Parameter induction in continuous univariate distributions: Well-established G families

Tahir, Muhammad H.; Nadarajah, Saralees

doi:10.1590/0001-3765201520140299

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Mathematical Sciences • An. Acad. Bras. Ciênc. 87 (2) • Apr-Jun 2015 • https://doi.org/10.1590/0001-3765201520140299 linkcopy

Parameter induction in continuous univariate distributions: Well-established G families

Authorship SCIMAGO INSTITUTIONS RANKINGS

Abstract

The art of parameter(s) induction to the baseline distribution has received a great deal of attention in recent years. The induction of one or more additional shape parameter(s) to the baseline distribution makes the distribution more flexible especially for studying the tail properties. This parameter(s) induction also proved helpful in improving the goodness-of-fit of the proposed generalized family of distributions. There exist many generalized (or generated) G families of continuous univariate distributions since 1985. In this paper, the well-established and widely-accepted G families of distributions like the exponentiated family, Marshall-Olkin extended family, beta-generated family, McDonald-generalized family, Kumaraswamy-generalized family and exponentiated generalized family are discussed. We provide lists of contributed literature on these well-established G families of distributions. Some extended forms of the Marshall-Olkin extended family and Kumaraswamy-generalized family of distributions are proposed.

Beta-distribution; exponentiated family; Kumaraswamy distribution; Marshall-Olkin family; McDonald distribution; reliability properties

S.No.	Pioneer year	Distribution	Author(s)
1	1967	Exponentiated exponential distribution	Ahuja and Nash (1967)
			Gupta et al. (1998)
			Gupta and Kundu (1999, 2001a, b, 2007)
			Nadarajah (2011)
			Venkatesan and Sundaram (2011)
2	1993	Exponential Weibull distribution	Mudholkar and Srivastava (1993)
			Mudholkar et al. (1995)
			Mudholkar and Hutson (1996)
			Gupta et al. (1998)
			Jiang and Murthy (1999)
			Nassar and Eissa (2003)
			Choudhury (2005)
			Nadarajah and Gupta (2005)
			Singh et al. (2005)
			Pal et al. (2006)
			Ahmed et al. (2008)
			Saleem and Abo-Kasem (2011)
			Mazucheli et al. (2012)
			Qian (2012)
			Barrios and Dios (2012)
			Nadarajah et al. (2013b)
3	1998	Exponentiated gamma distribution	Gupta et al. (1998)
			Nadarajah and Kotz (2006a)
			Nadarajah and Gupta (2007)
			Shawky and Bakoban (2008, 2009, 2012)
4	1998	Exponentiated Pareto distribution	Gupta et al. (1998)
			Nadarajah (2005a)
			Shawky and Abu-Zinadah (2009)
			Afify (2010)
5	2001	Exponentiated Rayleigh distribution	Surles and Padgett (1998, 2001, 2005)
			Raqab (1998)
			Kundu and Raqab (2005)
			Raqab and Kundu (2006)
			Raqab and Madi (2009, 2011)
			Abd-Elfattah (2011)
6	2003	Exponentiated Fréchet distribution	Nadarajah and Kotz (2003)
			Nadarajah and Kotz (2006a)
			Abd-Elfattah and Omima (2009)
			Abd-Elfattah et al. (2010)
			Jamjoom and Al-Saiary (2012)
			Al-Nasser and Al-Omari (2013)
			Marwa et al. (2013)
7	2004	Exponentiated generalized Pareto distribution	Adeyemi and Adebanji (2004)
8	2005	Exponentiated beta distribution	Nadarajah (2005b)
9	2006	Exponentiated generalized extreme value distribution	Adeyemi and Adebanji (2006)
10	2006	Exponentiated log-logistic distribution	Rosaiah et al. (2006)
			Aslam and Jun (2010)
			Rao et al. (2012, 2013)
11	2006	Exponentiated Gumbel distribution	Nadarajah (2006)
			Nadarajah and Kotz (2006a)
			Shirke and Kakade (2007)
			Kakade et al. (2008)
			Persson and Rydén (2010)
12	2006	Exponentiated log-normal distribution	Shirke and Kakade (2006)
			Raja and Mir (2011)
13	2008	Exponential modified Weibull distribution	Carrasco et al. (2008)
			Elbatal (2011)
14	2009	Exponentiated extreme value distribution	Cho et al. (2009)
15	2011	Exponentiated Lindley distribution	Nadarajah et al. (2011)
16	2011	Extended generalized exponential distribution	Kundu and Gupta (2011)
17	2011	Exponentiated Burr XII distribution	Al-Hussaini and Hussein (2011a, b)
			Maswadah (2013)
18	2011	Exponentiated generalized gamma distribution	Cordeiro et al. (2011a)
19	2011	Exponentiated generalized inverse Gaussian distribution	Lemonte and Cordeiro (2011)
20	2012	Exponentiated inverted Weibull distribution	Flaih et al. (2012)
			Kim et al. (2012)
			Hassan (2013)
			Aljuaid (2013)
21	2012	Exponentiated Kumaraswamy distribution	Kumar (2012)
22	2012	Exponentiated Lomax distribution	Abdul-Moniem and Abdel-Hameed (2012)
23	2012	Exponentiated Gompertz distribution	El-Gohary (2012)
24	2013	Exponentiated modified Weibull extension distribution	Sarhan and Apaloo (2013)
25	2013	Exponentiated generalized linear exponential distribution	Sarhan et al. (2013)
26	2013	Exponentiated Dagum distribution	Khan (2013)
27	2013	Exponentiated sinh Cauchy distribution	Cooray (2013)
28	2015	Exponentiated geometric distribution	Chakraborty and Gupta (2015)

S.No.	Pioneer year	Distribution	Author(s)
1	1997	MOE exponential distribution	Marshall and Olkin (1997)
			Alice and Jose (2004b)
			Parikh et al. (2008)
			Salah et al. (2009)
			Bdair (2011)
			Gopal and Damodaran (2011)
			Krishna (2011)
			Rao et al. (2011)
			Salah (2012)
			Pushkarna et al. (2013)
2	1997	MOE Weibull distribution	Marshall and Olkin (1997)
			Jose and Alice (2001)
			Hirose (2002)
			Ghitany et al. (2005)
			Zhang and Xie (2007)
			Caroni (2010)
			Srivastava and Kumar (2011)
			Athar et al. (2012)
			Cordeiro and Lemonte (2013)
3	2003	MOE Pareto-III distribution	Alice and Jose (2003)
4	2003	MOE Pareto-III distribution	Alice and Jose (2003)
5	2004	MOE Pareto-I distribution	Alice and Jose (2004a)
			Ghitany (2005)
6	2005	MOE logistic distribution	Alice and Jose (2005a)
			Kumar (2013)
7	2005	MO semi-logistic distribution	Alice and Jose (2005a)
8	2005	MO semi-Weibull distribution	Alice and Jose (2005b)
			Thomas and Jose (2005)
9	2005	MOE Fréchet distribution	Jose and Alice (2005)
			Krishna (2011)
			Krishna et al. (2013)
10	2007	MOE Lomax distribution	Ghitany et al. (2007)
			Gupta et al. (2010)
11	2007	MOE linear failure rate distribution	Ghitany and Kotz (2007)
12	2007	MOE gamma distribution	Ristić et al. (2007)
			Jose (2009)
13	2008	MOE q-Weibull distribution	Naik et al. (2008)
			Jose et al. (2010)
14	2008	MOE Burr distribution	Jayakumar and Mathew (2008)
			El-Bassiouny and Abdo (2010)
15	2008	MOE semi-Burr distribution†	Jayakumar and Mathew (2008)
16	2008	MOE semi-Pareto III distribution†	Jayakumar and Mathew (2008)
17	2009	MOE Linnik distribution†	Jose and Uma (2009)
18	2009	MOE Mittag-Leffler distribution†	Jose and Uma (2009)
19	2009	MOE beta distribution	Jose et al. (2009)
20	2011	MOE uniform distribution	Krishna (2011)
			Jose and Krishna (2011)
21	2011	MOE Gumbel distribution	Jose (2011)
22	2011	MOE generalized asymmetric Laplace distribution†	Krishna (2011)
			Krishna and Jose (2011)
24	2013	MOE Zipf distribution	P´erez-Casany and Casellas (2013)
25	2013	MOE power log-normal distribution	Gui (2013a)
26	2013	MOE log-logistic distribution	Gui (2013b)
27	2013	MOE quasi-Lindley distribution	Gui (2013c)
28	2013	MOE Esscher transformed Laplace distribution†	George and George (2013)

S.No.	Pioneer year	Distribution	Author(s)
1	2002	Beta normal distribution	Eugene et al. (2002)
			Famoye et al. (2002)
			Eugene (2004)
			Famoye et al. (2004)
			Gupta and Nadarajah (2004)
			Jones (2004b)
			Rˇego et al. (2012)
2	2003	Beta exponential distribution	Maynard (2003)
			Nadarajah and Kotz (2006b)
3	2004	Beta gamma distribution	Kong (2004), Kong et al. (2007)
4	2004	Beta Gumbel distribution	Nadarajah and Kotz (2004)
5	2004	Beta Fréchet distribution	Nadarajah and Gupta (2004)
			Barreto-Souza et al. (2011)
6	2005	Beta Weibull distribution	Famoye et al. (2005)
			Lee et al. (2007)
			Zografos (2008)
			Cordeiro et al. (2011b, c)
			Sun (2011), Mdziniso (2012)
			Mahmoud and Mandouh (2012a, b, c)
7	2006	Beta Bessel distribution	Gupta and Nadarajah (2006)
8	2008	Beta Pareto distribution	Akinsete et al. (2008)
9	2008	Beta Rayleigh distribution	Akinsete and Lowe (2009)
10	2008	Beta Laplace distribution	Kozubowski and Nadarajah (2008)
			Cordeiro and Lemonte (2011a)
11	2009	Beta generalized logistic-IV distribution	Morais (2009)
			Morais et al. (2013)
12	2010	Beta modified Weibull distribution	Silva et al. (2010)
			Nadarajah et al. (2012b)
13	2010	Beta generalized half-normal distribution	Pescim et al. (2010)
14	2010	Beta generalized exponential distribution	Barreto-Souza et al. (2010)
15	2010	Beta Maxwell distribution	Amusan (2010)
16	2010	Beta hyperbolic secant distribution	Fischer and Vaughan (2010)
17	2010	Beta inverse Weibull distribution	Kersey (2010)
			Hanook et al. (2013)
18	2011	Beta Cauchy distribution	Alshawarbeh (2011)
			Alshawarbeh et al. (2012)
19	2011	Beta half-Cauchy distribution	Cordeiro and Lemonte (2011b)
20	2011	Beta Burr XII distribution	Parana´ıba et al. (2011)
21	2011	Beta generalized Pareto distribution	Mahmoudi (2011)
			Nassar and Nada (2011)
22	2011	Beta Birnbaum-Sanders distribution	Cordeiro and Lemonte (2011c)
23	2012	Beta skew-normal distribution	Mameli (2012)
24	2012	Beta exponential-geometric distribution	Bidram (2012)
25	2012	Beta Moyal distribution	Cordeiro et al. (2012d)
26	2012	Beta generalized Weibull distribution	Singla et al. (2012)
27	2012	Beta exponentiated Pareto distribution	Zea et al. (2012)
28	2012	Beta power distribution	Cordeiro and Brito (2012)
29	2012	Beta linear failure rate distribution	Jafari and Mahmoudi (2012)
30	2012	Beta extended Weibull distribution	Cordeiro et al. (2012f)
31	2012	Beta truncated Pareto distribution	Lourenzutti et al. (2012)
32	2013	Beta Weibull-geometric distribution	Cordeiro et al. (2013f)
			Bidram et al. (2013)
33	2013	Beta generalized gamma distribution	Cordeiro et al. (2013a)
34	2013	Beta log-normal distribution	Castellars et al. (2013)
35	2013	Beta generalized Rayleigh distribution	Cordeiro et al. (2013b)
36	2013	Beta generalized logistic distribution	Morais et al. (2013)
37	2013	Beta exponentiated Weibull distribution	Cordeiro et al. (2013c)
38	2013	Beta Nakagami distribution	Shittu and Adepoju (2013)
39	2013	Beta Burr III distribution	Gomes et al. (2013)
40	2013	Beta Dagum distribution	Domma and Condino (2013)
41	2013	Beta Stoppa distribution	Mansoor (2013)
42	2013	Beta inverse Rayleigh distribution	Le˜ao et al. (2013)
43	2014	Beta generalized inverse Weibull distribution	Baharith et al. (2014)
44	2014	Beta extended half-normal distribution	Cordeiro et al. (2014f)
45	2014	Beta log-logistic distribution	Lemonte (2014)

S.No.	Pioneer year	Distribution	Author(s)
1	2010	Kumaraswamy Weibull distribution	Cordeiro et al. (2010)
2	2011	Kumaraswamy generalized gamma distribution	de Pascoa et al. (2011)
3	2011	Kumaraswamy skew-normal distribution	Kazemi et al. (2011)
			Mameli (2012)
			Mameli and Musio (2013)
4	2011	Kumaraswamy Gumbel minimum distribution	El-Sherpieny and Ahmed (2011)
5	2012	Kumaraswamy log-logistic distribution	de Santana et al. (2012)
			Muthulakshmi and Selvi (2013)
6	2012	Kumaraswamy Gumbel distribution	Cordeiro et al. (2012c)
7	2012	Kumaraswamy Birnbaum-Sanders distribution	Saulo et al. (2012)
8	2012	Kumaraswamy generalized half-normal distribution	Cordeiro et al. (2012e)
9	2012	Kumaraswamy inverse Weibull distribution	Shahbaz et al. (2012)
10	2012	Kumaraswamy normal distribution	Correa et al. (2012)
11	2012	Kumaraswamy generalized inverse Weibull distribution	Yang (2012)
12	2013	Kumaraswamy Pareto distribution	Bourguignion et al. (2013)
13	2013	Kumaraswamy generalized Pareto distribution	Nadarajah and Eljabri (2013)
14	2013	Kumaraswamy Burr XII distribution	Parana´ıba et al. (2013)
15	2013	Kumaraswamy generalized extreme value distribution	Eljabri (2013)
16	2013	Kumaraswamy linear exponential distribution	Elbatal (2013a)
17	2013	Kumaraswamy generalized linear failure rate distribution	Elbatal (2013b)
18	2013	Kumaraswamy exponentiated Pareto distribution	Elbatal (2013c)
19	2013	Kumaraswamy Lomax distribution	Shams (2013)
20	2013	Exponentiated Fisk (log-logistic) distribution	Zubair (2013)
21	2014	Exponentiated generalized Burr III distribution	Zubair (2013)
22	2014	Kumaraswamy modified Weibull distribution	Cordeiro et al. (2014d)
23	2014	Kumaraswamy generalized Rayleigh distribution	Gomes et al. (2014)

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[TFN02] † MO extension based on cf.

S.No.	Pioneer year	Distribution	Author(s)
1	2010	McDonald Kumaraswamy distribution	Carrasco et al. (2010)
2	2012	McDonald exponential distribution	Cordeiro et al. (2012f)
3	2012	McDonald gamma distribution	Marciano et al. (2012)
4	2012	McDonald inverted beta distribution	Cordeiro and Lemonte (2012)
5	2012	McDonald normal distribution	Cordeiro et al. (2012a)
6	2012	McDonald extended exponential distribution	Cordeiro et al. (2012b)
7	2013	McDonald Burr XII distribution	Gomes et al. (2015)
8	2013	McDonald Burr III distribution	Gomes et al. (2015)
9	2013	McDonald Lomax distribution	Lemonte and Cordeiro (2013)
10	2013	McDonald Birnbaum-Sanders distribution	Cordeiro et al. (2013d)
11	2013	McDonald Fisk (log-logistic) distribution	Zubair (2013)
12	2013	McDonald Dagum distribution	Rajasooriya (2013)
			Oluyede and Rajasooriya (2013)
13	2013	McDonald modified Weibull distribution	Merovci and Elbatal (2013)
14	2014	McDonald arcsine distribution	Cordeiro and Lemonte (2014)
15	2014	McDonald Weibull distribution	Cordeiro et al. (2014b)
16	2015	McDonlad Burr distribution	Cordeiro et al. (In press)