變異數的兩樣本統計推論以及指數分佈的速率。M-統計還應用於離散、二項和泊松分佈。新的估計量消除了當結果為零時，經典無偏估計量的悖論。

精確最佳統計推論適用於相關分析，包括皮爾森相關係數、平方相關係數和決定係數。我們開發了新的MC和MO估計量以及最佳統計檢定，並附帶相應的功效函數。

M-統計擴展到多維參數，並以同時統計推論的平均值和標準差、beta分佈的形狀參數、兩樣本二項分佈以及非線性回歸為例進行了說明。

我們的新發展伴隨著相應的算法和R代碼，可在GitHub上獲得，因此可以方便地應用。

《M-統計學》適用於專業人士和學生。對於理論統計學家和教師、研究人員和數據科學分析師來說，它是一種替代傳統和近似統計推論的高度有用的工具。

Eugene Demidenko is Professor of Biomedical Data Science at the Geisel School of Medicine and Mathematics at Dartmouth. He is a member of the American Statistical Association (ASA) and the Society of Industrial and Applied Mathematics (SIAM). In statistics, Professor Demidenko's research includes statistical methodology, mixed models, and nonlinear regression. In applied mathematics, he contributed to existence and uniqueness of global minimum, tumor regrowth theory, shape and image analysis, and solving ill-posed problems via mixed boundary partial differential equations. He is the author of two books published by Wiley in 2013 and 2020 "Mixed Models: Theory and Applications" and "Advanced Statistics with Applications in R." The latter book received a prestigious Ziegel Book Award in Statistics from Technometrics/ASA journal in 2022.

尤金·德米登科（Eugene Demidenko）是達特茅斯醫學院（Geisel School of Medicine）和數學系的生物醫學數據科學教授。他是美國統計學會（ASA）和工業與應用數學學會（SIAM）的成員。在統計學方面，德米登科教授的研究包括統計方法論、混合模型和非線性回歸。在應用數學方面，他對全局最小值的存在性和唯一性、腫瘤再生理論、形狀和圖像分析以及通過混合邊界偏微分方程解決不适定問題做出了貢獻。他是Wiley出版社於2013年和2020年出版的兩本書的作者，《混合模型：理論與應用》和《R語言應用高級統計學》。後者於2022年獲得了Technometrics/ASA期刊頒發的聞名的Ziegel統計學圖書獎。