1
$\begingroup$

I'm working on a research project for one of my professors. He wants to know whether a variable that takes on different values over a large period of time (say 1,000,000 different values) is i.i.d. and would like me to design an independence test for it.

I've been learning about independence tests such as Chi-square, McNemar and seen a bunch of research papers floating around about specific cases, but none seem to fit this case. The thing I'm most hung up on is that with all the examples I see, you test one variable against another. I suppose I could assume that all 1,000,000 instances of this variable are different random variables and construct a test that way, but I assume there has to be a better way.

I'd appreciate if you can point me in the right direction and recommend some good textbooks/other reference materials I can refer to! Thanks.

$\endgroup$
3
  • $\begingroup$ Can be that you don't need an Independence test, but rather a change-of-point test for your time-distributed variable. $\endgroup$ – Match Maker EE Aug 9 '20 at 14:51
  • 5
    $\begingroup$ With a simple expedient, you don't need to design any tests (which is a fraught and huge subject: read Knuth's Art of Computer Programming for the details). Instead, you can capitalize on what the experts have learned simply by applying an empirical probability integral transform to create a series of uniformly distributed values and running a standard test of a pseudorandom number generator, such as the Diehard suite. $\endgroup$ – whuber Aug 9 '20 at 14:54
  • $\begingroup$ Start out with some simpler methods, plotting, compute the autocorrelation function, ... $\endgroup$ – kjetil b halvorsen Aug 9 '20 at 19:32
0
$\begingroup$

Here is some 'low-hanging fruit' where auto-correlation plots and perhaps a runs test reveal marked lack of IID behavior of a sequence. (See @kjetilbhalvorsen's Comment.)

Data from the late 1970s show that eruptions of Old Faithful geyser in Yellowstone National Park were of short (0) or long (1) duration (less or more than 3 min in length) approximately according to a 2-state Markov chain--with no occurrences of two short eruptions in a row. Over the long run, the proportion of long eruptions is about 69%. The R code below simulates 2000 eruptions x according to this Markov chain.

set.seed(2020)
m = 2000;  n = 1:m;  x = numeric(n);  x[1]=0
a = 1;  b = 0.44
for (i in 2:m) {
  if (x[i-1]==0) x[i] = rbinom(1,1,a)
  else           x[i] = rbinom(1,1,1-b)
  }
mean(x==1)
[1] 0.7005

By contrast, the sequence y has 2000 independent Bernoulli observations with success probability $p=0.7.$

set.seed(809)
y = rbinom(2000, 1, .7)

ACF plots show significant autocorrelations with lags 2, 3 and 4 (outside the dotted bounds) for the Old Faithful chain (left). The Markov dependence "decays" after a few steps.

By contrast, there are no significant autocorrelations for the IID Bernoulli observations.

enter image description here

par(mfrow=c(1,2))
 acf(x, main="Old Faithful")
 acf(y, main="Bernoulli")
par(mfrow=c(1,1))

Here is a link to a recent discussion of runs tests on this site.

Note: The ACF plot for lengths of Old Faithful eruptions is similar to one in Suess (2010) p146, Springer.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.