3.1. The Testing Procedure
Let
be iid variables that are distributed identically as random variable
. In this section, statistic
given with (2) is used to test the null hypothesis that variable
is normally distributed, i.e.,
against the alternative
(
is not normally distributed).
The critical region is two-sided interval
, determined by the critical condition given with
where
is the level of significance.
If the empirical value of test-statistic is inside the critical region , the null hypothesis is rejected.
The
p-value for this test is calculated as usual by
and then could be compared to the level of significance
. If
, the null hypothesis is rejected.
All of the advantages of the Quantile-Zone test discussed in [
4] hold for the Zone test due to the similar definition of the zone function. For instance, the method used in constructing the test provides tools for assessing the frequency of sample elements within a selected interval. On some level, it is even sensitive to outliers, because more of them will increase the likelihood of hypothesis rejection, and few of them, especially for large-sized samples, will not affect hypothesis acceptance. That is important since, in theory, normal variates are not bounded.
An additional advantage is that sample (1) consists of iid random variables, which make the distribution of the test statistic available in both formal and simulated variants. That also means that the values of the order statistic do not affect the value of test statistic , which is the case for many other test statistics, especially ones based on the EDF; hence, test statistic is not sensitive to repetitive values, i.e., it does not result in false negative decisions, which can occur very frequently in the case of such an event. Specifically, theoretical consideration allows for the existence of repetitive values (given a reasonable number of repetitions) within our normality hypothesis; however, empirically, these repetitions do not hinder the overall alignment of the sample structure with the fundamental characteristics of the normality hypothesis. The presence of such invariance properties proves advantageous in the context of normality tests
The testing procedure for the Quantile-Zone test is not very complicated to perform or implement in a technical solution, but the Zone test is even less complicated and gives the possibility of fast performance and implementation. This is an important observation because, despite new, more powerful tests being developed [
19,
26], these tests are not yet widely accepted as an alternative, probably due to the complexity of their application, or the fact that they can only produce high power on certain occasions, etc.
3.2. Power Analysis
For power analysis, we use 10,000 runs of statistic
Monte Carlo simulations. Statistic
samples are modelled for various alternative distributions, divided into symmetric and asymmetric ones. Monte Carlo simulations, as well as the alternative distribution modelling, have been performed via MATLAB codes. Some modelling algorithms available for MATLAB functions, such as ‘normrnd’, ‘chi2rnd’, and ‘trnd’, etc., are used [
30]. For the alternative distributions with no available MATLAB sampling function, we coded algorithms using the inverse function (CDF) method [
31]. The numbers of simulations, 10,000 and 100,000, have both proven to be satisfactory [
32]. Additionally, in our simulation study, we used both for null distribution, and the results are asymptotically identical.
The null distribution is . We have used the level of significance . Various sample sizes have been discussed and are available in the following tables. The selection of parameters for alternative distributions has been done in a way that maximizes alignment with the null distribution, within reasonable bounds.
The power of the Zone test is calculated by the following algorithm:
Modelling the sample of the chosen alternative distribution for the observed sample size ;
Calculating
and the empirical value of the test statistic
Repeating the first two steps times and thus obtaining the sample ;
Determining the EDF
of the sample obtained in the third step (
is the event indicator; it equals 1 if the event had occurred and 0 if it had not);
Calculating the power
of the test by
where
and
are the critical values of statistic
for
.
In several other papers [
19,
20,
21,
22], power calculations encompass multiple distributions; however, for many, such analyses might be considered unnecessary, as preliminary methods, such as a histogram, suffice for normality assessment. Furthermore, this situation inflates the average power value, concealing alternative distributions where the test exhibits low power. In essence, the power value is artificially boosted by large but irrelevant data. To address this, alternative distributions are thoughtfully selected in this study to provide a more accurate representation of the test’s applicability. The following figure (
Figure 5) presents a comprehensive list of all alternative distributions for which power calculations are conducted.
When examining symmetric alternative distributions, it becomes evident that the Zone test exhibits excellent performance, especially when the parameters of the null hypothesis distribution are known. The test achieves the highest performance with the uniform distribution and the lowest with the Laplace (0, 1) distribution. The test is powerful even for small sample sizes, though there are some exceptions. For instance, when the alternative distribution is Laplace (0, 1), it is the best to use the Zone test for
n ≈ 150 or higher; when the alternative distribution is it is the best to use the Zone test for
n > 50 (see
Table 3). Observing
Figure 5a, we can see that the discrepancy of the chosen symmetric alternative distributions from the null distribution is not as large as it usually is in the power analysis studies [
19,
20]. Given this observation, and noting that the Zone test performs better for alternative distributions that deviate more from the null hypothesis distribution, we can confidently regard our test as a suitable choice for normality testing against symmetric alternative distributions.
The scenario remains largely consistent in the context of asymmetric alternative distributions. Specifically, when examining all the alternative distributions presented in
Figure 5b, in this case, the same conclusions as those drawn in the previous paragraph hold true. There is a slight difference considering the performance of the test for small sample sizes where the alternative distributions are the
distribution and the Gumbel (0, 1) distribution; however, the difference noted does not have a significant impact on the final conclusion (see
Table 4).
In the following tables, power analysis for the same levels of significance, sample sizes, and alternative distributions is given; however, it is the case of the estimated parameters that is considered here. Now, the distribution of statistic
is given in
Table 2.
Zone test performance (and that of any other test) gets better when parameters are estimated because the empirical mean and standard deviation , practically, are better representatives of average value and average deviation than the assumed values and . If the sample is truly drawn from the distribution, and hold; otherwise, the Zone test will register a larger discrepancy. The performance of the test has improved for all of the symmetric alternative distributions. The biggest improvement can be noticed in the Laplace (0, 1) distribution. An interesting observation is that, through parameter estimation, the Zone test demonstrates improved performance for the Logistic (0, 1) distribution compared to the Cauchy (0, 1) distribution. This outcome differs from the one with known parameters.
Figure 6 provides a visual comparison of the Zone test’s performance for symmetric alternative distributions, considering both specified and estimated parameters. This graphical representation corresponds to the information presented in
Table 3 and
Table 5.
Figure 7 provides a visual comparison of the Zone test’s performance for symmetric alternative distributions, considering both specified and estimated parameters. This graphical representation corresponds to the information presented in
Table 4 and
Table 6.
This conclusion holds for asymmetric alternative distributions as well. The most noticeable improvements in the power of the test are identified for the distribution and Gumbel (0, 1) distribution. When the parameters are specified, the Zone test shows better performance for Pareto (0.1, 1) compared to the Burr (3, 1) distribution, but for estimated parameters the opposite conclusion holds. The test performs better for symmetric alternative distributions; however, it has been shown to be very powerful in both variants.
The following section delves into a more detailed discussion of consistency properties, addressing both the advantages and disadvantages of the Zone test in comparison to other commonly employed normality tests.
3.3. Comparative Analysis
Here, the obtained power values are used for calculating the average ones, which are then compared to the average power values for other well-known and widely used normality tests. The tests that the Zone test is compared to are the Kolmogorov–Smirnov test [
33] with its variant for estimated parameters (Lilliefors test) [
34], Chi-square test [
28], Shapiro–Wilk test [
35], and Anderson–Darling test [
36]. For most of these tests, the same data obtained in [
4] are used.
Table 7 reveals that, with known parameters, the Zone test outperforms other tests in the case of symmetric alternative distributions. This conclusion holds even for small sample sizes. The power of the Zone test is the largest for all sample sizes. Even for large sample sizes, such as
, the Zone test is still noticeably more powerful than other analyzed tests.
In
Table 8, results indicate a different outcome; namely, that the Zone test has not performed as well as the competitor tests when the parameters of the normal distribution are known and in cases involving asymmetric alternative distributions. There is an exception when
, where this test has a higher power value than the Kolmogorov–Smirnov test. The power of this test is still very high, and the differences are not an argument against using the Zone test for known parameters of the normal null distribution.
When the parameters of the null distribution are estimated, the results show that the Zone test is more powerful than all considered competitor tests. This result holds for all of the sample sizes observed. The variant of the test for estimated parameters is more powerful because a better fit with the Zone distribution in
Table 2 is obtained.
The following figure (
Figure 8) provides a graphical interpretation of the previous results, offering a comparative analysis of power function graphs for both symmetric and asymmetric alternative distributions.
The results highlight the Zone test as a good choice for normality testing. In the cases of both the known and estimated parameters of the null hypothesis distribution, the Zone test exhibits similar or even better performance compared to the tests included in this comparative analysis. The alternative distributions utilized are likely among the most representative of commonly used distributions [
19,
21,
22]. Furthermore, upon reviewing power analysis results for tests not covered in this comparative analysis, an additional advantage of the Zone test becomes apparent. These observations can therefore be generalized when comparing the Zone test to the majority of well-known normality tests [
19,
20,
21,
22,
23,
24,
25,
26].
Finally, Zone test and Quantile-Zone test performances are compared since, in the authors’ previous paper, it was shown that, in a similar analysis, the Quantile-Zone test yielded the best results.
Table 9 illustrates that, for symmetric alternative distributions, the Quantile-Zone test demonstrates significantly better performance in the case of known parameters; however, in the case of estimated parameters, the Zone test is the one with better performance. For large sample sizes, the differences are of no significance.
Table 10 indicates that, for asymmetric alternative distributions, the Quantile-Zone test outperforms the Zone test, establishing it as the most powerful normality test based on our results. In
Figure 9, the results of this comparison are graphically illustrated.
Despite the best performance of the Quantile-Zone test, there are advantages to the Zone test that might make it a better choice, on some occasions, for the following reasons:
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 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.