Given X 1 , . . X n observations of a population with densitiy f ( x ) = { 1 b − a if x ∈ ( a , b ) 0 else where interval limits a and b are unknown. Determine the maximum likelihood estimations for a and b.This is example task from exam and I like to know how solve it correctly.When I understand formula of maximum likelihood estimation correct,This is maximize when ( b − a ) is as small as possible but also important that (a,b) include all the data. For this reason we have a = min ( x 1 , . . , x n )and b = max ( x 1 , . . , x n )

Question

Given       X    1    ,  .  .      X    n   observations of a population with densitiy  f  (  x  )  =      {                                        1                          b              −              a                                 if           x          ∈          (          a          ,          b          )                                      0                                                                                                                                                                                               else                          where interval limits a and b are unknown. Determine the maximum likelihood estimations for a and b.This is example task from exam and I like to know how solve it correctly.When I understand formula of maximum likelihood estimation correct,This is maximize when   (  b  −  a  ) is as small as possible but also important that (a,b) include all the data. For this reason we have  a  =  min  (      x    1    ,  .  .  ,      x    n    )and  b  =  max  (      x    1    ,  .  .  ,      x    n    )

kjakesHB · Accepted Answer

Step 1You are right. Recall the definition of the MLE, that is            θ      ^        =  arg  ⁡      max          (      a      ,      b      )      ∈      Θ            1          (      b      −      a              )        n                ,and   a  ≤      X    i    ≤  b for all i, thus you have to choose the closest statistics to b and a over   Θ in order to maximize the likelihood (over the parametric space). Namely,            a      ^        =  min  {      X    1    ,  .  .  .  ,      X    n    }    ,              b      ^        =  max  {      X    1    ,  .  .  .  ,      X    n    }    .Step 2But note that you need the equality in   a  ≤      X    i    ≤  b, otherwise the MLE is not in the parametric space.

Given X_1,..X_n observations of a population with densitiy f(x)={(1/(b-a),if,x in (a,b),,),(0,,else,,):} else where interval limits a and b are unknown. Determine the maximum likelihood estimations for a and b.

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