Date: Mar 10, 2013 8:52 PM Author: David Bernier Subject: Re: problem on record-breaking values in probability On 03/01/2013 08:41 AM, David Bernier wrote:

> On 02/27/2013 10:24 PM, David Bernier wrote:

>> On 02/27/2013 04:05 PM, James Waldby wrote:

>>> On Wed, 27 Feb 2013 07:10:08 -0500, David Bernier wrote:

>>>> On 02/27/2013 05:49 AM, David Bernier wrote:

>>>>> On 02/27/2013 05:31 AM, David Bernier wrote:

>>>>>> I used Marsaglia's 64-bit SUPER KISS pseudo-random number generator

>>>>>> to simulate uniform r.v.s on [0, 1] that are independent, as

>>>>>> X_1, X_2, X_3, ad infinitum

>>>>>>

>>>>>> For each go, (or sequence) I define its 1st record-breaking value

>>>>>> as R(1) as X_1, its 2nd record-breaking value R(2) as the

>>>>>> value taken by X_n for the smallest n with X_n > X_1, and in general

>>> [ R(k+1) is the value taken by X_n for the smallest n with X_n > R(k)]

>>> ...

>>>>>> In my first simulation I get: R(20) = 0.999999999945556

>>>>>> or about 5.4E-11 less than 1 , a one in 18 billion event.

>>> ...

>>>>>> In fact, R(20) is about 1 - (0.307)^20 ...

>>>>

>>>> I finally got another 20th record-breaking value, on my

>>>> second sequence, but it took much longer. The values

>>>> 1 - R(20), one per sequence, are what I call "p-values"

>>>>

>>>> from the simulations so far, a lot of variance in orders

>>>> of magnitude:

>>>>

>>>> the corresponding p-value is 0.000000000054 // seq. 1

>>>> the corresponding p-value is 0.000000000001 // seq. 2

>>>> the corresponding p-value is 0.000000002463

>>>> the corresponding p-value is 0.000000000782

>>>> the corresponding p-value is 0.000000106993

>>>> the corresponding p-value is 0.000000342142

>>>> the corresponding p-value is 0.000000001978

>>> [etc]

>>>

>>> It would be useful to report the number of trials each simulation

>>> took to find its 20th RBV. If a simulation takes m trials, the

>>> variance of the value X_m is approximately 1/(m^2), where X_m is

>>> the mth smallest (ie the largest) number among m trials. (See

>>> <http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29#Order_statistics>)

>>>

>>>

>>> Your simulation data implies there is a wide variance among the

>>> values of m required to find a 20th RBV.

>>>

>>> In following, let L(n) = Pr(n'th item of n is lowest). (Distribution

>>> of the lowest item should be similar to distribution of 1-(highest

>>> item).) I suppose that L(n) = 1/n and that the expected value of the

>>> number of record-low-values (RLV's) in m trials is sum{i=1 to m}(1/i),

>>> or about H_m, the m'th harmonic number, which can be approximated by

>>> log(m) + gamma, with gamma = Euler-Mascheroni constant, about 0.5772.

>>> When H_m ~ 20, log(m) ~ 19.42 and m is about 272 million. That is,

>>> simulations to find the 20th RLV or RBV will take about 272 million

>>> steps, on average. (I don't know which kind of average applies, or

>>> what the variance might be.)

>>>

>>

>> Yes, I hadn't thought of using expectations and expected values,

>> leading to partial sums of the harmonic series.

>>

>> You wrote:

>> "simulations to find the 20th RLV or RBV will take about 272 million

>> steps".

>>

>> Right, that looks like an interesting way of looking at

>> record values.

>>

>> Since it's like a stopping time, the number of the simulation,

>> I suppose it could be denoted S({X_i}_{i=1, ... oo}, 20)

>> or S(X_{BOLD_overligned}, 20).

>>

>> Electricity should go off tomorrow morning for repair.

>>

>> I think your estimate

>> E(log(S_20)) ~= 20 - gamma

>> should be very good.

>

> In the literature, a remarkable article, which may have

> appeared in the Am. Math. Monthly, can be found by

> searching for:

> Breaking Records and Breaking Boards. Ned Glick

>

> The author is Ned Glick, at some time at UBC in Canada.

>

> David Bernier

I did long simulations for 12th Record-Breaking Values.

With MatLab, I constructed a histogram of the natural

logarithms of the 76,000 values:

< http://img521.imageshack.us/img521/7702/records12log.jpg > .

The mean should be close to 12 - EulerGamma:

http://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant

David Bernier

--

$apr1$LJgyupye$GZQc9jyvrdP50vW77sYvz1