FitiingNote

MLE analysis

Exponential

Probability density function (PDF):

Likelihood function:

Log Likelihood function:

To estimate \lambda which maximizes L, differentiate L with respect to \lambda and find the solutions for =0.

Power-law

Probability density function (PDF):

Likelihood function:

Log Likelihood function:

To estimate \alpha which maximizes L, differentiate L with respect to \alpha and find the solutions for =0.

Truncated Power-law

Probability density function (PDF):

Stretched exponential

Probability density function (PDF):

References

Clauset A, Shalizi CR, Newman MEJ. 2009 Power-law distributions in empirical data. SIAM Rev. 51, 661–703. (doi:10.1137/070710111)

Bartumeus F, et al. 2010 Fishery discards impact on seabird movement patterns at regional scales. Curr. Biol. 20, 215–222. (doi:10.1016/j.cub.2009.11.073)