Strange behavior of RandomVariate and PascalDistribution

Question

Observe:

sr = 666;

d = PascalDistribution[1, 1/50];
od = OrderDistribution[{d, 10}, 1];

Column[{Mean[od] // N,
  SeedRandom[sr];
  Dimensions@(t1 = RandomVariate[d, {1000, 10}]),
  SeedRandom[sr];
  Dimensions@(t2 = Table[RandomVariate[d, 10], 1000]),
  SeedRandom[sr];
  Dimensions@(t3 = Partition[RandomVariate[d, 10000], 10]),
  SeedRandom[sr];
  Dimensions@(t4 = 
     ArrayReshape[Table[RandomVariate[d, 100], 100], {1000, 10}]),
  Min /@ t1 // Mean // N,
  Min /@ t2 // Mean // N,
  Min /@ t3 // Mean // N,
  Min /@ t4 // Mean // N}]

5.46666

{1000,10}

{1000,10}

{1000,10}

{1000,10}

5.463

6.484

5.463

5.463

The results from Min/@t...//Mean//N should be the same for all four cases.

Note the result (using the 666 seed) of 6.484 for the t2 case. This is consistently wacky (~+1 from actual expectation shown by the mean of the order distribution).

Changing the inner cardinality for the t4 case to anything 80 or above (e.g. ArrayReshape[Table[RandomVariate[d, 80], 125], {1000, 10}]keeps it consistent, but lowering it to say 50 in the RV generation results in also off-kilter results.

I'd venture there is some kind of heuristic switching of methods going on based on requested number of samples, and perhaps a bug in the low-count algorithm.

On 10.3 Windows, same results on 9.X Windows.

I'd appreciate verification (and explanation if I've pulled a DOH and there's a reason for this behavior).

Answer 1

Just an extended comment: Rather than a consistent shift for the values in the minimum values for t2, it appears that the distribution is completely different than for t1, t3, and t4. Here's a figure showing that:

h[x_, label_] := Histogram[Min /@ x, {1}, "PDF", PlotRange -> {{0, 30}, {0, 0.20}}, 
  PlotLabel -> Style[label, Bold, Larger]]
GraphicsGrid[{{h[t1, "t1"], h[t2, "t2"]}, {h[t3, "t3"], h[t4, "t4"]}}, ImageSize -> Large] 

Answer 2

In the t1 case

Random`DistributionVector[PascalDistribution[1, 1/50], 10000, ∞]

is evaluated and in the t2 case

Random`DistributionVector[PascalDistribution[1, 1/50], 10, ∞]

The definition of this function contains a Which statement. Its first test is True for 10000 and False for 10. Its second test (there are only two) is True.

For your parameters for the PascalDistribution the switch happens between 62 and 63.

Answer 3

This appears to be a bug.

Histogram shows result of 100K RV on PascalDistribution[1,1/50] for 100k batch using RandomVariate[...,100000] (blue) and generating a table of 100k single calls (default chicken-poop-and-mud jaundice beige):

Thanks to Karsten for verification and Jim for histogram idea.