## 1. Introduction

## 2. MD for Factor–Structured Variables

#### 2.1. MDs with Known Mean and Variance

**Definition**

**1.**

**μ**and covariance matrix

**Σ**. The factor model of ${\mathit{x}}_{p\times 1}$ is

**Proposition**

**1.**

**Ψ**is a diagonal matrix, and $\mathit{F}\sim N(\mathbf{0},\mathit{I})$ be distributed independently, which leads to $Cov(\epsilon ,\mathit{F})=\mathbf{0}$; the covariance structure for $\mathit{x}$ is given as follows:

**Definition**

**2.**

**μ**is

**Definition**

**3.**

#### 2.2. MDs with Unknown Mean and Unknown Covariance Matrix

**Proposition**

**2.**

**Definition**

**4.**

## 3. Distributional Properties of Factor Model MDs

**Proposition**

**3.**

**Proof.**

**Proposition**

**4.**

**Ψ**and $\mathit{L}$ are identified by ${\mathit{L}}^{\prime}\Psi \mathit{L}$ being diagonal with different and ordered elements and $\widehat{\epsilon}$ is homogeneity, as $n\to \infty $, the first two moments of the MDs under a factor model structure convergence in probability as follows:

**Proof.**

## 4. Contaminated Data

**Definition**

**5.**

**Proposition**

**5.**

**μ**. Let $\mathit{S}={n}^{-1}{\sum}_{i=1}^{n}{\mathit{x}}_{i}{\mathit{x}}_{i}^{\prime}$ be the sample covariance matrix and set $\mathit{W}=n\mathit{S}$ and ${\mathit{W}}_{\left(k\right)}=\mathit{W}-{\mathit{x}}_{k}{\mathit{x}}_{k}^{\prime}$. The expressions based on the contaminated and uncontaminated parts of the MD, up to ${o}_{p}\left({n}^{-\frac{1}{2}}\right)$, are given as follows:

**Proof.**

## 5. Empirical Example

## 6. Conclusions and Discussion

## Acknowledgments

## Conflicts of Interest

## Appendix A. Proof of Propositions

**Proof of Proposition**

**2.**

**Proof of Proposition**

**4.**

**Proof of Proposition**

**5.**

**Linear invariance of factor models MD (covariance matrix):**Let the linear transformation of data be ${\mathit{y}}_{i}=\mathit{a}+\mathit{B}{\mathit{x}}_{i}$ where $\mathit{B}$ is the symmetric invertible matrix, then the new MD ${D}_{ii,f}^{*}$ under the linear transformation is given as follows:

**The MLE of factor loadings:**The maximum likelihood estimator for the mean vector $\mu $, the factor loadings $\mathit{L}$ and the specific variances $\Psi $ are obtained by finding $\widehat{\mu}$, $\widehat{\mathit{L}}$, and $\widehat{\Psi}$ that maximises the log likelihood, which is given by the following expression:

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**Figure 1.**The box–plot of $D\left({\mathit{x}}_{i},\overline{\mathit{x}},\mathit{S}\right)$ for the stock data.

**Figure 3.**The box–plot of $D\left({\mathit{F}}_{i},\overline{\mathit{x}},{\widehat{\Sigma}}_{f}\right)$.

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