On Thu, 15 Dec 2005, Phineas wrote:

> How many distinct values can rnorm return?

2^32-1. This is described in help(Random)

> I assume that rnorm manipulates runif in some way, runif uses the Mersenne

> Twister, which has a period of 2^19937 - 1. Given that runif returns a 64

> bit precision floating point number in [0,1], the actual period of the

> Mersenne Twister in a finite precision world must be significantly less.

No. Not at all. Consider a sequence of 1-bit numbers: individual values

will repeat fairly frequently but the sequence need not be periodic with

any period

1101001000100001000001

is the start of one fairly obvious non-periodic sequence.

There are reasons that a longer period than 2^32 is useful. The most

obvious is that you can construct higher-resolution numbers from several

runif()s. The Mersenne Twister was designed so that quite long

subsequences (623 elements) would be uniformly distributed.

Less obvious is that fact that a periodic pseudorandom sequence is likely

to show a frequency distribution of repeat values that differs from the

random sequence once you get beyond about the square root of the period.

This means that a 32-bit PRNG should really have a period of at least

2^64.

The randaes package provides a runif() that uses 64 bits to construct a

double, providing about 53 bits of randomness.

> One of the arguments for Monte Carlo over the bootstrap is that for a sample

> size n the bootstrap can return at most 2^n distinct resamples, however for

> even for relatively small sample sizes there may be no increase in precision

> in using Monte Carlo.

I don't get this at all. What technique are you comparing to the bootstrap

and for what purpose?

-thomas

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