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

From: Models of deletion for visualizing bacterial variation: an application to tuberculosis spoligotypes

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@