# Some Stochastic Orders over an Interval with Applications

## Abstract

**:**

## 1. Introduction

- 1.

- 2.
- Utility: The usefulness of money may not be evaluated solely on a monetary scale. Thus, the usefulness of €x for an individual or a company is a function $u\left(x\right)$, the utility of €x. The expected utility hypothesis serves as a reference guide for decision makers with utility u, where the random future incomes modeled by the r.v.’s X and Y, i.e., $Eu\left(X\right)\le Eu\left(Y\right)$, provided the expectations exist. In this framework, the Laplace transform order represents preferences of decision makers with a negative exponential utility function given by$$u\left(x\right)=1-{e}^{hx},\phantom{\rule{1.em}{0ex}}h<0.$$

- 3.
- Life Insurance: The topic of PV is central to actuarial science. Let $\left(x\right)$ denote a person aged x, where $x\ge 0$. We denote his or her remaining lifetime of $\left(x\right)$ by a continuous r.v. ${T}_{x}$ considering that the death of $\left(x\right)$ can occur at any age greater than x. We typically assume that the interest rate is constant and fixed. This is appropriate, for example, if the premiums for an insurance policy are invested in risk-free bonds, all yielding the same interest rate, so that the term structure is flat. The whole life insurance plan pays EUR 1 at death. Since the PV of a future payment depends on the payment date, the PV of the benefit payment is a function of the time of death, and is therefore modeled as a r.v. For our $\left(x\right)$, the PV of a benefit of EUR 1 payable immediately on death is represented by a r.v. denoted as Z. This r.v. is defined as $Z={e}^{-\delta {T}_{x}}$, where $\delta $ is known as the continuously compounded rate of interest. The expected present value (EPV) of the whole life insurance benefit payment with a sum insured of EUR 1 euro is $E{e}^{-\delta {T}_{x}}\equiv {\overline{A}}_{x}$.

- 4.
- Reliability: Let a device (or a system) have survival function $\overline{F}\left(t\right)$ and s be the discount rate. If it produces one unit per minute when functioning and the PV of one unit produced at time t is $1\xb7{e}^{-st}$, then the EPV of total output produced during the life of the device is ${\int}_{0}^{\infty}{e}^{-st}\overline{F}\left(t\right)dt$, which is the Laplace transform of survival function $\overline{F}\left(t\right)$. Also, a different interpretation of ${\int}_{0}^{\infty}{e}^{-st}\overline{F}\left(t\right)dt$ may be seen as the EPV of the total maintenance cost of a device (or a system); see Shaked and Wong (1997).

## 2. Preliminaries

**Definition**

**1.**

- (i)
- stochastic dominance order over $[a,b]$(denoted by $X{\le}_{st[a,b]}Y)$ if ${\overline{F}}_{X}\left(t\right)\le {\overline{F}}_{Y}\left(t\right)$ for $a\le t\le b$;
- (ii)
- hazard rate order over $[a,b]$(denoted by $X{\le}_{hr[a,b]}Y)$ if ${r}_{X}\left(t\right)\ge {r}_{Y}\left(t\right)$ for $a\le t\le b$;
- (iii)
- mean residual lifetime order over $[a,b]$(denoted by $X{\le}_{mrl[a,b]}Y$) if ${m}_{X}\left(t\right)\le {m}_{Y}\left(t\right)$ for $a\le t\le b$;
- (iv)
- the harmonic mean residual lifetime order over $[a,b]$(denoted by $X{\le}_{hmrl[a,b]}Y)$ if $\{(1/t){\int}_{0}^{t}$$1/{m}_{X}{\left(z\right)dz\}}^{-1}\le {\{(1/t){\int}_{0}^{t}1/{m}_{Y}\left(z\right)dz\}}^{-1}$ for $a\le t\le b$.

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

## 3. Stochastic Orders over an Interval $[\mathit{a},\mathit{b}]$

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Corollary**

**1.**

**Theorem**

**4.**

**Proof.**

**Example**

**1.**

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

**Proof.**

**Example**

**2.**

**Example**

**3.**

## 4. Applications

#### 4.1. Upper Bound on DFR Reliability Function

**Example**

**4.**

**Example**

**5.**

**Example**

**6.**

**Remark**

**1.**

#### 4.2. Classical Risk Model

**Example**

**7.**

**Example**

**8.**

## 5. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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t | $\overline{\mathit{F}}\left(\mathit{t}\right)$ | ${\mathit{K}}_{1}\left(\mathit{t}\right)$ | ${\mathit{U}}_{\mathbf{LK}}\left(\mathit{t}\right)$ |
---|---|---|---|

(a) Pareto (4, 3) | |||

$0.05$ | 0.936 | 0.951 | 0.944 |

$0.1$ | 0.877 | 0.904 | 0.891 |

$0.2$ | 0.772 | 0.818 | 0.794 |

$0.5$ | 0.539 | 0.606 | 0.562 |

$0.33$ | 0.448 | 0.513 | 0.464 |

$0.75$ | 0.409 | 0.472 | 0.421 |

1 | 0.316 | 0.367 | 0.316 |

(b) Mixture of two exponentials | |||

$0.05$ | 0.961 | 0.985 | 0.984 |

$0.1$ | 0.926 | 0.971 | 0.968 |

$0.2$ | 0.863 | 0.944 | 0.938 |

$0.5$ | 0.725 | 0.866 | 0.854 |

1 | 0.590 | 0.751 | 0.729 |

$3.2$ | 0.352 | 0.402 | 0.364 |

$3.5$ | 0.331 | 0.367 | 0.331 |

(c) Weibull (2, 1/3) | |||

1/10 | 0.5572 | 0.9672 | 0.9412 |

1/5 | 0.4786 | 0.9355 | 0.8859 |

1/2 | 0.3678 | 0.8464 | 0.7387 |

1 | 0.2836 | 0.7165 | 0.5456 |

1.5 | 0.2363 | 0.6065 | 0.4031 |

2 | 0.2044 | 0.5134 | 0.2977 |

2.5 | 0.1808 | 0.4345 | 0.2199 |

2.9 | 0.1658 | 0.3803 | 0.1726 |

3 | 0.1624 | 0.3678 | 0.1624 |

u | ${\mathit{U}}_{\mathbf{LK}}\left(\mathit{u}\right)$ | ${\mathit{U}}_{\mathit{PP}1}\left(\mathit{u}\right)$ | ${\mathit{U}}_{\mathit{PP}2}\left(\mathit{u}\right)$ | ${\mathit{U}}_{\mathit{CP}}\left(\mathit{u}\right)$ |
---|---|---|---|---|

(a) Pareto(3, 1) | ||||

$0.2$ | $0.8865$ | $0.8808$ | $0.8806$ | $0.8888$ |

$0.25$ | $0.8809$ | $0.8750$ | $0.8747$ | $0.8844$ |

$0.5$ | $0.8536$ | $0.8509$ | $0.8493$ | $0.8635$ |

$0.75$ | $0.8271$ | $0.8322$ | $0.8283$ | $0.8437$ |

$0.9$ | $0.8116$ | $0.8225$ | $0.8170$ | $0.8322$ |

1 | $0.8014$ | $0.8166$ | $0.8099$ | $0.8247$ |

(b) Pareto (3, 2) | ||||

$0.5$ | $0.8809$ | $0.8750$ | $0.8747$ | $0.8844$ |

1 | $0.8536$ | $0.8509$ | $0.8493$ | $0.8635$ |

$1.25$ | $0.8402$ | $0.8411$ | $0.8384$ | $0.8535$ |

$1.5$ | $0.8271$ | $0.8322$ | $0.8283$ | $0.8437$ |

$1.75$ | $0.8142$ | $0.8241$ | $0.8189$ | $0.8341$ |

2 | $0.8014$ | $0.8166$ | $0.8099$ | $0.8247$ |

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**MDPI and ACS Style**

Kanellopoulos, L.
Some Stochastic Orders over an Interval with Applications. *Risks* **2023**, *11*, 161.
https://doi.org/10.3390/risks11090161

**AMA Style**

Kanellopoulos L.
Some Stochastic Orders over an Interval with Applications. *Risks*. 2023; 11(9):161.
https://doi.org/10.3390/risks11090161

**Chicago/Turabian Style**

Kanellopoulos, Lazaros.
2023. "Some Stochastic Orders over an Interval with Applications" *Risks* 11, no. 9: 161.
https://doi.org/10.3390/risks11090161