# On Partial Stochastic Comparisons Based on Tail Values at Risk

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## Abstract

**:**

## 1. Motivation and Preliminaries

**Definition**

**1.**

## 2. Properties and Relationships with Other Stochastic Orders

**Proposition**

**1.**

**Proof.**

**Lemma**

**1.**

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

**Proof.**

**Remark**

**1.**

**Remark**

**2.**

**Proposition**

**4.**

- (i)
- If $X{\le}_{{p}_{0}-\mathrm{tvar}}Y$, then ${\int}_{x}^{+\infty}\overline{F}\left(t\right)dt\le {\int}_{x}^{+\infty}\overline{G}\left(t\right)dt$, for all $x\ge {F}^{-1}\left({p}_{0}\right)$.
- (ii)
- If ${\int}_{x}^{+\infty}\overline{F}\left(t\right)\phantom{\rule{4pt}{0ex}}dt\le {\int}_{x}^{+\infty}\overline{G}\left(t\right)\phantom{\rule{4pt}{0ex}}dt$, for all $x\ge {x}_{0}$, then $X{\le}_{G\left({x}_{0}\right)-tvar}Y$.

**Proof.**

**Corollary**

**1.**

- (i)
- $E\left[{\left(X-x\right)}_{+}\right]\le E\left[{\left(Y-x\right)}_{+}\right]$, for all $x\ge {F}^{-1}\left({p}_{0}\right)$.
- (ii)
- ${\int}_{x}^{+\infty}\overline{F}\left(t\right)dt\le {\int}_{x}^{+\infty}\overline{G}\left(t\right)dt$, for all $x\ge {F}^{-1}\left({p}_{0}\right)$.

**Proof.**

**Theorem**

**1.**

**Example**

**1.**

- (i)
- Let $X\sim N({\mu}_{1},{\sigma}_{1})$ and $Y\sim N({\mu}_{2},{\sigma}_{2})$ be two normal random variables such that ${\mu}_{1}>{\mu}_{2}$ and ${\sigma}_{1}<{\sigma}_{2}$. Then, $X{\le}_{{p}_{0}-\mathrm{tvar}}Y$, where ${p}_{0}={F}_{Z}\left(\frac{{\mu}_{1}-{\mu}_{2}}{{\sigma}_{2}-{\sigma}_{1}}\right)$ and $Z\sim N(0,1)$.
- (ii)
- Let $X\sim \mathrm{Logistic}({\mu}_{1},{\sigma}_{1})$ and $Y\sim \mathrm{Logistic}({\mu}_{2},{\sigma}_{2})$ be two logistic random variables such that ${\mu}_{1}>{\mu}_{2}$ and ${\sigma}_{1}<{\sigma}_{2}$. Then, $X{\le}_{{p}_{0}-\mathrm{tvar}}Y$, where ${p}_{0}={F}_{Z}\left(\frac{{\mu}_{1}-{\mu}_{2}}{{\sigma}_{2}-{\sigma}_{1}}\right)$ and $Z\sim \mathrm{Logistic}(0,1)$.
- (iii)
- Let $X\sim W({\lambda}_{1},{k}_{1})$ and $Y\sim W({\lambda}_{2},{k}_{2})$ be two Weibull random variables such that $\mathrm{E}\left[X\right]>\mathrm{E}\left[Y\right]$ and ${k}_{2}<{k}_{1}$. Then, $X{\le}_{{p}_{0}-\mathrm{tvar}}Y$, where ${p}_{0}={F}_{Z}\left({a}_{0}^{{b}_{0}}\right)$, ${a}_{0}={\lambda}_{1}/{\lambda}_{2}$, ${b}_{0}={k}_{1}{k}_{2}/({k}_{1}-{k}_{2})$ and $Z\sim W(1,1)$.
- (iv)
- Let $X\sim P({a}_{1},{k}_{1})$ and $P\sim P({a}_{2},{k}_{2})$ be two Pareto random variables such that $\mathrm{E}\left[X\right]>\mathrm{E}\left[Y\right]$ and ${k}_{2}<{k}_{1}$. Then, $X{\le}_{{p}_{0}-\mathrm{tvar}}Y$, where ${p}_{0}={F}_{Z}\left({a}_{0}^{{b}_{0}}\right)$, ${a}_{0}={a}_{1}/{a}_{2}$, ${b}_{0}={k}_{1}{k}_{2}/({k}_{1}-{k}_{2})$ and $Z\sim P(1,1)$.

**Remark**

**3.**

**Remark**

**4.**

**Proposition**

**5.**

**Proof.**

## 3. A Real Data Example

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**Histograms of the log returns with a normal density estimate (solid line) and a logistic density estimate (dashed line) superimposed. The left-hand panel corresponds to ${R}^{MMX}$ and right-hand panel corresponds to ${R}^{HSI}$.

K-S Goodness of Fit Test (p-Value) | ||
---|---|---|

Index | Normal | Logistic |

MXX | $0.0512$ | $0.0347$ |

HSI | $0.0525$ | $0.0557$ |

Normal | Logistic | |||
---|---|---|---|---|

Index | $\mathit{\mu}$ | $\mathit{\sigma}$ | $\mathit{\mu}$ | $\mathit{\sigma}$ |

MXX | $-0.001486327$ | $0.01556418$ | $-0.001226663$ | $0.008706823$ |

HSI | $-0.005058756$ | $0.02017290$ | $-0.005402021$ | $0.011434130$ |

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**MDPI and ACS Style**

Bello, A.J.; Mulero, J.; Sordo, M.A.; Suárez-Llorens, A.
On Partial Stochastic Comparisons Based on Tail Values at Risk. *Mathematics* **2020**, *8*, 1181.
https://doi.org/10.3390/math8071181

**AMA Style**

Bello AJ, Mulero J, Sordo MA, Suárez-Llorens A.
On Partial Stochastic Comparisons Based on Tail Values at Risk. *Mathematics*. 2020; 8(7):1181.
https://doi.org/10.3390/math8071181

**Chicago/Turabian Style**

Bello, Alfonso J., Julio Mulero, Miguel A. Sordo, and Alfonso Suárez-Llorens.
2020. "On Partial Stochastic Comparisons Based on Tail Values at Risk" *Mathematics* 8, no. 7: 1181.
https://doi.org/10.3390/math8071181