Abstract

Taylor expansion is a vital mathematical tool applied across various scientific disciplines. However, there is a widespread misuse of this popular technique in the statistical literature. While mean-value forms for real-valued functions rely on a scalar intermediate point, directly extending this single-intermediate-point representation (Lagrange form) to vectorand matrix-valued functions is mathematically invalid, as a single common evaluation point generally does not exist. This paper addresses the issue by presenting accurate forms of Taylor expansions and highlighting that the commonly used single-point versions by some statisticians are non-existent. We distinguish between exact finite-sample equality errors and the validity of theoretical asymptotic properties (such as consistency and asymptotic normality), demonstrating that theoretical conclusions typically remain sound when valid component wise or integral-form expansions are employed. By bringing clarity and exact mathematical precision to the proper use of Taylor expansions, the statistical community can enhance the integrity of scientific research and impart rigorous theoretical standards to future generations of statisticians.

Keywords

estimating equation, integral-form taylor expansion, matrix-valued function, mean-value theorem, vector-valued function