I'm trying to compute the exact covariance matrix of normal order statistics.

Well known formulas (listed in Zakkula Govindarajulu, 1962) allow us to compute moments of order statistics, as well as their products, i.e., $E[X_iX_j], E[X_i]E[X_j]$

Does the explicit form of the covariance matrix of normal order statistics exist ? I mean, how to compute the covariance matrix from here ?

~~UPDATE~~ I did not see this : Order statistics of equal correlated continuous random variables

I am on it. -> UPDATE -> I already know this. That's exactly what one can read in Govindarajulu (1962). Excepted for moment products.

UPDATE :Mark L. Stone answered my question.

So, if i want to compute my covariance matrix from "here" : i "just" do $cov(X_i,X_j) = E[X_iX_j] - E[X_i]E[X_j]$

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    $\begingroup$ Can you put a proper citation instead of "Zakkula Govindarajulu, 1962"? $\endgroup$
    – user603
    Jun 19, 2015 at 18:07

1 Answer 1


This is spelled out in Section 3.1 of H.A. David and H.N. Nagaraja, "Order Statistics, 3rd edition" (2003), with specialization for the Normal distribution in section 3.2.

Equations 3.1.2 through 3.1.4, which are included in the linked Google book preview on p. 34 tell you exactly what to do from your "here" to get the covariance matrix. Also see equations 3.1.5 and 3.1.5' on p. 35 if you still need that part.

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    $\begingroup$ In response to the box below my answer "We're looking for long answers that provide some explanation and context. Don't just give a one-line answer; explain why your answer is right, ideally with citations. Answers that don't include explanations may be removed." Length does not equate to thoroughness or goodness of answer. A complete answer in 4 lines is better than 40 lines of rambling which doesn't really answer the questions, isn't it? $\endgroup$ Jun 19, 2015 at 19:32
  • $\begingroup$ That provided here is an eqn for deriving moments of order stats, for any arbitrary parent distribution i.e. the integral of this and/or that. Which leaves unresolved: what is the answer to those integrals for the OP's problem? $\endgroup$
    – wolfies
    Jun 20, 2015 at 6:04
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    $\begingroup$ Per the OP's original question, the OP already had the moments from the 1962 publication. The OP wrote "how to compute the covariance matrix from here ?", and the equations I linked to in David and Nagrajan provide the how to get from the "here" to the covariance matrix. Hence, I answered the question as stated. $\endgroup$ Jun 20, 2015 at 9:06
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    $\begingroup$ I should have stated that the equations I referenced in chapter 3 of David and Nagranjan only address the i.i.d. case. Section 5.3 has some material on Multivariate Normal. $\endgroup$ Jun 20, 2015 at 10:22

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