Assuming iid samples x ~ N(theta, 1), we have a sample of 5 observations with maximum value = 3. How to calculate the likelihood?

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    $\begingroup$ The sample is the maximum, which has a well-defined distribution, hence the likelihood. $\endgroup$
    – Xi'an
    Aug 24 at 8:36
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    $\begingroup$ You can find the CDF for the maximum, take its derivative for the pdf for the maximum, and use this (or something proportional to this) as the likelihood for $\theta$ given the maximum of the observations. The maximum in this case will be at about $x_{(5)}-1.0615$ $\endgroup$
    – Henry
    Aug 24 at 10:17
  • $\begingroup$ This is a special case of the situation analyzed (and solved, with working code) at stats.stackexchange.com/a/130174/919. $\endgroup$
    – whuber
    Aug 24 at 13:09


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