# Inverse Unit Teissier Distribution: Theory and Practical Examples

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## Abstract

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## 1. Introduction

## 2. Distribution Formulation

#### 2.1. Cumulative Distribution and Probability Density Functions

**Proposition 1.**

**Proof.**

#### 2.2. Survival and Hazard Rate Functions

**Proposition 2.**

**Proof.**

- For any $\theta \ge 1$, we clearly have $dh\left(x\right)/\left(dx\right)>0$, implying that $h\left(x\right)$ is increasing.
- For any $\theta \in (0,1)$, the equation $dh\left(x\right)/\left(dx\right)=0$ has the following solution: $x={(1-\theta )}^{-1/\theta}$, which belongs to $[1,+\infty )$. Since ${lim}_{x\to 1}h\left(x\right)={lim}_{x\to +\infty}h\left(x\right)=0$ and $h\left(x\right)\ge 0$, it is a maximum point. In other words, in this case, $h\left(x\right)$ is unimodal, ending the proof.

## 3. Mathematical Properties

#### 3.1. Quantile Function

**Proposition 3.**

**Proof.**

#### 3.2. Heavy-Tailed Nature

**Proposition 4.**

**Proof.**

- For $\theta \in (0,1)$, any $t>0$, and $x\to +\infty $, we have ${x}^{\theta -1}-t\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\sim \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}-t$, implying that the term ${e}^{tx}$ is dominant in the integrated function; by the Riemann integral rules, we have $I=+\infty $, meaning that the UTD is heavy-tailed.
- For $\theta =1$ and any $t>1$, the same argument holds: the UTD is heavy-tailed.
- For $\theta >1$, any $t>0$, and $x\to +\infty $, we have ${x}^{\theta -1}-t\to +\infty $, implying that the term ${e}^{-{x}^{\theta}}$ is dominant in the integrated function; by the Riemann integral rules, we have $I<+\infty $, meaning that the UTD is not heavy-tailed.

#### 3.3. Stochastic Dominance

**Proposition 5.**

- Let us set $F(x;\theta )=F\left(x\right)$. Then, for any $x\in \mathbb{R}$ and ${\theta}_{2}\ge {\theta}_{1}>0$, we have$$F(x;{\theta}_{1})\le F(x;{\theta}_{2}).$$
- Let us set $h(x;\theta )=h\left(x\right)$. Then, for any $x\in \mathbb{R}$ and ${\theta}_{2}\ge {\theta}_{1}>0$, we have$$h(x;{\theta}_{1})\le h(x;{\theta}_{2}).$$

**Proof.**

- For any $x<1$, it is clear that $F(x;{\theta}_{1})=F(x;{\theta}_{2})=0$. Let us investigate the monotonicity of $F(x;\theta )$ with respect to $\theta $ for any $x\ge 1$. We have$$\frac{d}{d\theta}F(x;\theta )={e}^{1-{x}^{\theta}}{x}^{\theta}({x}^{\theta}-1)log\left(x\right)\ge 0.$$Therefore, $F(x;\theta )$ is increasing with respect to $\theta $, and as a result, for any ${\theta}_{2}\ge {\theta}_{1}>0$, we have $F(x;{\theta}_{1})\le F(x;{\theta}_{2})$.
- Similarly, for any $x<1$, it is clear that $h(x;{\theta}_{1})=h(x;{\theta}_{2})=0$. Let us investigate the monotonicity of $h(x;\theta )$ with respect to $\theta $ for any $x\ge 1$. We have$$\frac{d}{d\theta}h(x;\theta )=\frac{{x}^{\theta}-1+a{x}^{a}log\left(x\right)}{x}\ge 0.$$Therefore, $h(x;\theta )$ is increasing with respect to $\theta $, and as a result, for any ${\theta}_{2}\ge {\theta}_{1}>0$, we have $h(x;{\theta}_{1})\le h(x;{\theta}_{2})$.

#### 3.4. Incomplete and Complete Moments

**Proposition 6.**

**Proof.**

#### 3.5. Order Statistics

## 4. Estimation of the Parameter

#### 4.1. Methods

#### 4.2. Numerical Simulation

- Average of bias:$$Bias=\frac{1}{H}\sum _{i=1}^{H}|{\widehat{\theta}}_{i}-\theta |,$$
- Mean squared error:$$MSE=\frac{1}{H}\sum _{i=1}^{H}{({\widehat{\theta}}_{i}-\theta )}^{2}.$$
- Mean relative error:$$MRE=\frac{1}{H}\sum _{i=1}^{H}\frac{|{\widehat{\theta}}_{i}-\theta |}{\theta}.$$
- Average absolute difference:$${D}_{abs}=\frac{1}{nH}\sum _{i=1}^{H}\sum _{j=1}^{n}|F({x}_{ij};\theta )-F({x}_{ij};\widehat{\theta})|,$$
- Maximum absolute difference:$${D}_{max}=\frac{1}{H}\sum _{i=1}^{H}\underset{j=1,\dots ,n}{max}|F({x}_{ij};\theta )-F({x}_{ij};\widehat{\theta})|.$$
- Average squared absolute error:$$ASAE=\frac{1}{H}\sum _{i=1}^{H}\frac{|{x}_{\left(i\right)}-{\widehat{x}}_{\left(i\right)}|}{{x}_{\left(n\right)}-{x}_{\left(1\right)}},$$

- It should be noted that all the parameter estimation methods are quite reliable and very close to their actual values; they are precise.
- All the calculated measures for every scenario under consideration decrease as n rises.
- The performance of each estimation method is excellent in finding the parameter.
- Table 7 displays the overall ranks for all estimation strategies, and we see that the MLE is the best for the values of the parameters assessed in our study (total score of 65.0).

n | Est. | MLE | ADE | CVME | MPSE | OLSE | RTADE | WLSE | LTADE | MSADE | MSALDE |
---|---|---|---|---|---|---|---|---|---|---|---|

20 | BIAS | $0.{01801}^{\left\{2\right\}}$ | $0.{01924}^{\left\{5\right\}}$ | $0.{02233}^{\left\{9\right\}}$ | $0.{01751}^{\left\{1\right\}}$ | $0.{0221}^{\left\{8\right\}}$ | $0.{01921}^{\left\{4\right\}}$ | $0.{02125}^{\left\{6\right\}}$ | $0.{02474}^{\left\{10\right\}}$ | $0.{021835}^{\left\{7\right\}}$ | $0.{018933}^{\left\{3\right\}}$ |

MSE | $0.{00054}^{\left\{2\right\}}$ | $6e-{04}^{\left\{4.5\right\}}$ | $0.{00085}^{\left\{8\right\}}$ | $0.{00049}^{\left\{1\right\}}$ | $0.{00079}^{\left\{7\right\}}$ | $6e-{04}^{\left\{4.5\right\}}$ | $0.{00074}^{\left\{6\right\}}$ | $0.{00101}^{\left\{10\right\}}$ | $0.{000869}^{\left\{9\right\}}$ | $0.{000566}^{\left\{3\right\}}$ | |

MRE | $0.{07202}^{\left\{2\right\}}$ | $0.{07695}^{\left\{5\right\}}$ | $0.{08931}^{\left\{9\right\}}$ | $0.{07005}^{\left\{1\right\}}$ | $0.{08842}^{\left\{8\right\}}$ | $0.{07684}^{\left\{4\right\}}$ | $0.{08499}^{\left\{6\right\}}$ | $0.{09896}^{\left\{10\right\}}$ | $0.{087339}^{\left\{7\right\}}$ | $0.{075733}^{\left\{3\right\}}$ | |

CIL | $0.{08885}^{\left\{2\right\}}$ | $0.{0941}^{\left\{5\right\}}$ | $0.{10581}^{\left\{7\right\}}$ | $0.{0821}^{\left\{1\right\}}$ | $0.{10606}^{\left\{8\right\}}$ | $0.{08899}^{\left\{3\right\}}$ | $0.{10071}^{\left\{6\right\}}$ | $0.{16145}^{\left\{10\right\}}$ | $0.{11374}^{\left\{9\right\}}$ | $0.{09233}^{\left\{4\right\}}$ | |

${D}_{abs}$ | $0.{04326}^{\left\{2\right\}}$ | $0.{04742}^{\left\{5\right\}}$ | $0.{0533}^{\left\{9\right\}}$ | $0.{0432}^{\left\{1\right\}}$ | $0.{05299}^{\left\{8\right\}}$ | $0.{04613}^{\left\{3\right\}}$ | $0.{05037}^{\left\{6\right\}}$ | $0.{05692}^{\left\{10\right\}}$ | $0.{0516}^{\left\{7\right\}}$ | $0.{04736}^{\left\{4\right\}}$ | |

${D}_{max}$ | $0.{06332}^{\left\{1\right\}}$ | $0.{06925}^{\left\{4\right\}}$ | $0.{0789}^{\left\{9\right\}}$ | $0.{0637}^{\left\{2\right\}}$ | $0.{07855}^{\left\{8\right\}}$ | $0.{06834}^{\left\{3\right\}}$ | $0.{07481}^{\left\{6\right\}}$ | $0.{08607}^{\left\{10\right\}}$ | $0.{07764}^{\left\{7\right\}}$ | $0.{06937}^{\left\{5\right\}}$ | |

ASAE | $0.{04918}^{\left\{3\right\}}$ | $0.{06303}^{\left\{7\right\}}$ | $0.{06803}^{\left\{8\right\}}$ | $0.{04772}^{\left\{2\right\}}$ | $0.{06938}^{\left\{9\right\}}$ | $0.{05354}^{\left\{5\right\}}$ | $0.{06091}^{\left\{6\right\}}$ | $0.{07825}^{\left\{10\right\}}$ | $0.{05227}^{\left\{4\right\}}$ | $0.{04759}^{\left\{1\right\}}$ | |

$\sum Ranks$ | ${14}^{\left\{2\right\}}$ | $40.{5}^{\left\{6\right\}}$ | ${57}^{\left\{9\right\}}$ | ${9}^{\left\{1\right\}}$ | ${54}^{\left\{8\right\}}$ | $31.{5}^{\left\{4\right\}}$ | ${40}^{\left\{5\right\}}$ | ${68}^{\left\{10\right\}}$ | ${48}^{\left\{7\right\}}$ | ${23}^{\left\{3\right\}}$ | |

40 | BIAS | $0.{01305}^{\left\{2\right\}}$ | $0.{01377}^{\left\{6\right\}}$ | $0.{01403}^{\left\{7\right\}}$ | $0.{01226}^{\left\{1\right\}}$ | $0.{01488}^{\left\{8\right\}}$ | $0.{01314}^{\left\{3\right\}}$ | $0.{01364}^{\left\{4\right\}}$ | $0.{01716}^{\left\{10\right\}}$ | $0.{016157}^{\left\{9\right\}}$ | $0.{013705}^{\left\{5\right\}}$ |

MSE | $0.{00026}^{\left\{2\right\}}$ | $3e-{04}^{\left\{5.5\right\}}$ | $0.{00033}^{\left\{7\right\}}$ | $0.{00024}^{\left\{1\right\}}$ | $0.{00037}^{\left\{8\right\}}$ | $0.{00027}^{\left\{3\right\}}$ | $3e-{04}^{\left\{5.5\right\}}$ | $0.{00048}^{\left\{10\right\}}$ | $0.{00043}^{\left\{9\right\}}$ | $0.{000297}^{\left\{4\right\}}$ | |

MRE | $0.{05218}^{\left\{2\right\}}$ | $0.{05508}^{\left\{6\right\}}$ | $0.{05613}^{\left\{7\right\}}$ | $0.{04903}^{\left\{1\right\}}$ | $0.{05951}^{\left\{8\right\}}$ | $0.{05257}^{\left\{3\right\}}$ | $0.{05457}^{\left\{4\right\}}$ | $0.{06864}^{\left\{10\right\}}$ | $0.{064626}^{\left\{9\right\}}$ | $0.{054822}^{\left\{5\right\}}$ | |

CIL | $0.{05996}^{\left\{2\right\}}$ | $0.{06619}^{\left\{5\right\}}$ | $0.{07239}^{\left\{8\right\}}$ | $0.{05766}^{\left\{1\right\}}$ | $0.{07208}^{\left\{7\right\}}$ | $0.{0622}^{\left\{3\right\}}$ | $0.{06784}^{\left\{6\right\}}$ | $0.{08669}^{\left\{10\right\}}$ | $0.{07696}^{\left\{9\right\}}$ | $0.{06583}^{\left\{4\right\}}$ | |

${D}_{abs}$ | $0.{03146}^{\left\{2\right\}}$ | $0.{03337}^{\left\{5\right\}}$ | $0.{03394}^{\left\{7\right\}}$ | $0.{03017}^{\left\{1\right\}}$ | $0.{03642}^{\left\{8\right\}}$ | $0.{03185}^{\left\{3\right\}}$ | $0.{03318}^{\left\{4\right\}}$ | $0.{04111}^{\left\{10\right\}}$ | $0.{03921}^{\left\{9\right\}}$ | $0.{03375}^{\left\{6\right\}}$ | |

${D}_{max}$ | $0.{04689}^{\left\{2\right\}}$ | $0.{04969}^{\left\{5\right\}}$ | $0.{05045}^{\left\{7\right\}}$ | $0.{0447}^{\left\{1\right\}}$ | $0.{05386}^{\left\{8\right\}}$ | $0.{04752}^{\left\{3\right\}}$ | $0.{04932}^{\left\{4\right\}}$ | $0.{06139}^{\left\{10\right\}}$ | $0.{05848}^{\left\{9\right\}}$ | $0.{05027}^{\left\{6\right\}}$ | |

ASAE | $0.{02731}^{\left\{2\right\}}$ | $0.{03399}^{\left\{7\right\}}$ | $0.{03657}^{\left\{8\right\}}$ | $0.{02722}^{\left\{1\right\}}$ | $0.{04011}^{\left\{9\right\}}$ | $0.{03068}^{\left\{3\right\}}$ | $0.{03297}^{\left\{6\right\}}$ | $0.{04125}^{\left\{10\right\}}$ | $0.{03155}^{\left\{5\right\}}$ | $0.{03134}^{\left\{4\right\}}$ | |

$\sum Ranks$ | ${14}^{\left\{2\right\}}$ | $43.{5}^{\left\{6\right\}}$ | ${49}^{\left\{7\right\}}$ | ${7}^{\left\{1\right\}}$ | ${54}^{\left\{8\right\}}$ | ${21}^{\left\{3\right\}}$ | $37.{5}^{\left\{5\right\}}$ | ${68}^{\left\{10\right\}}$ | ${57}^{\left\{9\right\}}$ | ${34}^{\left\{4\right\}}$ | |

80 | BIAS | $0.{00844}^{\left\{1.5\right\}}$ | $0.{00892}^{\left\{3\right\}}$ | $0.{01007}^{\left\{5\right\}}$ | $0.{00925}^{\left\{4\right\}}$ | $0.{0111}^{\left\{9\right\}}$ | $0.{00844}^{\left\{1.5\right\}}$ | $0.{01008}^{\left\{6\right\}}$ | $0.{01218}^{\left\{10\right\}}$ | $0.{010728}^{\left\{8\right\}}$ | $0.{010322}^{\left\{7\right\}}$ |

MSE | $0.{00012}^{\left\{1.5\right\}}$ | $0.{00013}^{\left\{3.5\right\}}$ | $0.{00017}^{\left\{7\right\}}$ | $0.{00013}^{\left\{3.5\right\}}$ | $2e-{04}^{\left\{9\right\}}$ | $0.{00012}^{\left\{1.5\right\}}$ | $0.{00016}^{\left\{5\right\}}$ | $0.{00024}^{\left\{10\right\}}$ | $0.{000187}^{\left\{8\right\}}$ | $0.{000168}^{\left\{6\right\}}$ | |

MRE | $0.{03377}^{\left\{2\right\}}$ | $0.{03568}^{\left\{3\right\}}$ | $0.{04027}^{\left\{5\right\}}$ | $0.{03701}^{\left\{4\right\}}$ | $0.{0444}^{\left\{9\right\}}$ | $0.{03376}^{\left\{1\right\}}$ | $0.{04031}^{\left\{6\right\}}$ | $0.{04874}^{\left\{10\right\}}$ | $0.{042913}^{\left\{8\right\}}$ | $0.{041289}^{\left\{7\right\}}$ | |

CIL | $0.{04187}^{\left\{2\right\}}$ | $0.{04649}^{\left\{4\right\}}$ | $0.{05062}^{\left\{8\right\}}$ | $0.{0404}^{\left\{1\right\}}$ | $0.{05052}^{\left\{7\right\}}$ | $0.{04376}^{\left\{3\right\}}$ | $0.{04683}^{\left\{5\right\}}$ | $0.{05675}^{\left\{10\right\}}$ | $0.{05412}^{\left\{9\right\}}$ | $0.{04697}^{\left\{6\right\}}$ | |

${D}_{abs}$ | $0.{02047}^{\left\{2\right\}}$ | $0.{02177}^{\left\{3\right\}}$ | $0.{02423}^{\left\{5\right\}}$ | $0.{02278}^{\left\{4\right\}}$ | $0.{02665}^{\left\{9\right\}}$ | $0.{02038}^{\left\{1\right\}}$ | $0.{02453}^{\left\{6\right\}}$ | $0.{02942}^{\left\{10\right\}}$ | $0.{02594}^{\left\{8\right\}}$ | $0.{02536}^{\left\{7\right\}}$ | |

${D}_{max}$ | $0.{03046}^{\left\{1\right\}}$ | $0.{03239}^{\left\{3\right\}}$ | $0.{03629}^{\left\{5\right\}}$ | $0.{03392}^{\left\{4\right\}}$ | $0.{03998}^{\left\{9\right\}}$ | $0.{03056}^{\left\{2\right\}}$ | $0.{03652}^{\left\{6\right\}}$ | $0.{04391}^{\left\{10\right\}}$ | $0.{03893}^{\left\{8\right\}}$ | $0.{03772}^{\left\{7\right\}}$ | |

ASAE | $0.{01595}^{\left\{2\right\}}$ | $0.{02098}^{\left\{7\right\}}$ | $0.{02129}^{\left\{9\right\}}$ | $0.{01589}^{\left\{1\right\}}$ | $0.{02108}^{\left\{8\right\}}$ | $0.{01745}^{\left\{4\right\}}$ | $0.{01872}^{\left\{6\right\}}$ | $0.{02531}^{\left\{10\right\}}$ | $0.{01798}^{\left\{5\right\}}$ | $0.{01737}^{\left\{3\right\}}$ | |

$\sum Ranks$ | ${12}^{\left\{1\right\}}$ | $26.{5}^{\left\{4\right\}}$ | ${44}^{\left\{7\right\}}$ | $21.{5}^{\left\{3\right\}}$ | ${61}^{\left\{9\right\}}$ | ${14}^{\left\{2\right\}}$ | ${40}^{\left\{5\right\}}$ | ${69}^{\left\{10\right\}}$ | ${54}^{\left\{8\right\}}$ | ${43}^{\left\{6\right\}}$ | |

120 | BIAS | $0.{00718}^{\left\{4\right\}}$ | $0.{00715}^{\left\{3\right\}}$ | $0.{00867}^{\left\{8\right\}}$ | $0.{00702}^{\left\{1\right\}}$ | $0.{00834}^{\left\{7\right\}}$ | $0.{00703}^{\left\{2\right\}}$ | $0.{0076}^{\left\{5\right\}}$ | $0.{00911}^{\left\{10\right\}}$ | $0.{009037}^{\left\{9\right\}}$ | $0.{008234}^{\left\{6\right\}}$ |

MSE | $8e-{05}^{\left\{2.5\right\}}$ | $8e-{05}^{\left\{2.5\right\}}$ | $0.{00012}^{\left\{8\right\}}$ | $8e-{05}^{\left\{2.5\right\}}$ | $0.{00011}^{\left\{7\right\}}$ | $8e-{05}^{\left\{2.5\right\}}$ | $9e-{05}^{\left\{5\right\}}$ | $0.{00013}^{\left\{9\right\}}$ | $0.{000138}^{\left\{10\right\}}$ | $0.{000103}^{\left\{6\right\}}$ | |

MRE | $0.{02874}^{\left\{4\right\}}$ | $0.{02861}^{\left\{3\right\}}$ | $0.{03469}^{\left\{8\right\}}$ | $0.{02808}^{\left\{1\right\}}$ | $0.{03335}^{\left\{7\right\}}$ | $0.{02813}^{\left\{2\right\}}$ | $0.{0304}^{\left\{5\right\}}$ | $0.{03646}^{\left\{10\right\}}$ | $0.{036149}^{\left\{9\right\}}$ | $0.{032934}^{\left\{6\right\}}$ | |

CIL | $0.{03381}^{\left\{2\right\}}$ | $0.{03767}^{\left\{4\right\}}$ | $0.{04106}^{\left\{7\right\}}$ | $0.{03335}^{\left\{1\right\}}$ | $0.{04106}^{\left\{8\right\}}$ | $0.{03531}^{\left\{3\right\}}$ | $0.{03808}^{\left\{5\right\}}$ | $0.{04615}^{\left\{10\right\}}$ | $0.{04361}^{\left\{9\right\}}$ | $0.{03858}^{\left\{6\right\}}$ | |

${D}_{abs}$ | $0.{01738}^{\left\{3\right\}}$ | $0.{01748}^{\left\{4\right\}}$ | $0.{02094}^{\left\{8\right\}}$ | $0.{01718}^{\left\{1\right\}}$ | $0.{02029}^{\left\{7\right\}}$ | $0.{0172}^{\left\{2\right\}}$ | $0.{01851}^{\left\{5\right\}}$ | $0.{02223}^{\left\{10\right\}}$ | $0.{02191}^{\left\{9\right\}}$ | $0.{02021}^{\left\{6\right\}}$ | |

${D}_{max}$ | $0.{02594}^{\left\{3\right\}}$ | $0.{02598}^{\left\{4\right\}}$ | $0.{03134}^{\left\{8\right\}}$ | $0.{02565}^{\left\{2\right\}}$ | $0.{03026}^{\left\{7\right\}}$ | $0.{02554}^{\left\{1\right\}}$ | $0.{02759}^{\left\{5\right\}}$ | $0.{0331}^{\left\{10\right\}}$ | $0.{03273}^{\left\{9\right\}}$ | $0.{0301}^{\left\{6\right\}}$ | |

ASAE | $0.{01145}^{\left\{1\right\}}$ | $0.{01427}^{\left\{7\right\}}$ | $0.{0148}^{\left\{8\right\}}$ | $0.{01167}^{\left\{2\right\}}$ | $0.{01482}^{\left\{9\right\}}$ | $0.{01354}^{\left\{5\right\}}$ | $0.{0137}^{\left\{6\right\}}$ | $0.{01627}^{\left\{10\right\}}$ | $0.{01346}^{\left\{4\right\}}$ | $0.{01198}^{\left\{3\right\}}$ | |

$\sum Ranks$ | $24.{5}^{\left\{3\right\}}$ | $32.{5}^{\left\{4\right\}}$ | ${50}^{\left\{8\right\}}$ | $15.{5}^{\left\{1\right\}}$ | ${47}^{\left\{7\right\}}$ | $22.{5}^{\left\{2\right\}}$ | ${41}^{\left\{6\right\}}$ | ${64}^{\left\{10\right\}}$ | ${54}^{\left\{9\right\}}$ | ${34}^{\left\{5\right\}}$ | |

160 | BIAS | $0.{00631}^{\left\{3\right\}}$ | $0.{00661}^{\left\{4\right\}}$ | $0.{00703}^{\left\{7\right\}}$ | $0.{00593}^{\left\{1\right\}}$ | $0.{0072}^{\left\{8\right\}}$ | $0.{00624}^{\left\{2\right\}}$ | $0.{00686}^{\left\{5\right\}}$ | $0.{00842}^{\left\{10\right\}}$ | $0.{007673}^{\left\{9\right\}}$ | $0.{006958}^{\left\{6\right\}}$ |

MSE | $6e-{05}^{\left\{2\right\}}$ | $7e-{05}^{\left\{4.5\right\}}$ | $8e-{05}^{\left\{7.5\right\}}$ | $6e-{05}^{\left\{2\right\}}$ | $8e-{05}^{\left\{7.5\right\}}$ | $6e-{05}^{\left\{2\right\}}$ | $7e-{05}^{\left\{4.5\right\}}$ | $0.{00011}^{\left\{10\right\}}$ | $9.7e-{05}^{\left\{9\right\}}$ | $7.5e-{05}^{\left\{6\right\}}$ | |

MRE | $0.{02523}^{\left\{3\right\}}$ | $0.{02643}^{\left\{4\right\}}$ | $0.{0281}^{\left\{7\right\}}$ | $0.{02372}^{\left\{1\right\}}$ | $0.{02879}^{\left\{8\right\}}$ | $0.{02496}^{\left\{2\right\}}$ | $0.{02743}^{\left\{5\right\}}$ | $0.{03366}^{\left\{10\right\}}$ | $0.{030693}^{\left\{9\right\}}$ | $0.{02783}^{\left\{6\right\}}$ | |

CIL | $0.{02926}^{\left\{2\right\}}$ | $0.{03273}^{\left\{4\right\}}$ | $0.{03541}^{\left\{8\right\}}$ | $0.{0289}^{\left\{1\right\}}$ | $0.{03531}^{\left\{7\right\}}$ | $0.{03076}^{\left\{3\right\}}$ | $0.{03311}^{\left\{5\right\}}$ | $0.{03995}^{\left\{10\right\}}$ | $0.{03748}^{\left\{9\right\}}$ | $0.{03365}^{\left\{6\right\}}$ | |

${D}_{abs}$ | $0.{01533}^{\left\{3\right\}}$ | $0.{01603}^{\left\{4\right\}}$ | $0.{01706}^{\left\{6\right\}}$ | $0.{01451}^{\left\{1\right\}}$ | $0.{01756}^{\left\{8\right\}}$ | $0.{01517}^{\left\{2\right\}}$ | $0.{01664}^{\left\{5\right\}}$ | $0.{02045}^{\left\{10\right\}}$ | $0.{01871}^{\left\{9\right\}}$ | $0.{01709}^{\left\{7\right\}}$ | |

${D}_{max}$ | $0.{02287}^{\left\{3\right\}}$ | $0.{02395}^{\left\{4\right\}}$ | $0.{02546}^{\left\{7\right\}}$ | $0.{02161}^{\left\{1\right\}}$ | $0.{02612}^{\left\{8\right\}}$ | $0.{02267}^{\left\{2\right\}}$ | $0.{02488}^{\left\{5\right\}}$ | $0.{03049}^{\left\{10\right\}}$ | $0.{02786}^{\left\{9\right\}}$ | $0.{02542}^{\left\{6\right\}}$ | |

ASAE | $0.{00903}^{\left\{1\right\}}$ | $0.{01064}^{\left\{6\right\}}$ | $0.{01171}^{\left\{9\right\}}$ | $0.{00911}^{\left\{2\right\}}$ | $0.{01132}^{\left\{8\right\}}$ | $0.{00994}^{\left\{4\right\}}$ | $0.{01088}^{\left\{7\right\}}$ | $0.{01276}^{\left\{10\right\}}$ | $0.{00998}^{\left\{5\right\}}$ | $0.{00946}^{\left\{3\right\}}$ | |

$\sum Ranks$ | ${18}^{\left\{2.5\right\}}$ | $32.{5}^{\left\{4\right\}}$ | $52.{5}^{\left\{7\right\}}$ | ${10}^{\left\{1\right\}}$ | $55.{5}^{\left\{8\right\}}$ | ${18}^{\left\{2.5\right\}}$ | $38.{5}^{\left\{5\right\}}$ | ${61}^{\left\{10\right\}}$ | ${60}^{\left\{9\right\}}$ | ${39}^{\left\{6\right\}}$ | |

200 | BIAS | $0.{00537}^{\left\{1\right\}}$ | $0.{00624}^{\left\{4\right\}}$ | $0.{0067}^{\left\{7\right\}}$ | $0.{00576}^{\left\{3\right\}}$ | $0.{00697}^{\left\{9\right\}}$ | $0.{00548}^{\left\{2\right\}}$ | $0.{00627}^{\left\{5\right\}}$ | $0.{0076}^{\left\{10\right\}}$ | $0.{006803}^{\left\{8\right\}}$ | $0.{006282}^{\left\{6\right\}}$ |

MSE | $5e-{05}^{\left\{2\right\}}$ | $6e-{05}^{\left\{4.5\right\}}$ | $7e-{05}^{\left\{7.5\right\}}$ | $5e-{05}^{\left\{2\right\}}$ | $7e-{05}^{\left\{7.5\right\}}$ | $5e-{05}^{\left\{2\right\}}$ | $6e-{05}^{\left\{4.5\right\}}$ | $9e-{05}^{\left\{10\right\}}$ | $7.1e-{05}^{\left\{9\right\}}$ | $6.1e-{05}^{\left\{6\right\}}$ | |

MRE | $0.{0215}^{\left\{1\right\}}$ | $0.{02496}^{\left\{4\right\}}$ | $0.{02681}^{\left\{7\right\}}$ | $0.{02304}^{\left\{3\right\}}$ | $0.{02789}^{\left\{9\right\}}$ | $0.{02191}^{\left\{2\right\}}$ | $0.{02509}^{\left\{5\right\}}$ | $0.{03039}^{\left\{10\right\}}$ | $0.{02721}^{\left\{8\right\}}$ | $0.{025129}^{\left\{6\right\}}$ | |

CIL | $0.{02602}^{\left\{2\right\}}$ | $0.{02927}^{\left\{4\right\}}$ | $0.{03171}^{\left\{7\right\}}$ | $0.{02569}^{\left\{1\right\}}$ | $0.{03177}^{\left\{8\right\}}$ | $0.{02762}^{\left\{3\right\}}$ | $0.{02954}^{\left\{5\right\}}$ | $0.{03568}^{\left\{10\right\}}$ | $0.{03346}^{\left\{9\right\}}$ | $0.{03001}^{\left\{6\right\}}$ | |

${D}_{abs}$ | $0.{01305}^{\left\{1\right\}}$ | $0.{01518}^{\left\{5\right\}}$ | $0.{01628}^{\left\{7\right\}}$ | $0.{01404}^{\left\{3\right\}}$ | $0.{01693}^{\left\{9\right\}}$ | $0.{01332}^{\left\{2\right\}}$ | $0.{01515}^{\left\{4\right\}}$ | $0.{01834}^{\left\{10\right\}}$ | $0.{01661}^{\left\{8\right\}}$ | $0.{01537}^{\left\{6\right\}}$ | |

${D}_{max}$ | $0.{01948}^{\left\{1\right\}}$ | $0.{02265}^{\left\{4\right\}}$ | $0.{0243}^{\left\{7\right\}}$ | $0.{02103}^{\left\{3\right\}}$ | $0.{02531}^{\left\{9\right\}}$ | $0.{01989}^{\left\{2\right\}}$ | $0.{02273}^{\left\{5\right\}}$ | $0.{02748}^{\left\{10\right\}}$ | $0.{02479}^{\left\{8\right\}}$ | $0.{02294}^{\left\{6\right\}}$ | |

ASAE | $0.{00748}^{\left\{1\right\}}$ | $0.{00894}^{\left\{6\right\}}$ | $0.{00954}^{\left\{8\right\}}$ | $0.{00779}^{\left\{2\right\}}$ | $0.{0098}^{\left\{9\right\}}$ | $0.{00831}^{\left\{4\right\}}$ | $0.{00944}^{\left\{7\right\}}$ | $0.{01096}^{\left\{10\right\}}$ | $0.{00877}^{\left\{5\right\}}$ | $0.{0081}^{\left\{3\right\}}$ | |

$\sum Ranks$ | ${9}^{\left\{1\right\}}$ | $32.{5}^{\left\{4\right\}}$ | $51.{5}^{\left\{7\right\}}$ | ${17}^{\left\{2.5\right\}}$ | $61.{5}^{\left\{9\right\}}$ | ${17}^{\left\{2.5\right\}}$ | $36.{5}^{\left\{5\right\}}$ | ${70}^{\left\{10\right\}}$ | ${53}^{\left\{8\right\}}$ | ${37}^{\left\{6\right\}}$ | |

250 | BIAS | $0.{00463}^{\left\{1\right\}}$ | $0.{00588}^{\left\{8\right\}}$ | $0.{00574}^{\left\{6\right\}}$ | $0.{00497}^{\left\{2\right\}}$ | $0.{00585}^{\left\{7\right\}}$ | $0.{00506}^{\left\{3\right\}}$ | $0.{00551}^{\left\{4\right\}}$ | $0.{00643}^{\left\{9\right\}}$ | $0.{006569}^{\left\{10\right\}}$ | $0.{005732}^{\left\{5\right\}}$ |

MSE | $3e-{05}^{\left\{1\right\}}$ | $5e-{05}^{\left\{5\right\}}$ | $5e-{05}^{\left\{5\right\}}$ | $4e-{05}^{\left\{2.5\right\}}$ | $6e-{05}^{\left\{8\right\}}$ | $4e-{05}^{\left\{2.5\right\}}$ | $5e-{05}^{\left\{5\right\}}$ | $7e-{05}^{\left\{10\right\}}$ | $6.7e-{05}^{\left\{9\right\}}$ | $5.2e-{05}^{\left\{7\right\}}$ | |

MRE | $0.{01851}^{\left\{1\right\}}$ | $0.{02353}^{\left\{8\right\}}$ | $0.{02297}^{\left\{6\right\}}$ | $0.{01989}^{\left\{2\right\}}$ | $0.{02341}^{\left\{7\right\}}$ | $0.{02024}^{\left\{3\right\}}$ | $0.{02203}^{\left\{4\right\}}$ | $0.{0257}^{\left\{9\right\}}$ | $0.{026276}^{\left\{10\right\}}$ | $0.{022928}^{\left\{5\right\}}$ | |

CIL | $0.{02325}^{\left\{2\right\}}$ | $0.{02613}^{\left\{4\right\}}$ | $0.{02836}^{\left\{8\right\}}$ | $0.{02309}^{\left\{1\right\}}$ | $0.{02826}^{\left\{7\right\}}$ | $0.{02461}^{\left\{3\right\}}$ | $0.{02622}^{\left\{5\right\}}$ | $0.{03169}^{\left\{10\right\}}$ | $0.{02997}^{\left\{9\right\}}$ | $0.{02705}^{\left\{6\right\}}$ | |

${D}_{abs}$ | $0.{01122}^{\left\{1\right\}}$ | $0.{01422}^{\left\{8\right\}}$ | $0.{01393}^{\left\{5\right\}}$ | $0.{01212}^{\left\{2\right\}}$ | $0.{01421}^{\left\{7\right\}}$ | $0.{0123}^{\left\{3\right\}}$ | $0.{01339}^{\left\{4\right\}}$ | $0.{01556}^{\left\{9\right\}}$ | $0.{01598}^{\left\{10\right\}}$ | $0.{01394}^{\left\{6\right\}}$ | |

${D}_{max}$ | $0.{01677}^{\left\{1\right\}}$ | $0.{02129}^{\left\{8\right\}}$ | $0.{02082}^{\left\{5\right\}}$ | $0.{01813}^{\left\{2\right\}}$ | $0.{02123}^{\left\{7\right\}}$ | $0.{01837}^{\left\{3\right\}}$ | $0.{02001}^{\left\{4\right\}}$ | $0.{02329}^{\left\{9\right\}}$ | $0.{02387}^{\left\{10\right\}}$ | $0.{02084}^{\left\{6\right\}}$ | |

ASAE | $0.{00651}^{\left\{2\right\}}$ | $0.{00769}^{\left\{6\right\}}$ | $0.{00815}^{\left\{8\right\}}$ | $0.{00635}^{\left\{1\right\}}$ | $0.{00831}^{\left\{9\right\}}$ | $0.{00652}^{\left\{3\right\}}$ | $0.{0077}^{\left\{7\right\}}$ | $0.{00914}^{\left\{10\right\}}$ | $0.{00746}^{\left\{5\right\}}$ | $0.{00665}^{\left\{4\right\}}$ | |

$\sum Ranks$ | ${9}^{\left\{1\right\}}$ | ${48}^{\left\{7\right\}}$ | ${44}^{\left\{6\right\}}$ | $12.{5}^{\left\{2\right\}}$ | ${53}^{\left\{8\right\}}$ | $20.{5}^{\left\{3\right\}}$ | ${34}^{\left\{4\right\}}$ | ${66}^{\left\{10\right\}}$ | ${62}^{\left\{9\right\}}$ | ${36}^{\left\{5\right\}}$ | |

350 | BIAS | $0.{00405}^{\left\{1\right\}}$ | $0.{00494}^{\left\{6\right\}}$ | $0.{0047}^{\left\{5\right\}}$ | $0.{0044}^{\left\{3\right\}}$ | $0.{00495}^{\left\{7\right\}}$ | $0.{00432}^{\left\{2\right\}}$ | $0.{0046}^{\left\{4\right\}}$ | $0.{00575}^{\left\{10\right\}}$ | $0.{005383}^{\left\{9\right\}}$ | $0.{004994}^{\left\{8\right\}}$ |

MSE | $3e-{05}^{\left\{3\right\}}$ | $4e-{05}^{\left\{7.5\right\}}$ | $3e-{05}^{\left\{3\right\}}$ | $3e-{05}^{\left\{3\right\}}$ | $4e-{05}^{\left\{7.5\right\}}$ | $3e-{05}^{\left\{3\right\}}$ | $3e-{05}^{\left\{3\right\}}$ | $5e-{05}^{\left\{10\right\}}$ | $4.4e-{05}^{\left\{9\right\}}$ | $3.7e-{05}^{\left\{6\right\}}$ | |

MRE | $0.{01619}^{\left\{1\right\}}$ | $0.{01974}^{\left\{6\right\}}$ | $0.{0188}^{\left\{5\right\}}$ | $0.{0176}^{\left\{3\right\}}$ | $0.{01981}^{\left\{7\right\}}$ | $0.{01727}^{\left\{2\right\}}$ | $0.{01838}^{\left\{4\right\}}$ | $0.{02302}^{\left\{10\right\}}$ | $0.{021532}^{\left\{9\right\}}$ | $0.{019977}^{\left\{8\right\}}$ | |

CIL | $0.{01968}^{\left\{2\right\}}$ | $0.{02225}^{\left\{5\right\}}$ | $0.{02384}^{\left\{8\right\}}$ | $0.{01951}^{\left\{1\right\}}$ | $0.{02373}^{\left\{7\right\}}$ | $0.{02084}^{\left\{3\right\}}$ | $0.{02215}^{\left\{4\right\}}$ | $0.{02674}^{\left\{10\right\}}$ | $0.{02507}^{\left\{9\right\}}$ | $0.{0229}^{\left\{6\right\}}$ | |

${D}_{abs}$ | $0.{00985}^{\left\{1\right\}}$ | $0.{012}^{\left\{6\right\}}$ | $0.{01145}^{\left\{5\right\}}$ | $0.{01075}^{\left\{3\right\}}$ | $0.{01207}^{\left\{7\right\}}$ | $0.{0105}^{\left\{2\right\}}$ | $0.{01118}^{\left\{4\right\}}$ | $0.{01397}^{\left\{10\right\}}$ | $0.{01311}^{\left\{9\right\}}$ | $0.{01217}^{\left\{8\right\}}$ | |

${D}_{max}$ | $0.{01469}^{\left\{1\right\}}$ | $0.{01793}^{\left\{6\right\}}$ | $0.{01707}^{\left\{5\right\}}$ | $0.{01605}^{\left\{3\right\}}$ | $0.{01799}^{\left\{7\right\}}$ | $0.{01568}^{\left\{2\right\}}$ | $0.{01669}^{\left\{4\right\}}$ | $0.{02089}^{\left\{10\right\}}$ | $0.{01959}^{\left\{9\right\}}$ | $0.{0182}^{\left\{8\right\}}$ | |

ASAE | $0.{00486}^{\left\{2\right\}}$ | $0.{00585}^{\left\{7\right\}}$ | $0.{00595}^{\left\{8\right\}}$ | $0.{00467}^{\left\{1\right\}}$ | $0.{00605}^{\left\{9\right\}}$ | $0.{00507}^{\left\{3\right\}}$ | $0.{00559}^{\left\{6\right\}}$ | $0.{00639}^{\left\{10\right\}}$ | $0.{00535}^{\left\{5\right\}}$ | $0.{00529}^{\left\{4\right\}}$ | |

$\sum Ranks$ | ${12}^{\left\{1\right\}}$ | $44.{5}^{\left\{7\right\}}$ | ${40}^{\left\{5\right\}}$ | ${18}^{\left\{2.5\right\}}$ | $52.{5}^{\left\{8\right\}}$ | ${18}^{\left\{2.5\right\}}$ | ${30}^{\left\{4\right\}}$ | ${70}^{\left\{10\right\}}$ | ${57}^{\left\{9\right\}}$ | ${43}^{\left\{6\right\}}$ |

n | Est. | MLE | ADE | CVME | MPSE | OLSE | RTADE | WLSE | LTADE | MSADE | MSALDE |
---|---|---|---|---|---|---|---|---|---|---|---|

20 | BIAS | $0.{0551}^{\left\{3\right\}}$ | $0.{05964}^{\left\{5\right\}}$ | $0.{0598}^{\left\{6\right\}}$ | $0.{05469}^{\left\{2\right\}}$ | $0.{06143}^{\left\{7\right\}}$ | $0.{05456}^{\left\{1\right\}}$ | $0.{0575}^{\left\{4\right\}}$ | $0.{07596}^{\left\{10\right\}}$ | $0.{065278}^{\left\{9\right\}}$ | $0.{062733}^{\left\{8\right\}}$ |

MSE | $0.{00531}^{\left\{3\right\}}$ | $0.{00565}^{\left\{5\right\}}$ | $0.{00568}^{\left\{6\right\}}$ | $0.{00488}^{\left\{2\right\}}$ | $0.{00624}^{\left\{7\right\}}$ | $0.{00478}^{\left\{1\right\}}$ | $0.{00552}^{\left\{4\right\}}$ | $0.{01016}^{\left\{10\right\}}$ | $0.{00696}^{\left\{9\right\}}$ | $0.{006363}^{\left\{8\right\}}$ | |

MRE | $0.{07347}^{\left\{3\right\}}$ | $0.{07951}^{\left\{5\right\}}$ | $0.{07973}^{\left\{6\right\}}$ | $0.{07291}^{\left\{2\right\}}$ | $0.{0819}^{\left\{7\right\}}$ | $0.{07275}^{\left\{1\right\}}$ | $0.{07667}^{\left\{4\right\}}$ | $0.{10128}^{\left\{10\right\}}$ | $0.{087037}^{\left\{9\right\}}$ | $0.{083644}^{\left\{8\right\}}$ | |

CIL | $0.{26393}^{\left\{2\right\}}$ | $0.{28314}^{\left\{5\right\}}$ | $0.{31971}^{\left\{8\right\}}$ | $0.{24616}^{\left\{1\right\}}$ | $0.{31274}^{\left\{7\right\}}$ | $0.{26627}^{\left\{3\right\}}$ | $0.{29891}^{\left\{6\right\}}$ | $0.{5048}^{\left\{10\right\}}$ | $0.{3367}^{\left\{9\right\}}$ | $0.{27678}^{\left\{4\right\}}$ | |

${D}_{abs}$ | $0.{04402}^{\left\{1\right\}}$ | $0.{04867}^{\left\{6\right\}}$ | $0.{04772}^{\left\{5\right\}}$ | $0.{04538}^{\left\{3\right\}}$ | $0.{04973}^{\left\{7\right\}}$ | $0.{04417}^{\left\{2\right\}}$ | $0.{0459}^{\left\{4\right\}}$ | $0.{05825}^{\left\{10\right\}}$ | $0.{05287}^{\left\{9\right\}}$ | $0.{05199}^{\left\{8\right\}}$ | |

${D}_{max}$ | $0.{0647}^{\left\{1\right\}}$ | $0.{07124}^{\left\{6\right\}}$ | $0.{07083}^{\left\{5\right\}}$ | $0.{06682}^{\left\{3\right\}}$ | $0.{07317}^{\left\{7\right\}}$ | $0.{06499}^{\left\{2\right\}}$ | $0.{06834}^{\left\{4\right\}}$ | $0.{0878}^{\left\{10\right\}}$ | $0.{0782}^{\left\{9\right\}}$ | $0.{07636}^{\left\{8\right\}}$ | |

ASAE | $0.{05296}^{\left\{5\right\}}$ | $0.{05049}^{\left\{2\right\}}$ | $0.{05358}^{\left\{6\right\}}$ | $0.{04916}^{\left\{1\right\}}$ | $0.{05407}^{\left\{7\right\}}$ | $0.{05055}^{\left\{3\right\}}$ | $0.{05121}^{\left\{4\right\}}$ | $0.{06318}^{\left\{10\right\}}$ | $0.{05553}^{\left\{9\right\}}$ | $0.{05523}^{\left\{8\right\}}$ | |

$\sum Ranks$ | ${18}^{\left\{3\right\}}$ | ${34}^{\left\{5\right\}}$ | ${42}^{\left\{6\right\}}$ | ${14}^{\left\{2\right\}}$ | ${49}^{\left\{7\right\}}$ | ${13}^{\left\{1\right\}}$ | ${30}^{\left\{4\right\}}$ | ${70}^{\left\{10\right\}}$ | ${63}^{\left\{9\right\}}$ | ${52}^{\left\{8\right\}}$ | |

40 | BIAS | $0.{03611}^{\left\{1\right\}}$ | $0.{04285}^{\left\{5\right\}}$ | $0.{04051}^{\left\{3\right\}}$ | $0.{03706}^{\left\{2\right\}}$ | $0.{0437}^{\left\{6\right\}}$ | $0.{04089}^{\left\{4\right\}}$ | $0.{04428}^{\left\{7\right\}}$ | $0.{05109}^{\left\{10\right\}}$ | $0.{045078}^{\left\{9\right\}}$ | $0.{044319}^{\left\{8\right\}}$ |

MSE | $0.{00224}^{\left\{2\right\}}$ | $0.{00281}^{\left\{5\right\}}$ | $0.{00279}^{\left\{4\right\}}$ | $0.{00214}^{\left\{1\right\}}$ | $0.{00301}^{\left\{7\right\}}$ | $0.{00258}^{\left\{3\right\}}$ | $0.{00327}^{\left\{8\right\}}$ | $0.{00423}^{\left\{10\right\}}$ | $0.{003659}^{\left\{9\right\}}$ | $0.{00297}^{\left\{6\right\}}$ | |

MRE | $0.{04815}^{\left\{1\right\}}$ | $0.{05713}^{\left\{5\right\}}$ | $0.{05402}^{\left\{3\right\}}$ | $0.{04941}^{\left\{2\right\}}$ | $0.{05827}^{\left\{6\right\}}$ | $0.{05451}^{\left\{4\right\}}$ | $0.{05904}^{\left\{7\right\}}$ | $0.{06813}^{\left\{10\right\}}$ | $0.{060104}^{\left\{9\right\}}$ | $0.{059092}^{\left\{8\right\}}$ | |

CIL | $0.{17945}^{\left\{2\right\}}$ | $0.{19924}^{\left\{5\right\}}$ | $0.{21873}^{\left\{8\right\}}$ | $0.{17214}^{\left\{1\right\}}$ | $0.{21482}^{\left\{7\right\}}$ | $0.{18583}^{\left\{3\right\}}$ | $0.{2057}^{\left\{6\right\}}$ | $0.{26987}^{\left\{10\right\}}$ | $0.{23098}^{\left\{9\right\}}$ | $0.{19657}^{\left\{4\right\}}$ | |

${D}_{abs}$ | $0.{02902}^{\left\{1\right\}}$ | $0.{03473}^{\left\{5\right\}}$ | $0.{03244}^{\left\{3\right\}}$ | $0.{0304}^{\left\{2\right\}}$ | $0.{03543}^{\left\{7\right\}}$ | $0.{03345}^{\left\{4\right\}}$ | $0.{03522}^{\left\{6\right\}}$ | $0.{04079}^{\left\{10\right\}}$ | $0.{03667}^{\left\{9\right\}}$ | $0.{03652}^{\left\{8\right\}}$ | |

${D}_{max}$ | $0.{04312}^{\left\{1\right\}}$ | $0.{05159}^{\left\{5\right\}}$ | $0.{04854}^{\left\{3\right\}}$ | $0.{04543}^{\left\{2\right\}}$ | $0.{05263}^{\left\{6\right\}}$ | $0.{04943}^{\left\{4\right\}}$ | $0.{0527}^{\left\{7\right\}}$ | $0.{06104}^{\left\{10\right\}}$ | $0.{0542}^{\left\{8\right\}}$ | $0.{05428}^{\left\{9\right\}}$ | |

ASAE | $0.{03247}^{\left\{4\right\}}$ | $0.{03371}^{\left\{5\right\}}$ | $0.{03376}^{\left\{6\right\}}$ | $0.{03176}^{\left\{2\right\}}$ | $0.{03539}^{\left\{8\right\}}$ | $0.{03142}^{\left\{1\right\}}$ | $0.{03233}^{\left\{3\right\}}$ | $0.{03799}^{\left\{10\right\}}$ | $0.{03659}^{\left\{9\right\}}$ | $0.{03444}^{\left\{7\right\}}$ | |

$\sum Ranks$ | ${12}^{\left\{1.5\right\}}$ | ${35}^{\left\{5\right\}}$ | ${30}^{\left\{4\right\}}$ | ${12}^{\left\{1.5\right\}}$ | ${47}^{\left\{7\right\}}$ | ${23}^{\left\{3\right\}}$ | ${44}^{\left\{6\right\}}$ | ${70}^{\left\{10\right\}}$ | ${62}^{\left\{9\right\}}$ | ${50}^{\left\{8\right\}}$ | |

80 | BIAS | $0.{02748}^{\left\{2\right\}}$ | $0.{02818}^{\left\{4\right\}}$ | $0.{03037}^{\left\{7\right\}}$ | $0.{02615}^{\left\{1\right\}}$ | $0.{03061}^{\left\{8\right\}}$ | $0.{02929}^{\left\{5\right\}}$ | $0.{02753}^{\left\{3\right\}}$ | $0.{03618}^{\left\{10\right\}}$ | $0.{033685}^{\left\{9\right\}}$ | $0.{029896}^{\left\{6\right\}}$ |

MSE | $0.{00116}^{\left\{2\right\}}$ | $0.{00125}^{\left\{4\right\}}$ | $0.{00153}^{\left\{8\right\}}$ | $0.{00105}^{\left\{1\right\}}$ | $0.{00149}^{\left\{7\right\}}$ | $0.{00135}^{\left\{5\right\}}$ | $0.{00121}^{\left\{3\right\}}$ | $0.{00214}^{\left\{10\right\}}$ | $0.{00184}^{\left\{9\right\}}$ | $0.{001414}^{\left\{6\right\}}$ | |

MRE | $0.{03664}^{\left\{2\right\}}$ | $0.{03757}^{\left\{4\right\}}$ | $0.{0405}^{\left\{7\right\}}$ | $0.{03486}^{\left\{1\right\}}$ | $0.{04082}^{\left\{8\right\}}$ | $0.{03905}^{\left\{5\right\}}$ | $0.{0367}^{\left\{3\right\}}$ | $0.{04823}^{\left\{10\right\}}$ | $0.{044913}^{\left\{9\right\}}$ | $0.{039861}^{\left\{6\right\}}$ | |

CIL | $0.{12496}^{\left\{2\right\}}$ | $0.{13917}^{\left\{4\right\}}$ | $0.{15085}^{\left\{7.5\right\}}$ | $0.{12219}^{\left\{1\right\}}$ | $0.{15085}^{\left\{7.5\right\}}$ | $0.{13121}^{\left\{3\right\}}$ | $0.{13998}^{\left\{5\right\}}$ | $0.{17188}^{\left\{10\right\}}$ | $0.{16158}^{\left\{9\right\}}$ | $0.{14146}^{\left\{6\right\}}$ | |

${D}_{abs}$ | $0.{02224}^{\left\{2\right\}}$ | $0.{02292}^{\left\{4\right\}}$ | $0.{0246}^{\left\{7\right\}}$ | $0.{02143}^{\left\{1\right\}}$ | $0.{02487}^{\left\{8\right\}}$ | $0.{02373}^{\left\{5\right\}}$ | $0.{02241}^{\left\{3\right\}}$ | $0.{02897}^{\left\{10\right\}}$ | $0.{02743}^{\left\{9\right\}}$ | $0.{0243}^{\left\{6\right\}}$ | |

${D}_{max}$ | $0.{0331}^{\left\{2\right\}}$ | $0.{03411}^{\left\{4\right\}}$ | $0.{03669}^{\left\{7\right\}}$ | $0.{03197}^{\left\{1\right\}}$ | $0.{03696}^{\left\{8\right\}}$ | $0.{03528}^{\left\{5\right\}}$ | $0.{0334}^{\left\{3\right\}}$ | $0.{04337}^{\left\{10\right\}}$ | $0.{04091}^{\left\{9\right\}}$ | $0.{03634}^{\left\{6\right\}}$ | |

ASAE | $0.{01998}^{\left\{1\right\}}$ | $0.{02058}^{\left\{5\right\}}$ | $0.{02167}^{\left\{7\right\}}$ | $0.{02017}^{\left\{3\right\}}$ | $0.{02151}^{\left\{6\right\}}$ | $0.{02008}^{\left\{2\right\}}$ | $0.{02021}^{\left\{4\right\}}$ | $0.{02492}^{\left\{10\right\}}$ | $0.{02392}^{\left\{9\right\}}$ | $0.{02321}^{\left\{8\right\}}$ | |

$\sum Ranks$ | ${13}^{\left\{2\right\}}$ | ${29}^{\left\{4\right\}}$ | $50.{5}^{\left\{7\right\}}$ | ${9}^{\left\{1\right\}}$ | $52.{5}^{\left\{8\right\}}$ | ${30}^{\left\{5\right\}}$ | ${24}^{\left\{3\right\}}$ | ${70}^{\left\{10\right\}}$ | ${63}^{\left\{9\right\}}$ | ${44}^{\left\{6\right\}}$ | |

120 | BIAS | $0.{01999}^{\left\{1\right\}}$ | $0.{0249}^{\left\{5\right\}}$ | $0.{02591}^{\left\{9\right\}}$ | $0.{02165}^{\left\{2\right\}}$ | $0.{02512}^{\left\{6\right\}}$ | $0.{02218}^{\left\{3\right\}}$ | $0.{02305}^{\left\{4\right\}}$ | $0.{03016}^{\left\{10\right\}}$ | $0.{025691}^{\left\{8\right\}}$ | $0.{025568}^{\left\{7\right\}}$ |

MSE | $0.{00062}^{\left\{1\right\}}$ | $0.{00102}^{\left\{7\right\}}$ | $0.{00105}^{\left\{8\right\}}$ | $0.{00071}^{\left\{2\right\}}$ | $0.{001}^{\left\{6\right\}}$ | $0.{00078}^{\left\{3\right\}}$ | $0.{00085}^{\left\{4\right\}}$ | $0.{00141}^{\left\{10\right\}}$ | $0.{001066}^{\left\{9\right\}}$ | $0.{00099}^{\left\{5\right\}}$ | |

MRE | $0.{02665}^{\left\{1\right\}}$ | $0.{0332}^{\left\{5\right\}}$ | $0.{03455}^{\left\{9\right\}}$ | $0.{02887}^{\left\{2\right\}}$ | $0.{03349}^{\left\{6\right\}}$ | $0.{02957}^{\left\{3\right\}}$ | $0.{03074}^{\left\{4\right\}}$ | $0.{04021}^{\left\{10\right\}}$ | $0.{034255}^{\left\{8\right\}}$ | $0.{03409}^{\left\{7\right\}}$ | |

CIL | $0.{10143}^{\left\{2\right\}}$ | $0.{11339}^{\left\{4\right\}}$ | $0.{12219}^{\left\{7\right\}}$ | $0.{10007}^{\left\{1\right\}}$ | $0.{12315}^{\left\{8\right\}}$ | $0.{10657}^{\left\{3\right\}}$ | $0.{1147}^{\left\{5\right\}}$ | $0.{13903}^{\left\{10\right\}}$ | $0.{12994}^{\left\{9\right\}}$ | $0.{11566}^{\left\{6\right\}}$ | |

${D}_{abs}$ | $0.{01616}^{\left\{1\right\}}$ | $0.{02012}^{\left\{5\right\}}$ | $0.{021}^{\left\{9\right\}}$ | $0.{01765}^{\left\{2\right\}}$ | $0.{02029}^{\left\{6\right\}}$ | $0.{018}^{\left\{3\right\}}$ | $0.{01866}^{\left\{4\right\}}$ | $0.{02429}^{\left\{10\right\}}$ | $0.{02086}^{\left\{7\right\}}$ | $0.{02091}^{\left\{8\right\}}$ | |

${D}_{max}$ | $0.{02416}^{\left\{1\right\}}$ | $0.{03004}^{\left\{5\right\}}$ | $0.{03132}^{\left\{9\right\}}$ | $0.{02632}^{\left\{2\right\}}$ | $0.{03033}^{\left\{6\right\}}$ | $0.{02682}^{\left\{3\right\}}$ | $0.{02784}^{\left\{4\right\}}$ | $0.{03639}^{\left\{10\right\}}$ | $0.{03126}^{\left\{8\right\}}$ | $0.{03117}^{\left\{7\right\}}$ | |

ASAE | $0.{01555}^{\left\{4\right\}}$ | $0.{01585}^{\left\{5\right\}}$ | $0.{01622}^{\left\{6\right\}}$ | $0.{01528}^{\left\{2\right\}}$ | $0.{01641}^{\left\{7\right\}}$ | $0.{01505}^{\left\{1\right\}}$ | $0.{0154}^{\left\{3\right\}}$ | $0.{01871}^{\left\{10\right\}}$ | $0.{01833}^{\left\{9\right\}}$ | $0.{01765}^{\left\{8\right\}}$ | |

$\sum Ranks$ | ${11}^{\left\{1\right\}}$ | ${36}^{\left\{5\right\}}$ | ${57}^{\left\{8\right\}}$ | ${13}^{\left\{2\right\}}$ | ${45}^{\left\{6\right\}}$ | ${19}^{\left\{3\right\}}$ | ${28}^{\left\{4\right\}}$ | ${70}^{\left\{10\right\}}$ | ${58}^{\left\{9\right\}}$ | ${48}^{\left\{7\right\}}$ | |

160 | BIAS | $0.{01777}^{\left\{1\right\}}$ | $0.{02113}^{\left\{6\right\}}$ | $0.{02332}^{\left\{9\right\}}$ | $0.{01865}^{\left\{2\right\}}$ | $0.{02131}^{\left\{7\right\}}$ | $0.{02097}^{\left\{5\right\}}$ | $0.{02006}^{\left\{4\right\}}$ | $0.{025}^{\left\{10\right\}}$ | $0.{022948}^{\left\{8\right\}}$ | $0.{019984}^{\left\{3\right\}}$ |

MSE | $5e-{04}^{\left\{1\right\}}$ | $0.{00069}^{\left\{6\right\}}$ | $0.{00087}^{\left\{9\right\}}$ | $0.{00055}^{\left\{2\right\}}$ | $0.{00071}^{\left\{7\right\}}$ | $0.{00068}^{\left\{5\right\}}$ | $0.{00062}^{\left\{3\right\}}$ | $0.{00105}^{\left\{10\right\}}$ | $0.{000848}^{\left\{8\right\}}$ | $0.{000634}^{\left\{4\right\}}$ | |

MRE | $0.{02369}^{\left\{1\right\}}$ | $0.{02818}^{\left\{6\right\}}$ | $0.{03109}^{\left\{9\right\}}$ | $0.{02486}^{\left\{2\right\}}$ | $0.{02842}^{\left\{7\right\}}$ | $0.{02796}^{\left\{5\right\}}$ | $0.{02674}^{\left\{4\right\}}$ | $0.{03334}^{\left\{10\right\}}$ | $0.{030597}^{\left\{8\right\}}$ | $0.{026646}^{\left\{3\right\}}$ | |

CIL | $0.{08749}^{\left\{2\right\}}$ | $0.{09803}^{\left\{4\right\}}$ | $0.{10602}^{\left\{7\right\}}$ | $0.{08654}^{\left\{1\right\}}$ | $0.{10607}^{\left\{8\right\}}$ | $0.{09229}^{\left\{3\right\}}$ | $0.{09887}^{\left\{5\right\}}$ | $0.{12021}^{\left\{10\right\}}$ | $0.{11287}^{\left\{9\right\}}$ | $0.{10122}^{\left\{6\right\}}$ | |

${D}_{abs}$ | $0.{01439}^{\left\{1\right\}}$ | $0.{01721}^{\left\{6\right\}}$ | $0.{01885}^{\left\{9\right\}}$ | $0.{01519}^{\left\{2\right\}}$ | $0.{01725}^{\left\{7\right\}}$ | $0.{01694}^{\left\{5\right\}}$ | $0.{01624}^{\left\{3\right\}}$ | $0.{02012}^{\left\{10\right\}}$ | $0.{01853}^{\left\{8\right\}}$ | $0.{01626}^{\left\{4\right\}}$ | |

${D}_{max}$ | $0.{02148}^{\left\{1\right\}}$ | $0.{02564}^{\left\{6\right\}}$ | $0.{02817}^{\left\{9\right\}}$ | $0.{02267}^{\left\{2\right\}}$ | $0.{02578}^{\left\{7\right\}}$ | $0.{02534}^{\left\{5\right\}}$ | $0.{02425}^{\left\{3\right\}}$ | $0.{03014}^{\left\{10\right\}}$ | $0.{0278}^{\left\{8\right\}}$ | $0.{02428}^{\left\{4\right\}}$ | |

ASAE | $0.{01259}^{\left\{2\right\}}$ | $0.{01342}^{\left\{4\right\}}$ | $0.{01414}^{\left\{7\right\}}$ | $0.{01243}^{\left\{1\right\}}$ | $0.{01368}^{\left\{6\right\}}$ | $0.{01264}^{\left\{3\right\}}$ | $0.{01357}^{\left\{5\right\}}$ | $0.{01556}^{\left\{10\right\}}$ | $0.{01546}^{\left\{9\right\}}$ | $0.{0145}^{\left\{8\right\}}$ | |

$\sum Ranks$ | ${18}^{\left\{2\right\}}$ | ${37}^{\left\{6\right\}}$ | ${58}^{\left\{9\right\}}$ | ${11}^{\left\{1\right\}}$ | ${48}^{\left\{7\right\}}$ | ${30}^{\left\{4\right\}}$ | ${26}^{\left\{3\right\}}$ | ${69}^{\left\{10\right\}}$ | ${57}^{\left\{8\right\}}$ | ${31}^{\left\{5\right\}}$ | |

200 | BIAS | $0.{01608}^{\left\{1\right\}}$ | $0.{01773}^{\left\{4\right\}}$ | $0.{01997}^{\left\{8\right\}}$ | $0.{01637}^{\left\{3\right\}}$ | $0.{01947}^{\left\{7\right\}}$ | $0.{01629}^{\left\{2\right\}}$ | $0.{01907}^{\left\{5\right\}}$ | $0.{02335}^{\left\{10\right\}}$ | $0.{02151}^{\left\{9\right\}}$ | $0.{01925}^{\left\{6\right\}}$ |

MSE | $4e-{04}^{\left\{1\right\}}$ | $0.{00049}^{\left\{4\right\}}$ | $0.{00059}^{\left\{7\right\}}$ | $0.{00042}^{\left\{2\right\}}$ | $0.{00059}^{\left\{7\right\}}$ | $0.{00043}^{\left\{3\right\}}$ | $0.{00059}^{\left\{7\right\}}$ | $0.{00083}^{\left\{10\right\}}$ | $0.{000728}^{\left\{9\right\}}$ | $0.{00057}^{\left\{5\right\}}$ | |

MRE | $0.{02144}^{\left\{1\right\}}$ | $0.{02364}^{\left\{4\right\}}$ | $0.{02663}^{\left\{8\right\}}$ | $0.{02182}^{\left\{3\right\}}$ | $0.{02596}^{\left\{7\right\}}$ | $0.{02172}^{\left\{2\right\}}$ | $0.{02542}^{\left\{5\right\}}$ | $0.{03113}^{\left\{10\right\}}$ | $0.{02868}^{\left\{9\right\}}$ | $0.{025667}^{\left\{6\right\}}$ | |

CIL | $0.{07846}^{\left\{2\right\}}$ | $0.{08753}^{\left\{4\right\}}$ | $0.{09448}^{\left\{7\right\}}$ | $0.{07747}^{\left\{1\right\}}$ | $0.{09493}^{\left\{8\right\}}$ | $0.{08178}^{\left\{3\right\}}$ | $0.{08824}^{\left\{5\right\}}$ | $0.{10679}^{\left\{10\right\}}$ | $0.{10026}^{\left\{9\right\}}$ | $0.{09046}^{\left\{6\right\}}$ | |

${D}_{abs}$ | $0.{01297}^{\left\{1\right\}}$ | $0.{01437}^{\left\{4\right\}}$ | $0.{01616}^{\left\{8\right\}}$ | $0.{0133}^{\left\{3\right\}}$ | $0.{01572}^{\left\{7\right\}}$ | $0.{01319}^{\left\{2\right\}}$ | $0.{01535}^{\left\{5\right\}}$ | $0.{01887}^{\left\{10\right\}}$ | $0.{01756}^{\left\{9\right\}}$ | $0.{01565}^{\left\{6\right\}}$ | |

${D}_{max}$ | $0.{01946}^{\left\{1\right\}}$ | $0.{02144}^{\left\{4\right\}}$ | $0.{02413}^{\left\{8\right\}}$ | $0.{01991}^{\left\{3\right\}}$ | $0.{02353}^{\left\{7\right\}}$ | $0.{01974}^{\left\{2\right\}}$ | $0.{02302}^{\left\{5\right\}}$ | $0.{0282}^{\left\{10\right\}}$ | $0.{02623}^{\left\{9\right\}}$ | $0.{02336}^{\left\{6\right\}}$ | |

ASAE | $0.{0108}^{\left\{2\right\}}$ | $0.{01127}^{\left\{4\right\}}$ | $0.{01158}^{\left\{6\right\}}$ | $0.{01104}^{\left\{3\right\}}$ | $0.{01181}^{\left\{7\right\}}$ | $0.{01076}^{\left\{1\right\}}$ | $0.{01135}^{\left\{5\right\}}$ | $0.{01355}^{\left\{10\right\}}$ | $0.{01338}^{\left\{9\right\}}$ | $$ |