# Parameter Estimation in Stable Law

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

## 3. Main Results

**Theorem**

**1.**

**Proof.**

**Definition**

**2.**

**Proposition**

**1.**

**Proof.**

**Definition**

**3.**

**Proposition**

**2.**

**Proof.**

**cumulant estimates**, denoted by ${\widehat{\alpha}}_{n},{\widehat{\beta}}_{n},{\widehat{\gamma}}_{n},{\widehat{\delta}}_{n}$ (not in bold), i.e., the non-random values computed on a particular realization ${x}_{1},\cdots ,{x}_{n}$ of a sample ${X}_{1},\cdots ,{X}_{n}$ i.i.d. random variables.

## 4. Empirical Search for the Optimal Arguments of Cumulant Estimators

**replicates**) of the sample ${X}_{1},...,{X}_{n}$, $n={10}^{5}$ i.i.d stable random variables, ${X}_{i}\sim S(\alpha $, β, $\gamma =1,\delta =0;1)$. For each replicate, we calculate the squared errors of cumulant and reduced values’ cumulant estimates at several selections of ${u}_{1},{u}_{2}\in \mathbb{R},{u}_{1}>0,{u}_{2}>0,{u}_{1}\ne {u}_{2}$. We assess the quality of estimates to be the mean of squared errors (of 200 estimates), denoted by MSE (${\widehat{\alpha}}_{n}$), MSE(${\widehat{\beta}}_{n}$), MSE(${\widehat{\gamma}}_{n}$), and MSE(${\widehat{\delta}}_{n}$).

**Remark**

**1.**

## 5. Simulations on the Effectiveness of Cumulant Estimates at ${\mathit{u}}_{\mathbf{1}}\mathbf{=}\mathbf{0.03}\mathbf{,}{\mathit{u}}_{\mathbf{2}}\mathbf{=}\mathbf{0.09}$

**replicates**). We simulate samples of sizes of $n={10}^{2},{10}^{3},{10}^{4},{10}^{5}$, while, to save space, present the mean squared errors, MSE(${\widehat{\alpha}}_{n}$), MSE(${\widehat{\beta}}_{n}$), MSE(${\widehat{\gamma}}_{n}$), MSE(${\widehat{\delta}}_{n})$, for $n={10}^{5}$ only. We note that the quality of estimates strongly depended on the sample size, as the smaller the sample, the lower the quality of estimates. All simulations are carried out with package “stabledist” [18] in the open-source environment for statistical computing and graphics R [19].

#### 5.1. Simulations for $\alpha =0.25,0.5,0.75,1.25,1.5,1.75$

**Remark**

**2.**

**Remark**

**3.**

#### 5.2. Simulations in the Neighbourhood of $\alpha =1$

#### 5.3. Simulations for $\alpha =1$

#### 5.4. Simulations for $\alpha \downarrow 0$ and $\alpha \uparrow 2$

**Remark**

**4.**

**Remark**

**5.**

## 6. Application in Non-Life Insurance

#### 6.1. Cumulant Estimates for Claim Sizes

#### 6.2. Reduced Values’ Cumulant Estimates for Claim Sizes

#### 6.3. Comparison to Other Estimation Methods

## 7. Summary

- show that the parameters of stable law can be expressed through cumulant function of one pair of arguments, and hence
- propose the method of Press [6] at one pair of arguments only;
- suggest data scaling by median, i.e., introduce reduced values’ cumulant estimates;
- perform an empirical search for the selection of two arguments;
- carry out simulation experiments over parameter space at arguments of ${u}_{1}=0.03$ and ${u}_{2}=0.09$;
- present an application to non-life insurance losses;

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Table A1.**The MSEs of reduced values’ cumulant estimates for the selection of ${\mathit{u}}_{\mathbf{1}},{\mathit{u}}_{\mathbf{2}}$ for 200 replicates (sample size $\mathit{n}={\mathbf{10}}^{\mathbf{5}}$) from $\mathit{S}(\mathbf{\alpha},\mathbf{\beta},\mathbf{\gamma}=\mathbf{1},\mathbf{\delta}=\mathbf{0}\mathbf{;}\mathbf{1})$.

$\mathit{S}(\mathit{\alpha}=0.5,\mathit{\beta}=0,\mathit{\gamma}=1,\mathit{\delta}=0;1)$ | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

${u}_{1}$ | 0.03 | 0.03 | 0.03 | 0.03 | 0.3 | 0.3 | 0.3 | 0.3 | 3 | 3 | 3 | 3 |

${u}_{2}$ | 0.09 | 0.9 | 9 | 90 | 0.09 | 0.9 | 9 | 90 | 0.09 | 0.9 | 9 | 90 |

MSE (${\widehat{\alpha}}_{n}$) | 0.0000 | 0.0000 | 0.0059 | 0.0392 | 0.0000 | 0.0002 | 0.0163 | 0.0755 | 0.0007 | 0.0045 | 0.1520 | 0.2151 |

MSE (${\widehat{\beta}}_{n}$) | 0.0002 | 0.0001 | 0.0012 | 0.0091 | 0.0002 | 0.0007 | 0.0086 | 0.0311 | 0.0005 | 0.0231 | 4 × 10^{1} | 3 × 10^{1} |

MSE (${\widehat{\gamma}}_{n}$) | 0.0001 | 0.0002 | 0.0425 | 0.3183 | 0.0001 | 0.0011 | 0.2077 | 7.7057 | 0.0002 | 0.1267 | Inf | Inf |

MSE (${\widehat{\delta}}_{n}$) | 0.0003 | 0.0000 | 0.0006 | 0.0000 | 0.0001 | 0.0002 | 0.0006 | 0.0000 | 0.0008 | 0.0030 | 0.0014 | 0.0000 |

$\mathit{S}(\mathbf{\alpha}=\mathbf{0.5},\mathbf{\beta}=\mathbf{1},\mathbf{\gamma}=\mathbf{1},\mathbf{\delta}=\mathbf{0};\mathbf{1})$ | ||||||||||||

${u}_{1}$ | 0.03 | 0.03 | 0.03 | 0.03 | 0.3 | 0.3 | 0.3 | 0.3 | 3 | 3 | 3 | 3 |

${u}_{2}$ | 0.09 | 0.9 | 9 | 90 | 0.09 | 0.9 | 9 | 90 | 0.09 | 0.9 | 9 | 90 |

MSE (${\widehat{\alpha}}_{n}$) | 0.0001 | 0.0000 | 0.0000 | 0.0003 | 0.0000 | 0.0000 | 0.0000 | 0.0005 | 0.0000 | 0.0000 | 0.0001 | 0.0014 |

MSE (${\widehat{\beta}}_{n}$) | 0.0003 | 0.0001 | 0.0001 | 0.0043 | 0.0003 | 0.0002 | 0.0001 | 0.0146 | 0.0001 | 0.0002 | 0.0004 | 0.0993 |

MSE (${\widehat{\gamma}}_{n}$) | 0.0036 | 0.0002 | 0.0002 | 0.0183 | 0.0012 | 0.0005 | 0.0002 | 0.0081 | 0.0001 | 0.0001 | 0.0003 | 0.0009 |

MSE (${\widehat{\delta}}_{n}$) | 0.0230 | 0.0006 | 0.0001 | 0.0218 | 0.0047 | 0.0013 | 0.0001 | 0.0235 | 0.0002 | 0.0004 | 0.0003 | 0.0318 |

$\mathit{S}(\mathbf{\alpha}=\mathbf{1.5},\mathbf{\beta}=\mathbf{0},\mathbf{\gamma}=\mathbf{1},\mathbf{\delta}=\mathbf{0};\mathbf{1})$ | ||||||||||||

${u}_{1}$ | 0.03 | 0.03 | 0.03 | 0.03 | 0.3 | 0.3 | 0.3 | 0.3 | 3 | 3 | 3 | 3 |

${u}_{2}$ | 0.09 | 0.9 | 9 | 90 | 0.09 | 0.9 | 9 | 90 | 0.09 | 0.9 | 9 | 90 |

MSE (${\widehat{\alpha}}_{n}$) | 0.0001 | 0.3339 | 0.8957 | 1.2256 | 0.0933 | 2.2773 | 2.2475 | 2.2518 | 1.1632 | 2.2708 | 2.2755 | 2.2477 |

MSE (${\widehat{\beta}}_{n}$) | 0.0002 | 0.0414 | 0.0369 | 0.0594 | 0.2565 | 2 × 10^{2} | 1 × 10^{2} | 4 × 10^{2} | 0.0653 | 2 × 10^{2} | 2 × 10^{2} | 4 × 10^{2} |

MSE (${\widehat{\gamma}}_{n}$) | 0.0000 | 0.1838 | 0.6175 | 0.8490 | 0.0056 | Inf | Inf | Inf | 1.0659 | Inf | Inf | Inf |

MSE (${\widehat{\delta}}_{n}$) | 0.0001 | 5 × 10^{3} | 0.0003 | 0.0000 | 8 × 10^{1} | 0.1019 | 0.0004 | 0.0000 | 0.0029 | 0.0090 | 0.0010 | 0.0000 |

$\mathit{S}(\mathbf{\alpha}=\mathbf{1.5},\mathbf{\beta}=\mathbf{1},\mathbf{\gamma}=\mathbf{1},\mathbf{\delta}=\mathbf{0};\mathbf{1})$ | ||||||||||||

${u}_{1}$ | 0.03 | 0.03 | 0.03 | 0.03 | 0.3 | 0.3 | 0.3 | 0.3 | 3 | 3 | 3 | 3 |

${u}_{2}$ | 0.09 | 0.9 | 9 | 90 | 0.09 | 0.9 | 9 | 90 | 0.09 | 0.9 | 9 | 90 |

MSE (${\widehat{\alpha}}_{n}$) | 0.0004 | 0.0001 | 0.1231 | 0.4640 | 0.0001 | 0.0000 | 0.3508 | 0.9181 | 0.0114 | 0.0928 | 2.2727 | 2.2629 |

MSE (${\widehat{\beta}}_{n}$) | 0.0011 | 0.0003 | 1.2741 | 1.9049 | 0.0002 | 0.0002 | 2.4033 | 4.4211 | 1.6320 | 5.1234 | 2 × 10^{2} | 4 × 10^{2} |

MSE (${\widehat{\gamma}}_{n}$) | 0.0008 | 0.0000 | 0.3852 | 0.8615 | 0.0000 | 0.0000 | 0.1876 | 0.6170 | 0.0216 | 0.0039 | Inf | Inf |

MSE (${\widehat{\delta}}_{n}$) | 0.0003 | 0.0002 | 0.1416 | 0.0045 | 0.0002 | 0.0002 | 8.5030 | 0.0030 | 0.2452 | 5 × 10^{1} | 0.0973 | 0.0005 |

## Appendix B

**Table B1.**The MSEs of reduced values’ cumulant estimates (RVCE) and cumulant estimates (CE) at ${\mathit{u}}_{\mathbf{1}}=\mathbf{0.03},{\mathit{u}}_{\mathbf{2}}=\mathbf{0.09}$ for 200 replicates (sample size $\mathit{n}={\mathbf{10}}^{\mathbf{5}}$) from $\mathbf{S}(\mathbf{\alpha},\mathbf{\beta},\mathbf{\gamma}=\mathbf{1},\mathbf{\delta}=\mathbf{0};\mathbf{1})$.

α | β | Method | MSE (${\widehat{\mathit{\alpha}}}_{\mathit{n}}$) | MSE (${\widehat{\mathit{\beta}}}_{\mathit{n}}$) | MSE (${\widehat{\mathit{\gamma}}}_{\mathit{n}}$) | MSE (${\widehat{\mathit{\delta}}}_{\mathit{n}}$) |
---|---|---|---|---|---|---|

0.25 | 0.1 | RVCE | 5.7 × 10^{−6} | 6.9 × 10^{−5} | 1.5 × 10^{−4} | 1.7 × 10^{−6} |

0.25 | 0.1 | CE | 3.8 × 10^{−5} | 6.3 × 10^{−4} | 5.5 × 10^{−3} | 5.5 × 10^{−3} |

0.25 | 0.25 | RVCE | 4.1 × 10^{−6} | 4.9 × 10^{−5} | 6.6 × 10^{−5} | 1.4 × 10^{−5} |

0.25 | 0.25 | CE | 3.5 × 10^{−5} | 4.6 × 10^{−4} | 4.6 × 10^{−3} | 4.6 × 10^{−3} |

0.25 | 0.5 | RVCE | 4.3 × 10^{−6} | 5.2 × 10^{−5} | 3.8 × 10^{−4} | 1.7 × 10^{−4} |

0.25 | 0.5 | CE | 3.9 × 10^{−5} | 6.1 × 10^{−4} | 5.8 × 10^{−3} | 5.8 × 10^{−3} |

0.25 | 0.75 | RVCE | 3.4 × 10^{−6} | 5.9 × 10^{−5} | 5.8 × 10^{−4} | 8.9 × 10^{−4} |

0.25 | 0.75 | CE | 3.6 × 10^{−5} | 6.1 × 10^{−4} | 5.4 × 10^{−3} | 5.4 × 10^{−3} |

0.25 | 1 | RVCE | 4.2 × 10^{−6} | 1.0 × 10^{−4} | 1.1 × 10^{−3} | 4.3 × 10^{−3} |

0.25 | 1 | CE | 4.3 × 10^{−5} | 7.1 × 10^{−4} | 6.3 × 10^{−3} | 6.3 × 10^{−3} |

0.5 | 0.1 | RVCE | 4.1 × 10^{−6} | 1.8 × 10^{−5} | 1.5 × 10^{−5} | 3.0 × 10^{−5} |

0.5 | 0.1 | CE | 4.6 × 10^{−5} | 3.2 × 10^{−4} | 4.3 × 10^{−3} | 4.3 × 10^{−3} |

0.5 | 0.25 | RVCE | 3.7 × 10^{−6} | 2.5 × 10^{−5} | 5.5 × 10^{−5} | 1.2 × 10^{−4} |

0.5 | 0.25 | CE | 6.7 × 10^{−5} | 2.4 × 10^{−4} | 3.6 × 10^{−3} | 3.6 × 10^{−3} |

0.5 | 0.5 | RVCE | 5.5 × 10^{−6} | 2.4 × 10^{−5} | 1.5 × 10^{−4} | 3.2 × 10^{−4} |

0.5 | 0.5 | CE | 5.8 × 10^{−5} | 2.6 × 10^{−4} | 4.4 × 10^{−3} | 4.4 × 10^{−3} |

0.5 | 0.75 | RVCE | 7.4 × 10^{−6} | 3.6 × 10^{−5} | 2.9 × 10^{−4} | 1.1 × 10^{−3} |

0.5 | 0.75 | CE | 5.5 × 10^{−5} | 2.8 × 10^{−4} | 6.1 × 10^{−3} | 6.1 × 10^{−3} |

0.5 | 1 | RVCE | 8.5 × 10^{−6} | 3.9 × 10^{−5} | 4.5 × 10^{−4} | 2.5 × 10^{−3} |

0.5 | 1 | CE | 5.5 × 10^{−5} | 3.0 × 10^{−4} | 8.4 × 10^{−3} | 8.4 × 10^{−3} |

0.75 | 0.1 | RVCE | 4.5 × 10^{−6} | 1.6 × 10^{−5} | 2.0 × 10^{−5} | 1.9 × 10^{−4} |

0.75 | 0.1 | CE | 9.9 × 10^{−5} | 3.1 × 10^{−4} | 7.3 × 10^{−3} | 7.3 × 10^{−3} |

0.75 | 0.25 | RVCE | 8.3 × 10^{−6} | 2.5 × 10^{−5} | 8.8 × 10^{−5} | 6.7 × 10^{−4} |

0.75 | 0.25 | CE | 1.2 × 10^{−4} | 3.1 × 10^{−4} | 9.6 × 10^{−3} | 9.6 × 10^{−3} |

0.75 | 0.5 | RVCE | 1.2 × 10^{−5} | 3.9 × 10^{−5} | 2.0 × 10^{−4} | 2.5 × 10^{−3} |

0.75 | 0.5 | CE | 9.9 × 10^{−5} | 2.5 × 10^{−4} | 1.7 × 10^{−2} | 1.7 × 10^{−2} |

0.75 | 0.75 | RVCE | 2.0 × 10^{−5} | 4.0 × 10^{−5} | 4.2 × 10^{−4} | 8.3 × 10^{−3} |

0.75 | 0.75 | CE | 9.5 × 10^{−5} | 2.3 × 10^{−4} | 2.8 × 10^{−2} | 2.8 × 10^{−2} |

0.75 | 1 | RVCE | 2.1 × 10^{−5} | 5.4 × 10^{−5} | 5.3 × 10^{−4} | 1.6 × 10^{−2} |

0.75 | 1 | CE | 9.4 × 10^{−5} | 2.3 × 10^{−4} | 4.3 × 10^{−2} | 4.3 × 10^{−2} |

$1.25$ | 0.1 | RVCE | 5.8 × 10^{−6} | 1.5 × 10^{−5} | 5.7 × 10^{−6} | 5.7 × 10^{−5} |

$1.25$ | 0.1 | CE | 3.2 × 10^{−4} | 8.5 × 10^{−4} | 1.4 × 10^{−3} | 1.4 × 10^{−3} |

$1.25$ | 0.25 | RVCE | 1.8 × 10^{−5} | 3.9 × 10^{−5} | 4.3 × 10^{−5} | 1.3 × 10^{−4} |

$1.25$ | 0.25 | CE | 4.1 × 10^{−4} | 1.0 × 10^{−3} | 2.7 × 10^{−3} | 2.7 × 10^{−3} |

$1.25$ | 0.5 | RVCE | 3.8 × 10^{−5} | 7.8 × 10^{−5} | 1.5 × 10^{−4} | 4.0 × 10^{−4} |

$1.25$ | 0.5 | CE | 4.3 × 10^{−4} | 7.7 × 10^{−4} | 4.8 × 10^{−3} | 4.8 × 10^{−3} |

$1.25$ | 0.75 | RVCE | 5.6 × 10^{−5} | 1.4 × 10^{−4} | 3.0 × 10^{−4} | 8.9 × 10^{−4} |

$1.25$ | 0.75 | CE | 4.0 × 10^{−4} | 7.2 × 10^{−4} | 7.9 × 10^{−3} | 7.9 × 10^{−3} |

$1.25$ | 1 | RVCE | 8.8 × 10^{−5} | 1.3 × 10^{−4} | 5.5 × 10^{−4} | 1.9 × 10^{−3} |

$1.25$ | 1 | CE | 3.8 × 10^{−4} | 6.0 × 10^{−4} | 1.2 × 10^{−2} | 1.2 × 10^{−2} |

$1.5$ | 0.1 | RVCE | 3.8 × 10^{−6} | 1.8 × 10^{−5} | 1.2 × 10^{−6} | 1.2 × 10^{−5} |

$1.5$ | 0.1 | CE | 5.9 × 10^{−4} | 1.9 × 10^{−3} | 2.2 × 10^{−4} | 2.2 × 10^{−4} |

$1.5$ | 0.25 | RVCE | 5.0 × 10^{−6} | 2.2 × 10^{−5} | 2.9 × 10^{−6} | 1.2 × 10^{−5} |

$1.5$ | 0.25 | CE | 6.0 × 10^{−4} | 2.0 × 10^{−3} | 2.4 × 10^{−4} | 2.4 × 10^{−4} |

$1.5$ | 0.5 | RVCE | 1.6 × 10^{−5} | 4.7 × 10^{−5} | 1.7 × 10^{−5} | 1.8 × 10^{−5} |

$1.5$ | 0.5 | CE | 6.2 × 10^{−4} | 1.9 × 10^{−3} | 2.6 × 10^{−4} | 2.6 × 10^{−4} |

$1.5$ | 0.75 | RVCE | 2.4 × 10^{−5} | 7.4 × 10^{−5} | 4.0 × 10^{−5} | 2.3 × 10^{−5} |

$1.5$ | 0.75 | CE | 5.6 × 10^{−4} | 1.9 × 10^{−3} | 3.3 × 10^{−4} | 3.3 × 10^{−4} |

$1.5$ | 1 | RVCE | 3.4 × 10^{−5} | 9.8 × 10^{−5} | 6.9 × 10^{−5} | 3.2 × 10^{−5} |

$1.5$ | 1 | CE | 5.9 × 10^{−4} | 1.7 × 10^{−3} | 3.9 × 10^{−4} | 3.9 × 10^{−4} |

$1.75$ | 0.1 | RVCE | 8.3 × 10^{−4} | 4.7 × 10^{−3} | 4.6 × 10^{−5} | 4.6 × 10^{−5} |

$1.75$ | 0.1 | CE | 4.5 × 10^{−2} | 4.6 × 10^{−1} | 1.2 × 10^{−3} | 4.2 × 10^{−1} |

$1.75$ | 0.25 | RVCE | 4.0 × 10^{−6} | 6.3 × 10^{−5} | 1.2 × 10^{−6} | 6.3 × 10^{−6} |

$1.75$ | 0.25 | CE | 8.0 × 10^{−4} | 6.1 × 10^{−3} | 7.1 × 10^{−5} | 7.1 × 10^{−5} |

$1.75$ | 0.5 | RVCE | 3.0 × 10^{−6} | 3.1 × 10^{−5} | 1.0 × 10^{−6} | 4.2 × 10^{−6} |

$1.75$ | 0.5 | CE | 7.8 × 10^{−4} | 7.1 × 10^{−3} | 6.5 × 10^{−5} | 6.5 × 10^{−5} |

$1.75$ | 0.75 | RVCE | 5.7 × 10^{−6} | 4.4 × 10^{−5} | 2.0 × 10^{−6} | 4.6 × 10^{−6} |

$1.75$ | 0.75 | CE | 7.6 × 10^{−4} | 7.4 × 10^{−3} | 5.8 × 10^{−5} | 5.8 × 10^{−5} |

$1.75$ | 1 | RVCE | 8.1 × 10^{−6} | 9.1 × 10^{−5} | 3.4 × 10^{−6} | 5.2 × 10^{−6} |

$1.75$ | 1 | CE | 9.7 × 10^{−4} | 9.4 × 10^{−3} | 7.3 × 10^{−5} | 7.3 × 10^{−5} |

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Argument | Values | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

${u}_{1}$ | 0.03 | 0.03 | 0.03 | 0.03 | 0.3 | 0.3 | 0.3 | 0.3 | 3 | 3 | 3 | 3 |

${u}_{2}$ | 0.09 | 0.9 | 9 | 90 | 0.09 | 0.9 | 9 | 90 | 0.09 | 0.9 | 9 | 90 |

Formula | (8) | (10) | (9) | (11) | |||
---|---|---|---|---|---|---|---|

$\mathbf{\alpha}$ | $\mathbf{\beta}$ | MSE (${\widehat{\mathbf{\alpha}}}_{\mathit{n}}$) | MSE (${\widehat{\mathbf{\beta}}}_{\mathit{n}}$) | MSE (${\widehat{\mathbf{\beta}}}_{\mathit{n}}$) | MSE (${\widehat{\mathbf{\gamma}}}_{\mathit{n}}$) | MSE (${\widehat{\mathbf{\delta}}}_{\mathit{n}}$) | MSE (${\widehat{\mathbf{\delta}}}_{\mathit{n}}$) |

0.95 | 0.1 | 0.0000 | 0.0001 | 0.0003 | 0.0000 | 0.0458 | 1.6567 |

0.95 | 1 | 0.0003 | 0.0008 | 0.0688 | 0.0023 | 9 × 10^{1} | 2 × 10^{2} |

0.96 | 0.1 | 0.0000 | 0.0001 | 0.0002 | 0.0001 | 0.1537 | 2.6231 |

0.96 | 1 | 0.0005 | 0.0012 | 0.0496 | 0.0044 | 5 × 10^{4} | 3 × 10^{2} |

0.98 | 0.1 | 0.0001 | 0.0002 | 0.0003 | 0.0003 | 7 × 10^{1} | 1 × 10^{1} |

0.98 | 1 | 0.0009 | 0.0027 | 0.0309 | 0.0135 | 4 × 10^{5} | 1 × 10^{3} |

0.99 | 0.1 | 0.0002 | 0.0005 | 0.0005 | 0.0007 | 1 × 10^{6} | 4 × 10^{1} |

0.99 | 1 | 0.0017 | 0.0045 | 0.0553 | 0.0333 | 2 × 10^{4} | 4 × 10^{3} |

1.01 | 0.1 | 0.0002 | 0.0004 | 0.0004 | 0.0007 | 2 × 10^{5} | 4 × 10^{1} |

1.01 | 1 | 0.0016 | 0.0060 | 0.0383 | 0.0278 | 4 × 10^{6} | 4 × 10^{3} |

1.02 | 0.1 | 0.0001 | 0.0002 | 0.0002 | 0.0002 | 1 × 10^{3} | 1 × 10^{1} |

1.02 | 1 | 0.0009 | 0.0027 | 0.0202 | 0.0108 | 4 × 10^{4} | 9 × 10^{2} |

1.04 | 0.1 | 0.0001 | 0.0001 | 0.0002 | 0.0001 | 0.1101 | 2.4367 |

1.04 | 1 | 0.0006 | 0.0015 | 0.0340 | 0.0046 | 2 × 10^{4} | 2 × 10^{2} |

1.05 | 0.1 | 0.0000 | 0.0001 | 0.0001 | 0.0000 | 0.0458 | 1.5814 |

1.05 | 1 | 0.0005 | 0.0013 | 0.0392 | 0.0032 | 3 × 10^{2} | 1 × 10^{2} |

Formula | (8) | (10) | (9) | (11) | ||
---|---|---|---|---|---|---|

$\mathbf{\beta}$ | MSE (${\widehat{\mathbf{\alpha}}}_{\mathit{n}}$) | MSE (${\widehat{\mathbf{\beta}}}_{\mathit{n}}$) | MSE (${\widehat{\mathbf{\beta}}}_{\mathit{n}}$) | MSE (${\widehat{\mathbf{\gamma}}}_{\mathit{n}}$) | MSE (${\widehat{\mathbf{\delta}}}_{\mathit{n}}$) | MSE (${\widehat{\mathbf{\delta}}}_{\mathit{n}}$) |

0.1 | 0.0002 | 0.0004 | 0.0004 | 0.0013 | 5 × 10^{1} | 0.0012 |

0.25 | 0.0002 | 0.0609 | 0.0609 | 0.0216 | 3 × 10^{3} | 0.0072 |

0.5 | 0.0002 | 0.2421 | 0.2422 | 0.0348 | 2 × 10^{1} | 0.0430 |

0.75 | 0.0002 | 0.5476 | 0.5473 | 0.0443 | 1 × 10^{3} | 0.0938 |

1 | 0.0002 | 0.9717 | 0.9713 | 0.3854 | 5 × 10^{3} | 0.1925 |

Min | 1st Quartile | Median | Mean | 3rd Quartile | Maximum |
---|---|---|---|---|---|

15.3 | 358.0 | 955.0 | 6703.0 | 2781.0 | 1166000.0 |

${\mathit{u}}_{1}$ | ${\mathit{u}}_{2}$ | Mean (${\widehat{\mathit{\alpha}}}_{\mathit{n}}$) | Mean (${\widehat{\mathit{\beta}}}_{\mathit{n}}$) | Mean (${\widehat{\mathit{\gamma}}}_{\mathit{n}}$) | Mean (${\widehat{\mathit{\delta}}}_{\mathit{n}}$) |
---|---|---|---|---|---|

0.03 | 0.09 | 0.13 | 1.62 | 9 × 10^{77} | –3.77 |

0.03 | 0.9 | 0.01 | 6.70 | Inf | 1.53 |

0.03 | 9 | 0.03 | 2.98 | Inf | –0.10 |

0.03 | 90 | 0.02 | –0.76 | Inf | –0.02 |

0.3 | 0.09 | 0.04 | –0.42 | Inf | 3.05 |

0.3 | 0.9 | –0.17 | –1.22 | Inf | 1.93 |

0.3 | 9 | –0.01 | –1.46 | Inf | –0.17 |

0.3 | 90 | –0.01 | 0.79 | Inf | –0.02 |

3 | 0.09 | 0.01 | –2.35 | Inf | 0.12 |

3 | 0.9 | 0.14 | 2.18 | 3 × 10^{125} | –0.87 |

3 | 9 | –0.04 | 0.36 | 5× 10^{184} | –0.29 |

3 | 90 | –0.03 | 0.59 | Inf | –0.02 |

${\mathit{u}}_{1}$ | ${\mathit{u}}_{2}$ | Mean | CV | Mean | CV | Mean | CV | Mean | CV |
---|---|---|---|---|---|---|---|---|---|

(${\widehat{\mathit{\alpha}}}_{\mathit{n}}$) | (${\widehat{\mathit{\alpha}}}_{\mathit{n}}$) | (${\widehat{\mathit{\beta}}}_{\mathit{n}}$) | (${\widehat{\mathit{\beta}}}_{\mathit{n}}$) | (${\widehat{\mathit{\gamma}}}_{\mathit{n}}$) | (${\widehat{\mathit{\gamma}}}_{\mathit{n}}$) | (${\widehat{\mathit{\delta}}}_{\mathit{n}}$) | (${\widehat{\mathit{\delta}}}_{\mathit{n}}$) | ||

0.03 | 9 | 0.71 | 0.030 | 1.19 | 0.059 | 382.46 | 0.073 | –432.25 | 0.206 |

3 | 0.09 | 0.72 | 0.030 | 1.12 | 0.042 | 444.48 | 0.045 | –574.15 | 0.209 |

0.3 | 9 | 0.67 | 0.032 | 1.17 | 0.046 | 410.33 | 0.059 | –335.48 | 0.181 |

0.03 | 0.9 | 0.77 | 0.039 | 1.05 | 0.057 | 568.07 | 0.057 | –1113.55 | 0.328 |

0.03 | 90 | 0.56 | 0.039 | 1.78 | 0.079 | 119.88 | 0.192 | –103.29 | 0.181 |

0.3 | 0.9 | 0.80 | 0.048 | 1.06 | 0.055 | 578.91 | 0.059 | –1460.33 | 0.342 |

0.3 | 90 | 0.48 | 0.058 | 1.87 | 0.085 | 181.45 | 0.175 | –85.05 | 0.183 |

3 | 0.9 | 0.60 | 0.064 | 1.18 | 0.061 | 475.25 | 0.046 | –283.09 | 0.345 |

0.3 | 0.09 | 0.75 | 0.069 | 1.00 | 0.072 | 523.66 | 0.145 | –819.13 | 0.793 |

0.03 | 0.09 | 0.78 | 0.099 | 1.09 | 0.101 | 581.48 | 0.304 | –1989.91 | 1.272 |

3 | 9 | 0.60 | 0.107 | 0.94 | 0.133 | 484.41 | 0.092 | –139.00 | 0.608 |

3 | 90 | 0.33 | 0.151 | 2.00 | 0.157 | 690.79 | 0.194 | –60.62 | 0.204 |

Fitting Method | Estimated Stable Distribution |
---|---|

Reduced values’ cumulant estimates | $S(\alpha =0.72,\beta =1,\gamma =444,\delta =-574;1)$ |

Characteristic function based [23] | $S(\alpha =0.78,\beta =1,\gamma =581,\delta =-1117;1)$ |

Maximum likelihood based [23] | $S(\alpha =0.60,\beta =1,\gamma =606,\delta =-189;1)$ |

Quantile based [23] | $S(\alpha =0.82,\beta =1,\gamma =1213,\delta =-3258;1)$ |

© 2016 by the author; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Krutto, A.
Parameter Estimation in Stable Law. *Risks* **2016**, *4*, 43.
https://doi.org/10.3390/risks4040043

**AMA Style**

Krutto A.
Parameter Estimation in Stable Law. *Risks*. 2016; 4(4):43.
https://doi.org/10.3390/risks4040043

**Chicago/Turabian Style**

Krutto, Annika.
2016. "Parameter Estimation in Stable Law" *Risks* 4, no. 4: 43.
https://doi.org/10.3390/risks4040043