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{* 25 May 2007}{...}
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help for {hi:kaplansky}
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{title:Graph examples of distributions of varying kurtosis}
{p 8 17 2}
{cmd:kaplansky}
[{cmd:,}
{cmd:show(}{it:numlist}{cmd:)}
{cmdab:ra:nge(}{it:min max}{cmd:)}
{it:twoway_function_options}
]
{title:Description}
{p 4 4 2}
{cmd: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.
{title:Remarks}
{p 4 4 2}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.
{p 4 4 2}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)
{p 4 4 2}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.
{p 4 4 2}By default, densities are shown by {cmd:kaplansky} for the range
{bind: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 {help twoway function}, regardless of whatever data may be in
memory.
{title:Vignette}
{p 4 4 2}Irving Kaplansky (1917{c -}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. Kaplansky was a member of the U.S. National Academy of
Sciences. See also some other contributions to probability and statistics
by Kaplansky (1943, 1945b) and Kaplansky and Riordan (1945).
{title:Options}
{p 4 8 2}{cmd: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
{cmd:show(2 4)} selects 2 and 4 from the list. The Gaussian is shown
regardless.
{p 4 8 2}{cmd:range()} selects a numeric range for x other than the default.
Minimum and maximum should be given as two numbers, such as {cmd:range(5 10)}.
{p 4 8 2}{it:twoway_function_options} are options of {help twoway function}.
{title:Examples}
{p 4 8 2}{cmd:. kaplansky}{p_end}
{p 4 8 2}{cmd:. kaplansky , ysc(log) yla(.3 .03 .003 .0003 0.00003) legend(pos(7))}
{title:Author}
{p 4 4 2}Nicholas J. Cox, Durham University{break}
n.j.cox@durham.ac.uk
{title:References}
{p 4 8 2}Albers, D.J., Alexanderson, G.L. and Reid, C. (Eds) 1990.
{it:More Mathematical People.} Boston: Harcourt Brace Jovanovich,
pp. 119{c -}136 on Irving Kaplansky.
{p 4 8 2}Kaplansky, I.
1943. A characterization of the normal distribution.
{it:Annals of Mathematical Statistics} 14: 197{c -}198.
{p 4 8 2}Kaplansky, I.
1945a. A common error concerning kurtosis.
{it:Journal, American Statistical Association} 40: 259 only.
{p 4 8 2}Kaplansky, I.
1945b. The asymptotic distribution of runs of consecutive elements.
{it:Annals of Mathematical Statistics} 16: 200{c -}203.
{p 4 8 2}Kaplansky, I. and Riordan, J.
1945.
Multiple matching and runs by the symbolic method.
{it:Annals of Mathematical Statistics} 16: 272{c -}277.