## regression on order statistics

Occasional Contributor
Posts: 6

# regression on order statistics

In my problem, n bidders place \$ bids for unopened lots.  They are basing their bids on their judgement of the value of the contents.  With bids sorted in order, we have n order statistics, repeated for m lots.  I am assuming that a single distribution type generates the bidding, e.g. lognormal or the like. In reality the mean and variance would likely be different for each lot, however to simplify initially I am willing to divide each lot's bids by its high bid.  E.g. lot A might have "standardized" order statistics high to low {1, .9, .8, .75, .70. .5,...}.  I wish to select a distribution type and its parameters based on these "standardized" sets or order statistics.

PROC UNIVARIATE provides Q-Q plots and methods that readily identify the best distribution and good parameter estimates for each lot.  e.g., the Q-Q plot of lot A might be shown to have the best fit to lognormal(.5,1.1)..  Of course, if I apply this method to each lot in turn, I get m different results.  What I wish is the parameter estimates that are the best fit to data for all lots.  Is there a built-in procedure in SAS that does that?  or am I on my own to specify a nonlinear regression?

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## Re: regression on order statistics

Dividing by the maximum bid might not be the best strategy, as extreme value statistics have high variance, especially with long-tailed distributions. I would rather try the transformation Yij = log(Bij/Bmj) where Bij is the bid from bidder i on lot j and Bmj is the median bid for lot j. Then I would give it a shot at fitting a single distribution to the whole set of Y's.

PG

PG
Occasional Contributor
Posts: 6

## Re: regression on order statistics

Thanks, your approach helps a lot.

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