On Posterior Properties of the Two Parameter Gamma Family of Distributions

RAMOS, PEDRO L.; DEY, DIPAK K.; LOUZADA, FRANCISCO; RAMOS, EDUARDO

doi:10.1590/0001-3765202120190826

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Sumário

MATHEMATICAL SCIENCES • An. Acad. Bras. Ciênc. 93 (suppl 3) • 2021 • https://doi.org/10.1590/0001-3765202120190826 copy

On Posterior Properties of the Two Parameter Gamma Family of Distributions

Authorship SCIMAGO INSTITUTIONS RANKINGS

Abstract

The gamma distribution has been extensively used in many areas of applications. In this paper, considering a Bayesian analysis we provide necessary and sufficient conditions to check whether or not improper priors lead to proper posterior distributions. Further, we also discuss sufficient conditions to verify if the obtained posterior moments are finite. An interesting aspect of our findings are that one can check if the posterior is proper or improper and also if its posterior moments are finite by looking directly in the behavior of the proposed improper prior. To illustrate our proposed methodology these results are applied in different objective priors.

Key words
Gamma distribution; improper prior; objective prior; posterior property

$𝛉$	n	Uniform		Jeffreys’ Rule		Jeffreys’ Prior		Miller		Reference $α$		Reference $β$		MDIP		Tibshirani		Consensus GM		Consensus AM
$𝛉$	n	$μ$	$θ$	$μ$	$θ$	$μ$	$θ$	$μ$	$θ$	$μ$	$θ$	$μ$	$θ$	$μ$	$θ$	$μ$	$θ$	$μ$	$θ$	$μ$	$θ$
	10	0.892	0.891	0.950	0.953	0.942	0.948	0.928	0.936	0.948	0.953	0.943	0.948	0.969	0.964	0.950	0.957	0.949	0.953	0.922	0.928
	20	0.907	0.908	0.946	0.949	0.941	0.945	0.938	0.937	0.944	0.949	0.942	0.946	0.960	0.956	0.947	0.951	0.945	0.949	0.931	0.932
$α = 2$	30	0.918	0.916	0.948	0.950	0.944	0.945	0.937	0.939	0.948	0.951	0.946	0.946	0.958	0.949	0.950	0.950	0.949	0.949	0.933	0.936
	40	0.923	0.921	0.948	0.946	0.944	0.944	0.938	0.942	0.947	0.947	0.943	0.944	0.953	0.948	0.949	0.948	0.948	0.947	0.937	0.938
	50	0.929	0.928	0.950	0.950	0.948	0.947	0.942	0.944	0.951	0.949	0.948	0.948	0.955	0.948	0.950	0.951	0.949	0.949	0.941	0.941
	60	0.929	0.928	0.947	0.946	0.943	0.944	0.941	0.940	0.944	0.946	0.944	0.943	0.948	0.945	0.945	0.946	0.946	0.946	0.938	0.938
$β = 0.5$	70	0.934	0.931	0.946	0.948	0.943	0.947	0.942	0.943	0.946	0.948	0.944	0.947	0.948	0.948	0.944	0.948	0.946	0.947	0.938	0.938
	80	0.934	0.934	0.948	0.947	0.947	0.948	0.943	0.944	0.946	0.949	0.946	0.945	0.948	0.946	0.949	0.948	0.946	0.948	0.940	0.942
	90	0.938	0.936	0.949	0.949	0.948	0.948	0.944	0.943	0.948	0.948	0.948	0.947	0.950	0.945	0.948	0.949	0.949	0.947	0.944	0.943
	100	0.938	0.933	0.945	0.945	0.945	0.944	0.943	0.940	0.944	0.944	0.944	0.943	0.946	0.942	0.946	0.945	0.944	0.944	0.943	0.941
	110	0.941	0.939	0.948	0.951	0.950	0.950	0.945	0.945	0.946	0.948	0.947	0.950	0.949	0.948	0.948	0.948	0.948	0.947	0.944	0.944
	120	0.942	0.943	0.950	0.951	0.949	0.950	0.948	0.950	0.948	0.953	0.948	0.950	0.950	0.950	0.950	0.952	0.949	0.951	0.946	0.947
	10	0.886	0.885	0.948	0.951	0.942	0.942	0.928	0.931	0.949	0.952	0.942	0.942	0.920	0.942	0.951	0.954	0.949	0.949	0.918	0.920
$α = 4$	20	0.907	0.905	0.947	0.946	0.942	0.941	0.935	0.933	0.947	0.944	0.943	0.940	0.923	0.940	0.950	0.946	0.947	0.944	0.928	0.928
	30	0.917	0.914	0.952	0.950	0.948	0.947	0.939	0.939	0.952	0.949	0.949	0.947	0.921	0.938	0.950	0.949	0.951	0.948	0.934	0.933
	40	0.923	0.920	0.947	0.946	0.946	0.943	0.939	0.939	0.947	0.948	0.945	0.944	0.925	0.939	0.946	0.948	0.948	0.947	0.935	0.935
	50	0.924	0.926	0.947	0.947	0.944	0.945	0.939	0.940	0.945	0.947	0.946	0.944	0.929	0.943	0.948	0.946	0.945	0.947	0.936	0.937
	60	0.932	0.930	0.950	0.950	0.948	0.946	0.942	0.943	0.951	0.949	0.947	0.947	0.933	0.940	0.952	0.950	0.950	0.948	0.940	0.941
	70	0.929	0.929	0.945	0.946	0.945	0.944	0.940	0.938	0.946	0.946	0.943	0.946	0.931	0.939	0.946	0.947	0.945	0.944	0.938	0.938
$β = 2$	80	0.936	0.936	0.950	0.952	0.949	0.949	0.944	0.946	0.950	0.951	0.949	0.949	0.937	0.942	0.950	0.952	0.948	0.951	0.943	0.945
	90	0.934	0.938	0.946	0.948	0.945	0.948	0.941	0.945	0.944	0.949	0.945	0.947	0.931	0.943	0.944	0.949	0.944	0.948	0.940	0.943
	100	0.941	0.939	0.949	0.952	0.949	0.949	0.947	0.948	0.951	0.951	0.951	0.951	0.936	0.942	0.949	0.950	0.950	0.952	0.946	0.945
	110	0.940	0.939	0.949	0.949	0.949	0.949	0.947	0.947	0.949	0.950	0.947	0.948	0.937	0.943	0.949	0.950	0.950	0.951	0.944	0.946
	120	0.940	0.937	0.949	0.945	0.947	0.944	0.944	0.943	0.947	0.945	0.947	0.944	0.936	0.941	0.946	0.947	0.947	0.945	0.944	0.942

$𝛉$	n	Uniform	Jeffreys’ Rule	Jeffreys’ Prior	Miller	Reference $α$	Reference $β$	MDIP	Tibshirani	Consensus GM	Consensus AM
$α = 2$	10	1.336(1.305)	1.130(0.655)	1.175(0.768)	1.232(0.928)	1.124(0.646)	1.169(0.762)	1.082(0.304)	1.067(0.542)	1.131(0.664)	1.252(1.014)
	20	1.209(0.598)	1.080(0.354)	1.109(0.397)	1.144(0.457)	1.076(0.351)	1.105(0.394)	1.070(0.230)	1.041(0.312)	1.081(0.358)	1.156(0.487)
	30	1.153(0.360)	1.059(0.234)	1.080(0.256)	1.106(0.288)	1.056(0.232)	1.077(0.254)	1.061(0.179)	1.030(0.211)	1.060(0.236)	1.114(0.302)
	40	1.122(0.256)	1.048(0.177)	1.064(0.191)	1.084(0.211)	1.045(0.176)	1.061(0.190)	1.054(0.149)	1.025(0.164)	1.048(0.179)	1.091(0.220)
	50	1.100(0.195)	1.039(0.142)	1.053(0.151)	1.070(0.164)	1.038(0.141)	1.051(0.151)	1.049(0.124)	1.021(0.133)	1.040(0.143)	1.075(0.170)
	60	1.084(0.157)	1.032(0.120)	1.044(0.126)	1.058(0.136)	1.031(0.119)	1.042(0.125)	1.042(0.109)	1.016(0.113)	1.032(0.120)	1.062(0.140)
	70	1.071(0.127)	1.026(0.100)	1.036(0.105)	1.049(0.111)	1.025(0.099)	1.034(0.104)	1.036(0.093)	1.012(0.095)	1.026(0.100)	1.052(0.114)
	80	1.067(0.110)	1.027(0.088)	1.036(0.092)	1.047(0.097)	1.026(0.087)	1.035(0.092)	1.038(0.083)	1.015(0.083)	1.028(0.088)	1.050(0.100)
	90	1.057(0.094)	1.022(0.077)	1.030(0.080)	1.040(0.084)	1.021(0.077)	1.029(0.080)	1.032(0.074)	1.011(0.074)	1.022(0.077)	1.043(0.086)
	100	1.055(0.086)	1.022(0.071)	1.030(0.074)	1.039(0.078)	1.021(0.071)	1.028(0.074)	1.032(0.069)	1.012(0.068)	1.023(0.071)	1.041(0.079)
	110	1.047(0.074)	1.018(0.063)	1.024(0.065)	1.033(0.068)	1.017(0.062)	1.023(0.065)	1.027(0.061)	1.009(0.061)	1.018(0.063)	1.035(0.069)
	120	1.045(0.069)	1.018(0.059)	1.024(0.060)	1.031(0.063)	1.017(0.058)	1.023(0.060)	1.027(0.058)	1.009(0.057)	1.018(0.059)	1.033(0.064)
$β = 0.5$	10	1.395(0.107)	1.157(0.053)	1.204(0.062)	1.262(0.074)	1.151(0.053)	1.198(0.062)	1.160(0.033)	1.093(0.044)	1.159(0.054)	1.283(0.081)
	20	1.246(0.049)	1.098(0.029)	1.127(0.032)	1.163(0.037)	1.094(0.028)	1.123(0.032)	1.121(0.023)	1.058(0.025)	1.099(0.029)	1.175(0.039)
	30	1.181(0.030)	1.073(0.019)	1.094(0.021)	1.121(0.023)	1.070(0.019)	1.091(0.021)	1.101(0.017)	1.044(0.018)	1.074(0.020)	1.129(0.025)
	40	1.143(0.022)	1.059(0.015)	1.075(0.016)	1.096(0.018)	1.056(0.015)	1.072(0.016)	1.085(0.014)	1.035(0.014)	1.059(0.015)	1.102(0.018)
	50	1.118(0.016)	1.048(0.011)	1.061(0.012)	1.079(0.013)	1.046(0.011)	1.059(0.012)	1.074(0.011)	1.029(0.011)	1.048(0.012)	1.084(0.014)
	60	1.100(0.013)	1.041(0.010)	1.052(0.010)	1.067(0.011)	1.039(0.010)	1.050(0.010)	1.065(0.010)	1.025(0.009)	1.041(0.010)	1.071(0.011)
	70	1.085(0.011)	1.034(0.008)	1.044(0.009)	1.056(0.009)	1.032(0.008)	1.042(0.009)	1.057(0.008)	1.020(0.008)	1.034(0.008)	1.060(0.009)
	80	1.080(0.009)	1.034(0.007)	1.043(0.008)	1.054(0.008)	1.033(0.007)	1.042(0.008)	1.055(0.007)	1.021(0.007)	1.034(0.007)	1.057(0.008)
	90	1.068(0.008)	1.027(0.006)	1.035(0.007)	1.045(0.007)	1.026(0.006)	1.034(0.007)	1.047(0.006)	1.016(0.006)	1.027(0.006)	1.048(0.007)
	100	1.064(0.007)	1.027(0.006)	1.034(0.006)	1.043(0.006)	1.026(0.006)	1.033(0.006)	1.046(0.006)	1.017(0.006)	1.027(0.006)	1.046(0.006)
	110	1.055(0.006)	1.022(0.005)	1.028(0.005)	1.036(0.005)	1.021(0.005)	1.027(0.005)	1.039(0.005)	1.012(0.005)	1.022(0.005)	1.039(0.006)
	120	1.053(0.006)	1.022(0.005)	1.028(0.005)	1.036(0.005)	1.021(0.005)	1.027(0.005)	1.039(0.005)	1.014(0.005)	1.022(0.005)	1.038(0.005)
$α = 4$	10	1.348(5.558)	1.128(2.743)	1.179(3.256)	1.238(3.930)	1.124(2.724)	1.177(3.243)	0.841(0.779)	1.066(2.290)	1.134(2.810)	1.270(4.420)
	20	1.217(2.522)	1.079(1.467)	1.111(1.659)	1.148(1.909)	1.077(1.458)	1.109(1.651)	0.879(0.616)	1.040(1.302)	1.083(1.493)	1.167(2.086)
	30	1.159(1.535)	1.059(0.986)	1.082(1.084)	1.109(1.217)	1.057(0.982)	1.081(1.081)	0.902(0.507)	1.030(0.897)	1.062(1.000)	1.123(1.308)
	40	1.127(1.136)	1.048(0.787)	1.067(0.850)	1.088(0.934)	1.047(0.784)	1.066(0.849)	0.919(0.452)	1.026(0.730)	1.050(0.796)	1.099(0.993)
	50	1.105(0.832)	1.040(0.601)	1.055(0.642)	1.072(0.698)	1.039(0.599)	1.054(0.642)	0.929(0.380)	1.021(0.563)	1.042(0.606)	1.081(0.736)
	60	1.088(0.673)	1.033(0.508)	1.045(0.537)	1.060(0.577)	1.032(0.506)	1.045(0.536)	0.936(0.346)	1.017(0.480)	1.034(0.511)	1.068(0.603)
	70	1.077(0.562)	1.029(0.436)	1.040(0.459)	1.053(0.489)	1.028(0.435)	1.040(0.458)	0.943(0.311)	1.015(0.416)	1.030(0.440)	1.060(0.510)
	80	1.069(0.475)	1.027(0.376)	1.037(0.394)	1.048(0.418)	1.026(0.375)	1.036(0.393)	0.949(0.277)	1.015(0.359)	1.028(0.379)	1.054(0.433)
	90	1.061(0.410)	1.023(0.333)	1.031(0.346)	1.041(0.365)	1.022(0.332)	1.031(0.345)	0.952(0.255)	1.011(0.319)	1.023(0.334)	1.047(0.377)
	100	1.053(0.350)	1.018(0.290)	1.027(0.301)	1.036(0.315)	1.018(0.290)	1.026(0.300)	0.954(0.232)	1.009(0.280)	1.019(0.291)	1.040(0.325)
	110	1.049(0.320)	1.018(0.269)	1.025(0.278)	1.034(0.291)	1.017(0.268)	1.025(0.278)	0.958(0.218)	1.009(0.261)	1.019(0.270)	1.038(0.299)
	120	1.046(0.292)	1.017(0.247)	1.024(0.256)	1.032(0.267)	1.017(0.247)	1.023(0.255)	0.962(0.203)	1.009(0.240)	1.018(0.249)	1.035(0.273)
$β = 2$	10	1.377(1.616)	1.142(0.798)	1.194(0.942)	1.253(1.130)	1.138(0.793)	1.191(0.939)	0.877(0.198)	1.079(0.667)	1.148(0.817)	1.285(1.265)
	20	1.235(0.728)	1.088(0.423)	1.120(0.476)	1.157(0.545)	1.086(0.420)	1.118(0.474)	0.903(0.164)	1.049(0.375)	1.092(0.430)	1.177(0.593)
	30	1.174(0.446)	1.066(0.286)	1.090(0.313)	1.117(0.350)	1.064(0.285)	1.088(0.313)	0.921(0.138)	1.037(0.260)	1.069(0.290)	1.131(0.375)
	40	1.138(0.327)	1.053(0.226)	1.072(0.243)	1.093(0.266)	1.052(0.225)	1.071(0.243)	0.933(0.124)	1.030(0.210)	1.055(0.228)	1.104(0.282)
	50	1.115(0.240)	1.045(0.172)	1.060(0.184)	1.078(0.200)	1.044(0.172)	1.059(0.184)	0.942(0.104)	1.026(0.161)	1.047(0.174)	1.087(0.210)
	60	1.096(0.193)	1.037(0.145)	1.050(0.153)	1.065(0.164)	1.036(0.145)	1.049(0.153)	0.947(0.095)	1.021(0.137)	1.038(0.146)	1.072(0.171)
	70	1.083(0.160)	1.032(0.124)	1.043(0.130)	1.056(0.138)	1.031(0.123)	1.042(0.130)	0.952(0.086)	1.018(0.118)	1.033(0.125)	1.063(0.144)
	80	1.076(0.136)	1.030(0.108)	1.040(0.113)	1.051(0.119)	1.029(0.107)	1.039(0.112)	0.958(0.077)	1.018(0.103)	1.031(0.108)	1.057(0.123)
	90	1.067(0.116)	1.026(0.094)	1.035(0.098)	1.045(0.103)	1.025(0.094)	1.034(0.098)	0.961(0.070)	1.015(0.090)	1.027(0.094)	1.050(0.106)
	100	1.058(0.100)	1.021(0.083)	1.029(0.086)	1.038(0.090)	1.020(0.083)	1.028(0.086)	0.961(0.064)	1.011(0.080)	1.022(0.083)	1.043(0.092)
	110	1.053(0.091)	1.020(0.076)	1.027(0.079)	1.035(0.082)	1.019(0.076)	1.027(0.079)	0.964(0.060)	1.011(0.074)	1.020(0.077)	1.040(0.084)
	120	1.051(0.085)	1.020(0.071)	1.026(0.074)	1.034(0.077)	1.019(0.071)	1.026(0.074)	0.968(0.057)	1.011(0.069)	1.020(0.072)	1.038(0.079)

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[1] Correspondence to: Pedro L. Ramos E-mail: pedro.ramos@mat.uc.cl