Statistical tolerance limits for posterior distributions based on partially specified prior distributions  Page 42 
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I where I I I = v(i  y )// 2 —a—nd r" = rm + r' . Proof: We first note that r^ replaces r to give r" « r^ + r*. Now, from the expression for L we obtain through conjugacy a lower confidence point where w is a lower bound to w. From independence, we obtain a probability S « v (l  y )/2. The upper statistical tolerance limit can be found in a similar way, replacing p by 1  p and v by 1  v and (l  y )/2 by (l+ y )/2 . We can obtain a lower bound for the lower tolerance limit for the complementary case as stated below. (An upper bound can be derived analogously. ) PROPOSITION 7: When the prior distribution is gamma with unknown parameter r* and known parameter t* and Poisson conditional* then the lower & statistical tolerance limit L"* for a proportion p with r^ (r^) failures in the jth (n+ l)batch has a lower bound L"' > where g * w/t' and t" = t + t". Proof: The proof follows from the definition of y, the monotonicity of g (*), and the definition of t". 37
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Title  Statistical tolerance limits for posterior distributions based on partially specified prior distributions  Page 42 
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Full text  I where I I I = v(i  y )// 2 —a—nd r" = rm + r' . Proof: We first note that r^ replaces r to give r" « r^ + r*. Now, from the expression for L we obtain through conjugacy a lower confidence point where w is a lower bound to w. From independence, we obtain a probability S « v (l  y )/2. The upper statistical tolerance limit can be found in a similar way, replacing p by 1  p and v by 1  v and (l  y )/2 by (l+ y )/2 . We can obtain a lower bound for the lower tolerance limit for the complementary case as stated below. (An upper bound can be derived analogously. ) PROPOSITION 7: When the prior distribution is gamma with unknown parameter r* and known parameter t* and Poisson conditional* then the lower & statistical tolerance limit L"* for a proportion p with r^ (r^) failures in the jth (n+ l)batch has a lower bound L"' > where g * w/t' and t" = t + t". Proof: The proof follows from the definition of y, the monotonicity of g (*), and the definition of t". 37 