Appendix 1
Let f
0 be the probability density function of standard normal distribution and f be the truncated probability density function defined as follows.
$$f{\left( z \right)} = \frac{{f_{0} {\left( z \right)}}} {{\Phi {\left( u \right)} - \Phi {\left( l \right)}}},\;\;l \leqslant z \leqslant u$$
Then
$$E{\left( z \right)} = {\int_{{\text{ }}l}^{{\text{ }}u} {\frac{{zf_{0} {\left( z \right)}}} {{\Phi {\left( u \right)} - \Phi {\left( l \right)}}}dz} } = {\int_{{\text{ }}l}^{{\text{ }}u} {\frac{{\frac{1} {{{\sqrt {2\pi } }}}ze^{{ - z^{2} /2}} }} {{\Phi {\left( u \right)} - \Phi {\left( l \right)}}}} }dz = \frac{{f_{0} {\left( l \right)} - f_{0} {\left( u \right)}}} {{\Phi {\left( u \right)} - \Phi {\left( l \right)}}},$$
which implies
$${\int_{{\text{ }}l}^{{\text{ }}u} {zf_{0} {\left( z \right)}dz} } = f_{0} {\left( l \right)} - f_{0} {\left( u \right)}$$
(8)
In addition,
$$\begin{array}{*{20}l} {{E{\left( {z^{2} } \right)}} \hfill} & {{ = {\int_{{\text{ }}l}^{{\text{ }}u} {\frac{{z^{2} f_{0} {\left( z \right)}}} {{\Phi {\left( u \right)} - \Phi {\left( l \right)}}}dz} } = {\int_{{\text{ }}l}^{{\text{ }}u} {\frac{{\frac{1} {{{\sqrt {2\pi } }}}z^{2} e^{{ - z^{2} /2}} }} {{\Phi {\left( u \right)} - \Phi {\left( l \right)}}}} }dz} \hfill} & {{} \hfill} \\ {{} \hfill} & {{ = \frac{{lf_{0} {\left( l \right)} - uf_{0} {\left( u \right)} + {\left( {\Phi {\left( u \right)} - \Phi {\left( l \right)}} \right)}}} {{\Phi {\left( u \right)} - \Phi {\left( l \right)}}}} \hfill} & {{{\left( {{\text{Integration}}\;{\text{by}}\;{\text{part}}} \right)}} \hfill} \\ {{} \hfill} & {{ = \frac{{lf_{0} {\left( l \right)} - uf_{0} {\left( u \right)}}} {{\Phi {\left( u \right)} - \Phi {\left( l \right)}}} + 1} \hfill} & {{} \hfill} \\ \end{array} $$
which implies
$${\int_l^u {z^{2} f_{0} {\left( z \right)}dz} } = lf_{0} {\left( l \right)} - uf_{0} {\left( u \right)} + \Phi {\left( u \right)} - \Phi {\left( l \right)}$$
(9)
Appendix 2: Derivation of Eq. 5
$$\begin{array}{*{20}l} {{E{\left( {H_{{0i}} } \right)}} \hfill} & {{ = {\int_{ - k}^{{\text{ }}k} {h{\left( z \right)}\frac{{f_{0} {\left( z \right)}}} {{1 - q_{0} }}dz} }} \hfill} \\ {{} \hfill} & {{ = \frac{1} {{1 - q_{0} }}{\int_{ - k}^{{\text{ }}k} {h{\left( z \right)}f_{0} {\left( z \right)}dz} }} \hfill} \\ {{} \hfill} & {{ = \frac{1} {{1 - q_{0} }}{\left\{ {{\text{ }}{\int_{ - k}^{ - b} {mf_{0} {\left( z \right)}dz} } + {\int_{ - b}^{ - a} {{\left( {Sz + I} \right)}f_{0} {\left( z \right)}dz} } + {\int_{ - a}^{{\text{ }}a} {Mf_{0} {\left( z \right)}dz} } + {\int_{{\text{ }}a}^{{\text{ }}b} {{\left( { - Sz + I} \right)}f_{0} {\left( z \right)}dz} } + {\int_{{\text{ }}b}^{{\text{ }}k} {mf_{0} {\left( z \right)}dz{\text{ }}} }} \right\}}} \hfill} \\ \end{array} $$
Merging the first term with the last term and the second term with the fourth term results in
$$E{\left( {H_{{0i}} } \right)} = \frac{1} {{1 - q_{0} }}{\left\{ {2m{\left[ {\Phi {\left( k \right)} - \Phi {\left( b \right)}} \right]} + 2{\left[ {S{\left( {f_{0} {\left( b \right)} - f_{0} {\left( a \right)}} \right)} + I{\left( {\Phi {\left( b \right)} - \Phi {\left( a \right)}} \right)}} \right]} + M{\left( {\Phi {\left( a \right)} - \Phi {\left( { - a} \right)}} \right)}} \right\}}$$
Appendix 3: Derivation of Eq. 6
Let f
1 be the probability density function of standard normal distribution under μ=μ
1, so
f
1(z)=f
0(z−δ).
$$\begin{array}{*{20}l} {{E{\left( {H_{{1i}} } \right)}} \hfill} & {{ = {\int_{ - k}^{{\text{ }}k} {h{\left( z \right)}\frac{{f_{1} {\left( z \right)}}} {{1 - q_{1} }}dz} }} \hfill} \\ {{} \hfill} & {{ = \frac{1} {{1 - q_{1} }}{\int_{ - k}^{{\text{ }}k} {h{\left( z \right)}f_{1} {\left( z \right)}dz} }} \hfill} \\ {{} \hfill} & {{ = \frac{1} {{1 - q_{1} }}{\left\{ {{\text{ }}{\int_{ - k}^{ - b} {mf_{1} {\left( z \right)}dz} } + {\int_{ - b}^{ - a} {{\left( {Sz + I} \right)}f_{1} {\left( z \right)}dz} } + {\int_{ - a}^{{\text{ }}a} {Mf_{1} {\left( z \right)}dz} } + {\int_{{\text{ }}a}^{{\text{ }}b} {{\left( { - Sz + I} \right)}f_{1} {\left( z \right)}dz} } + {\int_{{\text{ }}b}^{{\text{ }}k} {mf_{1} {\left( z \right)}dz{\text{ }}} }} \right\}}} \hfill} \\ \end{array} $$
Let z′=z−δ. This results in
$$E{\left( {H_{{1i}} } \right)} = \frac{1} {{1 - q_{1} }}{\left\{ {\begin{array}{*{20}l} {{m{\int_{ - k - \delta }^{ - b - \delta } {f_{0} {\left( {{z}\ifmmode{'}\else$'$\fi} \right)}} }d{z}\ifmmode{'}\else$'$\fi + {\int_{ - b - \delta }^{ - a - \delta } {{\left[ {S{\left( {{z}\ifmmode{'}\else$'$\fi + \delta } \right)} + I} \right]}f_{0} {\left( {{z}\ifmmode{'}\else$'$\fi} \right)}d{z}\ifmmode{'}\else$'$\fi + M} }{\int_{ - a - \delta }^{a - \delta } {f_{0} {\left( {{z}\ifmmode{'}\else$'$\fi} \right)}d{z}\ifmmode{'}\else$'$\fi} }} \hfill} \\ {{ + {\int_{a - \delta }^{b - \delta } {{\left[ { - S{\left( {{z}\ifmmode{'}\else$'$\fi + \delta } \right)} + I} \right]}f_{0} {\left( {{z}\ifmmode{'}\else$'$\fi} \right)}d{z}\ifmmode{'}\else$'$\fi} } + m{\int_{b - \delta }^{k - \delta } {f_{0} {\left( {{z}\ifmmode{'}\else$'$\fi} \right)})d{z}\ifmmode{'}\else$'$\fi} }} \hfill} \\ \end{array} } \right\}}$$
Applying the Eq. 9 from Appendix 1 to the second and fourth terms yields
$$E{\left( {H_{{1i}} } \right)} = \frac{1} {{1 - q_{1} }}{\left\{ {\begin{array}{*{20}l} {{m{\left[ {\Phi {\left( { - b - \delta } \right)} - \Phi {\left( { - k - \delta } \right)}} \right]} + S{\left[ {f_{0} {\left( { - b - \delta } \right)} - f_{0} {\left( { - a - \delta } \right)}} \right]}} \hfill} \\ {{ + {\left( {\delta S + I} \right)}{\left[ {\Phi {\left( { - a - \delta } \right)} - \Phi {\left( { - b - \delta } \right)}} \right]}} \hfill} \\ {{ + M{\left[ {\Phi {\left( {a - \delta } \right)} - \Phi {\left( { - a - \delta } \right)}} \right]} - S{\left[ {f_{0} {\left( {a - \delta } \right)} - f_{0} {\left( {b - \delta } \right)}} \right]}} \hfill} \\ {{ + {\left( { - \delta S + I} \right)}{\left[ {\Phi {\left( {b - \delta } \right)} - \Phi {\left( {a - \delta } \right)}} \right]} + m{\left[ {\Phi {\left( {k - \delta } \right)} - \Phi {\left( {b - \delta } \right)}} \right]}} \hfill} \\ \end{array} } \right\}}$$
Appendix 4: Derivation of Eq. 7
$$\begin{array}{*{20}l} {{E{\left( {H^{2}_{{oi}} } \right)}} \hfill} & {{ = {\int_{ - k}^{{\text{ }}k} {h^{2} {\left( z \right)}\frac{{f_{0} {\left( z \right)}}} {{1 - q_{0} }}dz} }} \hfill} \\ {{} \hfill} & {{ = \frac{1} {{1 - q_{0} }}{\int_{ - k}^{{\text{ }}k} {h^{2} {\left( z \right)}f_{0} {\left( z \right)}dz} }} \hfill} \\ {{} \hfill} & {{ = \frac{1} {{1 - q_{0} }}{\left\{ {\begin{array}{*{20}l} {{{\int_{ - k}^{ - b} {m^{2} f_{0} {\left( z \right)}dz + {\int_{ - b}^{ - a} {{\left( {S^{2} z^{2} - 2SIz + I^{2} } \right)}f_{0} {\left( z \right)}dz} }} }} \hfill} \\ {{ + {\int_{ - a}^{{\text{ }}a} {M^{2} f_{0} {\left( z \right)}dz} } + {\int_a^{{\text{ }}b} {{\left( {S^{2} z^{2} - 2SIz + I^{2} } \right)}} }f_{0} {\left( z \right)}dz + {\int_b^{{\text{ }}k} {m^{2} f_{0} {\left( z \right)}dz} }} \hfill} \\ \end{array} } \right\}}} \hfill} \\ \end{array} \begin{array}{*{20}l} {{} \hfill} \\ {{} \hfill} \\ {{} \hfill} \\ \end{array} $$
Applying Eqs. 9 and 10 from Appendix 1 to the second and fourth terms yields
$$\begin{array}{*{20}l} {{E{\left( {H^{2}_{{oi}} } \right)}} \hfill} & {{ = \frac{1} {{1 - q_{0} }}{\left\{ {\begin{array}{*{20}l} {{m^{2} {\left[ {\Phi {\left( { - b} \right)} - \Phi {\left( { - k} \right)}} \right]} + S^{2} {\left[ { - bf_{0} {\left( { - b} \right)} + af_{0} {\left( { - a} \right)} + \Phi {\left( { - a} \right)} - \Phi {\left( { - b} \right)}} \right]}} \hfill} \\ {{ + 2SI{\left[ {f_{0} {\left( { - b} \right)} - f_{0} {\left( { - a} \right)}} \right]} + I^{2} {\left[ {\Phi {\left( { - a} \right)} - \Phi {\left( { - b} \right)}} \right]}} \hfill} \\ {{ + M^{2} {\left[ {\Phi {\left( a \right)} - \Phi {\left( { - a} \right)}} \right]} + S^{2} {\left[ {af_{0} {\left( a \right)} - bf_{0} {\left( b \right)} + \Phi {\left( b \right)} - \Phi {\left( a \right)}} \right]}} \hfill} \\ {{ - 2SI{\left[ {f_{0} {\left( a \right)} - f_{0} {\left( b \right)}} \right]} + I^{2} {\left[ {\Phi {\left( b \right)} - \Phi {\left( a \right)}} \right]}} \hfill} \\ {{ + m^{2} {\left[ {\Phi {\left( k \right)} - \Phi {\left( b \right)}} \right]}} \hfill} \\ \end{array} } \right\}}} \hfill} \\ {{} \hfill} & {{ = \frac{1} {{1 - q_{0} }}{\left\{ {\begin{array}{*{20}l} {{2m^{2} {\left[ {\Phi {\left( k \right)} - \Phi {\left( b \right)}} \right]} + 2S^{2} {\left[ {af_{0} {\left( a \right)} - bf_{0} {\left( b \right)} + \Phi {\left( b \right)} - \Phi {\left( a \right)}} \right]}} \hfill} \\ {{ + 4SI{\left[ {f_{0} {\left( b \right)} - f_{0} {\left( a \right)}} \right]} + 2I^{2} {\left[ {\Phi {\left( b \right)} - \Phi {\left( a \right)}} \right]} + M^{2} {\left[ {\Phi {\left( a \right)} - \Phi {\left( { - a} \right)}} \right]}} \hfill} \\ \end{array} } \right\}}} \hfill} \\ \end{array} $$
Appendix 5: Derivation of Eq. 8
$$\begin{array}{*{20}l} {{E{\left( {H_{{1i}} ^{2} } \right)}} \hfill} & {{ = {\int_{ - k}^{{\text{ }}k} {h^{2} {\left( z \right)}\frac{{f_{1} {\left( z \right)}}} {{1 - q_{1} }}dz} }} \hfill} \\ {{} \hfill} & {{ = \frac{1} {{1 - q_{1} }}{\int_{ - k}^{{\text{ }}k} {h^{2} {\left( z \right)}f_{1} {\left( z \right)}d} }z} \hfill} \\ {{} \hfill} & {{ = \frac{1} {{1 - q1}}{\left\{ {\begin{array}{*{20}l} {{{\int_{ - k}^{ - b} {m^{2} f_{1} {\left( z \right)}dz} } + {\int_{ - b}^{{\text{ }}a} {{\left( {S^{2} z^{2} + 2SIz + I^{2} } \right)}f_{0} {\left( z \right)}dz} }} \hfill} \\ {{ + {\int_{ - a}^{{\text{ }}a} {M^{2} f_{1} {\left( z \right)}dz} } + {\int_a^{{\text{ }}b} {{\left( {S^{2} z^{2} - 2SIz + I^{2} } \right)}f_{1} {\left( z \right)}dz + {\int_{{\text{ }}b}^{{\text{ }}k} {m^{2} f_{1} {\left( z \right)}dz} }} }} \hfill} \\ \end{array} } \right\}}} \hfill} \\ \end{array} $$
Let z′=z−δ. It results that
$$E{\left( {H_{{1i}} ^{2} } \right)} = \frac{1} {{1 - q_{1} }}{\left\{ {\begin{array}{*{20}l} {{m^{2} {\int_{ - k - \delta }^{ - b - \delta } {f_{0} {\left( {{z}\ifmmode{'}\else$'$\fi} \right)}dz'} } + {\int_{ - b - \delta }^{ - a - \delta } {S^{2} {\left( {{z}\ifmmode{'}\else$'$\fi + \delta } \right)}^{2} f_{0} {\left( {{z}\ifmmode{'}\else$'$\fi} \right)}d{z}\ifmmode{'}\else$'$\fi} }} \hfill} \\ {{ + 2SI{\int_{ - k - \delta }^{ - a - \delta } {{\left( {{z}\ifmmode{'}\else$'$\fi + \delta } \right)}f_{0} {\left( {{z}\ifmmode{'}\else$'$\fi} \right)}d{z}\ifmmode{'}\else$'$\fi + I^{2} {\int_{ - b - \delta }^{ - a - \delta } {f_{0} {\left( {{z}\ifmmode{'}\else$'$\fi} \right)}d{z}\ifmmode{'}\else$'$\fi} }} }} \hfill} \\ {{ + M^{2} {\int_{ - a - \delta }^{{\text{ }}a - \delta } {f_{0} {\left( {{z}\ifmmode{'}\else$'$\fi} \right)}d{z}\ifmmode{'}\else$'$\fi} } + {\int_{{\text{ }}a - \delta }^{{\text{ }}b - \delta } {S^{2} {\left( {{z}\ifmmode{'}\else$'$\fi + \delta } \right)}^{2} f_{0} {\left( {{z}\ifmmode{'}\else$'$\fi} \right)}d{z}\ifmmode{'}\else$'$\fi} }} \hfill} \\ {{ - 2SI{\int_{{\text{ }}a - \delta }^{{\text{ }}b - \delta } {{\left( {{z}\ifmmode{'}\else$'$\fi + \delta } \right)}f_{0} {\left( {{z}\ifmmode{'}\else$'$\fi} \right)}d{z}\ifmmode{'}\else$'$\fi} } + I^{2} {\int_{{\text{ }}a - \delta }^{{\text{ }}b - \delta } {f_{0} {\left( {{z}\ifmmode{'}\else$'$\fi} \right)}d{z}\ifmmode{'}\else$'$\fi} } + m^{2} {\int_{{\text{ }}b - \delta }^{{\text{ }}k - \delta } {f_{0} {\left( {{z}\ifmmode{'}\else$'$\fi} \right)}d{z}\ifmmode{'}\else$'$\fi} }} \hfill} \\ \end{array} } \right\}}$$
Applying the Eqs. 9 and 10 of Appendix 1 to the second and sixth terms and Eq. 9 to the third and seventh terms yields
$$E{\left( {H_{{1i}} ^{2} } \right)} = \frac{1} {{1 - q_{1} }}{\left\{ {\begin{array}{*{20}l} {{m^{2} {\left[ {\Phi {\left( { - b - \delta } \right)} - \Phi {\left( { - k - \delta } \right)} + \Phi {\left( {b - \delta } \right)}} \right]}} \hfill} \\ {{ + M^{2} {\left[ {\Phi {\left( {a - \delta } \right)} - \Phi {\left( { - a - \delta } \right)}} \right]}} \hfill} \\ {{ + S^{2} {\left\{ {\begin{array}{*{20}l} {{{\left( { - b - \delta } \right)}f_{0} {\left( { - b - \delta } \right)} - {\left( { - a - \delta } \right)}f_{0} {\left( { - a - \delta } \right)} + \Phi {\left( { - a - \delta } \right)} - \Phi {\left( { - b - \delta } \right)}} \hfill} \\ {{ + 2\delta {\left[ {f_{0} {\left( { - b - \delta } \right)} - f_{0} {\left( { - a - \delta } \right)}} \right]} + \delta ^{2} {\left[ {\Phi {\left( { - a - \delta } \right)} - \Phi {\left( { - b - \delta } \right)}} \right]}} \hfill} \\ \end{array} } \right\}}} \hfill} \\ {{ + 2SI{\left[ {f_{0} {\left( { - b - \delta } \right)} - f_{0} {\left( { - a - \delta } \right)}} \right]}} \hfill} \\ {{ + {\left( {2SI\delta + I^{2} } \right)}{\left[ {\Phi {\left( { - a - \delta } \right)} - \Phi {\left( { - b - \delta } \right)}} \right]}} \hfill} \\ {{ + S^{2} {\left\{ {\begin{array}{*{20}l} {{{\left( {a - \delta } \right)}f_{0} {\left( {a - \delta } \right)} - {\left( {b - \delta } \right)}f_{0} {\left( {b - \delta } \right)} + \Phi {\left( {b - \delta } \right)} - \Phi {\left( {a - \delta } \right)}} \hfill} \\ {{ + 2\delta {\left[ {f_{0} {\left( {a - \delta } \right)} - f_{0} {\left( {b - \delta } \right)}} \right]} + \delta ^{2} {\left[ {\Phi {\left( {b - \delta } \right)} - \Phi {\left( {a - \delta } \right)}} \right]}} \hfill} \\ \end{array} } \right\}}} \hfill} \\ {{ - 2SI{\left[ {f_{0} {\left( {a - \delta } \right)} - f_{0} {\left( {b - \delta } \right)}} \right]}} \hfill} \\ {{ + {\left( {I^{2} - 2SI\delta } \right)}{\left[ {\Phi {\left( {b - \delta } \right)} - \Phi {\left( {a - \delta } \right)}} \right]}} \hfill} \\ \end{array} } \right\}}$$
