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Re: st: Simulate a skewed variable in stata, sample vs. population skewness


From   Austin Nichols <austinnichols@gmail.com>
To   statalist@hsphsun2.harvard.edu
Subject   Re: st: Simulate a skewed variable in stata, sample vs. population skewness
Date   Mon, 7 Dec 2009 09:11:35 -0500

Short answer:
it is the population property (e.g. skewness) that should be held
constant, not the sample property, within one set of simulations
designed to estimate small-sample performance at one level of that
property (and varied across sets to see the dependence on degree of
skewness).  But you should report the sample properties as well at
each level of the population property you simulate for.

On Mon, Dec 7, 2009 at 5:06 AM, Karl-Oskar Lindgren
<Karl-Oskar.Lindgren@statsvet.uu.se> wrote:
> Dear listusers,
>
> I have a question that I guess is partly statistical and partly
> philosphical. In a paper that uses Monte-Carlo simulations to study
> the small sample performance of an estimator I was asked by a referee
> to investigate how the estimator performs when the error terms are
> skewed.
>
> When trying to implement this suggestion I realized that sample
> skewness as reported by stata can differ considerably from the
> skewness of the underlying population (although both the sample mean
> and variance of the variable remain close to their population
> counterparts). My question is therefore if it is the sample skewness
> or the population skewness that should be kept constant when
> examining the small sample performance of a statistical estimator.
>
> In case my question is unclear the following simple example may help
> illustrate the gist of my problem. Let's assume that we want to study
> how the OLS-estimator perform in small samples when the error terms
> are skewed. In order to do this we decide to generate 10 error terms
> from a chi-square distribution with 1 degree-of-freedom. The
> population skewness should then be 2^(3/2), i.e., about 2.8. But if I
> generate 1000 samples from such a distribution in stata the average
> skewness across these 1000 samples turn out to be about 1.3 (see the
> example code below). I understand that the reason for the discrepancy
> is that measures of skewness tend to be biased in small samples when
> the variables are non-normal (indeed the sample skewness is
> approaching its theoretical level as we increases the number of
> observations in the example below).
>
> My question, however, concerns whether it is the sample skewness or
> the population skewness that I should keep constant in my
> replications when I vary the other parameters of the model. If it is
> the population skewness the implementation is straightforward since
> the skewness in the population is known. But if it is the sample
> skewness that  should be kept constant I would appreciate any hints
> of appropriate methods to accomplish this.
>
>
> **Example code to illustrate the bias of r(skewness)
> program define skewchi, rclass
>        version 9.2
>        drop _all
>        set obs 10
>        gen double x=invnorm(uniform())
>        gen double x2=x^2
>
>        sum x2, detail
>        return scalar mean=r(mean)
>        return scalar var=r(Var)
>        return scalar skew=r(skewness)
>
> end
>
>        simulate mean=r(mean) var=r(var) skew=r(skew), ///
>        reps(1000) seed(1)  dots: skewchi
>        sum
>
> Best wishes,
>
> Karl-Oskar Lindgren
> Department of Government
> Uppsala University

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