▁normal ▁log - lik elihood ▁function ▁has ▁infinitely ▁many ▁continuous ▁deriv ates ▁( i . e . ▁it ▁belongs ▁to ▁the ▁class ▁C ∞ ▁of ▁smooth ▁functions ) ▁only ▁if ▁ ▁is ▁a ▁positive , ▁even ▁integer . ▁ ▁Otherwise , ▁the ▁function ▁has ▁ ▁continuous ▁derivatives . ▁ ▁As ▁a ▁result , ▁the ▁standard ▁results ▁for ▁consist ency ▁and ▁asympt otic ▁normal ity ▁of ▁maximum ▁lik elihood ▁estimates ▁of ▁ ▁only ▁apply ▁when ▁. ▁ ▁Max imum ▁lik elihood ▁estim ator ▁ ▁It ▁is ▁possible ▁to ▁fit ▁the ▁generalized ▁normal ▁distribution ▁adopt ing ▁an ▁approximate ▁maximum ▁lik elihood ▁method . ▁With ▁ ▁initially ▁set ▁to ▁the ▁sample ▁first ▁moment ▁, ▁▁ ▁is ▁estimated ▁by ▁using ▁a ▁Newton – R aph son ▁iter ative ▁procedure , ▁starting ▁from ▁an ▁initial ▁guess ▁of ▁, ▁ ▁where ▁ ▁is ▁the ▁first ▁statistical ▁moment ▁of ▁the ▁absolute ▁values ▁and ▁ ▁is ▁the ▁second ▁statistical ▁moment . ▁The ▁iteration ▁is ▁ ▁where ▁ ▁and ▁▁▁▁ ▁and ▁where ▁ ▁and ▁ ▁are ▁the ▁dig amma ▁function ▁and ▁trig amma ▁function . ▁ ▁Given ▁a ▁value ▁for ▁, ▁it ▁is ▁possible ▁to ▁estimate ▁ ▁by ▁finding ▁the ▁minimum ▁of : ▁ ▁Finally ▁ ▁is ▁evaluated ▁as ▁ ▁For ▁, ▁median ▁is ▁a ▁more ▁appropriate ▁estim ator ▁of ▁ ▁. ▁Once ▁ ▁is ▁estimated , ▁ ▁and ▁ ▁can ▁be ▁estimated ▁as ▁described ▁above . ▁ ▁App lications ▁ ▁This ▁version ▁of ▁the ▁generalized ▁normal ▁distribution ▁has ▁been ▁used ▁in ▁model ing ▁when ▁the ▁concentration ▁of ▁values ▁around ▁the ▁mean