Topp–Leone distribution

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The Topp-Leone distribution, first proposed by Topp and Leone in 1955 [1], is a J-shaped alternative to the beta distribution. It has finite support and a bathtub shaped hazard function, which makes it especially suited for reliability studies [2].

Topp-Leone distribution properties

graph of the topp-leone distribution
Graph of the Topp-Leone distribution density [3].

The probability density function (PDF) for the Topp-Leone distribution is [4]

The cumulative distribution function (CDF) is

The hazard function can be expressed in closed form as [2]

𝜆(𝑥) = (2𝑣/𝑏)𝑦(1 − 𝑦2)𝑣 − 1/[1 − (1 − 𝑦2)𝑣] ,

where 𝑦 = 1 − (𝑥/𝑏) .

Extensions to the Topp-Leone distribution include:

  • The Type II generalized Topp– Leone–G (TIIGTL-G) family, which has a more flexible density function. In addition to J-shaped, its density function can be unimodel, left-skewed, right-skewed, and reversed-J shaped, with increasing, decreasing, upside-down, J or reversed-J hazard rates [5]. 
  •  The beta Topp-Leone generated family of distributions is a combination of the beta generated family and the Topp-Leone generated family [6]. 
  • The beta Topp-Leone Weibull distribution has flexible hazard shapes which can be monotonically increasing, monotonically de-creasing, bathtub-shaped, upside-down bathtub-shaped and modified bathtub-shaped [6].


[1] Topp, C. W. and F. C. Leone, 1955. A family of J-shaped frequency functions. Journal of the American Statistical Association, 50, 209-219.

[2] Giles, G. (2021). On the Estimation of the Topp-Leone Distribution. Retrieved April 29, 2023 from:

[3] Graph created with Desmos graphing calculator.

[4] Nadarajah, S. and Kotz, S. (2003). Moments of some J-shaped distributions.
Journal of Applied Statistics, 30, 311-317.

[5] Amal S. Hassan, M. Elgarhy, Zubair Ahmad, Type II Generalized Topp–Leone Family of Distributions: Properties and Applications, J. data sci. 17(2022), no. 4, 638-659, DOI 10.6339/JDS.201910_17(4).0001

[6] Watthanawisut, A. et al. The Beta Topp-Leone Generated Family of Distributions and Theirs [sic] Applications. Thailand Statistician July 2022; 20(3): 489-507. Retrieved April 29, 2023 from:

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