# Discrete Parameter-Free Zone Distribution and Its Application in Normality Testing

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Zone Distribution

#### 2.1. Motivation and Definition

**Definition 1.**

**Remark 1.**

**Proposition 1.**

**Definition 2.**

**Zone distribution with**$\mathit{n}$

**degrees of freedom**, which will be denoted by $A~{zone}_{n}$.

**Theorem 1.**

**Proof of Theorem 1.**

**Remark 2.**

**Corollary 1.**

#### 2.2. Basic Properties and Numerical Characteristics

**Proposition 2.**

**Proposition 3.**

**Proof of Proposition 3.**

**Remark 3.**

**Theorem 2.**

**Proof of Theorem 2.**

## 3. Zone Distribution in Normality Testing

#### 3.1. The Testing Procedure

#### 3.2. Power Analysis

- Modelling the sample ${x}_{1},{x}_{2},\dots ,{x}_{n}$ of the chosen alternative distribution for the observed sample size $n$;
- Calculating $zone\left({x}_{1}\right),\text{}zone\left({x}_{2}\right),\text{}\dots \text{},\text{}zone\left({x}_{n}\right)$ and the empirical value of the test statistic$$a=\frac{1}{n}\sum _{i=1}^{n}zone\left({x}_{i}\right);$$
- Repeating the first two steps $m=\mathrm{10,000}$ times and thus obtaining the sample ${a}_{1},{a}_{2},\dots ,{a}_{m}$;
- Determining the EDF$${F}_{m}^{*}\left(x\right)=\frac{1}{m}\sum _{i=1}^{m}I({a}_{i}\le x)$$
- Calculating the power $1-\beta $ of the test by$$1-\beta ={F}_{m}^{*}\left({c}_{1}\right)+\left(1-{F}_{m}^{*}\left({c}_{2}\right)\right)$$

#### 3.3. Comparative Analysis

- It is still more powerful than the other normality tests usually used;
- It is very simple to apply and program;
- It performs faster than the Quantile-Zone test, which is significant for big data;
- The elements of the sample $zone\left({X}_{j}\right);j=1,\dots ,n$ are not mutually dependent, which is not the case for zones in Quantile-Zone distribution. That makes the Zone distribution and many of its characteristics determinable theoretically (3);
- The invariance for outliers, to some extent, is the same as with the Quantile-Zone test, etc.

## 4. Examples

#### 4.1. Known Parameters Case

#### 4.2. Estimated Parameters Case

## 5. Conclusions

- Development of a novel discrete distribution named the Zone distribution, associated with normal distributions, including the presentation of functional characteristics, PMF, CDF, and corresponding graphical illustrations;
- Provision of quantile tables for the Zone distribution in the cases of both known and estimated parameters of related normal distributions;
- Computation of key numerical characteristics for the Zone distribution;
- Illustration of the application of the Zone distribution in normality testing, along with an exploration of its advantages and properties;
- Calculation of empirical power for the new Zone test, conducted separately for symmetric and asymmetric alternative distributions, providing power analysis results for both known and estimated parameters;
- Presentation of a highly illustrative graphical interpretation of the power analysis;
- Comparative power analysis of both variants of the Zone test (specified and estimated parameters) against other commonly used normality tests, accompanied by detailed results and graphical representations.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Proof of Theorem 1.**

**Proof of Proposition 3.**

**Proof of Theorem 2.**

## Appendix B

## References

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**Figure 3.**The graph of the CDF $F\left(x\right)=p$ for each $n$ (approximated values). (

**a**) Known parameters. (

**b**) Estimated parameters.

**Figure 4.**The histogram of the PMF $P\left(A=x\right)=p$ for each $n$ (approximated values). (

**a**) Known parameters. (

**b**) Estimated parameters.

**Figure 5.**Alternative distributions compared to the null distribution. (

**a**) Symmetric alternatives. (

**b**) Asymmetric alternatives.

**Figure 6.**Empirical power of the Zone test for various sample sizes with the level of significance $\alpha =0.05$—symmetric alternative distributions. (

**a**) Known parameters. (

**b**) Estimated parameters.

**Figure 7.**Empirical power of the Zone test for various sample sizes with the level of significance $\mathsf{\alpha}=0.05$—asymmetric alternative distributions. (

**a**) Known parameters. (

**b**) Estimated parameters.

**Figure 8.**Average empirical power values of the Zone test and some other normality tests for various sample sizes with the level of significance $\mathsf{\alpha}=0.05$. (

**a**) Symmetric alternative distributions. (

**b**) Asymmetric alternative distributions.

**Figure 9.**Average empirical power values of the Zone test and Quantile-Zone test for various sample sizes with the level of significance $\mathsf{\alpha}=0.05$. (

**a**) Symmetric alternative distributions. (

**b**) Asymmetric alternative distributions.

$\mathit{n}$ | $\mathit{p}$ | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0.01 | 0.025 | 0.05 | 0.1 | 0.15 | 0.2 | 0.5 | 0.8 | 0.85 | 0.9 | 0.95 | 0.975 | 0.99 | |

5 | * | * | * | * | 1.0011 | 1.0339 | 1.3840 | 1.5540 | 1.5651 | 1.6382 | 1.7421 | 1.8198 | 1.9201 |

10 | * | 1.0038 | 1.0315 | 1.0870 | 1.1840 | 1.1897 | 1.3393 | 1.4749 | 1.5080 | 1.5420 | 1.6270 | 1.6890 | 1.7518 |

20 | 1.0925 | 1.1391 | 1.1440 | 1.1914 | 1.2199 | 1.2385 | 1.3334 | 1.4303 | 1.4612 | 1.4940 | 1.5365 | 1.5730 | 1.6206 |

30 | 1.1280 | 1.1600 | 1.1918 | 1.2220 | 1.2447 | 1.2557 | 1.3370 | 1.4148 | 1.4359 | 1.4573 | 1.4970 | 1.5290 | 1.5647 |

50 | 1.1728 | 1.2016 | 1.2233 | 1.2428 | 1.2620 | 1.2744 | 1.3366 | 1.3990 | 1.4134 | 1.4325 | 1.4596 | 1.4846 | 1.5149 |

100 | 1.2228 | 1.2394 | 1.2552 | 1.2716 | 1.2840 | 1.2936 | 1.3366 | 1.3811 | 1.3912 | 1.4038 | 1.4235 | 1.4405 | 1.4602 |

200 | 1.2552 | 1.2680 | 1.2789 | 1.2916 | 1.3004 | 1.3072 | 1.3376 | 1.3684 | 1.3756 | 1.3849 | 1.3984 | 1.4108 | 1.4249 |

300 | 1.2703 | 1.2806 | 1.2897 | 1.2999 | 1.3070 | 1.3127 | 1.3375 | 1.3627 | 1.3686 | 1.3759 | 1.3872 | 1.3968 | 1.4084 |

500 | 1.2853 | 1.2933 | 1.3003 | 1.3084 | 1.3140 | 1.3184 | 1.3378 | 1.3572 | 1.3671 | 1.3675 | 1.3760 | 1.3835 | 1.3925 |

1000 | 1.3002 | 1.3061 | 1.3111 | 1.3170 | 1.3210 | 1.3241 | 1.3378 | 1.3515 | 1.3547 | 1.3588 | 1.3647 | 1.3699 | 1.3762 |

1500 | 1.3077 | 1.3121 | 1.3162 | 1.3209 | 1.3242 | 1.3267 | 1.3379 | 1.3490 | 1.3516 | 1.3548 | 1.3597 | 1.3641 | 1.3690 |

2000 | 1.3112 | 1.3154 | 1.3190 | 1.3231 | 1.3259 | 1.3281 | 1.3378 | 1.3476 | 1.3499 | 1.3527 | 1.3569 | 1.3605 | 1.3650 |

**Table 2.**$F\left(x\right)=P\left(A\le x\right);A~{zone}_{m}$ when parameters $\mu $ and ${\sigma}^{2}$ are estimated with ${\overline{X}}_{n}$ and ${\stackrel{~}{S}}_{n}^{2}$, respectively.

$\mathit{n}$ | $\mathit{p}$ | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0.01 | 0.025 | 0.05 | 0.1 | 0.15 | 0.2 | 0.5 | 0.8 | 0.85 | 0.9 | 0.95 | 0.975 | 0.99 | |

5 | * | 1.1920 | 1.1920 | 1.1920 | 1.1920 | 1.1920 | 1.1920 | 1.3840 | 1.3840 | 1.3840 | 1.3840 | 1.3840 | 1.3840 |

10 | 1.1580 | 1.1580 | 1.1580 | 1.1920 | 1.1920 | 1.1920 | 1.2880 | 1.3840 | 1.3840 | 1.3840 | 1.4800 | 1.4800 | 1.4800 |

20 | 1.1865 | 1.2230 | 1.2230 | 1.2400 | 1.2710 | 1.2710 | 1.3190 | 1.3785 | 1.3840 | 1.3840 | 1.4320 | 1.4320 | 1.4630 |

30 | 1.2203 | 1.2410 | 1.2540 | 1.2730 | 1.2843 | 1.2860 | 1.3293 | 1.3727 | 1.3820 | 1.3933 | 1.4047 | 1.4253 | 1.4480 |

50 | 1.2508 | 1.2608 | 1.2722 | 1.2868 | 1.2970 | 1.3050 | 1.3320 | 1.3636 | 1.3704 | 1.3828 | 1.3964 | 1.4088 | 1.4212 |

100 | 1.2750 | 1.2846 | 1.2925 | 1.3021 | 1.3089 | 1.3145 | 1.3366 | 1.3586 | 1.3636 | 1.3704 | 1.3800 | 1.3868 | 1.3987 |

200 | 1.2936 | 1.3008 | 1.3067 | 1.3134 | 1.3180 | 1.3216 | 1.3371 | 1.3530 | 1.3564 | 1.3611 | 1.3676 | 1.3735 | 1.3812 |

300 | 1.3019 | 1.3076 | 1.3125 | 1.3179 | 1.3217 | 1.3247 | 1.3375 | 1.3503 | 1.3533 | 1.3570 | 1.3623 | 1.3674 | 1.3731 |

500 | 1.3100 | 1.3144 | 1.3182 | 1.3225 | 1.3254 | 1.3277 | 1.3377 | 1.3477 | 1.3501 | 1.3528 | 1.3572 | 1.3608 | 1.3651 |

1000 | 1.3184 | 1.3213 | 1.3240 | 1.3270 | 1.3290 | 1.3307 | 1.3377 | 1.3448 | 1.3464 | 1.3484 | 1.3514 | 1.3541 | 1.3572 |

1500 | 1.3219 | 1.3244 | 1.3266 | 1.3291 | 1.3307 | 1.3320 | 1.3378 | 1.3436 | 1.3449 | 1.3466 | 1.3491 | 1.3512 | 1.3538 |

2000 | 1.3241 | 1.3263 | 1.3281 | 1.3302 | 1.3317 | 1.3328 | 1.3378 | 1.3428 | 1.3440 | 1.3455 | 1.3476 | 1.3495 | 1.3516 |

**Table 3.**Empirical power of the Zone test for various sample sizes with the level of significance $\alpha =0.05$ when the parameters are known—symmetric alternative distributions.

Distribution | $\mathit{n}$ | |||||
---|---|---|---|---|---|---|

10 | 20 | 30 | 50 | 100 | 200 | |

Laplace (0, 1) | 0.1441 | 0.2187 | 0.2692 | 0.3818 | 0.6030 | 0.8508 |

${\mathit{t}}_{\mathbf{2}}$ | 0.2679 | 0.4392 | 0.5574 | 0.7541 | 0.9489 | 0.9990 |

Tukey (0.14) | 0.3331 | 0.5776 | 0.7361 | 0.9029 | 0.9943 | 1 |

$\mathit{N}$(0, 1.5^{2}) | 0.3640 | 0.6172 | 0.7734 | 0.9283 | 0.9966 | 1 |

Logistic (0, 1) | 0.5106 | 0.7924 | 0.9068 | 0.9873 | 1 | 1 |

Cauchy (0, 1) | 0.5434 | 0.8027 | 0.9147 | 0.9865 | 1 | 1 |

$\mathit{N}$(0, 0.5^{2}) | 0.6321 | 0.9446 | 0.9980 | 1 | 1 | 1 |

$\mathit{U}$(−3.5, 3.5) | 0.9216 | 0.9974 | 0.9998 | 1 | 1 | 1 |

Average | 0.4646 | 0.6737 | 0.7694 | 0.8676 | 0.9428 | 0.9812 |

**Table 4.**Empirical power of the Zone test for various sample sizes with the level of significance $\alpha =0.05$ when the parameters are known—asymmetric alternative distributions.

Distribution | $\mathit{n}$ | |||||
---|---|---|---|---|---|---|

10 | 20 | 30 | 50 | 100 | 200 | |

${\mathit{\chi}}_{\mathbf{1}}^{\mathbf{2}}$ | 0.1476 | 0.1944 | 0.2139 | 0.2861 | 0.4226 | 0.6630 |

Gumbel (0, 1) | 0.1478 | 0.2087 | 0.2636 | 0.3784 | 0.6002 | 0.8517 |

Burr (3, 1) | 0.2574 | 0.4649 | 0.6216 | 0.8290 | 0.9816 | 1 |

Pareto (0.1, 1) | 0.3641 | 0.6064 | 0.6670 | 0.8504 | 0.9882 | 1 |

$\mathit{N}$(1, 1) | 0.3850 | 0.6215 | 0.7977 | 0.9378 | 0.9974 | 1 |

Lognormal (0, 1) | 0.4822 | 0.7294 | 0.8721 | 0.9701 | 0.9998 | 1 |

Weibull (1, 2) | 0.7867 | 0.9628 | 0.9932 | 0.9998 | 1 | 1 |

Gamma (2, 1) | 0.9382 | 0.9976 | 1 | 1 | 1 | 1 |

Beta (2, 1.5) | 1 | 1 | 1 | 1 | 1 | 1 |

Average | 0.5010 | 0.6429 | 0.7143 | 0.8057 | 0.8878 | 0.9461 |

**Table 5.**Empirical power of the Zone test for various sample sizes with the level of significance $\alpha =0.05$ when the parameters are estimated—symmetric alternative distributions.

Distribution | $\mathit{n}$ | |||||
---|---|---|---|---|---|---|

10 | 20 | 30 | 50 | 100 | 200 | |

Laplace (0, 1) | 0.4668 | 0.6013 | 0.6365 | 0.7150 | 0.8739 | 0.9716 |

${\mathit{t}}_{\mathbf{2}}$ | 0.6163 | 0.7812 | 0.8532 | 0.9269 | 0.9921 | 1 |

Tukey (0.14) | 0.6971 | 0.8795 | 0.9682 | 0.9827 | 0.9995 | 1 |

$\mathit{N}$(0, 1.5^{2}) | 0.7232 | 0.8946 | 0.9509 | 0.9905 | 1 | 1 |

Logistic (0, 1) | 0.8171 | 0.9606 | 0.9972 | 1 | 1 | 1 |

Cauchy (0, 1) | 0.8215 | 0.9522 | 0.9826 | 0.9986 | 1 | 1 |

$\mathit{N}$(0, 0.5^{2}) | 0.9271 | 0.9973 | 0.9998 | 1 | 1 | 1 |

$\mathit{U}$(−3.5, 3.5) | 0.9866 | 0.9997 | 1 | 1 | 1 | 1 |

Average | 0.7570 | 0.8833 | 0.9236 | 0.9517 | 0.9832 | 0.9965 |

**Table 6.**Empirical power of the Zone test for various sample sizes with the level of significance $\alpha =0.05$ when the parameters are estimated—asymmetric alternative distributions.

Distribution | $\mathit{n}$ | |||||
---|---|---|---|---|---|---|

10 | 20 | 30 | 50 | 100 | 200 | |

${\mathit{\chi}}_{\mathbf{1}}^{\mathbf{2}}$ | 0.4479 | 0.5499 | 0.5524 | 0.6078 | 0.7485 | 0.8865 |

Gumbel (0, 1) | 0.4548 | 0.5963 | 0.6246 | 0.7169 | 0.8745 | 0.9710 |

Burr (3, 1) | 0.6164 | 0.8248 | 0.9044 | 0.9691 | 0.9990 | 1 |

Pareto (0.1, 1) | 0.6026 | 0.8190 | 0.8941 | 0.9686 | 0.9991 | 1 |

$\mathit{N}$(1, 1) | 0.7305 | 0.8995 | 0.9581 | 0.9939 | 1 | 1 |

Lognormal (0, 1) | 0.7744 | 0.9312 | 0.9767 | 0.9950 | 1 | 1 |

Weibull (1, 2) | 0.9411 | 0.9967 | 0.9997 | 1 | 1 | 1 |

Gamma (2, 1) | 0.9900 | 0.9999 | 1 | 1 | 1 | 1 |

Beta (2, 1.5) | 1 | 1 | 1 | 1 | 1 | 1 |

Average | 0.7286 | 0.8464 | 0.8789 | 0.9168 | 0.9579 | 0.9842 |

**Table 7.**Average empirical power values of the Zone test and some other normality tests for various sample sizes with the level of significance $\mathsf{\alpha}=0.05$—symmetric alternative distributions.

Test | $\mathit{n}$ | |||||
---|---|---|---|---|---|---|

10 | 20 | 30 | 50 | 100 | 200 | |

Zone (EP ^{1}) | 0.7570 | 0.8833 | 0.9236 | 0.9517 | 0.9832 | 0.9965 |

Zone (KP ^{2}) | 0.4646 | 0.6737 | 0.7694 | 0.8676 | 0.9428 | 0.9812 |

Shapiro–Wilk | 0.2328 | 0.4548 | 0.6768 | 0.7712 | 0.8469 | 0.9005 |

Anderson–Darling | 0.2316 | 0.4522 | 0.6730 | 0.7644 | 0.8368 | 0.8901 |

${\mathit{\chi}}^{\mathbf{2}}$ | 0.2158 | 0.4208 | 0.6257 | 0.7307 | 0.8160 | 0.8672 |

Lilliefors | 0.2177 | 0.4245 | 0.6313 | 0.7222 | 0.7996 | 0.8605 |

Kolmogorov–Smirnov | 0.1917 | 0.3726 | 0.5534 | 0.6723 | 0.7628 | 0.8175 |

^{1}Estimated Parameters.

^{2}Known Parameters.

**Table 8.**Average empirical power values of the Zone test and some other normality tests for various sample sizes with the level of significance $\mathsf{\alpha}=0.05$—asymmetric alternative distributions.

Test | $\mathit{n}$ | |||||
---|---|---|---|---|---|---|

10 | 20 | 30 | 50 | 100 | 200 | |

Zone (EP ^{1}) | 0.7286 | 0.8464 | 0.8789 | 0.9168 | 0.9579 | 0.9842 |

Zone (KP ^{2}) | 0.5010 | 0.6429 | 0.7143 | 0.8057 | 0.8878 | 0.9461 |

Shapiro–Wilk | 0.6698 | 0.7714 | 0.8730 | 0.9191 | 0.9552 | 0.9759 |

Anderson–Darling | 0.6666 | 0.7649 | 0.8633 | 0.9087 | 0.9465 | 0.9702 |

${\mathit{\chi}}^{\mathbf{2}}$ | 0.6552 | 0.7423 | 0.8293 | 0.8841 | 0.9293 | 0.9615 |

Lilliefors | 0.6587 | 0.7493 | 0.8398 | 0.8859 | 0.9285 | 0.9586 |

Kolmogorov–Smirnov | 0.6467 | 0.7253 | 0.8038 | 0.8543 | 0.9040 | 0.9308 |

^{1}Estimated Parameters.

^{2}Known Parameters.

**Table 9.**Average empirical power values of the Zone test and Quantile-Zone test for various sample sizes with the level of significance $\mathsf{\alpha}=0.05$—symmetric alternative distributions.

Test | $\mathit{n}$ | |||||
---|---|---|---|---|---|---|

10 | 20 | 30 | 50 | 100 | 200 | |

Zone (EP ^{1}) | 0.7570 | 0.8833 | 0.9236 | 0.9517 | 0.9832 | 0.9965 |

Quantile-Zone (EP) | 0.7309 | 0.8365 | 0.8945 | 0.9466 | 0.9855 | 0.9962 |

Zone (KP ^{2}) | 0.4646 | 0.6737 | 0.7694 | 0.8676 | 0.9428 | 0.9812 |

Quantile-Zone (KP) | 0.4921 | 0.7294 | 0.8462 | 0.9295 | 0.9832 | 0.9962 |

^{1}Estimated Parameters.

^{2}Known Parameters.

**Table 10.**Average empirical power values of the Zone test and Quantile-Zone test for various sample sizes with the level of significance $\mathsf{\alpha}=0.05$—asymmetric alternative distributions.

Test | $\mathit{n}$ | |||||
---|---|---|---|---|---|---|

10 | 20 | 30 | 50 | 100 | 200 | |

Zone (EP ^{1}) | 0.7286 | 0.8464 | 0.8789 | 0.9168 | 0.9579 | 0.9842 |

Quantile-Zone (EP) | 0.8984 | 0.9294 | 0.9470 | 0.9664 | 0.9881 | 0.9978 |

Zone (KP ^{2}) | 0.5010 | 0.6429 | 0.7143 | 0.8057 | 0.8878 | 0.9461 |

Quantile-Zone (KP) | 0.5673 | 0.9066 | 0.9367 | 0.9623 | 0.9878 | 0.9977 |

^{1}Estimated Parameters.

^{2}Known Parameters.

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**MDPI and ACS Style**

Avdović, A.; Jevremović, V.
Discrete Parameter-Free Zone Distribution and Its Application in Normality Testing. *Axioms* **2023**, *12*, 1087.
https://doi.org/10.3390/axioms12121087

**AMA Style**

Avdović A, Jevremović V.
Discrete Parameter-Free Zone Distribution and Its Application in Normality Testing. *Axioms*. 2023; 12(12):1087.
https://doi.org/10.3390/axioms12121087

**Chicago/Turabian Style**

Avdović, Atif, and Vesna Jevremović.
2023. "Discrete Parameter-Free Zone Distribution and Its Application in Normality Testing" *Axioms* 12, no. 12: 1087.
https://doi.org/10.3390/axioms12121087