### Journals Information

**
Mathematics and Statistics Vol. 4(1), pp. 40 - 45 DOI: 10.13189/ms.2016.040105 Reprint (PDF) (114Kb) **

## The Constructive Implicit Function Theorem and Proof in Logistic Mixtures

**Xiao Liu ^{*}**

Methods in Empirical Educational Research, TUM School of Education and Centre for International Student Assessment (ZIB), TU Munchen, Arcisstr. 21, 80333 Munich, Germany

**ABSTRACT**

There is the work by Bridges et al (1999) on the key features of a constructive proof of the implicit function theorem, including some applications to physics and mechanics. For mixtures of logistic distributions such information is lacking, although a special instance of the implicit function theorem prevails therein. The theorem is needed to see that the ridgeline function, which carries information about the topography and critical points of a general logistic mixture problem, is well-defined [2]. In this paper, we express the implicit function theorem and related constructive techniques in their multivariate extension and propose analogs of Bridges and colleagues' results for the multivariate logistic mixture setting. In particular, the techniques such as the inverse of Lagrange's mean value theorem [4] allow to prove that the key concept of a logistic ridgeline function is well-defined in proper vicinities of its arguments.

**KEYWORDS**

Constructive Implicit Function Theorem, Logistic Mixture, Lagrange Mean Value Theorem, Ridgeline

**Cite This Paper in IEEE or APA Citation Styles**

(a). IEEE Format:

[1] Xiao Liu , "The Constructive Implicit Function Theorem and Proof in Logistic Mixtures," Mathematics and Statistics, Vol. 4, No. 1, pp. 40 - 45, 2016. DOI: 10.13189/ms.2016.040105.

(b). APA Format:

Xiao Liu (2016). The Constructive Implicit Function Theorem and Proof in Logistic Mixtures. Mathematics and Statistics, 4(1), 40 - 45. DOI: 10.13189/ms.2016.040105.