Hansmann’s Distributions


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Hansmann’s distributions [1], obtained from a generalized Pearson differential equation [2], are symmetric about zero.


They are derived from Pearson’s equation

Which becomes

The probability density functions [which Johnson et al [3] notes contain a correction by Pawula & Rice [4]) are:

Where

  • K= ½ {c2(b2 – a2)}-1
  • K, K1, K2 are normalizing constants.

K1 and K2 must satisfy the conditions

  • ∫P(x)dx = 1
  • ∫x2p(x)= σ2x.

References

[1] Hansmann, G. H. (1934). On certain non-normal symmetric frequency distributions, Biometrika, 26, 129-135

[2] Haight, F. (1958). Index to the Distributions of Mathematical Statistics. National Bureau of Standards Report.

[3] Johnson, Kotz, and Balakrishnan, (1994), Continuous Univariate Distributions, Volumes I and II, 2nd. Ed., John Wiley and Sons.

[4] Pawula, R. F., and Rice, S. 0. (1989). A note on Hansmann’s 1934 family of distributions, IEEE Transactions on Information Theory, 35, 910-91 1.

CITE THIS AS:
Stephanie Glen. "Hansmann’s Distributions" From StatisticsHowTo.com: Elementary Statistics for the rest of us! https://www.statisticshowto.com/hansmanns-distributions/
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