Lineo-Normal Distribution


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The Lineo-normal distribution is a member of the modified normal distributions constructed by Romanowski [1].

He states

“The lineo-normal distributions seem to be strikingly well confirmed by types of observations and measurements”.

M. Romanowski

It is a special case of the modified normal cumulative density function (CDF) [2]

with a = 1.

The above formula is a mixture distribution obtained by giving a power function distribution with density (a + 1)ta, 0 ≤ t ≤ 1 and a ≥ 1 to the standard normal distribution N(0, σ2)/ σ2.

When a tends to infinity, the curve becomes a normal distribution. When a = 3 the curve is approximately normal. The ratio between the peak of the modified normal curve and the peak of the corresponding normal curve—with equal variance—is 1.09 for the lineo-normal distribution [3].

Properties of the Lineo-Normal Distribution

  • Symmetric about 0.
  • Variance: σ2(a + 1)/(a + 2) = σ2(1 + 1)/(1 + 2) = 2σ2/3.   
  • Kurtosis: 3(a + 2)2/{(a + 1)(a + 3)} = 3(3 + 2)2/{(1 + 1)(1 + 3)} = 3(5)2/{(2)(4)} = 75/6 = 12.5.

See also:

References

[1] M. ROMANOWSKI: On the normal law of errors. Bull. Géodésique 73, 95 (1964).

[2] Kotz, S., and Johnson, N. L. (1985). Modified normal distributions, Encyclopedia of Statistical Sciences, 5, S. Kotz, N. L. Johnson, and C. B. Read (editors), 590-591, New York: Wiley.

[3] Maarek, A. & Konecny, G. Modulated Normal Distribution. Online: https://www.asprs.org/wp-content/uploads/pers/1973journal/sep/1973_sep_959-965.pdf