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Graph examples of distributions of varying kurtosis

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

Description

kaplanskygraphs 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.

RemarksIn 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

kaplanskyfor 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.

VignetteIrving 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. 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).

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. Thusshow(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 asrange(5 10).

twoway_function_optionsare options of twoway function.

Examples

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

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

ReferencesAlbers, D.J., Alexanderson, G.L. and Reid, C. (Eds) 1990.

MoreMathematical People.Boston: Harcourt Brace Jovanovich, pp. 119-136 on Irving Kaplansky.Kaplansky, I. 1943. A characterization of the normal distribution.

Annals of Mathematical Statistics14: 197-198.Kaplansky, I. 1945a. A common error concerning kurtosis.

Journal,American Statistical Association40: 259 only.Kaplansky, I. 1945b. The asymptotic distribution of runs of consecutive elements.

Annals of Mathematical Statistics16: 200-203.Kaplansky, I. and Riordan, J. 1945. Multiple matching and runs by the symbolic method.

Annals of Mathematical Statistics16: 272-277.