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Re: variance of difference sign test

• To: mathgroup at smc.vnet.net
• Subject: [mg65197] Re: variance of difference sign test
• From: ab_def at prontomail.com
• Date: Fri, 17 Mar 2006 05:45:37 -0500 (EST)
• References: <duudcl$hug$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

Suppose that we have n independent identically distributed random
variables {u[1], ..., u[n]} and P[u[i] == u[j]] == 0 for i != j. We
form another sequence {xi[1] = Boole[u[1] > u[2]], ..., xi[n - 1] =
Boole[u[n - 1] > u[n]]} and we're looking for the variance of the sum
of xi[i]:

D[N[n]] == Variance[Sum[xi[i], {i, n - 1}]] ==

  Variance[Sum[xi[i], {i, n - 2}] + xi[n - 1]] ==

  Variance[Sum[xi[i], {i, n - 2}]] + Variance[xi[n - 1]] +

    2*Covariance[Sum[xi[i], {i, n - 2}], xi[n - 1]] ==

  D[N[n - 1]] + 1/4 + 2*Sum[Covariance[xi[i], xi[n - 1]], {i, n - 2}]

For any pair of adjacent elements we have

Covariance[xi[1], xi[2]] ==

  P[xi[1] == 1 && xi[2] == 1] - P[xi[1] == 1]*P[xi[2] == 1] ==

  P[u[1] > u[2] > u[3]] - P[u[1] > u[2]]*P[u[2] > u[3]] ==

  1/6 - 1/4 == -1/12

because all permutations of {u[1], ..., u[n]} are equally probable. For
any non-adjacent elements Covariance[xi[i], xi[j]] == 0. Therefore,

D[N[n]] == D[N[n - 1]] + 1/4 + 2*(-1/12), D[N[2]] = 1/4

and D[N[n]] == (n + 1)/12 if n >= 2.

Here is a check for n = 6:

In[1]:= n = 6;

Lvalfreq = {First@ #, Length@ #}& /@ Split@ Sort@
  (Count[Sign[Most@ # - Rest@ #], 1]& /@
    Permutations@ Range@ n)

{Lval, Lp} = {Lvalfreq[[All, 1]], Lvalfreq[[All, 2]]/n!};
mu = Lval.Lp
sigma = ((Lval - mu)^2).Lp

Out[2]= {{0, 1}, {1, 57}, {2, 302}, {3, 302}, {4, 57}, {5, 1}}

Out[4]= 5/2

Out[5]= 7/12

And a numerical test:

In[6]:= Lcnt = Array[
  Count[Sign[Most@ # - Rest@ #]&@ Array[Random[]&, n], 1]&,
  10^5];

{Mean@ Lcnt, Variance@ Lcnt} - {mu, sigma} // N

Out[7]= {0.00262, 0.0033856695}

Maxim Rytin
m.r at inbox.ru

Darren Glosemeyer wrote:
>
> For the variance quoted on the TimeSeries page, I initially thought the
> same thing you did. Assuming the signs are independent and there are equal
> probabilities of getting positive and negative signs (and 0 probability of
> getting a 0 difference), the statistic would follow
> BinomialDistribution[n-1, 1/2], which would have a variance of
> (n-1)/4.  Simulations give a variance that appears to be (n+1)/12 (which
> would still indicate a typo in the TimeSeries documentation).  I haven't
> figured out why this should be the variance yet.  My best guess is that the
> assumption of independence is not valid given the differencing and as a
> result the distribution is something other than BinomialDistribution[n-1, 1/2].
>
>
> Darren Glosemeyer
> Wolfram Research
>
>
> At 05:15 AM 3/10/2006 -0500, john.hawkin at gmail.com wrote:
> >Hello,
> >
> >I have two questions.
> >
> >1.  Are there any resources of .nb files available on the internet
> >where I might find an implementation of the D'Agostino Pearson k^2 test
> >for normal variates?
> >
> >2.  In the mathematica time series package (an add-on), the
> >"difference-sign" test of residuals is mentioned (url:
> >http://documents.wolfram.com/applications/timeseries/UsersGuidetoTimeSeries/1.6.2.html).
> >  It says that the variance of this test is (n+1) / 2.  However, it
> >would seem to me that a simple calculation gives a variance of (n-1)/4.
> >  It goes as follows:
> >
> >If the series is differenced once, then the number of positive and
> >negative values in the difference should be approximately equal.  If Xi
> >denotes the sign of each value in the differenced series, then
> >Mean(Xi) = 0.5(1) + 0.5(0) = 0.5
> >Var(Xi) = Expectation( (Xi - Mean(Xi))^2 )
> >= Expectation( Xi^2 -Xi + 0.25 )
> >= 0.5 - 0.5 + 0.25
> >= 0.25
> >
> >And assuming independence of each sign from the others, the total
> >variance should be the sum of the individual variances, up to n-1 for n
> >data points (since there are only n-1 changes in sign), thus
> >
> >Variance = (n-1) / 4
> >
> >There is an equivalent problem in Lemon's "Stochastic Physics" about
> >coin flips, for which the answer is listed, without proof, as (n-1)/8.
> >Because of these three conficting results I am wondering if I have made
> >an error in my calculation, and if anyone can find one please let me
> >know.
> >
> >Thank you very much,
> >
> >-John Hawkin

• Follow-Ups:
  • Re: Re: variance of difference sign test
    • From: leigh pascoe <leigh@cephb.fr>