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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.
    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. 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.