What is the logarithmic distribution? (many questions)

[Vincent Vinh-Hung]

General question,
I've seen two descriptions of "logarithmic distribution".
One is related to the frequency of digits called Benford's law (digit 1
occurs more frequently than 2, 2 than 3, etc) whose explanation is that
it is the result of a mixture of distributions.
The other description is a 2-page paragraph The logarithmic distribution
in Kendall and Stuart (1977, The Advanced theory of statistics, Vol 1,
4th edition, pp 139-140), attributing the derivation to Fisher (1943).
Are these concepts of logarithmic distribution the same or not?

Second question I would like to ask: Kendall and Stuart give an
example of a distribution of the logarithmic type from Fisher (1943),
"distribution of butterflies in Malaya, with theoretical frequencies
given by the logarithmic distribution"
No. of species Theoretical frequency Observed frequency
1 135.05 118
2 67.33 74
3 44.75 44
4 33.46 24
5 26.69 29
6 22.17 22
7 18.95 20
etc ...
From what I've understood, the Theoretical frequency was generated
by
- ( q^r ) / ( r * ln(1-q) )
in which r is the No. of species, q is the probability of the presence
of an attribute.
How was, how can the fit be realized?

With thanks in advance,
Vincent Vinh-Hung

[Edzo Wisman]

isn't the lognormal distribution the same as logarithmic? Just guessing.
Else maybe you could look in the direction of exponential distributions.
I am just guessing though... :)
good luck!
Edzo

"Vincent Vinh-Hung" <conr...@az.vub.ac.be> wrote in message
news:3909DA49...@az.vub.ac.be...

[Vincent Vinh-Hung]

Lognormal I believe most often is used to describe a normal
distribution after logarithm transform, while logarithmic
distribution in the sense of Kendall-Stuart is else (I didn't
really grasp KS' formalism).

BTW, I queried how the fit was done because I can't find the same
figures as the Fisher 1943 example, assigning q=0.97293 I come
with 135.05 (ok), 65.7 (instead of the published 67.33),
42.6 (instead of 44.75), 31.1 (instead of 33.46), etc.

Thanks for your suggestion,
Vincent

[Graeme Byrne]

Vincent Vinh-Hung wrote:

> General question,
> I've seen two descriptions of "logarithmic distribution".
> One is related to the frequency of digits called Benford's law (digit 1

> occurs more frequently than 2, 2 than 3, etc) whose explanation is that
> it is the result of a mixture of distributions.
> The other description is a 2-page paragraph The logarithmic distribution
> in Kendall and Stuart (1977, The Advanced theory of statistics, Vol 1,
> 4th edition, pp 139-140), attributing the derivation to Fisher (1943).
> Are these concepts of logarithmic distribution the same or not?
>
> Second question I would like to ask: Kendall and Stuart give an
> example of a distribution of the logarithmic type from Fisher (1943),
> "distribution of butterflies in Malaya, with theoretical frequencies
> given by the logarithmic distribution"
> No. of species Theoretical frequency Observed frequency
> 1 135.05 118
> 2 67.33 74
> 3 44.75 44
> 4 33.46 24
> 5 26.69 29
> 6 22.17 22
> 7 18.95 20
> etc ...

> From what I've understood, the theoretical frequency was generated

> by
> - ( q^r ) / ( r * ln(1-q) )
> in which r is the No. of species, q is the probability of the presence
> of an attribute.
> How was, how can the fit be realized?

You will need a value of q first. This will either be estimated from the raw
data or assumed by some hypothesis. Once you have this just plug in the
value of r you want and multiply the resulting probability by the sum of
the observed frequencies.

You might also be able to use the theorectical mean q/((q - 1 )*Log[1 - q])
to estimate q by equating it to the sample mean and solving for q.

>
>
> With thanks in advance,
> Vincent Vinh-Hung

--
Dr Graeme Byrne
La Trobe University, Bendigo
PO Box 199, Bendigo, 3552
Phone: 61 3 5444 7263
Fax: 61 3 5444 7998
g.b...@bendigo.latrobe.edu.au

[Jack Tomsky]

In answer to your second question, if N is the total number of
butterflies and p(x) is the logarithmic probablity of x species, then
the fitted frequency of x species is N*p(x).

Jack

[Vincent Vinh-Hung]

Many thanks to Dr Byrne, the explicit expression of the mean
hinted at the correct direction, the error I made was confusing
natural logarithm and base 10 logarithm!

I apologize that I didn't post the complete example data,
which follows:

No. of species Theoretical frequency Observed frequency
1 135.05 118
2 67.33 74
3 44.75 44
4 33.46 24
5 26.69 29
6 22.17 22
7 18.95 20

8 16.53 19
9 14.65 20
10 13.14 15
11 11.91 12
12 10.89 14
13 10.02 6
14 9.28 12
15 8.63 6
16 8.07 9
17 7.57 9
18 7.13 6
19 6.74 10
20 6.38 10
21 6.06 11
22 5.77 5
23 5.5 3
24 5.25 3

(from Fisher 1943, Kendall-Stuart 1977)

With thanks,

Vincent Vinh-Hung
Oncologisch Centrum, AZ-VUB
101 Laarbeeklaan
B-1090 Jette

Graeme Byrne wrote:
(...)

>
> You will need a value of q first. This will either be estimated from the raw
> data or assumed by some hypothesis. Once you have this just plug in the
> value of r you want and multiply the resulting probability by the sum of
> the observed frequencies.
>
> You might also be able to use the theorectical mean q/((q - 1 )*Log[1 - q])
> to estimate q by equating it to the sample mean and solving for q.
>
>

[Vincent Vinh-Hung]

Thank you very much for the comment, I made the error of confusing
neperian log and base 10 log when applying p(x), therefore
couldn't find anything approaching the example results.

Vincent Vinh-Hung
Oncologisch Centrum, AZ-VUB

B-1090 Jette