B. KUMPHON 419

21

1

1

ln ln

11

1l

ni

i

n

ii

x

n

L

x

n

ii

xx

,0x

(25)

By Equation (21), a comparison of the equations of the

ME and the MLE immediately reveals that Equation (18)

is equivalent to Equation (23), Equation (19) to Equation

(24) and Equation (20) to Equation (25), where

and

01211kk

0,1,2,. Consequently,

the two methods become equivalent for discrete random

variables.

5. Conclusion

A positive skewness distribution, the three-parameter ka p pa

distribution, is considered. Parameter estimation by the

maximum likelihood method requires a certain cutoff in

the parameter space or a best starting value, for otherwise

the solution may appear under-determined instead of a

unique answer (there can exist a concave set). The prin-

ciple of maximum entropy is another tool to address this

problem under constraints that show the characteristic of

the distribution given the empirical evidence, using the

method of Lagrange multipliers. For

01211kk

,0x0,1,2,, and

the

principle of maximum entropy method is equivalent to

the maximum likelihood method for the discrete case.

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