An Alternative Measures of Moments Skewness Kurtosis and JB Test of Normality
- https://doi.org/10.2991/jsta.d.210525.002How to use a DOI?
- Robust moments, Robust skewness, Robust Kurtosis, Robust test of normality
If we know the statistics of central tendency and dispersion, we still cannot nature a complete design about the distribution. About these measures we should know more information's of skewness and kurtosis, which are enables us to have a design the distribution. However, there is evidence that they may response poorly in the presence of non-normality or when outliers arise in data. We examine the performances of popular and frequently used measures of skewness , kurtosis and Jarque–Bera test of normality that they may not perform and we anticipates in the existence of non-normality or outliers. In this paper, firstly, we develop robust measures of moments and we formulate a new statistics of skewness and kurtosis which we name robust skewness and robust kurtosis . Again, in this paper, we modify Jarque–Bera test of normality, which we label Robust Jarque–Bera (RJB). These measures should be fairly robust. The effectiveness of the proposed measures is investigated by simulation approach. The results demonstrate that the newly proposed skewness , kurtosis and RJB test outperform the skewness, kurtosis and Jarque–Bera test of normality when a small percentage of outliers are present or absent in the data.
- © 2021 The Authors. Published by Atlantis Press B.V.
- Open Access
- This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).
The learning of central tendency and dispersion provides us with variable information involving to the central value as well as the variability of the distribution. Unfortunately, these measures fail to exhibit how the observations are given and accumulated about the central value of the distribution. The arrangements and accumulation of the observations establish the characteristics of the distribution with respect to its shape and pattern . By shape characteristics of a distribution, we refer to the level of these two characteristics what is known as the measures of skewness and kurtosis respectively. Skewness and kurtosis can help us to visualize the asymmetry and peakedness of a frequency distribution. The theoretical and practical background of various measures of moments, skewness and kurtosis are documented in several books [2–11] and journal articles , to name but a few. Among them the absolute measures of skewness are not calculate for comparing two series. On the other hand, Prof. Karl Pearson's coefficient of skewness is baffled to calculate, if mode is ill-defined as well as skewness for moderately asymmetrical distribution give limit is . In practice, these limits are rarely attained. Again, Bowley's coefficient of skewness is depend only on the central 50% of the data as well as based upon moments, coefficient of skewness is depends on and , but if either or . Since, cannot negative, if and only if . However, the most popular location and scale estimators are the mean and standard deviation, which is known to be extremely sensitive to outliers. Although mean, variance and covariance are the most frequently used summary measures of univariate and multivariate PDFs, we occasionally need to consider higher moments of the PDFs, such as the second, third and the fourth moments. Now, the rth moment about the mean (μ) is defined as
The second, third and fourth moments of a distribution are often used in studying the “shape” of a probability distribution, in particular, its skewness, S (i.e., lack of symmetry) and kurtosis, K (i.e., tallness or flatness), are defined as
One measure of skewness is defined as
A commonly used measure of kurtosis is given by
Here we define a most popular and commonly used goodness of fit Robust Jarque and Bera  test for normality, which utilized the information of the skewness and kurtosis, is formulate by
In that case the value of the JB statistic is expected to be 0. Under the null hypothesis that the data set is normally distributed, JB showed that asymptotically (i.e., in large samples) the JB statistic follows the chi-square distribution with 2 degrees of freedom. If the computed p value of the JB statistic in an application is sufficiently low, which will happen if the value of JB is very different from 0, one can reject the hypothesis that the data are normally distributed. But if the p value is reasonably high, which will happen if the value of the statistic is close to 0, we do not reject the normality [13,14].
Since 's sensitive to outliers from the underlying distribution, the resulting skewness and kurtosis can be affected by outliers. In particular, the JB test is sensitive to outliers because of that. In this paper, to overcome the sensitivity of departures from normal distribution, we focus on finding a novel and straightforward measure of skewness, kurtosis and JB statistic. We label its robust skewness , robust kurtosis and robust Jarque-Bera test for normality which are introduced in Section 2. The properties of these new measures are illustrated in Section 3 with a real-life data. The performance of the proposed measures is investigated in Section 4 through a Monte Carlo simulation experiment.
2. PROPOSE ROBUST MODIFICATION OF MOMENTS SKEWNESS KURTOSIS AND JB STATISTIC
The presence of a small proportion of outliers in a sample can have a large distorting influence on the sample mean and the sample variance. It is well known that these classical estimators, optimal under the normality assumption, are extremely sensitive to atypical observations in the data. Since the measures of skewness and kurtosis are based on mean and variance, it's also sensitive to outliers. There exist several measures of robustness of an estimator [15,16], but in this paper, the decile mean (DM) will be used. This is rich tool that summarizes several aspects of the robustness of an estimator. A survey on DM is given by . Now we define DM aswhere , are 9 decile from grouped or ungrouped data. Therefore, we develop robust measures of moments, skewness, kurtosis and JB statistic of normality test.
2.1. Robust Moments
In statistics, moments are certain constant values in a given distribution, it's obviously fall under descriptive statistics. Because of this nature, the moments help us to establish the nature and form of the underlying distribution. Consider a variable X, assuming values , and then the raw moment of a variable X about any point A is defined by
The first-four raw moments about the value A are defined as
Replacing A by in the above expression, we will get the central moments are defined as
In general, the central moment is defined as
Thus, it is to be significant that you can compute an infinite number of moments for a given distribution, but in practice, we need only four moments to investigate the form and characteristics of a distribution.
2.1.1. Relation between raw moments and central moments
In general, ,
Thus the formula enable us to find the moments about any point, once the decile mean and the decile mean are known.
2.1.2. Effect of change of origin and scale on moments
Let , where A and h are origin and scale respectively.
Now the raw moments of x about any point A is given by
And the moment of x about decile mean is
Thus the moment of the variable x about decile mean is times the moment of the variable y about decile mean .
2.2. Robust Skewness and Robust Kurtosis
Literally, skewness means “lack of symmetry” as well as kurtosis means “convexity of curve.” We study skewness and kurtosis to have an idea about the shape and pattern of the curve. The robust measures of skewness and kurtosis may also be obtained by making use of the proposed robust moments. A relative measure of robust skewness denoted by , is define as follows:
The value of shall be zero for a perfectly symmetrical distribution. It is obvious from the above formula that a distribution will be positively or negatively skewed according as the value of is positive or negative.
The most important measure of robust kurtosis is , defined as
For normal distribution . In other words, if , the distribution is leptokurtic; if , the distribution is platykurtic; if , the distribution is mesokurtic.
2.2.1. Prove that and are invariant to the changes in origin and scale of measurement
Let and denote the values of and calculated from a set of observations pertaining to a variable X.
Let Y be a transformed variable assuming values .
Now, suppose , where A and h are origin and scale respectively.
The corresponding phi values are as follows:
Hence the proof.
2.2.2. For any set of values , prove that
Let us recall that
Consider the following expression
Choosing and , the above expression becomes
This completes the proof.
2.2.3. For any set of values , prove that
Let us recall that
Consider the following expression
Choosing and , the above expression becomes
2.3. Robust Jarque–Bera (RJB) Test of Normality
AS a result, and following the measures of robust skewness and robust kurtosis discussed earlier, for a normal PDF and , that is a normal distribution is symmetric and mesokurtic. Therefore, a simple test of normality is to find out whether the computed values of robust skewness and robust kurtosis depart from the norms of 0 and 3, is defined by-
It follows that the value of the statistic is estimated to be 0. Under the null hypothesis of normality, is distributed as a chi-square statistic with 2 degrees of freedom. If the p value is reasonably high, which will happen if the value of the statistic is close to 0, we do not reject the normality. But if the computed p value of the statistic in an application is sufficiently low, which will happen if the value of is very different from 0, one can reject the hypothesis that the data are normally distributed.
3. REAL DATA EXAMPLES
In this section, we apply some recognized graphs, classical and our newly proposed measures as well as tests on real data sets to make out the data are normal or not. Let us first consider the weight of a bag of carrots data, which is taken from . This data consists of 12 observations. When we apply usual outliers' detection method (Med-MAD) [18,19], we notice that this data does not hold any outlier. The outcome of graphical, classical and newly proposed measures and tests for this data are given below:
Since, the original data set is free from outliers we watch from Figure 1 that the type of the density plot is positively skewed distribution as well as QQ-plot are reasonably normal in shape.
From Table 1 reports the data are positively skewed and platykurtic normal shape based on both classical and proposes estimators. Any more notice that the inference of classical and propose JB tests results are same. But it is worth mentioning that Hogg and Tanis  used only graphical two tests and announced that the data is normal.
Four measures result of the weight of a bag of carrots data.
Now judge another data set, the diameter of individual grains of soil, such as porosity, data has taken as of , which contains 30 observations. In the beginning, we make sure outliers by usual method (Med-MAD) [18,19]; it detects 2 outliers (cases 6 and 14). Original data and deleting these outliers we verify the normality of the data set by graphical as well as analytical tests of normality, which results has publicized below:
In Figure 2 gives the two conclusions: one the density and QQ plots indicate the data set is positively skewed and normal characteristics when the data contain outliers, another graphs look negatively skewed and non-normal pattern because of free from contamination.
From Table 2 demonstrate that the classical measures , and suggest that the data set is positively skewed and platykurtic as well as the classical JB test declare that the data set is normal when outliers presence in the data set. On the other hand, when extreme values present in the data set my newly proposed robust estimators , and advice that the data set is negatively skewed and platykurtic as well as my newly proposed RJB test identify non-normality. It is to be important that both tests speak out non-normality when the data set free from unusual observations. But it is notified that Hogg and Tanis  utilized only graphical two tests and certified that the data is normal. Since the classical measures and JB test fail to discover the actual nature and shape of the data distribution, we recommend that our newly proposed measures and RJB test are more efficient and robust for correct inference.
|Proposed||−0.00001||−8.559||0.0075||0.0067||1.919||1.794||1.4606||1.6970||0.482||0.428||Not normal||Not normal|
Four measures of the diameter of individual grains of soil, such as porosity, data with outliers (WO) and without outliers (WOO).
Again, we assume a real data; the weight of packaged product data is taken from , which is consists of 100 observations. Initially, we confirm outliers by usual method (Med-MAD) [18,19]; it detects 6 outliers (cases 29, 50, 70, 71, 75 and 81). We test out the normality of the data set by graphical and analytical methods, which results have shown below:
From Figure 3 demonstrate that the data display positively skewed and non-normal because the points do fall far from a straight line in QQ-plot when the data contain extreme values. Conversely, we show that the data exhibit negatively skewed and normal for the reason that the points do drop over the straight line in QQ-plot when the data set is free from outliers.
From Table 3 shows that the classical statistics , and hint that the data are positively skewed and leptokurtic as well as the classical JB test recognize non-normal pattern when outliers present in the data set. Alternatively, our newly proposed statistics , and tell that the data is negatively skewed and platykurtic as well as RJB test has given exact identification when the data set hold outliers. Moreover, both the classical and proposed measures and tests have given right finding after removing outliers. But it is noteworthy that Hogg and Tanis  used only graphical two tests and licensed that the data is normal. Thus, we explain that the classical measures and JB test baffle to determine actual inference when extreme observations present in the data set. On the other hand, our newly proposed measures and RJB test have given correct inference when outliers present in the data set or absent.
Four measures of the weight of packaged product data with outliers (WO) and without outliers (WOO).
4. REPORT OF MONTE CARLO SIMULATION STUDY
In this section, we report a Monte Carlo simulation study which is aim to compare the performance of the newly proposed robust measure of moment , skewness and kurtosis with other popular and commonly used classical same measures. We also verify the sound power comparison of my newly proposed RJB test and classical JB test. We simulate data under not normal as well as normal from uniform distribution. In my simulation experiment, we have taken different sample sizes, n = 50, 100, 200, 500 and 1000. Each experiment is run 10,000 times and the tests consequences are given below.
From Table 4 shows that the classical measure , and give higher percentage for normal when the data sets are not normal. The classical normality test JB, the rejecting power of alternative hypothesis is very high when alternative hypothesis is true. Alternatively, the proposed measure , and give very low percentage for normal when the data sets are not normal. The proposed normality test RJB, the rejecting power of alternative hypothesis is very low when alternative hypothesis is true.
|Power (In Percentage)
|n = 50|
|n = 100|
|n = 200|
|n = 500|
|n = 1000|
Performance comparison under not normal.
From Table 5 reports that the classical measure , and give very low percentage for normal when the data sets are normal. The classical normality test JB, the rejecting power of null hypothesis is very high when null hypothesis is true. Conversely, the proposed measure , and give very high percentage for normal when the data sets are normal. The proposed normality test RJB, the rejecting power of null hypothesis is very low when null hypothesis is true.
|Power (In Percentage)
|n = 50|
|n = 100|
|n = 200|
|n = 500|
|n = 1000|
Performance comparison under normal.
Analyzing the above discussion, we demonstrate that the proposed measures and test give right outcome when the data set is normal and not normal. So over all we can say that the proposed measures and test are better than any other measures and tests to check the normality.
In this paper, to sum up the whole aforesaid discussion, our main objectives was to propose a new robust measures of moments, skewness, kurtosis which represent the data better than any others existing tools. We also propose a new statistic of Jarque–Bera test of normality, so that it can correctly identify right inference than any others existing tests. Both cases we have seen that irrespective of the presence of outliers or not, our newly proposed robust measures of moments, skewness, kurtosis and RJB test performs better than other classical measures and tests for different sample sizes. Note that our proposed measures fulfill the various properties and conditions which we proved in Appendices. Mention that all existing graphical and analytical measures and test of normality fail to identify appropriate outcome for real data sets and small to moderate sample sizes when outliers present in the data sets. Not only that, both the real-life data and simulation study demonstrate that our newly proposed robust moments, robust skewness, robust kurtosis and RJB test of normality have given more actual sound results in a variety of situations and hence can be recommended to use an effective measures and test.
CONFLICTS OF INTEREST
The authors declare they have no conflicts of interest.
Md.Siraj-Ud-Doulah conceived and designed the study, analyzed the data, interpretation of the data and wrote the manuscript. The final version of the manuscript was reviewed and approved by the author.
The author would like to thank the anonymous referees for their helpful remarks.
Cite this article
TY - JOUR AU - Md. Siraj-Ud-Doulah PY - 2021 DA - 2021/05/31 TI - An Alternative Measures of Moments Skewness Kurtosis and JB Test of Normality JO - Journal of Statistical Theory and Applications SP - 219 EP - 227 VL - 20 IS - 2 SN - 2214-1766 UR - https://doi.org/10.2991/jsta.d.210525.002 DO - https://doi.org/10.2991/jsta.d.210525.002 ID - Siraj-Ud-Doulah2021 ER -