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Assignment asks to explain why max of X will have the highest probability..

What is the reason why the probabilities of each random variable create a constant positive slope?

(max$_k$≤)=(1 ≤ ∩ 2 ≤ ∩ ⋯ n ≤ ) explain why

OR

=(1≤)(2≤)⋯ ( ≤) explain why


is this because of linearity and independence ?


This is the very first statistics class I've taken. I'm still learning.

I am suppose to find an unbiased estimator for a given pmf for a uniform distribution.

pmf: $() = \frac{1}\theta{} \ \ \ \ (0 < < )$

BruceET
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localplutonium
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  • The maximum of the observations is a good estimator of θ, It's the MLE. But it is obviously biased: The max must always be smaller than θ. A constant multiplier (depending on n) of the max can be used to get an unbiased estimator Let $X_{(n)}$ denote the max, Find its dist'n and then find $E[X_{(n)}].$. – BruceET 9 mins ago – BruceET Oct 31 '17 at 16:46
  • @BruceET semi duplicate more trying to figure out why the independent probabilities are ascending order. Just need an answer to the assignment question "explain why" ... I've proved / did the rest of showing the estimator to be unbiased. (n+1)/n *maxX was given. – localplutonium Oct 31 '17 at 17:53
  • The original observations are random. They are used to get the distribution of the max. Of course the sorted observations are not independent because the $i$th order statistic $X_{(i)}$ is smaller than the $(i+1)$st order statistic $X_{(i+1)}.$ – BruceET Oct 31 '17 at 21:09

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