Table 3 Overview of selected... | Emerald Publishing

Table 3

Overview of selected goodness-of-fit tests, associated test statistics and corresponding rejection criteria

S. no.	Test of normality	Test statistic	Criteria for rejection
1	Lilliefors Test (Lilliefors, 1967)	Test statistic: D_n = max \|F_x(x) – S_n(x)\| S_n = cumulative frequency function F_x = assumed theoretical CDFD_n = maximum difference $D_{n}^{α} = critical values at various α$	If $D_{n} < D_{n}^{α}$ ⁠, the proposed theoretical distribution is acceptable at the specified significance level α
2	J-B test (Jarque and Bera, 1987)	Test statistic: $J B = n (\frac{{(\sqrt{b_{1}})}^{2}}{6} + \frac{{(b_{2} - 3)}^{2}}{24})$ $\sqrt{b 1} = s k e w n e s s$ ⁠; b₂ = kurtosis	The test statistic is compared with 5.9915
3	A-D test (Anderson and Darling, 1954)	$A^{2} = - \sum_{i = 1}^{n} [(2 i - 1) {ln F_{x} (x_{i}) + ln [1 - F_{x} (X_{(n + 1 - i)})]} / n] - n$ $A^{} = A^{2} (1.0 + \frac{0.75}{n} + \frac{2.25}{n^{2}})$ $c_{α} = a_{α} (1 + \frac{b_{0}}{n} + \frac{b_{1}}{n^{2}})$ ⁠, n = no. of sample a_α, b₀, b₁are constants*	The test statistic is compared with the related critical value
4	DP test (D’Agostino and Pearson, 1973)	$D P = z^{2} (\sqrt{b_{1}}) + z^{2}$ (b₂) $\sqrt{b_{1}} = s k e w n e s s, b_{2} = k u r t o s i s$	p-value of DP value is compared with significance level (α)
5	CSQ test (Pearson, 1900)	$CSQ = \sum_{j = 1}^{c} \frac{{(O_{j} - E_{j})}^{2}}{E_{j} .}$ O_j = observed values, E_j = expected values	CSQ value is compared with 5.991
6	CVM test (Cramer, 1928; Smirnov, 1936; Stephens, 1974)	$C V M = \frac{1}{12 n} + {\sum_{i = 1}^{n} (z_{i} - \frac{2 i - 1}{2 n})}^{2}$ $Z_{i} = \frac{X (i) - \bar{X}}{S}$ Z_i = is the standardized observation $\bar{X} = i s t h e s a m p l e m e a n$	The calculated CVM statistics are compared with the critical value at a chosen significance level
7	S-W test (Shapiro and Wilk, 1965)	b = a1(x_n – x₁) + a2(x_n_–1 – x2) +… …, $w = \frac{b^{2}}{(n - 1) S^{2}}$ a1, a2,… are coefficients based on sample size, s = standard deviation of sample data	Calculated W is compared with table W

S. no.	Test of normality	Test statistic	Criteria for rejection
1	Lilliefors Test (Lilliefors, 1967)	Test statistic: D_n = max \|F_x(x) – S_n(x)\| S_n = cumulative frequency function F_x = assumed theoretical CDFD_n = maximum difference $D_{n}^{α} = critical values at various α$	If $D_{n} < D_{n}^{α}$ , the proposed theoretical distribution is acceptable at the specified significance level α
2	J-B test (Jarque and Bera, 1987)	Test statistic: $J B = n (\frac{{(\sqrt{b_{1}})}^{2}}{6} + \frac{{(b_{2} - 3)}^{2}}{24})$ $\sqrt{b 1} = s k e w n e s s$ ; b₂ = kurtosis	The test statistic is compared with 5.9915
3	A-D test (Anderson and Darling, 1954)	$A^{2} = - \sum_{i = 1}^{n} [(2 i - 1) {ln F_{x} (x_{i}) + ln [1 - F_{x} (X_{(n + 1 - i)})]} / n] - n$ $A^{*} = A^{2} (1.0 + \frac{0.75}{n} + \frac{2.25}{n^{2}})$ $c_{α} = a_{α} (1 + \frac{b_{0}}{n} + \frac{b_{1}}{n^{2}})$ , n = no. of sample a_α, b₀, b₁are constants	The test statistic is compared with the related critical value
4	DP test (D’Agostino and Pearson, 1973)	$D P = z^{2} (\sqrt{b_{1}}) + z^{2}$ (b₂) $\sqrt{b_{1}} = s k e w n e s s, b_{2} = k u r t o s i s$	p-value of DP value is compared with significance level (α)
5	CSQ test (Pearson, 1900)	$CSQ = \sum_{j = 1}^{c} \frac{{(O_{j} - E_{j})}^{2}}{E_{j} .}$ O_j = observed values, E_j = expected values	CSQ value is compared with 5.991
6	CVM test (Cramer, 1928; Smirnov, 1936; Stephens, 1974)	$C V M = \frac{1}{12 n} + {\sum_{i = 1}^{n} (z_{i} - \frac{2 i - 1}{2 n})}^{2}$ $Z_{i} = \frac{X (i) - \bar{X}}{S}$ Z_i = is the standardized observation $\bar{X} = i s t h e s a m p l e m e a n$	The calculated CVM statistics are compared with the critical value at a chosen significance level
7	S-W test (Shapiro and Wilk, 1965)	b = a1(x_n – x₁) + a2(x_n_–1 – x2) +… …, $w = \frac{b^{2}}{(n - 1) S^{2}}$ a1, a2,… are coefficients based on sample size, s = standard deviation of sample data	Calculated W is compared with table W

Source(s): Table by authors