Computational Statistics in Data Science. Группа авторов

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alt="upper Y Subscript i"/> be the BOD at time x Subscript i. Assume for epsilon Subscript i Baseline tilde Overscript i i d Endscripts upper N left-parenthesis 0 comma sigma squared right-parenthesis and upper Y Subscript i Baseline vertical-bar beta 1 comma beta 2 comma sigma squared equals beta 1 left-parenthesis 1 minus e Superscript minus beta 2 x Super Subscript i Superscript Baseline right-parenthesis plus epsilon Subscript i Baseline. Newton and Raftery [41] assume a default prior on sigma squared, normal pi left-parenthesis sigma squared right-parenthesis proportional-to sigma Superscript negative 2, and a transformation invariant design‐dependent prior for beta equals left-parenthesis beta 1 comma beta 2 right-parenthesis such that normal pi left-parenthesis beta right-parenthesis proportional-to StartAbsoluteValue upper V Superscript upper T Baseline upper V EndAbsoluteValue Superscript 1 slash 2, where upper V is an 8 times 2 where the left-parenthesis i comma j right-parenthesisth element of upper V equals partial-differential normal upper E left-bracket y Subscript i Baseline vertical-bar x Subscript i Baseline right-bracket slash partial-differential beta Subscript j. The resulting posterior distribution of left-parenthesis beta comma sigma squared right-parenthesis is intractable and up to normalization and can be written as

normal pi left-parenthesis beta comma sigma squared bar y 1 comma ellipsis comma y 8 right-parenthesis proportional-to left-parenthesis StartFraction 1 Over sigma squared EndFraction right-parenthesis Superscript n slash 2 plus 1 Baseline StartAbsoluteValue upper V Superscript upper T Baseline upper V EndAbsoluteValue Superscript 1 slash 2 Baseline exp left-brace minus StartFraction 1 Over 2 sigma squared EndFraction sigma-summation Underscript i equals 1 Overscript 8 Endscripts left-parenthesis y Subscript i Baseline minus beta 1 left-parenthesis 1 minus e Superscript minus beta 2 x Super Subscript i Superscript Baseline right-parenthesis right-parenthesis squared right-brace

      The goal is to estimate the posterior mean of left-parenthesis beta comma sigma squared right-parenthesis. We implement an MCMC algorithm to estimate the posterior mean and implement the relative‐standard deviation sequential stopping rule via effective sample size.

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from an initial run of n Superscript asterisk Baseline equals 8123 (dashed) and at termination (solid).

      At termination, the estimated posterior mean is left-parenthesis 2.5074 comma 0.2034 comma 0.00 654 right-parenthesis, and 80% credible intervals are left-parenthesis 2.357 comma 2.665 right-parenthesis, left-parenthesis 0.178 comma 0.229 right-parenthesis, and left-parenthesis 0.00 246 comma 0.01 200 right-parenthesis for beta 1, beta 2, and sigma squared, respectively.

      It is possible to run a more efficient linchpin sampler [42] by integrating out sigma squared from the posterior. That is, normal pi left-parenthesis beta comma sigma squared vertical-bar y right-parenthesis equals normal pi left-parenthesis sigma squared vertical-bar beta comma y right-parenthesis normal pi left-parenthesis beta vertical-bar y right-parenthesis, where

sigma squared vertical-bar beta comma y tilde Inverse Gamma left-parenthesis StartFraction n Over 2 EndFraction comma one half sigma-summation Underscript i equals 1 Overscript 8 Endscripts left-parenthesis y Subscript i Baseline minus beta 1 left-parenthesis 1 minus e Superscript minus beta 2 x Super Subscript i Superscript Baseline right-parenthesis 
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