Page 47 - 34-1

NTU Management Review Vol. 34 No. 1 Apr. 2024

 The skewness of ( ( ))
 The skewness of ( ( ))
��
��
 The skewness of ( ( )) �
�
� ( )�
��
� ( )�
( ) = � �
��
( ) =
��
 The skewness of ( ( �
�� �� � ( )�
�V( )�
 The skewness of ( ( )) )) � � �
�V( )��
( ) = � ( )�
��
�
�
� ( )� �
�
�) 3 ( ) ( ) + 2 ( )
(
( ) 3 ( ) ( ) + 2 ( )
�
( )
( ) == �V( )�� �
=
�� ��
� �
= � � �
� �V( )�� � �
�V( )� 2 ( )
( ) 3 ( ) ( ) +
�V( )��
= �V( )��
�
�
�
�
( ) 3 ( ) ( ) + 2 ( )
�
�
�
( ) 3 ( ) ( ) + 2 ( )

= � ( 1)� ( +2) sinh(3Ω) +3 sinh(Ω)�
�V( )��
=
� ( 1)� ( +2) sinh(3Ω) +3 sinh(Ω)�
�
,
= �
= �V( )�� �
�V( )��
�2� cosh(2Ω) +1�
� ( 1)� ( +2) sinh(3Ω) �+3 sinh(Ω)�
�2� cosh(2Ω) +1�
=
where the numerator part of the skewness is �
� ( 1)� ( +2) sinh(3Ω) +3 sinh(Ω)�
where the numerator part of the skewness is
� ( 1)� ( +2) sinh(3Ω) +3 sinh(Ω)�
�2� cosh(2Ω) +1�
=
=
�) 3E( )E( ) + 2 ( )

�2� cosh(2Ω) +1�
�
�
�
E(
where the numerator part of the skewness is � �

�
�
E( ) 3E( )E( ) + 2 ( ) +1�
�2� cosh(2Ω)
E( ) 3E( )E( ) + 2 ( )

where the numerator part of the skewness is � 3 �
where the numerator part of the skewness is 3 � �
�
�
3
�
1
�
�
�
3
1
E( ) 3E( )E( ) + 2 ( ) �cosh(2Ω) 1) � �sinh(Ω)�
=
� �sinh(3Ω) + �sinh(Ω) + (
�
�
= 4 1 �sinh(3Ω) + �sinh(Ω) + ( cosh(2Ω) 1) � �sinh(Ω)�
E( ) 3E( )E( ) + 2 ( )
2
�
� 4
�
4
4 3
�
2 3
�
�
�
= 1 �sinh(3Ω) + �sinh(Ω) + ( cosh(2Ω) 1) �� �sinh(Ω)�
3
3
�
�
�
�
3 4
�(Ω)
= 1 4 �sinh(3Ω) + � �sinh(Ω) + ( cosh(2Ω) 1) � �sinh(Ω)�
3 2
�
�
� �
2 �sinh
2 �sinh (Ω)
�
2
= 4 �sinh(3Ω) 4 + �sinh(Ω) + ( cosh(2Ω) 1) � �sinh(Ω)�
4 � 4 2
�
1
3 � 2 � sinh (Ω) 3 � 3 � 3 �
�
� �
2 �sinh (Ω)
3
�
�
�
3
�
3
�
= 3 �sinh(Ω) 1 � �sinh(3Ω) + �sinh(3Ω) �sinh(Ω) �sinh(Ω)
= 4 �sinh(Ω) �sinh(3Ω) + �sinh(3Ω) �sinh(Ω) �sinh(Ω)
4 sinh (Ω)
4
4
2
4
�
2 3
4 3
4 3
1
4 3
�
�
2 � �
�
�
�
�
3
�
1
�
�
3
3
= 3 �sinh(Ω) �sinh(3Ω) + �sinh(3Ω) �sinh(Ω) �sinh(Ω)
�
3
�
1
3 3 4 �sinh(Ω)
3 4
3 2
= 4 �sinh(Ω) 1 �sinh(3Ω) + �sinh(3Ω) �sinh(Ω) �sinh(Ω)
�
�
3
�
� �
� �
�sinh(3Ω) + �sinh(Ω)
1 4 �sinh(3Ω) +
4
4 4 4 2
= �sinh(Ω) �sinh(3Ω) + �sinh(3Ω) �sinh(Ω) �sinh(Ω)
2
2
�
�
2 4
4 1 1 � 3 3 � 4 2
2 4
� �sinh(3Ω) + �sinh(Ω)
�
1 �sinh(3Ω) + �sinh(Ω) �
3
� 3 + 2) sinh(Ω) �( 2 1)
1
= sinh(3Ω) �( 3 + 2) sinh(Ω) �( 2 1)
= sinh(3Ω) 1 2 �� 2 3 2 3 � � � �
2 �(
�sinh(3Ω) + �sinh(Ω)
4
4
4 1
4 3
�
�
�2
� �( 2 1)
3 2
1
�
�
�
= sinh(3Ω) �( 3 + 2) sinh(Ω) �( 2 1)
= sinh(3Ω) �( 3 + 2) sinh(Ω)
�
�
1
4 3 4
�
1 4
� �
�
= 41 �( 1) �� ( +2)sinh(3Ω) + 3sinh(Ω)� �
= �( 1) � ( +2)sinh(3Ω) + 3sinh(Ω)� 2 1)
�
= sinh(3Ω) �( 3 + 2) sinh(Ω) �(
4
1
4 � �
4
= �( 1) � ( +2)sinh(3Ω) + 3sinh(Ω)�
14
� �
where = �( 1) � ( +2)sinh(3Ω) + 3sinh(Ω)� ,
where
41 4
�
�
1
where
where = �( 1) � ( +2)sinh(3Ω) + 3sinh(Ω)�
�sinh(3Ω) 3sinh(Ω)�
1
�
4
�(Ω) =
sinh
�sinh(3Ω) 3sinh(Ω)�
4
where sinh (Ω) = �sinh(3Ω) 3sinh(Ω)�
1
4 1
�sinh(3Ω) 3sinh(Ω)�
� �
sinh (Ω) = = 1 = �sinh(3Ω) sinh(Ω)�
sinh (Ω)
1
1
4 4
sinh(Ω)cosh(2Ω)
�sinh(3Ω) 3sinh(Ω)� ,
�
sinh(Ω)cosh(2Ω) = �sinh(3Ω) sinh(Ω)�
1 2
sinh (Ω) =
2 1
4
sinh(Ω)cosh(2Ω) = �sinh(3Ω) sinh(Ω)�
and sinh(Ω)cosh(2Ω) = �sinh(3Ω) sinh(Ω)�
2
and
1 2
and � 1 sinh(Ω)cosh(2Ω) = �sinh(3Ω) sinh(Ω)� ,
�
and

1
�
� �� �
1

1

2
E(
�) =E �� �

1 2
and E( ) =E �� � � � �� �
� �
E( ) =E �� �
�
1 2
�� �
2 1
2 1
�
�� �
� �
�
E( ) =E 2 �� � 3 � 2 � � 3
1 2 3
1
3
3 1 2

3

3

�
= 1 � � � E � � 3 �� � � E � � �� + � � E � � ��
� E � � 3 � �
E( ) =E
��
� 3 �� �
� E � � �� + � � E � �
��
�� �

3
3
= 1

8
8

� E � � 3
8 3
= 1 2 23 8 ��

8 � 3
8
�� �
� E � � �� + � � E � �
= 8 � � E � � 3 �� 8 � � E � 8 ��
� �� + � � E � �

1
3
1 8 3 � 3 � E � � 3 �� 3 8
1

3
3 8
3
� 3
1
= � � E � � �� �� �� � E � � �� + � � E � � ��
8
� E � � �
8 � 3
� E � �
8 1 8 3 8

3
1 � � 3 � � E � � � 1 �� �
8 � �
�� � �

1
3
�
�
3
3 1 8��
��� � �
= 1 ��� � ��� � + � 1 1 3 �� �
� � �
� + � ��
3
3
=
8
��
= 8 ��� � 3 �� � 8 � � E � � 8 � �
8 1
8 3
�
�
�
� + � 1
8
8 8
8
8
8
�

= 8 1 ��� � �� � + � �� �
�
�
3
� �sinh(3Ω) + �sinh(Ω)
1
�
3
3
= 1 1 8 ��� � 3 8 �� 3� � � 8 � � 1 8 �� �
�sinh(3Ω) + �sinh(Ω)
�
� +
4
39
= = = 4 �sinh(3Ω) + �sinh(Ω) � �
1
4 3
�
�
8
8
4 4 8 �sinh(3Ω) + �sinh(Ω) 8
4
�).
=
�
⁄
⁄
Ω= , and = (1
Ω= , and = (1 ). �
1 4
Ω= , and = (1 ). 3 4
⁄ �
�
⁄ ⁄
⁄
�sinh(3Ω) + �sinh(Ω)
Ω= , and = (1 ).
=
�
⁄
⁄
4
Ω= , and = (1 ). 4
�
⁄ ⁄ 35 35
35

35

35