  Although Zipf's law is widespread in natural and social data, one often
encounters situations where one or both ends of the ranked data deviate from
the power-law function. Previously we proposed the Beta rank function to
improve the fitting of data which does not follow a perfect Zipf's law. Here we
show that when the two parameters in the Beta rank function have the same
value, the Lavalette rank function, the probability density function can be
derived analytically. We also show both computationally and analytically that
Lavalette distribution is approximately equal, though not identical, to the
lognormal distribution. We illustrate the utility of Lavalette rank function in
several datasets. We also address three analysis issues on the statistical
testing of Lavalette fitting function, comparison between Zipf's law and
lognormal distribution through Lavalette function, and comparison between
lognormal distribution and Lavalette distribution.
