Table 3 Probability mass functions and maximum likelihood estimators of the models considered

Model name support Probability mass function Maximum likelihood estimator
Geometric k [1, ∞) P(K = k) = P(k) = pk-1(1 - p) $p ^ = 1 − 1 x ¯ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiCaaNbaKaacqGH9aqpcqaIXaqmcqGHsisljuaGdaWcaaqaaiabigdaXaqaaiqbdIha4zaaraaaaaaa@3350@$
Negative binomial k [1, ∞) $P ( k ) = ( 1 − p ) r 1 − ( 1 − p ) r ( k + r − 1 r − 1 ) p k MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiuaaLaeiikaGIaem4AaSMaeiykaKIaeyypa0tcfa4aaSaaaeaacqGGOaakcqaIXaqmcqGHsislcqWGWbaCcqGGPaqkdaahaaqabeaacqWGYbGCaaaabaGaeGymaeJaeyOeI0IaeiikaGIaeGymaeJaeyOeI0IaemiCaaNaeiykaKYaaWbaaeqabaGaemOCaihaaaaadaqadaqaauaabeqaceaaaeaacqWGRbWAcqGHRaWkcqWGYbGCcqGHsislcqaIXaqmaeaacqWGYbGCcqGHsislcqaIXaqmaaaacaGLOaGaayzkaaGccqWGWbaCdaahaaWcbeqaaiabdUgaRbaaaaa@4E08@$
where k, r ≥ 1
$p ^ = 1 − r ^ x ^ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiCaaNbaKaacqGH9aqpcqaIXaqmcqGHsisljuaGdaWcaaqaaiqbdkhaYzaajaaabaGafmiEaGNbaKaaaaaaaa@33D5@$
Conditional Poisson k [1, ∞) $P ( k ) = e − λ λ k k ! ( 1 − e − λ ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiuaaLaeiikaGIaem4AaSMaeiykaKIaeyypa0Jaemyzau2aaWbaaSqabeaacqGHsislcqaH7oaBaaqcfa4aaSaaaeaacqaH7oaBdaahaaqabeaacqWGRbWAaaaabaGaem4AaSMaeiyiaeIaeiikaGIaeGymaeJaeyOeI0Iaemyzau2aaWbaaeqabaGaeyOeI0Iaeq4UdWgaaiabcMcaPaaaaaa@42D7@$
where k ≥ 1, λ > 0
Solution to $x ¯ = λ ^ / ( 1 − e − λ ^ ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiEaGNbaebacqGH9aqpcuaH7oaBgaqcaiabc+caViabcIcaOiabigdaXiabgkHiTiabdwgaLnaaCaaaleqabaGaeyOeI0Iafq4UdWMbaKaaaaGccqGGPaqkaaa@38E0@$
Logarithmic series k [1, ∞) $P ( k ) = − θ k k log ( 1 − θ ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiuaaLaeiikaGIaem4AaSMaeiykaKIaeyypa0JaeyOeI0scfa4aaSaaaeaacqaH4oqCdaahaaqabeaacqWGRbWAaaaabaGaem4AaSMagiiBaWMaei4Ba8Maei4zaCMaeiikaGIaeGymaeJaeyOeI0IaeqiUdeNaeiykaKcaaaaa@4099@$ Solution to $x ¯ = − θ ^ ( 1 − θ ^ ) log ( 1 − θ ^ ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiEaGNbaebacqGH9aqpjuaGdaWcaaqaaiabgkHiTiqbeI7aXzaajaaabaGaeiikaGIaeGymaeJaeyOeI0IafqiUdeNbaKaacqGGPaqkcyGGSbaBcqGGVbWBcqGGNbWzcqGGOaakcqaIXaqmcqGHsislcuaH4oqCgaqcaiabcMcaPaaaaaa@4085@$
Zeta k [1, ∞) $P ( k ) = k − ρ ∑ d = 1 ∞ d − ρ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiuaaLaeiikaGIaem4AaSMaeiykaKIaeyypa0tcfa4aaSaaaeaacqWGRbWAdaahaaqabeaacqGHsislcqaHbpGCaaaabaWaaabmaeaacqWGKbazdaahaaqabeaacqGHsislcqaHbpGCaaaabaGaemizaqMaeyypa0JaeGymaedabaGaeyOhIukacqGHris5aaaaaaa@40A3@$
where ρ > 1
Estimated numerically
Zipf k [1, 43] $P ( k ) = k − ρ ∑ d = 1 43 d − ρ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiuaaLaeiikaGIaem4AaSMaeiykaKIaeyypa0tcfa4aaSaaaeaacqWGRbWAdaahaaqabeaacqGHsislcqaHbpGCaaaabaWaaabmaeaacqWGKbazdaahaaqabeaacqGHsislcqaHbpGCaaaabaGaemizaqMaeyypa0JaeGymaedabaGaeGinaqJaeG4mamdacqGHris5aaaaaaa@411C@$
where ρ > 1
Estimated numerically
Uniform k [1, 43] $P ( k ) = 1 43 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiuaaLaeiikaGIaem4AaSMaeiykaKIaeyypa0tcfa4aaSaaaeaacqaIXaqmaeaacqaI0aancqaIZaWmaaaaaa@348D@$ $∏ k = 1 43 ( 1 a ) x k MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaebmaeaajuaGdaqadaqaamaalaaabaGaeGymaedabaGaemyyaegaaaGaayjkaiaawMcaaOWaaWbaaSqabeaacqWG4baEdaWgaaadbaGaem4AaSgabeaaaaaaleaacqWGRbWAcqGH9aqpcqaIXaqmaeaacqaI0aancqaIZaWma0Gaey4dIunaaaa@3AA3@$
Uniform k [1, a] $P ( k ) = 1 a MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiuaaLaeiikaGIaem4AaSMaeiykaKIaeyypa0tcfa4aaSaaaeaacqaIXaqmaeaacqWGHbqyaaaaaa@33EE@$
where 1 ≤ a ≤ 43
Estimated numerically
Empirical k [1, 43] $P ( k ) = x k m MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiuaaLaeiikaGIaem4AaSMaeiykaKIaeyypa0tcfa4aaSaaaeaacqWG4baEdaWgaaqaaiabdUgaRbqabaaabaGaemyBa0gaaaaa@360F@$ $∏ k = 1 43 P ( k ) x k MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaebmaeaajuaGcqWGqbaucqGGOaakcqWGRbWAcqGGPaqkdaahaaqabeaacqWG4baEdaWgaaqaaiabdUgaRbqabaaaaaWcbaGaem4AaSMaeyypa0JaeGymaedabaGaeGinaqJaeG4mamdaniabg+Givdaaaa@3AE8@$