Table 2 The GoF measures of the fitted EN, SN, normal (N) and beta normal (BN) distributions to two real data sets

First application

\(\hat {\theta }_{i}\)

(0.2490, 3.2023, 1.7709, 0.9768)

(4.746, 1.109, 0.016)

(4.7619, 1.1094)

(991.2293, 54.4065, -23.4249, 17.3049)

\(\text {Sd}(\hat {\theta }_{i})\)

(<0.0001, 0.0005, <0.0001, <0.0001)

(1.029, 0.039, 0.004)

(0.2420, 0.1711)

(1.1628, 1.3819, 0.4080, 0.1533)

W ^∗

0 . 0 6 3 0

1.5672

0.0800

0.4296

A ^∗

0 . 3 9 1 8

7.5418

0.4758

2.3953

KS

0 . 1 4 8 4

1.0000

0.1452

1.0000

Second application

\(\hat {\theta }_{i}\)

(0.4804, 100.0000, -141.4568, 75.5640)

(0.0009, 166.0232, 32.7894)

(166.0746, 46.3746)

(1001.6514, 4.5535, -565.4361, 276.6320)

\(\text {Sd}(\hat {\theta }_{i})\)

(0.0126, 0.0005, 0.0002, 0.0001)

(0.1082, 0.0001, <0.001)

(5.6655, 4.0061)

(149.8685, 0.6375, 0.2980, 0.1719)

W ^∗

0 . 0 2 5 9

0.1037

0.1185

0.3018

A ^∗

0 . 2 4 2 8

0.8134

0.8942

1.8874

KS

0 . 0 5 3 2

0.8519

0.0835

1.000