Tuesday, August 28, 2012

Topp-Leone Distribution

In May, I posted about bias-correcting maximum likelihood estimators (MLEs), and I referred to a series of related papers that I've been authoring/co-authoring in recent times.

I've just completed another such paper - this one relates to the estimation of the scale parameter in the so-called Topp-Leone (1955) distribution. You can access the paper here; and the abstract explains why this distribution is especially interesting, and summarizes the main results:

"The Topp-Leone distribution is attractive for reliability studies as it has finite support and a bathtub-shaped hazard function. We compare some properties of the method of moments, maximum likelihood, and bias-adjusted maximum likelihood estimators of its shape parameter. The last of these estimators is very simple to apply and dominates the method of moments estimator in terms of relative bias and mean squared error."

Here are some illustrative plots of the density function and the hazard function for the Topp-Leone distribution (also, see Nadarajah and Kotz, 2003):

One thing that's interesting about bias-correcting the MLE in this particular problem is that the "corrective" approach proposed by Cox and Snell (1968), and the "preventive" approach suggested by Firth (1993), give rise to exactly the same bias-adjusted MLE.


Cox, D. R. and E. J. Snell, 1968. A general definition of residuals. Journal of the Royal Statistical Society, B, 30, 248-275.

Firth, D., 1993. Bias reduction of maximum likelihood estimates. Biometrika, 80, 27-38.

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

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

© 2012, David E. Giles

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