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Graph examples of distributions of varying kurtosis

kaplansky [, show(numlist) range(min max) twoway_function_options ]

Description

kaplansky graphs some examples of distributions with various values of
kurtosis discussed by Kaplansky in 1945. The intent is cautionary: values
of kurtosis higher and lower than the Gaussian or normal value are
consistent with various peak and tail properties.

Remarks

In a short note Kaplansky (1945a) drew attention to four examples of
distributions with different values of kurtosis and behaviour not
consistent with some discussions of kurtosis.

The distributions all are symmetric with mean 0 and variance 1 and have
density functions, for variable x and c = sqrt(pi),

(1) (1 / 3c) (9/4 + x^4) exp(-x^2)

(2) (3 / (c sqrt(8))) exp(-x^2 / 2) - (1 / 6c) (9/4 + x^4) exp(-x^2)

(3) (1 / 6c) (exp(-x^2 / 4) + 4 exp(-x^2))

(4) (3 sqrt(3) / 16c) (2 + x^2) exp(-3x^2 / 4)

The kurtosis is (1) 2.75 (2) 3.125 (3) 4.5 (4) 8/3 ~ 2.667: compare the
Gaussian or normal value of 3. The density at the mean is (1) 0.423 (2)
0.387 (3) 0.470 (4) 0.366:  compare the Gaussian value of 0.399.

By default, densities are shown by kaplansky for the range 0 <= x <= 4.
The Gaussian or normal density with kurtosis 3 is shown as a reference.
Using a logarithmic scale for density is instructive. The graph is drawn
using twoway function, regardless of whatever data may be in memory.

Vignette

Irving Kaplansky (1917-2006) was a Canadian (later American)
mathematician.  He was born in Toronto shortly after his parents
immigrated from Poland and gained Bachelor's and Master's degrees from
the University of Toronto and a Ph.D. from Harvard in field theory (as
Saunders Mac Lane's first student).  He taught and researched at Harvard,
Chicago and Berkeley, with a wartime year in the Applied Mathematics
Group of the National Defense Council at Columbia University.  Kaplansky
made major contributions to group theory, ring theory, the theory of
operator algebras and field theory.  He was an accomplished pianist and
lyricist and an enthusiastic and lucid expositor of mathematics.
some other contributions to probability and statistics by Kaplansky
(1943, 1945b) and Kaplansky and Riordan (1945).

Options

show() selects distributions from the list 1 2 3 4 (see numbered
annotations above) and is to be used when you wish to show fewer than
the full set. Thus show(2 4) selects 2 and 4 from the list. The
Gaussian is shown regardless.

range() selects a numeric range for x other than the default.  Minimum
and maximum should be given as two numbers, such as range(5 10).

twoway_function_options are options of twoway function.

Examples

. kaplansky
. kaplansky , ysc(log) yla(.3 .03 .003 .0003 0.00003) legend(pos(7))

Author

Nicholas J. Cox, Durham University
n.j.cox@durham.ac.uk

References

Albers, D.J., Alexanderson, G.L. and Reid, C. (Eds) 1990.  More
Mathematical People. Boston: Harcourt Brace Jovanovich, pp. 119-136
on Irving Kaplansky.

Kaplansky, I.  1943. A characterization of the normal distribution.
Annals of Mathematical Statistics 14: 197-198.

Kaplansky, I.  1945a. A common error concerning kurtosis.  Journal,
American Statistical Association 40: 259 only.

Kaplansky, I.  1945b. The asymptotic distribution of runs of consecutive
elements.  Annals of Mathematical Statistics 16: 200-203.

Kaplansky, I. and Riordan, J.  1945.  Multiple matching and runs by the
symbolic method.  Annals of Mathematical Statistics 16: 272-277.

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