Hansmann’s Distributions

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What are Hansmann’s distributions?

Hansmann’s distributions [1] are a Pearson type symmetric family obtained from a generalized Pearson differential equation [2]. These distributions are symmetric about zero and flexible in shape; Hansmann gave explicit solutions for seven different forms.

Outside of a mention in the usual indices of probability distributions such as Johnson, Kotz, and Balakrishnan’s seminal work on continuous probability distributions [3], there are no recent mentions of the distribution in the literature, despite Hansmann’s original article being considered “important” in Deming’s 1962 publication Selected Bibliography of Statistical Literature 1930 to 1957 [4]. The fact that these distributions are seldom used today is perhaps because of errors in Hansmann’s original equations — later corrected by Pawula & Rice [5], but perhaps not forgiven.

Deriving Hansmann’s Distributions

Hansmann’s distributions are derived from Pearson’s equation

Which becomes

The probability density functions (which Johnson et al [3] notes contain a correction by Pawula & Rice [5]) are:

Where

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

K₁ and K₂ must satisfy the conditions

• ∫P(x)dx = 1
• ∫x²p(x)= σ²_x.

Where ∫ is an integral (from calculus), a.k.a. the area under the curve.

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. Deming, L. (1961). Selected Bibliography of Statistical Literature 1930 to 1957: V. Frequency Functions, Moments, and Graduation. JOURNAL OF RESEARCH of the National Bureau of Standards-B. Mathematics and Mathematical Physics Vol. 66B, No.1, January-March 1962
5. 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.