The heteroscedastic odd log-logistic generalized gamma regression - PowerPoint PPT Presentation

The heteroscedastic odd log-logistic generalized gamma regression model for censored data F´ abio Prataviera Edwin M. M. Ortega Gauss M. Cordeiro Altemir da Silva Braga VIII - Encontro dos Alunos da P´ os Gradua¸ c˜ ao em Estat´ ıstica e Experimenta¸ c˜ ao Agronˆ omica Piracicaba, november of 2018 (ESALQ-USP) OLLGG distribution Piracicaba, november of 2018 1 / 39

Structure Introduction 1 The odd log-logistic generalized gamma distribution 2 The log odd log-logistic generalized gamma distribution 3 The heteroscedastic LOLLGG regression model 4 Maximum Likelihood Estimation 5 Simulation study 6 Application 7 Concluding Remarks 8 Future research 9 10 References (ESALQ-USP) OLLGG distribution Piracicaba, november of 2018 2 / 39

Odd log-logistic family Following the same idea by Gleaton and Lynch (2006) - odd log-logistic- G (”OLL- G ”) family � G ( t ; ξ ) λ x λ − 1 G ( t ; ξ ) λ ¯ G ( t ; ξ ) F ( t ) = (1 + x λ ) 2 dx = G ( t ; ξ ) λ . (1) G ( t ; ξ ) λ + ¯ 0 We can write � � F ( t ) log ¯ F ( t ) ¯ � � λ = e G ( t ; ξ ) = 1 − G ( t ; ξ ) . G ( t ) log ¯ G ( t ) f ( t ) = λ g ( t ; ξ ) { G ( t ; ξ )[1 − G ( t ; ξ )] } λ − 1 . (2) � G ( t ; ξ ) λ + [1 − G ( t ; ξ )] λ � 2 (ESALQ-USP) OLLGG distribution Piracicaba, november of 2018 3 / 39

Generalized gamma distribution Stacy and Mihram (1965) The generalized gamma (GG) model has probability density function (pdf) and cumulative distribution function (cdf) given by � � τ � � t � τ k − 1 � t | τ | g ( t ; α, τ, k ) = exp − , α Γ( k ) α α and  � � t � τ � τ ) γ ( k , ( t α )   γ 1 k , = , if τ > 0 ,  α Γ( k ) G ( t ; α, τ, k ) = (3)  � � t � τ �  τ ) γ ( k , ( t α )  1 − γ 1 k , = 1 − , if τ < 0 , α Γ( k ) respectively, where τ � = 0 and k > 0 are the shape parameters, α > 0 is the scale parameter. (ESALQ-USP) OLLGG distribution Piracicaba, november of 2018 4 / 39

Odd log-logistic generalized gamma distribution Density function Further, the odd log-logistic generalized gamma (OLLGG) pdf (for t > 0), say f ( t ) = f ( t ; α, τ, k , λ ), becomes f ( t ) = λ | τ | ( t /α ) τ k − 1 exp[ − ( t /α ) τ ] { γ 1 ( k , ( t /α ) τ )[1 − γ 1 ( k , ( t /α ) τ )] } λ − 1 , (4) α Γ( k ) { γ λ 1 ( k , ( t /α ) τ ) + [1 − γ 1 ( k , ( t /α ) τ )] λ } 2 where τ is not zero and the other parameters are positive. Hazard function The hrf of T can be constant, decreasing, increasing, upside-down bathtub (unimodal), bathtub and bimodal shaped. It is given by  τ ) λ τ ( t /α ) τ k − 1 exp[ − ( t /α ) τ ] γ λ − 1 ( k , ( t α )  1 τ )] , if τ > 0 ,   τ ) + [ 1 − γ 1 ( k , ( t τ )] λ �  � 1 ( k , ( t α ) α ) [ 1 − γ 1 ( k , ( t α ) γ λ α Γ( k )  h ( t ) = (5)   τ )] λ − 1 λ ( − τ ) ( t /α ) τ k − 1 exp[ − ( t /α ) τ ] [ 1 − γ 1 ( k , ( t α )   τ ) , if τ < 0 .  τ ) + [ 1 − γ 1 ( k , ( t τ )] λ � � 1 ( k , ( t α ) α ) γ 1 ( k , ( t α ) γ λ α Γ( k ) (ESALQ-USP) OLLGG distribution Piracicaba, november of 2018 5 / 39

Plots of the OLLGG density function (a) (b) (c) 2.0 1.5 2.0 τ =−4.00;k=0.45; λ =0.15 τ =4.00;k=0.45; λ =0.15 τ =−8.00;k=30.00 τ =−4.50;k=0.40; λ =0.20 τ =4.50;k=0.40; λ =0.20 τ =−5.00;k=10.00 τ =−5.45;k=0.35; λ =0.25 τ =5.45;k=0.35; λ =0.25 τ =3.00;k=25.00 τ =−5.50;k=0.30; λ =0.30 τ =5.50;k=0.30; λ =0.30 τ =4.00;k=30.00 1.5 1.5 τ =−6.00;k=0.25; λ =0.35 τ =6.00;k=0.25; λ =0.35 τ =6.00;k=30.00 1.0 f(t) 1.0 f(t) f(t) 1.0 0.5 0.5 0.5 0.0 0.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 2.5 2 3 4 5 6 7 8 t t t Figura: Plots of the OLLGG density function for some parameter values. (a) For some values of τ < 0 with α = 1 fixed. (b) For some values of τ > 0 with α = 1 fixed. (c) For some values of τ < 0 and τ > 0 with α = 2 and λ = 0 . 15 fixed. (ESALQ-USP) OLLGG distribution Piracicaba, november of 2018 6 / 39

Plots of the OLLGG density function (a) (b) (c) 1.5 3.0 5 τ =−10.00;k=20.00 τ =−8.00;k=30.00 τ =−8.50;k=25.00 τ =−5.00;k=10.00 τ =−7.45;k=30.00 τ =3.00;k=25.00 2.5 τ =−6.50;k=35.00 τ =4.00;k=30.00 4 τ =−6.00;k=35.00 τ =6.00;k=30.00 1.0 2.0 3 h(t) h(t) h(t) 1.5 α =0.60; τ =−2.00;k=1.20; λ =0.30 2 α =0.35; τ =0.70;k=2.30; λ =1.10 0.5 1.0 α =1.55; τ =1.25;k=0.55; λ =1.00 α =1.00; τ =1.00;k=1.00; λ =1.00 α =1.50; τ =2.00;k=2.00; λ =1.35 1 0.5 0.0 0.0 0 0 1 2 3 4 5 6 1.4 1.6 1.8 2.0 2.2 2.4 2 3 4 5 6 7 8 t t t Figura: Plots of the OLLGG hrf for some parameter values. (a) For some values of τ < 0 and τ > 0, an k and α fixed. (b) For some values of τ < 0, α = 1, λ = 0 . 15 and k fixed. (c) For some values of τ < 0 and τ > 0, α = 2, λ = 0 . 15 and k fixed. (ESALQ-USP) OLLGG distribution Piracicaba, november of 2018 7 / 39

New models Tabela: Some OLL-G sub-models for τ > 0. Distribution α τ k λ OLL-Gamma α 1 λ k OLL-Weibull α τ 1 λ OLL-Exponential α 1 1 λ n OLL-Chi-square 2 1 λ √ 2 n OLL-Chi 2 2 λ √ 2 n OLL-Scaled Chi 2 σ 2 λ 2 OLL-Rayleigh α 2 1 λ 3 OLL-Maxwell α 2 λ √ 2 1 OLL-Folded normal 2 2 λ √ 2 OLL-Reciprocal Circular normal 2 2 1 λ √ 3 OLL-Spherical normal 2 2 λ 2 1 2 γ θ 1 OLL-Generalized half-normal 2 2 γ λ 2 1 2 θ 1 OLL-Half-normal 2 2 λ 2 (ESALQ-USP) OLLGG distribution Piracicaba, november of 2018 8 / 39

New models Tabela: Some OLL-G sub-models for τ < 0. Distribution α τ k λ OLL-Reciprocal Gamma α -1 λ k OLL-Reciprocal Weibull α - τ 1 λ OLL-Reciprocal Exponential α -1 1 λ n OLL-Reciprocal Chi-square 2 -1 λ √ 2 n OLL-Reciprocal Chi 2 -2 λ √ 2 n OLL-Reciprocal Scaled Chi 2 σ -2 λ 2 OLL-Reciprocal Rayleigh α -2 1 λ 3 OLL-Reciprocal Maxwell α -2 λ √ 2 1 OLL-Reciprocal Folded normal 2 -2 λ √ 2 OLL-Reciprocal Circular normal 2 -2 1 λ √ 3 OLL-Reciprocal Spherical normal 2 -2 λ 2 1 2 γ θ 1 OLL-Reciprocal Generalized half-normal 2 -2 γ λ 2 1 2 θ 1 OLL-Reciprocal Half-normal 2 -2 λ 2 (ESALQ-USP) OLLGG distribution Piracicaba, november of 2018 9 / 39

Useful expansions for OLLGG distribution First, we define the exponentiated-generalized gamma (“Exp-GG”) distribution, say W ∼ Exp c [ G ( α, τ, k )] with power parameter c > 0, if W has cdf and pdf given by � � τ � � t � τ k − 1 � t c | τ | H c ( t ) = G ( t ; α, τ, k ) c and h c ( t ) = G ( t ; α, τ, k ) c − 1 , exp − α Γ( k ) α α respectively. Second, we obtain an expansion for F ( t ) in (2) using a power series for G ( t ; α, τ, k ) λ ( λ > 0 real) � ∞ G ( t ; α, τ, k ) λ = a j G ( t ; α, τ, k ) j , (6) j =0 where � λ � � u � � ∞ ( − 1) u + j a j = a j ( λ ) = . u j j = u (ESALQ-USP) OLLGG distribution Piracicaba, november of 2018 10 / 39

Useful expansions for OLLGG distribution For any real λ > 0, we use generalized binomial expansion to obtain � λ � � ∞ [1 − G ( t ; α, τ, k )] λ = ( − 1) j G ( t ; α, τ, k ) j . (7) j j =0 Inserting (6) and (7) in equation for cdf gives the following expressions  � ∞ j =0 a j G ( t ; α,τ, k ) j   j =0 b j G ( t ; α,τ, k ) j , if τ > 0 , � ∞ F ( t ) = � ∞ j =0 a ∗ j G ( t ; α,τ, k ) j   j =0 b j G ( t ; α,τ, k ) j , if τ < 0 , � ∞ j = ( − 1) j � λ � and b j = a j + ( − 1) j � λ � where a ∗ for j ≥ 0. j j (ESALQ-USP) OLLGG distribution Piracicaba, november of 2018 11 / 39

Useful expansions for OLLGG distribution Thus, for τ < 0, the ratio of the two power series can be expressed as � ∞ c j G ( t ; α, τ, k ) j , F ( t ) = (8) j =0 where c 0 = a ∗ 0 / b 0 and the coefficients c j ’s (for j ≥ 1) are determined from the recurrence equation � � j � c j = b − 1 a ∗ b r a ∗ j − . 0 j − r r =1 By differentiating (8), the pdf of T follows as � ∞ f ( t ) = c j +1 h j +1 ( t ) , (9) j =0 where h j +1 ( t ) (for j ≥ 0) is the Exp-GG density function with power parameter j + 1 given by � t � τ k − 1 � � t � τ � h j +1 ( t ) = ( j + 1) | τ | γ 1 ( k , ( t /α ) τ ) j . exp − (10) α Γ( k ) α α (ESALQ-USP) OLLGG distribution Piracicaba, november of 2018 12 / 39

The LOLLGG distribution If the random variable T follows the OLLGG density function (4), we √ k ) − 1 and define Y = log( T ). Setting k = q − 2 , τ = ( σ � � µ − τ − 1 log( k ) α = exp , the density function of Y can be expressed as ( y ∈ R ) λ | q | ( q − 2 ) q − 2 � q − 1 � y − µ � � � y − µ ��� − q − 2 exp f ( y ) = exp × q σ Γ( q − 2 ) σ σ � � � � y − µ ��� � � � � y − µ ����� λ − 1 q − 2 , q − 2 exp q − 2 , q − 2 exp γ 1 1 − γ 1 × q q σ σ � ���� λ � − 2 � � � y − µ ��� � � � � y − µ γ λ q − 2 , q − 2 exp q − 2 , q − 2 exp q + 1 − γ 1 q , 1 σ σ where µ ∈ R , σ > 0, λ > 0 and q is different from zero. We consider an extended form including the case q = 0 (Lawless, 2003). (ESALQ-USP) OLLGG distribution Piracicaba, november of 2018 13 / 39