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29 lines
1.8 KiB
TeX
29 lines
1.8 KiB
TeX
As we are analyzing survival rates, a geometric mean is better used to describe average effects.
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By weighting the geometric mean with constellation sizes, we get:
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\begin{align}
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R_G = \exp \left[ \frac{1}{S_t} \sum^N_{j=1} s_t^j \ln(1-l^j(s^j_t,S_t,D_t)) \right]
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\end{align}
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The marginal effect is assumed to be negative, thus-
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\begin{align}
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0 > \parder{R_G}{s^i_t}{} =& \exp \left[ \frac{1}{S_t} \sum^N_{j=1} s_t^j \ln(1-l^j(s^j_t,S_t,D_t)) \right]
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\left[ \parder{}{s^i_t}{} \frac{1}{S_t} \sum^N_{j=1} s_t^j \ln(1-l^j(s^j_t,S_t,D_t)) \right] \\
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0 > \parder{R_G}{s^i_t}{} =& \frac{R_G}{S_t^2} \left[ S^t-
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\left( \ln(1-l^i)-
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- \frac{s^i_t}{1-l^i} \parder{l^i}{s^i_t}{}
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- \sum^N_{j=1} \frac{s^j_t}{1-l^j} \parder{l^j}{S_t}{}
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\right)-
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- \sum^N_{j=1} s_t^j \ln(1-l^j) \right] \\
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0 > \parder{R_G}{s^i_t}{} =& \frac{R_G}{S_t^2} \left[ S^t-
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\left( \ln(R_i)-
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- \frac{s^i_t}{1-l^i} \parder{l^i}{s^i_t}{}
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- \sum^N_{j=1} \frac{s^j_t}{1-l^j} \parder{l^j}{S_t}{}
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\right)-
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- \sum^N_{j=1} s_t^j \ln(R_j) \right] \\
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0 > & \ln R_i - \frac{s^i_t}{1-l^i} \parder{l^i}{s^i_t}{}
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- \sum^N_{j=1} \frac{s^j_t}{1-l^j} \parder{l^j}{S_t}{} - \sum^N_{j=1} \frac{s_t^j}{S_t} \ln(R_j) \\
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0 > & \ln R_i - \frac{s^i_t}{1-l^i} \parder{l^i}{s^i_t}{}
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- \sum^N_{j=1} \frac{s^j_t}{1-l^j} \parder{l^j}{S_t}{} - \ln R_G \\
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\ln \frac{R_G}{R_i} =& \ln R_G - \ln R_i > - \frac{s^i_t}{1-l^i} \parder{l^i}{s^i_t}{}
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- \sum^N_{j=1} \frac{s^j_t}{1-l^j} \parder{l^j}{S_t}{}--
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\end{align}
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