# Principal Curves for Statistical Divergences and an Application to Finance

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Statistical Divergences and Principal Curves

**Definition**

**1.**

**Theorem**

**1**

**.**Given $o,z,w\in \mathcal{M}$ such that that the dual affine geodesic connecting z and w is orthogonal to the affine geodesic connecting w and o, the following generalized Pythagorean relation holds

**Theorem**

**2.**

**Proof.**

**Corollary**

**1.**

**Proof.**

## 3. The Space of Financial Assets

#### 3.1. Deformed Exponentials and Portfolio Selection

#### 3.2. Mean-Divergence Efficient Frontier

**Theorem**

**3**

**.**Let $\mathcal{E}=\mathrm{span}\{{k}_{\mathsf{e}},{k}_{\mathsf{q}}\}$ the subspace in $\mathcal{M}$ spanned by the expectation and pricing kernels. Given $z\in \mathcal{M}$, we have

## 4. Generalized Beta Pricing Models and CAPM

## 5. Generalized Principal Components Analysis (PCA) and Applications to Finance

**Theorem**

**4.**

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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Rodrigues, A.F.P.; Cavalcante, C.C. Principal Curves for Statistical Divergences and an Application to Finance. *Entropy* **2018**, *20*, 333.
https://doi.org/10.3390/e20050333

**AMA Style**

Rodrigues AFP, Cavalcante CC. Principal Curves for Statistical Divergences and an Application to Finance. *Entropy*. 2018; 20(5):333.
https://doi.org/10.3390/e20050333

**Chicago/Turabian Style**

Rodrigues, Ana Flávia P., and Charles Casimiro Cavalcante. 2018. "Principal Curves for Statistical Divergences and an Application to Finance" *Entropy* 20, no. 5: 333.
https://doi.org/10.3390/e20050333