# New Classes of Distortion Risk Measures and Their Estimation

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Proposed Methods and Distortions

#### 2.1. Distortions via Exponentiated Exponential Distribution

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

#### 2.2. Distortions via Gompertz Distribution

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

## 3. Examples of Distortion Risk Measures

#### 3.1. Uniform Loss

#### 3.2. Exponential Loss

#### 3.3. Lomax Loss

#### 3.4. Weibull Loss

## 4. Numerical Analyses and Estimation

#### 4.1. Numerical Results for Distortion Risk Measures

#### 4.2. Estimation of Distortion Risk Measures

## 5. Concluding Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Abdelaziz, Rassoul. 2015. An improved estimator of the distortion risk measure for heavy-tailed claims. Hacettepe Journal of Mathematics and Statistics 44: 735–46. [Google Scholar]
- Acerbi, Carlo. 2002. Spectral measures of risk: A coherent representation of subjective risk aversion. Journal of Banking & Finance 26: 1505–18. [Google Scholar]
- Aldhufairi, Fadal A. A., and Jungsywan H. Sepanski. 2020. New families of bivariate copulas via unit Weibull distortion. Journal of Statistical Distributions and Applications 7: 8. [Google Scholar] [CrossRef]
- Aldhufairi, Fadal Abdullah-A, Ranadeera G. M. Samanthi, and Jungsywan H. Sepanski. 2020. New families of bivariate copulas via unit Lomax distortion. Risks 8: 106. [Google Scholar] [CrossRef]
- Artzner, Philippe, Freddy Delbaen, Jean-Marc Eber, and David Heath. 1997. Thinking coherently. Risk 10: 68–71. [Google Scholar]
- Artzner, Philippe, Freddy Delbaen, Jean-Marc Eber, and David Heath. 1999. Coherent measures of risk. Math Finance 9: 203–28. [Google Scholar] [CrossRef]
- Belles-Sampera, Jaume, Montserrat Guillen, and Miguel Santolino. 2016. What attitudes to risk underlie distortion risk measure choices? Insurance: Mathematics and Economics 68: 101–9. [Google Scholar] [CrossRef]
- Bihary, Zsolt, Peter Csoka, and David Zoltan Szabó. 2020. Spectral risk measure of holding stocks in the long run. Annals of Operations Research 2951: 75–89. [Google Scholar] [CrossRef]
- Brahimi, Brahimi, Fatima Meddi, and Abdelhakim Necir. 2012. Bias-corrected estimation in distortion risk premiums for heavy-tailed losses. Journal Afrika Statistika 7: 474–90. [Google Scholar] [CrossRef]
- Denneberg, Dieter. 1994. Non-Additive Measure and Integral. Theory and Decision Library. Dordrecht: Springer, vol. 27. [Google Scholar]
- Denuit, Michel, Jan Dhaene, Marc Goovaerts, and Rob Kaas. 2005. Measures, Orders and Models. In Actuarial Theory for Dependent Risks. New York: John Wiley & Sons, Ltd. [Google Scholar]
- Dowd, Kevin, John Cotter, and Ghulam Sorwar. 2008. Spectral risk measures: Properties and limitations. Journal of Financial Services Research 49: 121–31. [Google Scholar]
- Eugene, A. J., Mila Novita, and Siti Nurrohmah. 2021. Gini Shortfall: A Gini mean difference-based risk measure. Journal of Physics: Conference Series 1725: 012094. [Google Scholar] [CrossRef]
- Fischer, Matthias, Thorsten Moser, and Marius Pfeuffer. 2018. A discussion on recent risk measures with application to credit risk: Calculating risk contributions and identifying risk concentrations. Risks 6: 142. [Google Scholar] [CrossRef]
- Frees, Edward. 2018. Loss Data Analytics. Available online: https://openacttexts.github.io/Loss-Data-Analytics/ (accessed on 6 October 2021).
- Furman, Edward, Ruodu Wang, and Ričardas Zitikis. 2017. Gini-type measures of risk and variability: Gini shortfall, capital allocations, and heavy-tailed risks. Journal of Banking & Finance 3: 70–84. [Google Scholar]
- Jones, Bruce L., and Ricardas Zitikis. 2003. Empirical estimation of risk measures and related quantities. North American Actuarial Journal 7: 44–54. [Google Scholar] [CrossRef]
- Jones, Bruce L., and Ricardas Zitikis. 2007. Risk measures, distortion parameters, and their empirical estimation. Insurance: Mathematics and Economics 41: 279–97. [Google Scholar] [CrossRef]
- Kim, Joseph H. T. 2010. Bias correction for estimated distortion risk measure using the bootstrap. Insurance: Mathematics and Economics 47: 198–205. [Google Scholar] [CrossRef]
- Minasyan, Vigen. 2020. New ways to measure catastrophic financial risks: VaR to the power of t measures and how to calculate them. Finance: Theory and Practice 25: 165–84. [Google Scholar] [CrossRef]
- Minasyan, Vigen. 2021. New risk measures for variance distortion and catastrophic financial risk measures. Finance: Theory and Practice 25: 165–84. [Google Scholar] [CrossRef]
- Pflug, Georg. 2009. On distortion functionals. Statistics and Decisions 27: 201–9. [Google Scholar] [CrossRef]
- Rockafellar, R. Tyrrell, and Stanislav Uryasev. 2002. Conditional value-at-risk for general loss distributions. Journal of Banking & Finance 26: 1443–71. [Google Scholar]
- Samanthi, Ranadeera G. M., and Jungsywan Sepanski. 2018. Methods for generating coherent distortion risk measures. Annals of Actuarial Science 13: 400–16. [Google Scholar] [CrossRef]
- Sereda, Ekaterina N., Efim M. Bronshtein, Svetozar T. Rachev, Frank J. Fabozzi, Wei Sun, and Stoyan V. Stoyanov. 2010. Distortion Risk Measures in Portfolio Optimization. In Handbook of Portfolio Construction. Berlin: Springer. [Google Scholar] [CrossRef]
- Wang, Shaun. 1995. Insurance pricing and increased limits ratemaking by proportional hazards transforms. Insurance: Mathematics and Economics 17: 43–54. [Google Scholar] [CrossRef]
- Wang, Shaun. 2000. A class of distortion operations for pricing financial and insurance risks. Journal of Risk and Insurance 67: 15–36. [Google Scholar] [CrossRef]
- Wang, Wei, and Huifu Xu. 2023. Preference robust distortion risk measure and its application. Mathematical Finance 33: 389–434. [Google Scholar] [CrossRef]
- Wirch, Julia Lynn, and Mary R. Hardy. 1999. A synthesis of risk measures for capital adequacy. Insurance: Mathematics and Economics 25: 337–47. [Google Scholar]
- Yaari, Menahem E. 1987. The Dual Theory of Choice Under Risk. Econometrica 55: 95–115. [Google Scholar] [CrossRef]
- Yin, Chuancun, and Dan Zhu. 2018. New class of distortion risk measures and their tail asymptotic with emphasis on VaR. Journal of Financial Risk Management 7: 12–38. [Google Scholar] [CrossRef]

**Figure 1.**UG distortion cures are displayed in (

**a**) and (

**b**), and UGQ in (

**c**) and (

**d**) for varying $\theta $ or α.

Distortion | Function Form | Admissible Parameter Space |
---|---|---|

Kumaraswamy | ${H}_{e}\left(w\right)=1-{(1-{w}^{\alpha})}^{\theta}$ | $0<\alpha \le 1,\theta \ge 1$ |

UEE | ${K}_{e}\left(w\right)={(1-{(1-w)}^{\theta})}^{\alpha}$ | $0<\alpha \le 1,\theta \ge 1$ |

UG | ${K}_{o}\left(w\right)=1-{e}^{-\theta [{\left(1-w\right)}^{-\alpha}-1]}$ | $\alpha >0$, $\theta \ge 1+1/\alpha $ |

UGQ | ${H}_{o}^{-1}\left(w\right)={\left(1-{\theta}^{-1}lnw/\right)}^{-1/\alpha}$ | $\alpha >0$, $\theta \ge 1+1/\alpha $ |

Distortion | Parameter Space | Risk Measures |
---|---|---|

Power | $0<\alpha \le 1$ | $2b\alpha /(1+\alpha )$ |

Dual-power | $\theta \ge 1$ | $2b\theta /(\theta +1)$ |

Kumaraswamy | $0<\alpha \le 1,\theta \ge 1$ | $2b-(2b/\alpha )B(\theta +1,1/\alpha )]$ |

UEE | $0<\alpha \le 1,\theta \ge 1$ | $(2b/\theta )\left[B\right(\alpha +1,1/\theta \left)\right]$ |

UG | $\alpha >0,\theta >1+1/\alpha +1$ | $2b-(2b/\alpha ){e}^{\theta}E(1+1/\alpha ,\theta )$ |

UGQ | $\alpha >0,\theta >1+1/\alpha +1$ | $2b\theta {e}^{\theta}E(1/\alpha ,\theta )$ |

Distortion | Parameter Space | Risks Measure |
---|---|---|

Power | $0<\alpha \le 1$ | $b/\alpha $ |

Dual Power | $\theta \ge 1$ | $b\left[\mathsf{\Psi}\right(\theta +1)-\mathsf{\Psi}(1\left)\right]$ |

Kumaraswamy | $0<\alpha \le 1,\theta \ge 1$ | $(b/\alpha )\left[\mathsf{\Psi}\right(\theta +1)-\mathsf{\Psi}(1\left)\right]$ |

UEE | $0<\alpha \le 1,\theta \ge 1$ | $b\alpha {\sum}_{k=1}^{\infty}(1/k)B(1+(k/\theta ),\alpha )$ |

UG | $\alpha >0,\theta >\frac{1}{\alpha}+1$ | $b\theta {e}^{\theta}{\sum}_{k=1}^{\infty}(1/k)E(k/\alpha ,\theta )$ |

UGQ | $\alpha >0,\theta >\frac{1}{\alpha}+1$ | $b\alpha \theta /(1-\alpha )$ if $0<\alpha <1;$ undefined if $\alpha \ge 1$ |

Distortion | Parameter Space | Risk Measures |
---|---|---|

Power | $0<\alpha \le 1$ | $b/(a\alpha -1),\phantom{\rule{0.166667em}{0ex}}a\alpha \ne 1$ |

Dual Power | $\theta \ge 1$ | $b\theta \left[B\right(1-1/a,\theta )-B(1,\theta \left)\right]$ |

Kumaraswamy | $0<\alpha \le 1,\theta \ge 1$ | $b\theta \left[B\right(1-1/\left(a\alpha \right),\theta )-B(1,\theta \left)\right],\phantom{\rule{0.166667em}{0ex}}a\alpha \ne 1$ |

UEE | $0<\alpha \le 1,\theta \ge 1$ | ${\sum}_{k=0}^{\infty}\left(\genfrac{}{}{0pt}{}{\theta}{k}\right){(-1)}^{k}[1+{\sum}_{i=1}^{\infty}\left(\genfrac{}{}{0pt}{}{\alpha k}{i}\right)\frac{{(-1)}^{i}b}{ia-1}]$ |

UG | $\alpha >0,\theta >\frac{1}{\alpha}+1$ | $b\theta {e}^{\theta}{\sum}_{k=1}^{\infty}{(-1)}^{k}\left(\genfrac{}{}{0pt}{}{-1/a}{k}\right)E\left(k/\alpha ,\theta \right)$ |

UGQ | $\alpha >0,\theta >\frac{1}{\alpha}+1$ | Undefined |

Distortion | Parameter Space | Risk Measures |
---|---|---|

Power | $\theta \ge 1$ | $\mathsf{\Gamma}(1+1/c){\left(b\theta \right)}^{1/c}$ |

Dual-Power | $\theta \ge 1$ | $\mathsf{\Gamma}(1+1/c){b}^{1/c}{\sum}_{k=1}^{\infty}\left(\genfrac{}{}{0pt}{}{\theta}{k}\right){(-1)}^{k}{\left(k\right)}^{-1/c}$ |

Kumaraswamy | $0<\alpha \le 1,\theta >1$ | $\mathsf{\Gamma}(1+1/c){b}^{1/c}{\alpha}^{-1/c}{\sum}_{k=1}^{\infty}\left(\genfrac{}{}{0pt}{}{\theta}{k}\right){(-1)}^{k}{k}^{-1/c}$ |

UEE | $0<\alpha \le 1,\theta \ge 1$ | $\mathsf{\Gamma}(1+1/c){b}^{1/c}{\sum}_{k=0}^{\infty}\left(\genfrac{}{}{0pt}{}{\alpha}{k}\right){(-1)}^{k}{\sum}_{i=0}^{\infty}\left(\genfrac{}{}{0pt}{}{\theta k}{i}\right){(-1)}^{i}{i}^{-1/c}$ |

UG | $\alpha >0,\theta >\frac{1}{\alpha}+1$ | Finite |

UGQ | $\alpha >0,\theta >\frac{1}{\alpha}+1$ | $(1/c){\left(\theta b\right)}^{1/c}B(1/\alpha -1/c,1/c),\alpha <c$ |

**Table 6.**VaR and CTE values at levels of 0.25, 0.5, 0.75, 0.95, and 0.99 for uniform $(0,100)$, exponential $\left(0.02\right)$, Lomax $(12.61,580.40)$, Weibull $(5,0.5)$, and Weibull $(412.20,1.5)$ losses.

Level | ||||||
---|---|---|---|---|---|---|

Loss | 0.25 | 0.5 | 0.75 | 0.95 | 0.99 | |

Uniform | VaR | 25 | 50 | 75 | 95 | 99 |

CTE | 62.6 | 75 | 87.5 | 97.5 | 99.5 | |

Exponential | VaR | 14.38 | 34.66 | 69.31 | 149.79 | 230.26 |

CTE | 64.38 | 84.66 | 119.31 | 199.79 | 280.26 | |

Lomax | VaR | 13.39 | 32.80 | 67.45 | 155.64 | 255.84 |

CTE | 64.54 | 85.61 | 123.25 | 219.04 | 327.87 | |

Weibull (5, 0.5) | VaR | 2.07 | 12.01 | 48.05 | 224.36 | 530.19 |

CTE | 66.45 | 96.67 | 167.36 | 424.15 | 810.45 | |

Weibull (412.20, 1.5) | VaR | 24.14 | 43.38 | 68.86 | 115.10 | 153.31 |

CTE | 62.01 | 76.23 | 97.32 | 138.63 | 174.22 |

**Table 7.**The beta, Kumaraswamy, UEE risk measures for uniform $(0,100)$, exponential $\left(0.02\right)$, Lomax $(12.61,580.40)$, Weibull $(5,0.5)$, and Weibull $(412.20,1.5)$ losses.

Beta | Kumaraswamy | UEE | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

$\alpha $ | $\alpha $ | $\alpha $ | ||||||||

Loss | $\theta $ | 0.25 | 0.5 | 1 | 0.25 | 0.5 | 1 | 0.25 | 0.5 | 1 |

Uniform | 1 | 80 | 66.67 | 50 | 80 | 66.67 | 50 | 80 | 66.67 | 50 |

2 | 88.89 | 80 | 66.67 | 93.33 | 83.33 | 66.67 | 87.4 | 78.54 | 66.67 | |

10 | 97.56 | 95.24 | 90.91 | 99.90 | 98.48 | 90.91 | 96.76 | 94.36 | 90.91 | |

Exponential | 1 | 200 | 100 | 50 | 200 | 100 | 50 | 196.78 | 99.97 | 50 |

2 | 240 | 133.33 | 75 | 300 | 150 | 75 | 227.27 | 128.5 | 75 | |

10 | 325.26 | 213.33 | 146.45 | 585.79 | 292.9 | 146.45 | 302.62 | 203.32 | 146.65 | |

Lomax | 1 | 269.64 | 109.41 | 50 | 269.64 | 109.41 | 50 | 209.30 | 109.41 | 50 |

2 | 327.22 | 147.91 | 76.02 | 429.87 | 168.82 | 76.02 | 313.05 | 142.23 | 76.02 | |

10 | 460.15 | 247.46 | 155.91 | 1034.72 | 363.65 | 155.49 | 431.41 | 234.36 | 155.49 | |

Weibull (5, 0.5) | 1 | 800.00 | 200.00 | 50.00 | 800 | 200 | 50 | 799.30 | 200.00 | 50.00 |

2 | 992.00 | 288.89 | 87.50 | 1400 | 350 | 26.91 | 945.45 | 274.58 | 87.50 | |

10 | 1485.30 | 575.95 | 253.22 | 4051.45 | 1012.86 | 253.22 | 1373.67 | 532.94 | 253.22 | |

Weibull (412.20, 1.5) | 1 | 125.99 | 79.37 | 50.00 | 125.99 | 79.37 | 50.00 | 125.99 | 79.37 | 50.00 |

2 | 146.72 | 99.98 | 68.50 | 172.61 | 108.74 | 68.50 | 142.34 | 97.22 | 68.50 | |

10 | 185.73 | 141.92 | 111.28 | 280.41 | 176.65 | 111.28 | 178.52 | 137.52 | 111.28 |

**Table 8.**UG and UGQ risk measures for uniform $(0,100)$, exponential $\left(0.02\right)$, Lomax $(12.61,580.40)$ and Weibull $(5,0.5)$, and Weibull $(412.20,1.5)$ losses.

UG | UGQ | ||||||||
---|---|---|---|---|---|---|---|---|---|

$\alpha $ | $\alpha $ | ||||||||

Loss | $\theta $ | 0.25 | 0.5 | 1 | 5 | 0.25 | 0.5 | 1 | 5 |

Uniform | 5 | 58.10 | 73.94 | 85.21 | 96.69 | 58.10 | 79.94 | 85.21 | 96.09 |

10 | 72.78 | 84.37 | 91.56 | 98.20 | 72.78 | 84.37 | 91.56 | 98.20 | |

15 | 79.76 | 88.79 | 94.08 | 98.76 | 79.76 | 88.79 | 94.08 | 98.76 | |

20 | 83.88 | 91.26 | 95.44 | 99.05 | 83.88 | 91.26 | 95.44 | 99.05 | |

Exponential | 5 | 59.76 | 87.14 | 117.85 | 195.02 | 83.33 | 250 | 1252.69 | 16,897.55 |

10 | 85.64 | 116.10 | 148.57 | 227.24 | 166.67 | 493.38 | 2162.13 | 19,249.5 | |

15 | 102.56 | 134.26 | 167.40 | 246.63 | 250.00 | 735.20 | 2944.07 | 20,729.93 | |

20 | 115.16 | 147.54 | 181.03 | 260.56 | 333.33 | 973.87 | 3644.30 | 21,821.35 | |

Lomax | 5 | 59.92 | 88.74 | 122.41 | 214.18 | - | - | - | - |

10 | 87.24 | 120.58 | 157.79 | 256.06 | - | - | - | - | |

15 | 105.62 | 144.22 | 180.29 | 282.28 | - | - | - | - | |

20 | 119.60 | 156.67 | 196.99 | 301.59 | - | - | - | - | |

Weibull (c = 0.5) | 5 | 61.88 | 106.77 | 172.66 | 416.78 | 208.33 | 5022.05 | 180,021.20 | - |

10 | 105.13 | 169.86 | 257.64 | 554.89 | 833.33 | 16,688.58 | - | - | |

15 | 139.51 | 217.08 | 318.39 | 647.58 | 1875 | 33,135.06 | - | - | |

Weibull (c = 1.5) | 5 | 58.03 | 77.02 | 95.73 | 135.91 | 67.69 | 130.45 | 330.07 | 2279.65 |

10 | 75.87 | 94.59 | 112.54 | 150.81 | 107.46 | 206.83 | 499.95 | 2572.00 | |

15 | 86.47 | 104.79 | 122.24 | 159.43 | 140.81 | 270.65 | 462.31 | 2752.00 | |

20 | 93.96 | 111.93 | 129.01 | 165.47 | 170.58 | 327.39 | 744.17 | 2882.70 |

**Table 9.**Sample means, biases and standard deviations in parentheses of L-estimates and plug-in estimates from 500 simulations for exponential loss with mean 50.

Distortion | L-estimate | Plug-in | L-estimate | Plug-in | L-estimate | Plug-in | ||
---|---|---|---|---|---|---|---|---|

Beta (0.5, 3) | 153.33 | 153.33 | Beta (0.5, 6) | 187.82 | 187.82 | Beta (1, 3) | 91.67 | 91.67 |

$n=50$ | 133.92 (24.45) | 134.28 (24.51) | $n=50$ | 159.65 (32.04) | 159.61 (33.06) | $n=50$ | 90.53 (12.79) | 90.51 (13.16) |

mean bias | −19.42 | −19.00 | −28.17 | −28.21 | −1.14 | −1.16 | ||

$n=100$ | 140.11 (19.52) | 142.36 (18.98) | $n=100$ | 168.60 (25.88) | 166.54 (24.38) | $n=100$ | 91.28 (9.92) | 91.30 (4.54) |

mean bias | −13.22 | −12.65 | −19.22 | −21.28 | −0.38 | −0.29 | ||

$n=500$ | 147.08 (10.50) | 149.63 (9.89) | $n=500$ | 178.81 (14.23) | 180.29 (14.59) | $n=500$ | 91.40 (4.56) | 91.44 (4.54) |

mean bias | −6.26 | −5.97 | −9.01 | −7.53 | −0.27 | −0.25 | ||

$n=1000$ | 148.94 (7.62) | 152.67 (6.90) | $n=1000$ | 181.48 (10.40) | 183.32 (8.52) | $n=1000$ | 91.54 (3.01) | 91.92 (2.67) |

mean bias | −4.39 | −0.66 | −6.34 | −3.5 | −0.13 | 0.25 | ||

Kumar (0.5, 3) | 183.33 | 183.33 | Kumar (0.5, 6) | 245 | 245 | Kumar (1, 3) | 91.67 | 91.67 |

$n=50$ | 153.52 (30.90) | 149.16 (28.54) | $n=50$ | 189.24 (44.53) | 189.21 (40.51) | $n=50$ | 90.53 (12.19) | 92.77 (13.10) |

mean bias | −29.81 | −34.17 | −55.76 | −55.79 | −1.14 | 1.1 | ||

$n=100$ | 162.68 (25.22) | 166.45 (21.42) | $n=100$ | 205.39 (37.99) | 211.48 (33.99) | $n=100$ | 91.28 (9.92) | 92.41 (10.04) |

mean bias | −20.65 | −16.88 | −39.61 | −33.52 | −0.39 | 0.74 | ||

$n=500$ | 173.56 (14.22) | 180.52 (13.96) | $n=500$ | 226.17 (23.12) | 230.59 (21.46) | $n=500$ | 91.40 (4.56) | 91.63 (4.58) |

mean bias | −9.77 | −2.81 | −18.83 | −14.41 | −0.27 | −0.04 | ||

$n=1000$ | 176.74 (11.19) | 181.33 (10.01) | $n=1000$ | 231.92 (18.99) | 234.67 (17.93) | $n=1000$ | 91.71 (3.04) | 91.82 (3.04) |

mean bias | −6.59 | −2.00 | −13.08 | −10.33 | 0.04 | 0.15 | ||

UEE (0.5, 3): | 146.51 | 146.51 | UEE (0.5, 6) | 178.78 | 178.78 | UEE (1, 3) | 91.67 | 91.67 |

$n=50$ | 128.66 (23.05) | 130.89 (21.63) | $n=50$ | 153.37 (29.94) | 154.26 (27.26) | $n=50$ | 90.53 (12.79) | 92.77 (13.09) |

mean bias | −17.85 | −15.62 | −25.41 | −24.52 | −1.14 | 1.10 | ||

$n=100$ | 134.37 (18.37) | 139.65 (17.32) | $n=100$ | 161.48 (24.08) | 168.94 (22.45) | $n=100$ | 91.28 (9.92) | 92.41 (10.04) |

mean bias | −12.14 | −6.86 | −17.30 | −9.84 | −0.39 | 0.74 | ||

$n=500$ | 140.78 (9.83) | 143.20 (8.66) | $n=500$ | 170.69 (13.15) | 174.27 (12.59) | $n=500$ | 91.40 (4.56) | 91.63 (4.58) |

mean bias | −5.73 | −3.31 | −8.09 | −4.51 | −0.27 | −0.04 | ||

$n=1000$ | 142.67 (7.48) | 144.89 (6.21) | $n=1000$ | 173.43 (10.12) | 176.53 (10.05) | $n=1000$ | 91.71 (3.04) | 91.82 (3.04) |

mean bias | −3.75 | −1.62 | −5.35 | −2.25 | −0.04 | 0.15 | ||

UG (1, 3) | 96.90 | 96.90 | UG (1, 6) | 125.71 | 125.71 | UG (6, 3) | 181.26 | 181.26 |

$n=50$ | 95.75 (13.60) | 97.76 (13.85) | $n=50$ | 122.56 (19.12) | 129.45 (20.08) | $n=50$ | 171.92 (33.96) | 201.23 (39.16) |

mean bias | −1.15 | 0.86 | −3.15 | 3.74 | −9.34 | 19.97 | ||

$n=100$ | 96.51 (10.53) | 97.51 (10.63) | $n=100$ | 124.56 (14.73) | 127.86 (15.10) | $n=100$ | 177.20 (26.07) | 191.95 (28.05) |

mean bias | −0.39 | 0.61 | −1.15 | 2.15 | −4.06 | 10.66 | ||

$n=500$ | 96.63 (4.84) | 96.83 (4.85) | $n=500$ | 125.18 (6.84) | 125.83 (6.57) | $n=500$ | 179.86 (12.19) | 182.81 (12.38) |

mean bias | −0.27 | −0.07 | −0.53 | 0.12 | −1.4 | 1.55 | ||

$n=1000$ | 96.94 (3.22) | 97.04 (3.22) | $n=1000$ | 125.70 (4.54) | 126.03 (4.55) | $n=1000$ | 180.89 (8.34) | 182.37 (8.41) |

mean bias | 0.04 | 0.14 | −0.01 | 0.32 | −0.37 | 1.11 | ||

UGQ (0.5, 4) | 200 | 200 | UGQ (0.5, 8) | 400 | 400 | UGQ (0.25, 8) | 133.33 | 133.33 |

$n=50$ | 106.04 (20.39) | 111.20 (21.23) | $n=50$ | 143.76 (32.37) | 146.12 (29.49) | $n=50$ | 98.67 (17.67) | 99.11 (18.4) |

mean bias | −93.96 | −88.80 | −256.24 | −254.88 | −34.66 | −34.22 | ||

$n=100$ | 113.58 (17.54) | 114.67 (18.96) | $n=100$ | 158.15 (29.37) | 169.43 (26.40) | $n=100$ | 104.20 (14.66) | 110.76 (15.00) |

mean bias | −86.42 | −85.33 | −241.85 | −230.57 | −29.13 | −22.57 | ||

$n=500$ | 126.15 (12.16) | 130.04 (11.99) | $n=500$ | 184.32 (22.95) | 193.44 (19.98) | $n=500$ | 112.40 (9.09) | 123.13 (7.99) |

mean bias | −73.85 | −69.96 | −215.69 | 206.56 | −20.94 | −10.20 | ||

$n=1000$ | 130.88 (1128) | 133.81 (10.99) | $n=1000$ | 194.36 (22.41) | 196.27 (20.02) | $n=1000$ | 115.33 (7.87) | 125.04 (6.82) |

mean bias | −69.12 | −66.19 | −205.64 | −203.73 | −18.00 | −8.29 |

**Table 10.**Sample means, biases and standard deviations parentheses of L-estimates and plug-in estimates from 500 simulations for Lomax loss with parameters (5, 200).

Distortion | L-estimate | Plug-in | L-estimate | Plug-in | L-estimate | Plug-in | ||
---|---|---|---|---|---|---|---|---|

Beta (0.5, 3) | 218.06 | 218.06 | Beta (0.5, 6) | 281.53 | 281.53 | Beta (1, 3) | 97.62 | 97.62 |

$n=50$ | 156.33 (50.52) | 160.01 (51.77) | $n=50$ | 192.52 (68.41) | 191.22 (67.13) | $n=50$ | 94.78 (20.66) | 93.63 (20.10) |

mean bias | −61.73 | −58.05 | −89.01 | −90.31 | −3.14 | −3.85 | ||

$n=100$ | 179.79 (37.62) | 178.99 (38.82) | $n=100$ | 214.40 (51.68) | 215.30 (50.96) | $n=100$ | 97.20 (13.71) | 97.17 (13.45) |

mean bias | −46.27 | −39.07 | −67.13 | −66.23 | −0.42 | −0.45 | ||

$n=500$ | 188.99 (26.26) | 190.43 (27.11) | $n=500$ | 239.49 (36.98) | 240.15 (35.49) | $n=500$ | 97.45 (6.63) | 97.43 (6.06) |

mean bias | −29.07 | −27.63 | −42.05 | −41.38 | −0.17 | −0.19 | ||

$n=1000$ | 193.66 (21.78) | 194.59 (21.74) | $n=1000$ | 246.79 (33.98) | 248.15 (32.94) | $n=1000$ | 97.42 (5.04) | 97.58 (4.55) |

mean bias | −24.40 | −23.47 | −34.74 | −33.38 | −0.20 | −0.04 | ||

Kumar (0.5, 3) | 280.77 | 280.77 | Kumar (0.5, 6) | 422.11 | 422.11 | Kumar (1, 3) | 97.62 | 97.62 |

$n=50$ | 185.03 (66.45) | 185.16 (33.83) | $n=50$ | 239.38 (100.43) | 242.69 (55.64) | $n=50$ | 94.48 (20.66) | 96.76 (21.13) |

mean bias | −95.74 | −95.61 | −182.73 | −179.42 | −3.14 | −0.86 | ||

$n=100$ | 207.45 (51.01) | 217.63 (28.73) | $n=100$ | 278.24 (81.00) | 281.56 (49.53) | $n=100$ | 97.20 (13.71) | 97.68 (6.65) |

mean bias | −73.32 | −63.14 | −143.87 | −140.55 | −0.42 | 0.06 | ||

$n=500$ | 234.50 (37.86) | 240.52 (24.08) | $n=500$ | 330.43 (65.85) | 340.23 (28.26) | $n=500$ | 97.45 (6.63) | 97.68 (6.65) |

mean bias | −46.27 | −40.25 | −91.68 | −81.88 | −0.17 | 0.06 | ||

$n=1000$ | 242.66 (33.26) | 249.74 (22.81) | $n=1000$ | 346.24 (60.48) | 353.64 (24.77) | $n=1000$ | 97.57 (4.52) | 97.58 (4.50) |

mean bias | −38.11 | −31.03 | −75.87 | −68.47 | −0.05 | −0.04 | ||

UEE (0.5, 3): | 206.21 | 206.21 | UEE (0.5, 6) | 263.84 | 263.84 | UEE (1, 3) | 97.62 | 97.62 |

$n=50$ | 149.17 (47.26) | 155.63 (46.39) | $n=50$ | 183.26 (63.31) | 184.81 (60.52) | $n=50$ | 94.48 (20.66) | 96.76 (21.13) |

mean bias | −57.04 | −50.58 | −80.58 | −79.03 | −3.14 | −0.86 | ||

$n=100$ | 163.51 (35.10) | 159.99 (33.47) | $n=100$ | 203.19 (47.58) | 206.34 (49.08) | $n=100$ | 97.20 (13.71) | 98.36 (13.86) |

mean bias | −42.7 | −22.97 | −38.03 | −31.26 | −0.14 | 0.06 | ||

$n=500$ | 179.37 (24.38) | 183.24 (22.31) | $n=500$ | 225.81 (33.75) | 232.58 (34.25) | $n=500$ | 97.48 (6.63) | 97.68 (6.65) |

mean bias | −26.84 | −22.97 | −38.03 | −31.26 | −0.14 | 0.06 | ||

$n=1000$ | 184.16 (20.70) | 186.47 (19.05) | $n=1000$ | 232.63 (28.91) | 239.43 (28.93) | $n=1000$ | 97.58 (4.52) | 97.58 (4.50) |

mean bias | −22.05 | −19.74 | −31.21 | −24.41 | −0.04 | −0.04 | ||

UG (1, 3) | 103 | 103 | UG (1, 6) | 141.23 | 141.23 | UG (6, 3) | 226 | 226 |

$n=50$ | 99.84 (20.69) | 101.79 (22.03) | $n=50$ | 135.03 (31.33) | 142.03 (35.59) | $n=50$ | 207.08 (62.65) | 241.23 (79.61) |

mean bias | −3.16 | −1.21 | −6.20 | 0.8 | −18.92 | 15.23 | ||

$n=100$ | 102.47 (15.29) | 103.63 (14.51) | $n=100$ | 139.25 (23.97) | 143.50 (23.03) | $n=100$ | 217.35 (50.29) | 237.62 (50.37) |

mean bias | −0.53 | 0.63 | −1.98 | 2.27 | −8.65 | 11.62 | ||

$n=500$ | 102.67 (6.78) | 103.64 (6.95) | $n=500$ | 140.38 (10.84) | 141.44 (11.16) | $n=500$ | 223.12 (24.09) | 227.67 (24.57) |

mean bias | −0.33 | 0.64 | −0.85 | 0.21 | −2.88 | 1.67 | ||

$n=1000$ | 102.89 (4.77) | 102.89 (4.78) | $n=1000$ | 140.92 (7.76) | 141.56 (7.67) | $n=1000$ | 224.91 (17.55) | 228.35 (17.05) |

mean bias | −0.11 | −0.11 | −0.31 | 0.33 | −1.09 | 2.35 |

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## Share and Cite

**MDPI and ACS Style**

Sepanski, J.H.; Wang, X.
New Classes of Distortion Risk Measures and Their Estimation. *Risks* **2023**, *11*, 194.
https://doi.org/10.3390/risks11110194

**AMA Style**

Sepanski JH, Wang X.
New Classes of Distortion Risk Measures and Their Estimation. *Risks*. 2023; 11(11):194.
https://doi.org/10.3390/risks11110194

**Chicago/Turabian Style**

Sepanski, Jungsywan H., and Xiwen Wang.
2023. "New Classes of Distortion Risk Measures and Their Estimation" *Risks* 11, no. 11: 194.
https://doi.org/10.3390/risks11110194