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 The kurtosis of ( ( ))
��

( ) =
��
�
�V( )�
�
�
�
( ) 4 ( ) ( ) + 6 ( ) ( ) 3 ( )

=
�
�V( )��
�
�
�
�
�
( + 2 + 3 3) cosh(4Ω) + 4 ( +2) cosh(2Ω) +3(2 + )
=
�
2� cosh(2Ω) + �
where the numerator part of the skewness is

( ) 4 ( ) ( ) + 6 ( ) ( ) 3 ( )
�
�
�

�
�
�
�
= cosh(4Ω) cosh(2Ω) +
2
8
8
+ 3 sinh (Ω)cosh(2Ω) 3 sinh (Ω) 3 sinh (Ω)
�
�
�
�
�

3

�
�
�
�
cosh(4Ω) + cosh(2Ω)
= cosh(4Ω) cosh(2Ω) +
2
8
2
2
8
3
3
3
3
�
�
+ cosh(2Ω) cosh(4Ω) cosh(2Ω) +
4
2
4
2
2 � ( )� � � � 3 � � 3 sinh(3Ω)sinh(Ω) + 3 sinh (Ω) �

3 3 3 3 9
cosh(2Ω) + cosh(4Ω) + cosh(2Ω)
�
�
�
2 2 8 2 8
NTU Management Review Vol. 34 No. 1 Apr. 2024

�
�
�
�
�
= cosh(4Ω)( 4 + 6 3) + cosh(2Ω)( 3 +2)
8 2
3
+ (2 3 + )
�
�
8

�
�
�
�
�
�
= ( ) � ( + 2 + 3 3) cosh(4Ω) + 4 ( +2) cosh(2Ω)
8
+3(2 + )� ,
where

sinh (Ω) = �cosh(2Ω) � ,
�
2

�
sinh (Ω) = �cosh(4Ω) 4cosh(2Ω) +3� ,
8

sinh(3Ω)sinh(Ω) = �cosh(4Ω) cosh(2Ω)� ,
2

�
sinh (Ω)cosh(2Ω) = �cosh(4Ω) 2cosh(2Ω) + � ,
4
and
�
�
E( ) =E �� exp � � exp � �� �
2 2
1 −4 4 4 −2 2
= � � E � � �� − � � E � � ��
16 16 36

4 2 −2 1 4 −4 6
− � � E � � �� + � � E � � �� +
16 16 16
1 1 1 1 3
�
�
�
�
= ��� − ��� − �� + �� +
16 4 4 16 8
1 1 3
�
�
= cosh(4Ω) − cosh(2Ω) + , ,
8 2 8
Ω= , and = (1 ).
�
⁄
⁄

41
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