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Topic: problem on record-breaking values in probability
Replies: 14   Last Post: Apr 14, 2013 11:36 PM

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David Bernier

Posts: 3,222
Registered: 12/13/04
Re: problem on record-breaking values in probability
Posted: Apr 14, 2013 12:17 AM
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On 04/13/2013 11:51 PM, David Bernier wrote:
> On 03/11/2013 12:32 PM, David Bernier wrote:
>> On 03/11/2013 02:07 AM, James Waldby wrote:
>>> On Sun, 10 Mar 2013 21:47:27 -0400, David Bernier wrote:
>>>> On 03/10/2013 08:52 PM, David Bernier wrote:
>>>>> 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)]
>>>>>>>> ...

>>> [snip]
>>>>>>>> [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

>>> [snip]
>>>>>>>> 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.

>>> [snip]
>>>>>> 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

>>> ...
>>>>> 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 > .

>>> ...
>>>> S_12 is number of trials (steps) taken to find the 12th
>>>> Record-Breaking Value. On Average, log(S_12) is close
>>>> to 12 - gamma (gamma is the Euler-Mascheroni constant).
>>>>
>>>> A number of 76,000 sequences were generated, each being
>>>> continued until the 12th Record-Breaking Value for
>>>> that sequence was found. There is such variance from
>>>> one sample S_12 to another that I prefer the
>>>> quantities log(S_12) , for the histograms.
>>>>
>>>> Occasionally, an unusually high record is attained
>>>> in the 1st, 2nd, ... or 11th Record-Breaking Value.
>>>> That makes breaking the record all the more difficult.
>>>> In the simulations, the computer would pass (say) three
>>>> hours or more on the same sequence, with no new output
>>>> to the file for three or more hours.

>>>
>>> You may already have done so, but if not and if you are going to
>>> run more simulations, consider (a) profiling the code, and
>>> (b) trying different compilation options. (a) allows you to find
>>> out which lines of code use most of the hours of CPU time, so you
>>> can try alternate ways of coding them. Under (b), starting with
>>> the same random seed in each case, try optimization options -O1,
>>> -O2, -O3, -Ofast, timing the execution and also verifying the
>>> same results. From my interpretation of URL below, those 4
>>> optimization options are all you need to try.
>>> <http://gcc.gnu.org/onlinedocs/gcc/Optimize-Options.html>
>>>

>>
>> Maybe I've spent enough computer time on this distribution
>> problem. I'm grateful for your posts in reply.
>>
>> I think I found something known about the limiting distribution
>> of the k'th record-breaking trial number from the article of
>> Ned Glick:
>>
>> From Ned Glick's (UBC) 1978 article:
>>
>> N_r is the r'th record-breaking time (serial number
>> of the trial). N_1 = 1 always.
>>
>> "Also Renyi [30] stated a "central limit theorem" for the random
>> variables N_r: as r --> oo, the
>> distribution of (ln(N_r) - r)/sqrt(r) is
>> asymptotically normal with mean = 0 and variance= 1."
>> Ned Glick, "Breaking Records and Breaking Boards", 1978
>>
>>
>>
>> In Renyi's article,
>>
>> (log(nu_k) - k)/sqrt(k) ~ N(0,1) as k -> oo
>>
>> "Theorie des elements saillants d'une suite d'observations",
>> 1962.

> [...]
>
> "Advances in Combinatorial Methods and
> Applications to Probability and Statistics"
> edited by N. Balakrishnan,
> Google Books,
>
>
> http://books.google.ca/books/about/Advances_in_Combinatorial_Methods_and_Ap.html?id=WdJSxINF7VIC
>
>
> Part II, Applications to Probability Problems,
> Chapter 13, "Stirling Numbers and Records",
> pp. 189-201 approximately.
>
> In the "classical scheme" of records,
> alpha_1 = alpha_2 = ... alpha_n .
>
> Then Prob[xi_n = 1] = 1/n.
>
> xi_n is the indicator function for the n'th r.v.
> (n = 1, 2, 3, ... ).
> xi_n = 1 if X_n is a record, 0 otherwise.
>
> X_1, X_2, ... are i.i.d. uniform [0, 1] r.v.s (say).
>
> Formulas (13.19) and (13.19) give formulas
> for probability distributions related to
> record values.
>
> I think using Generalized Stirling numbers
> makes it harder to understand.
>
> But "Stirling Numbers and Records" is memorable.
>
> David Bernier
>


There is also the book "Records:mathematical theory"
by Valery B. Nevzorov.

Probably th Nevzorov who wrote a chapter in
"Advances in Combinatorial Methods and
Applications to Probability and Statistics"


"Records:mathematical theory":

http://books.google.ca/books/about/Records.html?id=B7ZpJKouceoC&redir_esc=y

dave

--
Jesus is an Anarchist. -- J.R.



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