# Testing Symmetry of Unknown Densities via Smoothing with the Generalized Gamma Kernels

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## Abstract

**:**

## 1. Introduction

## 2. Family of the GG Kernels: A Brief Review

**Definition 1.**

**Hirukawa and Sakudo [32], Definition 1**) Let $\left(\alpha ,\beta ,\gamma \right)=\left({\alpha}_{b}\left(x\right),{\beta}_{b}\left(x\right),{\gamma}_{b}\left(x\right)\right)\in {\mathbb{R}}_{+}^{3}$ be a continuous function of the design point x and the smoothing parameter b. For such$\left(\alpha ,\beta ,\gamma \right)$ , consider the pdf of $GG\left(\alpha ,\beta \Gamma \left(\alpha /\gamma \right)/\Gamma \left\{\left(\alpha +1\right)/\gamma \right\},\gamma \right)$ , i.e.,

**Condition 1.**

**Condition 2.**

**Condition 3.**

**Condition 4.**

**Condition 5.**

## 3. Tests for Symmetry and Conditional Symmetry Smoothed by the GG Kernels

#### 3.1. SSST as a Special Case of Two-Sample Goodness-of-Fit Tests

**Assumption 1.**

**Assumption 2.**

**Assumption 3.**

**Assumption 4.**

- (a)
- $E\left\{{K}_{{X}_{2}}\left({X}_{1}\right){K}_{{Y}_{2}}\left({X}_{1}\right)\right\}\sim E\left\{f\left(X\right)g\left(X\right)\right\}$; and $E\left\{{K}_{{Y}_{2}}\left({Y}_{1}\right){K}_{{X}_{2}}\left({Y}_{1}\right)\right\}\sim E\left\{f\left(Y\right)g\left(Y\right)\right\}$.
- (b)
- $E\left\{{K}_{{X}_{2}}\left({X}_{1}\right){K}_{{X}_{1}}\left({X}_{2}\right)\right\}\sim {b}^{-1/2}{V}_{I}\left(2\right)E\left\{{X}^{-1/2}f\left(X\right)\right\}$; $E\left\{{K}_{{X}_{2}}\left({Y}_{1}\right){K}_{{Y}_{1}}\left({X}_{2}\right)\right\}\sim $${b}^{-1/2}{V}_{I}\left(2\right)$$E\left\{{X}^{-1/2}g\left(X\right)\right\}$; $E\left\{{K}_{{Y}_{2}}\left({X}_{1}\right){K}_{{X}_{1}}\left({Y}_{2}\right)\right\}\sim {b}^{-1/2}{V}_{I}\left(2\right)E\left\{{Y}^{-1/2}f\left(Y\right)\right\}$;and $E\left\{{K}_{{Y}_{2}}\left({Y}_{1}\right){K}_{{Y}_{1}}\left({Y}_{2}\right)\right\}$ $\sim {b}^{-1/2}{V}_{I}\left(2\right)E\left\{{Y}^{-1/2}g\left(Y\right)\right\}$, where ${V}_{I}\left(2\right)$ is a kernel-specific constant given in Condition 5 of Definition 1.

**Lemma 1.**

**Theorem 1.**

- (i)
- Under ${H}_{0}$, $n{b}^{1/4}{I}_{n}\stackrel{d}{\to}N\left(0,{\sigma}^{2}\right)$ as $n\to \infty $, where$${\sigma}^{2}=2{V}_{I}\left(2\right)E\left[{X}^{-1/2}\left\{f\left(X\right)+g\left(X\right)\right\}+{Y}^{-1/2}\left\{f\left(Y\right)+g\left(Y\right)\right\}\right],$$
- (ii)
- A consistent estimator of ${\sigma}^{2}$ is given by$${\widehat{\sigma}}^{2}=2{V}_{I}\left(2\right)\frac{1}{n}\sum _{i=1}^{n}\left[{X}_{i}^{-1/2}\left\{\widehat{f}\left({X}_{i}\right)+\widehat{g}\left({X}_{i}\right)\right\}+{Y}_{i}^{-1/2}\left\{\widehat{f}\left({Y}_{i}\right)+\widehat{g}\left({Y}_{i}\right)\right\}\right]\phantom{\rule{4pt}{0ex}}.$$

**Proposition 1.**

#### 3.2. SSST When Two Sub-Samples Have Unequal Sample Sizes

**Theorem 2.**

- (i)
- Under ${H}_{0}$, ${n}_{1}{b}^{1/4}{I}_{{n}_{1},{n}_{2}}\stackrel{d}{\to}N\left(0,{\sigma}_{\lambda}^{2}\right)$ as ${n}_{1}\to \infty $, where$$\begin{array}{cc}\hfill {\sigma}_{\lambda}^{2}& =2{V}_{I}\left(2\right)\left[E\left\{{X}^{-1/2}f\left(X\right)\right\}+\lambda E\left\{{X}^{-1/2}g\left(X\right)\right\}\right.\hfill \\ & \left.+\lambda E\left\{{Y}^{-1/2}f\left(Y\right)\right\}+{\lambda}^{2}E\left\{{Y}^{-1/2}g\left(Y\right)\right\}\right],\hfill \end{array}$$
- (ii)
- A consistent estimator of ${\sigma}_{\lambda}^{2}$ is given by$$\begin{array}{cc}\hfill {\widehat{\sigma}}_{\lambda}^{2}& =2{V}_{I}\left(2\right)\left\{\frac{1}{{n}_{1}}\sum _{i=1}^{{n}_{1}}{X}_{i}^{-1/2}\widehat{f}\left({X}_{i}\right)+\left(\frac{{n}_{1}}{{n}_{2}}\right)\frac{1}{{n}_{1}}\sum _{i=1}^{{n}_{1}}{X}_{i}^{-1/2}\widehat{g}\left({X}_{i}\right)\right.\hfill \\ & \left.+\left(\frac{{n}_{1}}{{n}_{2}}\right)\frac{1}{{n}_{2}}\sum _{i=1}^{{n}_{2}}{Y}_{i}^{-1/2}\widehat{f}\left({Y}_{i}\right)+{\left(\frac{{n}_{1}}{{n}_{2}}\right)}^{2}\frac{1}{{n}_{2}}\sum _{i=1}^{{n}_{2}}{Y}_{i}^{-1/2}\widehat{g}\left({Y}_{i}\right)\right\}\phantom{\rule{4pt}{0ex}}.\hfill \end{array}$$

**Proposition 2.**

**Corollary 1.**

#### 3.3. Extension to a Test for Conditional Symmetry

**Assumption 5.**

**Theorem 3.**

## 4. Smoothing Parameter Selection

**Proposition 3.**

Step 1: Choose some $\delta \in \left(0,1\right)$ and specify $M=min\left\{\u230a{n}_{1}^{\delta}\u230b,\u230a{n}_{2}^{\delta}\u230b\right\}$. |

Step 2: Make M sub-samples of sizes $\left({k}_{1},{k}_{2}\right)=\left(\u230a{n}_{1}/M\u230b,\u230a{n}_{2}/M\u230b\right)$. |

Step 3: Pick two constants $0<\underline{H}<\overline{H}<1$ and define ${H}_{{k}_{1}}=\left[\underline{H},\overline{H}\right]$. |

Step 4: Set ${c}_{m}\left(\alpha \right)\equiv {z}_{\alpha}$ and find ${\widehat{b}}_{{k}_{1}}=inf\left\{arg{max}_{{b}_{{k}_{1}}\in {H}_{{k}_{1}}}{\widehat{\pi}}_{M}\left({b}_{{k}_{1}}\right)\right\}$ by a grid search. |

Step 5: Obtain $\widehat{B}={\widehat{b}}_{{k}_{1}}{k}_{1}^{q}$ and calculate ${\widehat{b}}_{{n}_{1}}=\widehat{B}{n}_{1}^{-q}$. |

## 5. Finite-Sample Performance

#### 5.1. Setup

#### 5.2. Simulation Results

## 6. Conclusions

## Appendix A. Appendix

#### Appendix A.1. Proof of Lemma 1

- Stirling’s formula (“SF”):$$\Gamma \left(z+1\right)=\sqrt{2\pi}{z}^{z+1/2}{e}^{-z}\left\{1+\frac{1}{12z}+\frac{1}{288{z}^{2}}+O\left({z}^{-3}\right)\right\}\phantom{\rule{4.pt}{0ex}}\text{as}\phantom{\rule{4.pt}{0ex}}z\to \infty .$$
- Legendre’s duplication formula (“LDF”):$$\Gamma \left(z\right)\Gamma \left(z+\frac{1}{2}\right)=\frac{\sqrt{\pi}}{{2}^{2z-1}}\Gamma \left(2z\right)\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}z>0.$$

**Lemma A1.**

**Proof of (A1)**.

Step 1: approximating $J\left(x,y\right):={\int}_{0}^{\infty}{L}_{b}\left(u;x,y\right)f\left(u\right)du.$ |

Step 2: approximating $J:={\int}_{{b}^{1-\u03f5}}^{\infty}\left\{{\int}_{{b}^{1-\u03f5}}^{\infty}J\left(x,y\right)g\left(y\right)dy\right\}f\left(x\right)dx.$ |

**Step 1:**Define

**Step 2:**For some $t\in \left(0,1\right)$, we split the interval for y into four subintervals as follows:

**Proof of (A2).**Again for some $\u03f5\in \left(0,1/2\right)$,

#### Appendix A.2. Proof of Theorem 1

**Lemma A2.**

- (a)
- $E\left\{{K}_{{X}_{2}}^{2}\left({X}_{1}\right)\right\}\sim {b}^{-1/2}{V}_{I}\left(2\right)E\left\{{X}^{-1/2}f\left(X\right)\right\}$; $E\left\{{K}_{{X}_{2}}^{2}\left({Y}_{1}\right)\right\}\sim {b}^{-1/2}{V}_{I}\left(2\right)E\left\{{X}^{-1/2}g\left(X\right)\right\}$;$E\left\{{K}_{{Y}_{2}}^{2}\left({X}_{1}\right)\right\}\sim {b}^{-1/2}{V}_{I}\left(2\right)E\left\{{Y}^{-1/2}f\left(Y\right)\right\}$; and $E\left\{{K}_{{Y}_{2}}^{2}\left({Y}_{1}\right)\right\}\sim {b}^{-1/2}{V}_{I}\left(2\right)E\left\{{Y}^{-1/2}g\left(Y\right)\right\},$ where ${V}_{I}\left(2\right)$ is given in Condition 5 of Definition 1.
- (b)
- $E\left\{{K}_{{X}_{2}}\left({X}_{1}\right){K}_{{Y}_{2}}\left({Y}_{1}\right)\right\}\sim E\left\{f\left(X\right)\right\}E\left\{g\left(Y\right)\right\}$; $E\left\{{K}_{{X}_{2}}\left({Y}_{1}\right){K}_{{Y}_{2}}\left({X}_{1}\right)\right\}\sim E\left\{g\left(X\right)\right\}E\left\{f\left(Y\right)\right\}$;$E\left\{{K}_{{X}_{2}}\left({Y}_{1}\right){K}_{{X}_{1}}\left({Y}_{2}\right)\right\}\sim {E}^{2}\left\{g\left(X\right)\right\}$; and $E\left\{{K}_{{Y}_{2}}\left({X}_{1}\right){K}_{{Y}_{1}}\left({X}_{2}\right)\right\}\sim {E}^{2}\left\{f\left(Y\right)\right\}$.
- (c)
- $E\left\{{K}_{{X}_{2}}\left({X}_{1}\right){K}_{{X}_{2}}\left({Y}_{1}\right)\right\}\sim E\left\{f\left(X\right)g\left(X\right)\right\}$; and $E\left\{{K}_{{Y}_{2}}\left({Y}_{1}\right){K}_{{Y}_{2}}\left({X}_{1}\right)\right\}\sim E\left\{g\left(Y\right)f\left(Y\right)\right\}$.
- (d)
- $E\left\{{K}_{{X}_{2}}\left({X}_{1}\right){K}_{{X}_{1}}\left({Y}_{2}\right)\right\}\sim E\left\{f\left(X\right)g\left(X\right)\right\}$; $E\left\{{K}_{{Y}_{1}}\left({X}_{2}\right){K}_{{X}_{2}}\left({X}_{1}\right)\right\}\sim E\left\{{f}^{2}\left(Y\right)\right\}$;$E\left\{{K}_{{X}_{1}}\left({Y}_{2}\right){K}_{{Y}_{2}}\left({Y}_{1}\right)\right\}\sim E\left\{{g}^{2}\left(X\right)\right\}$; and $E\left\{{K}_{{Y}_{2}}\left({Y}_{1}\right){K}_{{Y}_{1}}\left({X}_{2}\right)\right\}\sim E\left\{f\left(Y\right)g\left(Y\right)\right\}$.

**Lemma A3.**

**Lemma A4.**

#### Appendix A.2.1. Proof of Lemma A2

#### Appendix A.2.2. Proof of Lemma A3

#### Appendix A.2.3. Proof of Lemma A4

#### Appendix A.2.4. Proof of Theorem 1

#### Appendix A.3. Proof of Proposition 1

#### Appendix A.4. Proof of Theorem 3

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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Distribution | Skewness | Kurtosis | |
---|---|---|---|

S1 | $N\left(0,1\right)$ | $0.00$ | $3.00$ |

S2 | ${t}_{10}$ | $0.00$ | $4.00$ |

S3 | $DE\left(0,1\right)$ or Standard Laplace | $0.00$ | $24.00$ |

S4 | $U\left[-1,1\right]$ or GLD with $\left({\lambda}_{1},{\lambda}_{2},{\lambda}_{3},{\lambda}_{4}\right)=\left(0,1,1,1\right)$ | $0.00$ | $1.80$ |

A1 | $LN\left(0,1\right)-exp\left(1/2\right)$ | $6.18$ | $113.94$ |

A2 | ${\chi}_{3}^{2}-3$ | $1.63$ | $7.00$ |

A3 | GLD with $\left({\lambda}_{1},{\lambda}_{2},{\lambda}_{3},{\lambda}_{4}\right)=\left(12.601,-0.00980045,-0.11,-0.0001\right)$ | $-2.92$ | $19.52$ |

A4 | GLD with $\left({\lambda}_{1},{\lambda}_{2},{\lambda}_{3},{\lambda}_{4}\right)=\left(-9.7726,-0.0151878,-0.001,-0.13\right)$ | $3.16$ | $23.75$ |

(A) Size | (%) | |||||||||||||

N | Test | δ | Distribution | |||||||||||

S1 | S2 | S3 | S4 | |||||||||||

5% | 10% | 5% | 10% | 5% | 10% | 5% | 10% | |||||||

50 | FMS-G-O | − | 4.8 | 9.4 | 4.9 | 9.5 | 6.0 | 10.8 | 7.4 | 12.6 | ||||

FMS-G-AltVar | − | 4.7 | 9.3 | 4.4 | 8.9 | 5.8 | 10.9 | 6.9 | 12.6 | |||||

${T}_{{n}_{1},{n}_{2}}$-G | 0.3 | 3.5 | 7.5 | 3.1 | 7.0 | 4.5 | 8.7 | 5.2 | 9.6 | |||||

0.5 | 3.8 | 6.8 | 3.0 | 7.3 | 4.4 | 7.9 | 5.4 | 10.2 | ||||||

0.7 | 3.4 | 6.7 | 3.4 | 7.3 | 4.5 | 8.9 | 5.4 | 10.5 | ||||||

${T}_{{n}_{1},{n}_{2}}$-MG | 0.3 | 3.6 | 7.7 | 3.9 | 7.5 | 4.9 | 9.5 | 5.3 | 9.7 | |||||

0.5 | 4.0 | 7.2 | 3.7 | 7.2 | 4.9 | 8.7 | 5.4 | 10.2 | ||||||

0.7 | 3.3 | 6.8 | 3.8 | 7.3 | 4.7 | 9.1 | 5.3 | 10.7 | ||||||

${T}_{{n}_{1},{n}_{2}}$-NM | 0.3 | 3.4 | 7.3 | 3.1 | 6.7 | 4.3 | 8.3 | 5.2 | 9.3 | |||||

0.5 | 3.9 | 6.7 | 3.1 | 6.4 | 4.2 | 7.9 | 5.2 | 9.8 | ||||||

0.7 | 3.3 | 7.0 | 3.1 | 6.4 | 4.4 | 8.1 | 5.2 | 10.3 | ||||||

100 | FMS-G-O | − | 5.2 | 9.8 | 6.7 | 10.8 | 5.6 | 10.1 | 7.4 | 12.5 | ||||

FMS-G-AltVar | − | 4.9 | 8.9 | 6.4 | 10.6 | 5.3 | 9.9 | 6.6 | 12.6 | |||||

${T}_{{n}_{1},{n}_{2}}$-G | 0.3 | 4.0 | 7.0 | 6.1 | 9.3 | 5.1 | 8.8 | 7.3 | 11.9 | |||||

0.5 | 3.6 | 7.3 | 5.8 | 9.1 | 5.2 | 8.9 | 7.1 | 11.4 | ||||||

0.7 | 3.8 | 7.6 | 5.9 | 9.5 | 4.8 | 9.3 | 6.6 | 11.8 | ||||||

${T}_{{n}_{1},{n}_{2}}$-MG | 0.3 | 4.5 | 7.4 | 6.2 | 9.5 | 5.8 | 9.6 | 7.3 | 12.1 | |||||

0.5 | 3.7 | 7.7 | 6.1 | 9.6 | 5.7 | 9.0 | 7.2 | 11.6 | ||||||

0.7 | 3.8 | 8.1 | 5.8 | 9.6 | 5.1 | 9.4 | 6.7 | 12.1 | ||||||

${T}_{{n}_{1},{n}_{2}}$-NM | 0.3 | 4.2 | 6.8 | 5.3 | 8.9 | 4.3 | 8.4 | 7.1 | 12.4 | |||||

0.5 | 4.0 | 6.6 | 5.3 | 8.7 | 4.9 | 8.7 | 7.2 | 11.6 | ||||||

0.7 | 4.2 | 6.8 | 5.3 | 8.8 | 4.9 | 8.4 | 7.3 | 11.7 | ||||||

200 | FMS-G-O | − | 4.1 | 7.3 | 6.0 | 9.3 | 5.8 | 8.9 | 8.9 | 14.1 | ||||

FMS-G-AltVar | − | 4.1 | 7.0 | 6.1 | 8.9 | 5.4 | 9.3 | 8.7 | 15.1 | |||||

${T}_{{n}_{1},{n}_{2}}$-G | 0.3 | 3.3 | 6.3 | 4.9 | 8.0 | 4.9 | 8.7 | 8.9 | 12.9 | |||||

0.5 | 3.2 | 6.4 | 4.9 | 8.6 | 5.3 | 8.7 | 8.7 | 13.3 | ||||||

0.7 | 3.5 | 6.8 | 5.2 | 9.2 | 5.4 | 8.9 | 8.4 | 13.9 | ||||||

${T}_{{n}_{1},{n}_{2}}$-MG | 0.3 | 3.5 | 6.7 | 5.5 | 8.4 | 5.5 | 9.4 | 9.0 | 13.4 | |||||

0.5 | 3.5 | 6.8 | 5.2 | 9.0 | 5.7 | 8.8 | 8.7 | 13.6 | ||||||

0.7 | 3.5 | 7.0 | 5.5 | 9.8 | 5.7 | 9.1 | 8.6 | 13.7 | ||||||

${T}_{{n}_{1},{n}_{2}}$-NM | 0.3 | 3.5 | 6.0 | 4.0 | 7.5 | 4.4 | 8.5 | 8.7 | 13.3 | |||||

0.5 | 3.6 | 5.9 | 4.4 | 7.6 | 4.4 | 8.4 | 8.7 | 13.3 | ||||||

0.7 | 3.6 | 5.8 | 4.6 | 7.4 | 4.5 | 8.4 | 8.8 | 13.1 | ||||||

(B) Power | (%) | |||||||||||||

N | Test | δ | Distribution | |||||||||||

A1 | A2 | A3 | A4 | |||||||||||

5% | 10% | 5% | 10% | 5% | 10% | 5% | 10% | |||||||

50 | FMS-G-O | − | 42.0 | 51.6 | 21.4 | 30.3 | 26.3 | 37.2 | 28.8 | 40.8 | ||||

[43.7] | [52.4] | [22.7] | [31.0] | [27.6] | [38.1] | [30.5] | [42.0] | |||||||

FMS-G-AltVar | − | 24.9 | 37.0 | 13.3 | 22.4 | 39.8 | 52.2 | 43.1 | 55.3 | |||||

[27.5] | [39.0] | [13.9] | [24.1] | [41.5] | [53.6] | [44.9] | [57.4] | |||||||

${T}_{{n}_{1},{n}_{2}}$-G | 0.3 | 31.3 | 41.5 | 17.3 | 24.7 | 30.8 | 41.3 | 33.8 | 44.3 | |||||

[35.4] | [46.6] | [20.8] | [29.5] | [35.1] | [46.3] | [38.7] | [50.8] | |||||||

0.5 | 29.5 | 39.7 | 16.0 | 23.8 | 29.4 | 39.1 | 31.4 | 42.9 | ||||||

0.7 | 27.8 | 38.7 | 14.5 | 22.6 | 28.0 | 36.8 | 30.2 | 40.6 | ||||||

${T}_{{n}_{1},{n}_{2}}$-MG | 0.3 | 32.1 | 42.2 | 17.9 | 25.8 | 31.6 | 41.8 | 35.5 | 45.5 | |||||

[35.2] | [45.7] | [20.9] | [29.3] | [35.0] | [45.8] | [39.1] | [49.5] | |||||||

0.5 | 30.1 | 40.8 | 16.5 | 24.5 | 30.2 | 40.3 | 32.2 | 43.1 | ||||||

0.7 | 28.0 | 38.9 | 15.1 | 23.0 | 28.5 | 37.5 | 30.8 | 41.1 | ||||||

${T}_{{n}_{1},{n}_{2}}$-NM | 0.3 | 33.8 | 42.5 | 18.2 | 25.2 | 32.9 | 42.5 | 35.2 | 46.5 | |||||

[36.6] | [48.8] | [20.7] | [30.4] | [36.8] | [48.5] | [39.2] | [53.5] | |||||||

0.5 | 33.5 | 42.2 | 17.7 | 24.9 | 32.4 | 42.2 | 34.6 | 45.8 | ||||||

0.7 | 33.0 | 41.4 | 17.4 | 24.2 | 32.2 | 41.6 | 34.5 | 45.7 | ||||||

100 | FMS-G-O | − | 73.0 | 81.5 | 41.4 | 51.4 | 59.1 | 71.2 | 64.2 | 74.6 | ||||

[72.7] | [81.8] | [40.7] | [52.3] | [58.7] | [71.8] | [63.8] | [74.6] | |||||||

FMS-G-AltVar | − | 56.4 | 70.2 | 30.3 | 43.4 | 73.7 | 80.7 | 77.2 | 83.7 | |||||

[58.1] | [71.6] | [32.2] | [44.4] | [74.4] | [81.8] | [78.1] | [84.4] | |||||||

${T}_{{n}_{1},{n}_{2}}$-G | 0.3 | 72.3 | 80.5 | 37.9 | 48.6 | 70.3 | 78.4 | 74.1 | 80.5 | |||||

[75.4] | [84.4] | [40.2] | [54.9] | [72.8] | [82.3] | [76.1] | [84.1] | |||||||

0.5 | 67.1 | 77.2 | 33.9 | 45.0 | 65.2 | 75.7 | 69.5 | 78.2 | ||||||

0.7 | 62.9 | 73.6 | 31.9 | 42.5 | 61.7 | 72.3 | 65.2 | 75.8 | ||||||

${T}_{{n}_{1},{n}_{2}}$-MG | 0.3 | 73.1 | 80.2 | 38.3 | 49.3 | 70.5 | 78.3 | 73.6 | 81.0 | |||||

[74.8] | [83.6] | [40.9] | [53.5] | [72.4] | [80.7] | [75.4] | [83.3] | |||||||

0.5 | 67.4 | 77.5 | 34.6 | 45.4 | 65.6 | 75.4 | 69.5 | 77.9 | ||||||

0.7 | 62.9 | 73.5 | 32.0 | 42.3 | 62.2 | 72.5 | 65.3 | 75.2 | ||||||

${T}_{{n}_{1},{n}_{2}}$-NM | 0.3 | 76.8 | 84.0 | 41.7 | 51.9 | 75.5 | 82.1 | 76.8 | 84.0 | |||||

[79.6] | [87.2] | [44.8] | [58.0] | [77.6] | [85.0] | [79.7] | [86.9] | |||||||

0.5 | 77.0 | 84.0 | 40.1 | 51.0 | 75.1 | 82.0 | 75.7 | 83.0 | ||||||

0.7 | 76.4 | 83.7 | 39.6 | 50.1 | 74.9 | 81.9 | 75.6 | 82.9 | ||||||

200 | FMS-G-O | − | 97.4 | 98.3 | 71.5 | 80.7 | 95.6 | 98.1 | 97.1 | 97.8 | ||||

[97.8] | [98.6] | [75.0] | [84.2] | [96.9] | [98.6] | [97.4] | [98.4] | |||||||

FMS-G-AltVar | − | 93.4 | 96.3 | 60.4 | 72.8 | 98.7 | 99.0 | 98.4 | 99.1 | |||||

[95.2] | [97.5] | [69.1] | [78.5] | [98.8] | [99.2] | [99.0] | [99.2] | |||||||

${T}_{{n}_{1},{n}_{2}}$-G | 0.3 | 97.7 | 99.1 | 77.0 | 84.8 | 98.6 | 99.1 | 98.6 | 99.2 | |||||

[98.7] | [99.4] | [82.7] | [90.3] | [98.8] | [99.4] | [98.9] | [99.2] | |||||||

0.5 | 97.2 | 98.1 | 71.3 | 80.9 | 97.7 | 98.9 | 97.9 | 98.7 | ||||||

0.7 | 96.4 | 97.6 | 65.4 | 75.8 | 96.7 | 98.8 | 97.2 | 98.3 | ||||||

${T}_{{n}_{1},{n}_{2}}$-MG | 0.3 | 97.9 | 99.1 | 76.2 | 85.3 | 98.6 | 99.1 | 98.5 | 99.1 | |||||

[98.6] | [99.4] | [81.8] | [89.8] | [98.7] | [99.4] | [98.7] | [99.2] | |||||||

0.5 | 97.3 | 98.1 | 71.4 | 80.3 | 97.6 | 98.9 | 97.9 | 98.6 | ||||||

0.7 | 96.5 | 97.6 | 65.5 | 75.9 | 96.7 | 98.7 | 97.2 | 98.3 | ||||||

${T}_{{n}_{1},{n}_{2}}$-NM | 0.3 | 98.9 | 99.0 | 84.8 | 91.1 | 98.8 | 99.2 | 98.6 | 98.9 | |||||

[99.0] | [99.2] | [91.1] | [95.8] | [99.4] | [99.7] | [99.5] | [99.7] | |||||||

0.5 | 96.8 | 97.0 | 81.2 | 88.2 | 98.3 | 98.6 | 98.4 | 98.6 | ||||||

0.7 | 95.6 | 95.7 | 80.9 | 88.3 | 98.3 | 98.6 | 98.4 | 98.6 |

^{1}Hirukawa and Sakudo [32] present the Weibull kernel as yet another special case. However, it is not confirmed that this kernel satisfies Lemma 1 below, and thus the kernel is not investigated throughout.^{2}It is possible to use different asymmetric kernels and/or different smoothing parameters to estimate f and g. For convenience, however, we choose to employ the same asymmetric kernel function and a single smoothing parameter.^{3}Although the GLDs corresponding to A3 and A4 are used in Zheng [17] and FMS, they are found to have non-zero means. Therefore, we adjust the values of ${\lambda}_{1}$ and ${\lambda}_{2}$ with skewness and kurtosis maintained so that the resulting distributions have means of zero.

© 2016 by the authors; 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**

Hirukawa, M.; Sakudo, M. Testing Symmetry of Unknown Densities via Smoothing with the Generalized Gamma Kernels. *Econometrics* **2016**, *4*, 28.
https://doi.org/10.3390/econometrics4020028

**AMA Style**

Hirukawa M, Sakudo M. Testing Symmetry of Unknown Densities via Smoothing with the Generalized Gamma Kernels. *Econometrics*. 2016; 4(2):28.
https://doi.org/10.3390/econometrics4020028

**Chicago/Turabian Style**

Hirukawa, Masayuki, and Mari Sakudo. 2016. "Testing Symmetry of Unknown Densities via Smoothing with the Generalized Gamma Kernels" *Econometrics* 4, no. 2: 28.
https://doi.org/10.3390/econometrics4020028