"David C. Howell" wrote:
>
> My daughter just asked me a question that I should be able to answer,
> but can't. Any help would be appreciated.
>
> It is well know that the sum of normally distributed random variables
> is itself normally distributed. But I want to know the sampling
> distribution of the sum of lognormal distributions.
>
> My first thought would be that the central limit theorem would suggest
> that the sampling distribution would approach normal as N increases.
> On the other hand, when I generate multiple independent lognormal
> distributions and then take their sum, a P-P plot tells me that the
> sum is nicely lognormal, not normal.
Did you do the plot correctly?
Even for the sum of 2 lognormal distributions I get this (below) for
a q-q plot against another lognormal. It is far from straight:
-
21.0+
- *
2sort -
-
-
14.0+ *
- *
- 2
-
- **
7.0+ 2*2
- 34*
- 273
- 6+7
- ++
0.0+ 5*
+---------+---------+---------+---------+---------+------1sort
0.0 4.0 8.0 12.0 16.0 20.0
but also far from normal (plot against sorted normals below)
-
21.0+
- *
2sort -
-
-
14.0+ *
- *
- * *
-
- **
7.0+ 22*
- *4*2
- *2522
- 36324343
- 23 2 6**23452***
0.0+ 2* ** *
--------+---------+---------+---------+---------+--------Nsort
-1.60 -0.80 0.00 0.80 1.60
Adding 10 lognormals and plotting against an eleventh I get the even
less straight
-
- *
36+
- * *
10sort - 2
- ***
- 3**
24+ *4
- 5
- 67
- +3
- +3
12+ 98
- +
- 5
-
-
+---------+---------+---------+---------+---------+------1sort
0.0 4.0 8.0 12.0 16.0 20.0
Whereas plotting against sorted normals I get
-
- *
36+
- * *
10sort - * *
- 2*
- *22
24+ 3**
- *3*
- *425*
- 32433
- 2**46
12+ ***2345
- *23 2 5
- 2* **
-
-
--------+---------+---------+---------+---------+--------Nsort
-1.60 -0.80 0.00 0.80 1.60
which is considerably closer.
Repeating for n=100 I get (against sorted lognormals)
-
210+ * * *
- ** 2
100sort - 2*2
- *4*
- 75
175+ 39
- +
- +*
- 39
- 8
140+ 6
- *5
- 2
- 2
-
105+
+---------+---------+---------+---------+---------+------1sort
0.0 4.0 8.0 12.0 16.0 20.0
and against sorted normals:
-
210+ * * *
- *2 *
100sort - 22*
- 3*2
- *523*
175+ 344*
- 2324
- ***44
- 245*
- 3**2*
140+ * 2 3
- * *22
- * *
- 2
-
105+
--------+---------+---------+---------+---------+--------Nsort
-1.60 -0.80 0.00 0.80 1.60
The twitches at the ends are because I was lazy and used sorted
pseudorandom numbers rather than theoretical normal quantiles - but we
see that as n gets larger the sum clearly *does* approach normality. I
doubt if there is any
really nice form for the intermediate distributions, though.
-Robert Dawson
.
.
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