Applications of Normality Test in Statistical Analysis

In this study, to power comparison test, different univariate normality testing procedures are compared by using new algorithm. Different univariate and multivariate test are also analyzed here. And also review efficient algorithm for calculating the size corrected power of the test which can be used to compare the efficiency of the test. Also to test the randomness of generated random numbers. For this purpose, 1000 data sets with combinations of sample size n = 10, 20, 25, 30, 40, 50, 100, 200, 300 were generated from uniform distribution and tested by using different tests for randomness. The assessment of normality using statistical tests is sensitive to the sample size. Observed that with the increase of n, overall powers are increased but Shapiro Wilk (SW) test, Shapiro Francia (SF) test and Andeson Darling (AD) test are the most powerful test among other tests. Cramer-Von-Mises (CVM) test performs better than Pearson chi-square, Lilliefors test has better power than Jarque Bera (JB) Test. Jarque Bera (JB) Test is less powerful test among other tests.


Introduction
In parametric analysis assuming that population is normal. In this case, checking whether population is normal or not. Normality tests are used in different sectors. One application of normality tests is to the residuals from a linear regres- After testing normality if data are not normal, to apply Box Cox transformation method for transforming non-normality data to normal. So main aim is to discuss whether the goodness of fit of a statistical model describes how well it fits a set of observations. Measures of goodness of fit typically summarize the discrepancy between observed values and expected values under the model in question.
There is the use of size corrected method in the determination of power. Main aim is to propose a new algorithm for testing multivariate normality. A random number generator is a computational or physical device designed to generate a sequence of numbers or symbols. Nowadays, after the advent of computational random number generator, a growing number of government-run lotteries, and lottery games, are using RNGs instead of more traditional drawing methods, such as using ping-pong or rubber balls etc.
If normality is not a durable assumption, then one alternative is to ignore findings of the normality check and proceed as the data are normally distributed.
But this is not practically recommended because in many situations it could lead to incorrect calculations. Due to countless possible deviations from normality, Andrews et al. [1] concluded that multiple approaches for testing MVN would be needed. Conover et al. [2] observed that the Kolmogorov Smirnov statistic belongs to the supremum class of EDF statistics and this statistics is based on the largest vertical difference between hypothesized and empirical distribution.
Gray et al. [3] observed that the multivariate normality of stock returns is a crucial assumption in many tests of assets pricing models. This paper utilizes a multivariate test procedure, based on the generalized method of moments, to test whether residuals from market model regressions are multivariate normal.
Kankainen et al. [4] observed that classical multivariate analysis is based on the assumption that the data come from a multivariate normal distribution. Several tests for assessing multinormality, among them Mardia's popular multivariate skewness and kurtosis statistics, are based on standardized third and fourth moments. In that report, they investigate whether, in the test construction, it is advantageous to replace the regular sample mean vector and sample covariance matrix by their affine equivariant robust competitors.
Koizumi et al. [5] observed that some tests for the multivariate normality based on the sample measures of multivariate skewness and kurtosis. For univariate case, Jarque and Bera proposed bivariate test using skewness and kurtosis.
They propose some new bivariate tests for assessing multivariate normality which are natural extensions of Jarque-Bera test.
Major power studies done by Pearson et al. [6] have not arrived at a definite answer but a general consensus has been reached about which tests are powerful.
Richardson et al. [7] observed that a general procedure that takes account of correlation across assets that focus on both the marginal and joint distribution of Major power studies done by Shapiro et al. [8] have not arrived at a definite answer but a general consensus has been reached about which tests are powerful.
The performance of different univariate normality testing procedures for power comparison are compared by using the new algorithm and different univariate and multivariate test are analyzed and also review efficient algorithm for calculating the size corrected power of the test which can be used to compare the efficiency of the test. Also to test the randomness of generated random numbers.
Different datasets are generated from uniform distribution and tested by using different tests for randomness. And data were also generated from multivariate normal distribution to compare the performance of power of univariate test by using different new algorithms.

Materials and Methods
To complete this study 1000 data sets with combinations of sample size n = 10, data, which is a plot of the ordered data x j against the normal quantiles q j , is shown in the following Figure 1.
From the figure we see that the pairs of points (q j, x j ) very nearly along a straight line and we would not reject the notion that these data are normally distributed with sample size n = 40.

Univariate Normality Test Procedure
The purpose of this study is to focus on general goodness of fit tests and their  applications to test for normality. While we attempt to mention as broad range of tests as possible, we will be concerned mainly with those tests that have been shown to have decent power at detecting normality to decide which test is appropriate at which situation. In Chi-square test, a single random sample of size n is drawn from a population with unknown cdf F x . The test criterion suggested by Pearson et al. [6] is the random variable

Multivariate Test Procedure
The

Analysis of Univariate Normality Test
Null hypothesis: Data follows normal distribution.
Alternative hypothesis: Data does not follow normal distribution (Table 1).
From Table 2 observe that p-values for these tests are less than level of significance 0.05. To accept the null hypothesis i.e. the data may not come from normal distribution.

Comparative Study for Generated Random Number of Different Univariate Normality Test
In many power comparison problems, it was not clearly under stable from their pothesis will be 0.05 but under the alternative hypothesis, the power will be greater because of the greater deviation from null or departure from normality and a particular time it will be closer to 0 or 1. The higher the distance from null hypothesis, higher the power. The hypotheses exhibit the variation of power because of the sample size such as low cell frequency due to small sample size, contamination and change of parameter value. So it's better and convenient to use size corrected power (Table 3). Figure 2 is concerned with power curves. From Figure 2 and Table 3    chi-square Q-Q plot we can observe that the data are not scattered around the 45-degree line with a positive slope, the greater the departure from this line, the greater the evidence for the conclusion that the series is not normally distributed. Here the data departures from the line so we may conclude that the series is not normally distributed.

Limitations of the Study
In this case, we use R programming language which is not always best. Another limitation on sample size and dimension is imposed to keep the amount of computing time reasonable and to consider sample size that minimal for the multivariate analysis. These small sample sizes are likely to be the situation where the assumption of the multivariate normality is most critical to the researchers.