Bayesian Nonlinear Regression Models with Scale ... - Semantic Scholar
Bayesian Nonlinear Regression Models with Scale Mixtures of Skew Normal Distributions: Estimation and Case Influence Diagnostics Vicente G. Cancho (ICMC-USP), Victor H. Lachos (IMECC-UNICAMP) and Marinho Andrade (ICMC-USP)
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of Skew mar¸ (ICMC-USP) coNormal - 2010Distributions: 1 / 35
Motivation
Motivation The routine use of the normality in nonlinear models (N-NLM) has been recently questioned by many authors (see Cysneiros & Vanegas, 2008; Cordeiro et al., 2009, among others).
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of Skew mar¸ (ICMC-USP) coNormal - 2010Distributions: 2 / 35
Motivation
Motivation The routine use of the normality in nonlinear models (N-NLM) has been recently questioned by many authors (see Cysneiros & Vanegas, 2008; Cordeiro et al., 2009, among others). Thus, it is of practical interest to develop statistical model with considerable flexibility in the distributional assumptions of the random error term in NLM.
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of Skew mar¸ (ICMC-USP) coNormal - 2010Distributions: 2 / 35
Motivation
Motivation The routine use of the normality in nonlinear models (N-NLM) has been recently questioned by many authors (see Cysneiros & Vanegas, 2008; Cordeiro et al., 2009, among others). Thus, it is of practical interest to develop statistical model with considerable flexibility in the distributional assumptions of the random error term in NLM. Cancho et al. (2009) and Xie et al. (2009b,a) has shown the advantage of using the skew-normal distribution in the context of nonlinear regression models (SN-NLM).
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of Skew mar¸ (ICMC-USP) coNormal - 2010Distributions: 2 / 35
Motivation
Motivation The routine use of the normality in nonlinear models (N-NLM) has been recently questioned by many authors (see Cysneiros & Vanegas, 2008; Cordeiro et al., 2009, among others). Thus, it is of practical interest to develop statistical model with considerable flexibility in the distributional assumptions of the random error term in NLM. Cancho et al. (2009) and Xie et al. (2009b,a) has shown the advantage of using the skew-normal distribution in the context of nonlinear regression models (SN-NLM). We extend the SN-NLM by assuming that the model errors follow scale mixtures of skew-normal distributions (SMSN, Branco and Dey, 2001). (skew-normal,skew-t, skew-slash, skew-contaminated normal). Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of Skew mar¸ (ICMC-USP) coNormal - 2010Distributions: 2 / 35
SMSN Distributions
The Skew-Normal distribution
Z ∼ SN(µ, σ 2 , λ) , with pdf 2
f (z) = 2φ(z; µ, σ )Φ
λ(z − µ) σ
, (Azzalini, 1985)
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of Skew mar¸ (ICMC-USP) coNormal - 2010Distributions: 3 / 35
SMSN Distributions
The Skew-Normal distribution
Z ∼ SN(µ, σ 2 , λ) , with pdf 2
f (z) = 2φ(z; µ, σ )Φ
λ(z − µ) σ
, (Azzalini, 1985)
Marginal stochastic representation: Z = µ + ∆|T0 | + Γ1/2 T1 , where ∆ = σδ, δ =
√ λ , 1+λ2
Γ = (1 − δ 2 )σ 2 , T0 and T1 are
independent standard normal random variables.
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of Skew mar¸ (ICMC-USP) coNormal - 2010Distributions: 3 / 35
SMSN Distributions
Scale Mixtures of Skew-Normal distributions(SMSN) Y = µ + κ1/2 (U)Z ,
Z ∼ SN(0, σ 2 , λ) U is a positive random variable with cdf H(·; ν) (pdf h(·; ν)) κ(.) is weight function, this paper we restrict κ(u) = 1/u The pdf of Y is given by: Z f (y ) = 2
∞
φ(y ; µ, u
−1 2
σ )Φ
0
u 1/2 λ(y − µ) σ
! dH(u; ν),
Notation: Y ∼ SMSN(µ, σ 2 , λ; H) Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of Skew mar¸ (ICMC-USP) coNormal - 2010Distributions: 4 / 35
SMSN Distributions
Examples of SMSN distributions The skew–t: Y ∼ ST (µ, σ 2 , λ; ν), U ∼ Gamma(ν/2, ν/2). ! r ν+1 ) Γ( ν+1 d − 2 v +1 2 f (y ) = ν √ 1+ A; ν + 1 , y ∈ R, T ν d +ν Γ( 2 ) πνσ
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of Skew mar¸ (ICMC-USP) coNormal - 2010Distributions: 5 / 35
SMSN Distributions
Examples of SMSN distributions The skew–t: Y ∼ ST (µ, σ 2 , λ; ν), U ∼ Gamma(ν/2, ν/2). ! r ν+1 ) Γ( ν+1 d − 2 v +1 2 f (y ) = ν √ 1+ A; ν + 1 , y ∈ R, T ν d +ν Γ( 2 ) πνσ The skew–slash:Y ∼ SSL(µ, σ 2 , λ; ν),U ∼ Beta(ν, 1). Z f (y ) = 2ν
1
u ν−1 φ(y ; µ, u −1 σ 2 )Φ(u 1/2 A)du, y ∈ R,
0
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of Skew mar¸ (ICMC-USP) coNormal - 2010Distributions: 5 / 35
SMSN Distributions
Examples of SMSN distributions The skew–t: Y ∼ ST (µ, σ 2 , λ; ν), U ∼ Gamma(ν/2, ν/2). ! r ν+1 ) Γ( ν+1 d − 2 v +1 2 f (y ) = ν √ 1+ A; ν + 1 , y ∈ R, T ν d +ν Γ( 2 ) πνσ The skew–slash:Y ∼ SSL(µ, σ 2 , λ; ν),U ∼ Beta(ν, 1). Z f (y ) = 2ν
1
u ν−1 φ(y ; µ, u −1 σ 2 )Φ(u 1/2 A)du, y ∈ R,
0
Applications of the skew–slash distribution can be found in Wang & Genton (2006).
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of Skew mar¸ (ICMC-USP) coNormal - 2010Distributions: 5 / 35
SMSN Distributions
Examples of SMSN distributions
The skew contaminated normal:Y ∼ SCN(µ, σ 2 , λ; ν, γ). The pdf of U is given by h(u|ν) = νI(u=γ) + (1 − ν)I(u=1) , 0 < ν < 1, 0 < γ ≤ 1,
f (y ) = 2{νφ(y ; µ, γ −1 σ 2 )Φ(γ 1/2 A) + (1 − ν)φ(y ; µ, σ 2 )Φ(A)}.
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of Skew mar¸ (ICMC-USP) coNormal - 2010Distributions: 6 / 35
The SMSN nonlinear regression model
The SMSN nonlinear regression model
Yi = η(β, xi ) + εi , i = 1, . . . , n,
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of Skew mar¸ (ICMC-USP) coNormal - 2010Distributions: 7 / 35
The SMSN nonlinear regression model
The SMSN nonlinear regression model
Yi = η(β, xi ) + εi , i = 1, . . . , n,
η(.) is injective and twice continuously differentiable,
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of Skew mar¸ (ICMC-USP) coNormal - 2010Distributions: 7 / 35
The SMSN nonlinear regression model
The SMSN nonlinear regression model
Yi = η(β, xi ) + εi , i = 1, . . . , n,
η(.) is injective and twice continuously differentiable, q εi ∼ SMSN(− π2 k1 ∆, σ 2 , λ; H) where kr = E [U r |y ],
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of Skew mar¸ (ICMC-USP) coNormal - 2010Distributions: 7 / 35
The SMSN nonlinear regression model
The SMSN nonlinear regression model
Yi = η(β, xi ) + εi , i = 1, . . . , n,
η(.) is injective and twice continuously differentiable, q εi ∼ SMSN(− π2 k1 ∆, σ 2 , λ; H) where kr = E [U r |y ], q Yi ∼ SMSN(η(β, xi ) + b∆, σ 2 , λ; H), with b = − π2 k1 ,
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of Skew mar¸ (ICMC-USP) coNormal - 2010Distributions: 7 / 35
The SMSN nonlinear regression model
The SMSN nonlinear regression model
Yi = η(β, xi ) + εi , i = 1, . . . , n,
η(.) is injective and twice continuously differentiable, q εi ∼ SMSN(− π2 k1 ∆, σ 2 , λ; H) where kr = E [U r |y ], q Yi ∼ SMSN(η(β, xi ) + b∆, σ 2 , λ; H), with b = − π2 k1 , E [Yi ] = η( β, xi ), Var [Yi ] = k2 σ 2 − b 2 ∆2 .
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of Skew mar¸ (ICMC-USP) coNormal - 2010Distributions: 7 / 35
Bayesian inference
Bayesian inference
The hierarchical representation: Yi |Ti = ti ∼ N1 (η(β, xi ) + ∆ti , Ui−1 Γ) Ti |Ui ∼ TN1 (b, ui−1 )I (b, ∞) Ui ∼ H(.; ν), where i = 1, . . . , n Γ = (1 − δ 2 )σ 2 , and ∆ = σδ.
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of Skew mar¸ (ICMC-USP) coNormal - 2010Distributions: 8 / 35
Bayesian inference
Bayesian inference
Let y = (y1 , . . . , yn )> , x = (x1 , . . . , xn )> , t = (t1 , . . . , tn )> and u = (u1 , . . . , un )> . It follows that the complete likelihood function of θ associated with (y, x, t, u) is given by n Y Lc (θ|y, x, t, u) ∝ [φ1 (yi ; η(β, xi )+∆ti , ui−1 τ )φ1 (ti ; b, ui−1 )I(b,∞) h(ui |ν)]. i=1
(1)
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of Skew mar¸ (ICMC-USP) coNormal - 2010Distributions: 9 / 35
Bayesian inference
Prior distribution
βj = N(µβj , σβ2j ), j = 1, . . . , p, 2 ) and ∆ ∼ N1 (µ∆ , σ∆ ρ % τ −1 ∼ Gamma( , ). 2 2
ς For the skew-t model: ν ∼ exp( )I(2,∞) , 2 For the skew-slash model: ν ∼ Gamma(a, b) (b a), For the skew-contaminated normal model: ν ∼ U(0, 1) and γ Beta(a, b) (ν ⊥ γ).
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 10 / 35
Bayesian inference
The posterior distribution
The joint posterior density of all unobservable is is given by π(θ, t, u|y, x) ∝
Qn
i=1 [φ1 (yi ; η(β, xi )
+ ∆ti , ui−1 τ )
(2)
φ1 (ti ; b, ui−1 )I(b,∞) h(ui |ν)]π(θ). Distribution (2) is not tractable analytically but MCMC methods such as the Gibbs sampler, can be used to draw samples, from which features of marginal posterior distributions of interest can be inferred.
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 11 / 35
Bayesian inference
The full conditional distributions
Ti |β, ∆, τ, ν, y, u ∼ TN1 (µTi + b, ui−1 MT2 )I (b, ∞), i = 1, . . . , n; ∆ τ , µ Ti = 2 (yi − η(β, xi ) − ∆b), +τ ∆ +τ 2 2 +τ Bσ∆ + τ µ∆ Aσ∆ ∆|β, τ, ν, y, u, t ∼ N1 , , 2 +τ 2 Aσ∆ τ σ∆ P P where A = ni=1 ui ti and B = ni=1 ui ti (yi − η(β, xi )),
where MT2 =
∆2
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 12 / 35
Bayesian inference
The full conditional distributions
n
X 1 1 ui (yi − η(β, xi ) − ∆ti )2 )); τ −1 |β, ∆, ν, y, u, t ∼ Gamma( (ρ + n), (% + 2 2 i=1
β|∆, τ ν, y, u, t ∝ φp (β; µβ , D)
n Y
φ1 (η(β, xi ); (yi − ∆ti ), ui−1 τ ),
i=1
where µβ = (µβ1 , . . . , µβp )> , D = diagonal(σβ21 , . . . , σβ2p ).
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 13 / 35
Bayesian inference
The full conditional distributions
For each element of u, the density is:
1 ui (yi − η(β, xi ) − ∆ti )2 (3) 2τ 1 2 − ui (ti − b) h(ui |ν), 2
π(ui |β, ∆, τ, ν, y, x, t) ∝ ui exp −
for i = 1, . . . , n. For ν, the density is: π(ν|β, ∆, τ, y, u, t) ∝ π(ν)
n Y
h(ui |ν).
(4)
i=1
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 14 / 35
Bayesian inference
The full conditional distributions
(i) Skew-t. The density of the conditional posterior distribution in (3) takes the form: ui |θ(−ν) , y, t ∼ Gamma(2; ν/2 + Ci /2), where Ci = τ1 (yi − η(β, xi ) − ∆ti )2 + (ti − b)2 and the full conditional posterior density of ν is π(ν|θ(−ν) , y, t, u) ∝ π1 (ν) × Gamma(
1X nν − 1, (ui − log ui ))I(2,∞) , 2 2
where π1 (ν) = (2ν/2 Γ(ν/2))−n .
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 15 / 35
Bayesian inference
The full conditional distributions
• Skew-slash. In this case, the fully conditional posterior density of each ui is: ui |θ(−ν) , y, t ∼ Gamma((nu + 1)/2; Ci /2)I(0,1) . Further, the conditional posterior density of ν is ν|θ(−ν) , y, b, u, t ∼ Gamma(n + a, b −
X
log ui ).
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 16 / 35
Bayesian inference
The full conditional distributions • Skew-contaminated normal distribution The full conditional posterior density of the proportion of outliers ν is: P P ui − nγ n − ui ν|θ(−ν) , y, b, u, t ∼ Beta a + ;b + . 1−γ 1−γ The conditional posterior density of γ is: P P n − ui ui − nγ ) ) ( 1 − γ 1 −γ π(γ|θ(−γ) , y, b, u, t) ∝ ν × (1 − ν) . (
An interesting Metropolis–Hastings method to update from γ is described in Rosa et al. (2003). Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 17 / 35
Bayesian case influence diagnostics
Bayesian case influence diagnostics
Let K (P, P(−i) ) denote the Kullback-Leibler (K–L) divergence between P and P(−i) , where P denotes the posterior distribution of θ for full data, and P(−i) denotes the posterior distribution of θ without the ith case. Specifically, "
Z K (P, P(−i) ) =
π(θ|D) log
π(θ|D) π(θ|D (−i) )
# dθ.
(5)
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 18 / 35
Bayesian case influence diagnostics
Bayesian case influence diagnostics
As pointed by Peng & Dey (1995) and Cho et al. (2009), calibration of K (P, P(−i) ) can be done by solving for pi the equation K (P, P(−i) ) = K B(0.5), B(pi ) = − log 4pi (1 − pi ) /2, where B(p) denotes the Bernoulli distribution with success probability p. After some algebra it can be shown that q o 1n pi = 1 + 1 − exp − 2K (P, P(−i) ) . 2 This equation implies that 0.5 ≤ pi ≤ 1.
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 19 / 35
Bayesian case influence diagnostics
Bayesian case influence diagnostics
For our model in (7) it can be shown that (5) can be expressed as a posterior expectation K (P, P(−i) ) = log Eθ|D [g (yi |θ)]−1 + Eθ|D {log [g (yi |θ)]}
(6)
= − log(CPOi ) + Eθ|D {log [g (yi |θ)]} , where Eθ|D (·) denotes the expectation with respect to the joint posterior π(θ|D).
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 20 / 35
Bayesian case influence diagnostics
Bayesian case influence diagnostics
Thus (6) can be computed by sampling from the posterior distribution of θ via MCMC methods. Let θ1 , . . . , θQ be a sample of size Q of π(θ|D). Then, a Monte Carlo estimate of K (P, P(−i) ) is given by Q X \ [i) + 1 K (P, P(−i) ) = − log(CPO log [g (yi |θq )] . Q
(7)
q=1
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 21 / 35
Applications
Simulated data
Influence of outlying observations We consider the following logistic model: Yi =
β1 + εi , i = 1, · · · , 50, 1 + β2 exp(−β3 xi )
(8)
p where εi ∼ SN(− 2/π∆, σ 2 , λ), the variable xi ranging from 1 to 50 and we fixed the parameter values at: β1 = 30, β2 = 5, β3 = .7, σ 2 = 2 and λ = −3. We selected cases 4, 13 and 45 for perturbation. To create influential observation in the dataset, we choose one, two or tree of these selected cases and perturbed the response variable as follows y˜i = yi + 4Sy , i = 4, 13 and 45, where Sy is the standard deviations of the yi0 s. Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 22 / 35
Applications
Simulated data
Simulated data
1
The following independent priors were adopted in the Bayesian computations. βk ∼ log − N1 (0, 103 ), k = 1, 2, 3, ∆ ∼ N1 (0, 103 ), and 1/τ ∼ Gamma(1, 0.01).
2
we generated two parallel independent runs of the Gibbs sampler with size 60000 for each parameter, disregarding the first 10000 iterations to eliminate the effect of the initial values and, to avoid correlation problems, we considered a spacing of size 20, obtaining a sample of size 2500.
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 23 / 35
Applications
Simulated data
Influence of outlying observations
Table: Simulated SN data. Posterior means and standard deviations of the parameters from fitting a SN-NLM. Data
Perturbed
β1 = 30
β2 = 5
β3 = 0.7
σ2 = 2
λ = −3
names
case
Mean
SD
Mean
SD
Mean
SD
Mean
SD
Mean
SD
a
None
29.86
0.17
5.05
0.74
0.69
0.05
2.59
0.84
-2.81
1.67 1.18
b
4
30.24
0.35
10.56
6.93
0.99
0.21
10.65
3.07
1.14
c
13
30.48
0.32
5.91
2.74
0.73
0.12
11.04
2.55
3.60
1.11
d
45
30.51
0.32
6.13
3.11
0.74
0.12
11.20
2.67
3.59
1.06
e
{4,45}
30.93
0.43
10.06
7.25
0.94
0.21
19.60
4.33
4.41
1.34
f
{4,13,45}
31.55
0.50
10.68
8.74
0.93
0.24
30.18
6.86
5.07
1.41
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 24 / 35
Applications
Simulated data
Simulated SN data. Index plots of K (P, P(−i) ) from fitting a SN-NLM.
6
(c)
6
(b)
6
(a)
5
5
5
4
30
40
50
4 3 0
1
2
K−L divergence
4 3 1 0
20
0
10
20
30
40
50
0
10
(f)
5 4
45
40
50
1
13
0
1 0 30
Index
50
3
45
2
K−L divergence
5 4 3
4
2
K−L divergence
5 4 3 2 1 0
20
40
6
Index
(e)
4
10
30
Index
(d) 45
0
20
Index
6
10
6
0
K−L divergence
2
K−L divergence
4 3 2 0
1
K−L divergence
13
0
10
20
30
Index
40
50
0
10
20
30
40
50
Index
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 25 / 35
Applications
Simulated data
Influence of outlying observations
Table: Simulated SN data. Posterior means and standard deviations of the parameters from fitting a ST–NLM. Data
β1
β2
σ2
β3
λ
ν
names
Mean
SD
Mean
SD
Mean
SD
Mean
SD
Mean
SD
Mean
SD
a
29.87
0.17
5.03
0.76
0.69
0.05
1.88
0.76
-2.26
1.36
12.31
8.95
b
29.89
0.24
5.43
1.41
0.72
0.09
1.07
0.42
-0.55
0.96
2.88
0.79
c
29.92
0.23
5.090
0.85
0.69
0.05
0.99
0.43
-0.77
0.98
2.87
0.80
d
29.92
0.23
5.06
0.84
0.69
0.05
0.99
0.42
-0.77
0.98
2.87
0.80
e
30.05
0.26
5.31
1.29
0.71
0.08
1.01
0.47
0.04
0.78
2.43
0.42
f
30.22
0.28
5.19
1.32
0.70
0.08
1.17
0.58
0.47
0.72
2.31
0.31
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 26 / 35
Applications
Simulated data
Simulated SN data. Index plots of K (P, P(−i) ) from fitting a ST-NLM.
30
40
50
6 10
20
30
40
50
0
10
20
30
(f)
Index
40
50
5 4 3 1
4
45
13
45
0
4
0 30
50
2
K−L divergence
5 4 3 2 1
K−L divergence
5 4 3 2 1
20
40
6
Index
(e)
6
Index
(d)
0
10
4
5 0
Index
45
0
3 1 0
1 0 20
13
2
K−L divergence
5 4 3 4
2
K−L divergence
5 4 3 2
K−L divergence
1 0
10
6
0
K−L divergence
(c)
6
(b)
6
(a)
0
10
20
30
Index
40
50
0
10
20
30
40
50
Index
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 27 / 35
Applications
Simulated data
Influence of outlying observations
Table: Simulated data. Comparison between SN–NLM and ST–NLM fitting by using different Bayesian criteria. Data
SN–NLM
ST–NLM
names
B
DIC
EAIC
EBIC
B
DIC
EAIC
EBIC
a
-76,802
153.338
157.928
167.4881
-77.141
153.949
159.539
171.011
b
-116.794
218.119
223.519
233.079
-93.142
176.988
182.389
193.861
c
-112.506
216.516
221.35
230.910
-90.133
176.508
182.578
194.050 194.296
d
-113.861
216.51
221.594
231.154
-90.766
176.834
182.824
e
-126.484
240.758
245.334
254.894
-99.465
195.338
201.348
212.820
f
-134.148
259.939
264.85
274.4101
-106.483
210.077
215.897
227.369
where B =
n P
[ i ). log(CPO
i=1
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 28 / 35
Applications
The oil palm dataset
The oil palm dataset
We consider a Bayesian analysis of the data set presented in Foong (1999) that describe the oil palm yield. Assuming a nonlinear growth-curve model, we fit a NLM to the data as specified by Cancho et al. (2009) Yi =
β1 + εi , 1 + β2 exp(−β3 xi )
(9)
q where εi ∼ SMSN(− π2 k1 ∆, σ 2 , λ, H), for i = 1, . . . , 19. In our analysis iid
we assume SN-NLM, ST-NLM, SCN-NLM and SSL-NLM from the SMSN class for comparative purposes.
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 29 / 35
Applications
The oil palm dataset
The oil palm dataset
Table: Oil palm yield data set. Summary results from the posterior distribution, mean and standard deviation (SD) for parameter under SMSN distributions. SN-NLM
ST-NLM
SSL-NLM
SCN-NLM
Parameter
Mean
SD
Mean
SD
Mean
SD
Mean
SD
β1
37.251
0.491
37.565
0.469
37.461
0.538
37.421
0.478
β2
42.025
14.538
40.343
10.739
40.144
12.407
40.328
12.284
β3
0.730
0.065
0.717
0.050
0.712
0.057
0.720
0.058
σ2
6.447
2.890
1.966
1.482
2.674
1.907
2.484
2.022
λ
-2.603
2.070
-1.823
1.369
-3.423
2.108
-2.241
1.710
ν
-
-
3.415
1.267
1.967
0.539
0.562
0.197
γ
-
-
-
-
-
-
0.389
0.114
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 30 / 35
Applications
The oil palm dataset
The oil palm dataset
Table: Oil palm yield data set. Comparison between SMSN–NLM by using different Bayesian criteria.
criterion
SN–NLM
ST–NLM
SSL–NLM
SCN–NLM
B
-40.007
-38.949
-39.459
-39.146
DIC
75.469
72.626
73.561
73.183
EAIC
85.595
83.349
84.561
84.5612
EBIC
90.317
88.015
90.227
90.172
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 31 / 35
Applications
The oil palm dataset
The oil palm dataset
The K-L divergence and related calibration are computed. We do not find highly influential cases. The K (P, P(−i) ) is smaller that 0.71 and the corresponding calibrations are smaller than 1. However, for SN-NLM, cases 13,18, 10 and 15 had larger K (P, P(−i) ) when compared with the other cases.
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 32 / 35
Applications
The oil palm dataset
The oil palm dataset
Table: Oil palm yield data. Case influence diagnostics.
SN–NLM
ST–NLM
SSL-NLM
SCN–NLM
Case
K (P, P(−i) )
Cal.
K (P, P(−i) )
Cal.
K (P, P(−i) )
Cal.
K (P, P(−i) )
Cal.
13
0.709
0.935
0.549
0.908
0.699
0.934
0.571
0.912
18
0.703
0.934
0.290
0.832
0.307
0.818
0.561
0.911
10
0.306
0.838
0.304
0.836
0.297
0.835
0.214
0.795
15
0.272
0.823
0.244
0.810
0.258
0.817
0.256
0.816
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 33 / 35
Applications
The oil palm dataset
5
15
10
18
Index
(c)
(d)
15
1.5 10
15
18
13
0.5 15
18
10
15
10
15
0.0
10
0.0
0.5
13
1.0
1.0
K−L divergence
1.5
2.0
Index
2.0
5
K−L divergence
1.0
1.5 15
13 10
0.0 10
0.0
15
0.5
18
10
K−L divergence
1.0
13
0.5
K−L divergence
1.5
2.0
(b)
2.0
(a)
5
Index
5
Index
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 34 / 35
Conclusions
Conclusions
• We propose a interesting, useful and novel class of asymmetric heavy-tailed NLM.
• We discus the use of Markov Chain Monte Carlo methods as an alternative way to get Bayesian inference for the proposed model
• Bayesian case influence diagnostics based on the Kullback-Leibler divergence in order to study the sensitivity of the Bayesian estimates under perturbations in the model/data.
• Our analysis indicates that a ST-NLM
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 35 / 35
References Branco, M. D., Dey, D. K., 2001. A general class of multivariate skew-elliptical distributions. Journal of Multivariate Analysis 79, 99–113. Cancho, V. C., Lachos, V. H. & Ortega, E. M. M. (2009). A nonlinear regression model with skew-normal errors. Statistical Papers, doi:10.1007/s00362-008-0139-y. Cho, H., Ibrahim, J. G., Sinha, D. & Zhu, H. (2009). Bayesian case influence diagnostics for survival models. Biometrics, 65, 116–124. Cordeiro, G. M., Cysneiros, A. H. M. A., Cysneiros, F. J. A., 2009. Corrected maximum likelihood estimators im heteroscedastic symmetric nonlinear models. Journal of Statistical Computation and Simulation doi:10.1080/00949650802706420. Cysneiros, F. J. A., Vanegas, L. H., (2008). Residuals and their statistical properties in symmetrical nonlinear models. Statistics & Probability Letters 78, 3269–3273. Foong, F. S. (1999). Impact of mixture on potential evapotranspiration, growth and yield of palm oil. PORIM Interl. Palm Oil Cong. (Agric.), pages 265–287. Peng, F. & Dey, D. (1995). Bayesian analysis of outlier problems using divergence measures. The Canadian Journal of Statistics/La Revue Canadienne de Statistique, 23(2), 199–213. Rosa, G. J. M., Padovani, C. R. & Gianola, D. (2003). Robust linear mixed models with normal/independent distributions and bayesian mcmc implementation. Biometrical Journal, 45, 573–590. Spiegelhalter, D. J., Best, N. G., Carlin, B. P. & van der Linde, A. (2002). Bayesian measures of model complexity and fit. 64(4), 583–639. Wang, J. & Genton, M. G. (2006). The multivariate skew-slash distribution. Journal of Statistical Planning and Inference, 136, 209–220. Xie, F. C., Wei, B. C., Lin, J. G., 2009a. Homogeneity diagnostics for skew-normal nonlinear regression models. Statistics & Probability Letters 79, 821–827. Xie, F. C., Lin, J. G., Wei, B. C., 2009b. Diagnostics for skew-normal nonlinear regression models with ar(1) errors. Computational Statistics & Data Analysis, (in press). Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 35 / 35
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of Skew mar¸ (ICMC-USP) coNormal - 2010Distributions: 1 / 35
Motivation
Motivation The routine use of the normality in nonlinear models (N-NLM) has been recently questioned by many authors (see Cysneiros & Vanegas, 2008; Cordeiro et al., 2009, among others).
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of Skew mar¸ (ICMC-USP) coNormal - 2010Distributions: 2 / 35
Motivation
Motivation The routine use of the normality in nonlinear models (N-NLM) has been recently questioned by many authors (see Cysneiros & Vanegas, 2008; Cordeiro et al., 2009, among others). Thus, it is of practical interest to develop statistical model with considerable flexibility in the distributional assumptions of the random error term in NLM.
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of Skew mar¸ (ICMC-USP) coNormal - 2010Distributions: 2 / 35
Motivation
Motivation The routine use of the normality in nonlinear models (N-NLM) has been recently questioned by many authors (see Cysneiros & Vanegas, 2008; Cordeiro et al., 2009, among others). Thus, it is of practical interest to develop statistical model with considerable flexibility in the distributional assumptions of the random error term in NLM. Cancho et al. (2009) and Xie et al. (2009b,a) has shown the advantage of using the skew-normal distribution in the context of nonlinear regression models (SN-NLM).
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of Skew mar¸ (ICMC-USP) coNormal - 2010Distributions: 2 / 35
Motivation
Motivation The routine use of the normality in nonlinear models (N-NLM) has been recently questioned by many authors (see Cysneiros & Vanegas, 2008; Cordeiro et al., 2009, among others). Thus, it is of practical interest to develop statistical model with considerable flexibility in the distributional assumptions of the random error term in NLM. Cancho et al. (2009) and Xie et al. (2009b,a) has shown the advantage of using the skew-normal distribution in the context of nonlinear regression models (SN-NLM). We extend the SN-NLM by assuming that the model errors follow scale mixtures of skew-normal distributions (SMSN, Branco and Dey, 2001). (skew-normal,skew-t, skew-slash, skew-contaminated normal). Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of Skew mar¸ (ICMC-USP) coNormal - 2010Distributions: 2 / 35
SMSN Distributions
The Skew-Normal distribution
Z ∼ SN(µ, σ 2 , λ) , with pdf 2
f (z) = 2φ(z; µ, σ )Φ
λ(z − µ) σ
, (Azzalini, 1985)
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of Skew mar¸ (ICMC-USP) coNormal - 2010Distributions: 3 / 35
SMSN Distributions
The Skew-Normal distribution
Z ∼ SN(µ, σ 2 , λ) , with pdf 2
f (z) = 2φ(z; µ, σ )Φ
λ(z − µ) σ
, (Azzalini, 1985)
Marginal stochastic representation: Z = µ + ∆|T0 | + Γ1/2 T1 , where ∆ = σδ, δ =
√ λ , 1+λ2
Γ = (1 − δ 2 )σ 2 , T0 and T1 are
independent standard normal random variables.
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of Skew mar¸ (ICMC-USP) coNormal - 2010Distributions: 3 / 35
SMSN Distributions
Scale Mixtures of Skew-Normal distributions(SMSN) Y = µ + κ1/2 (U)Z ,
Z ∼ SN(0, σ 2 , λ) U is a positive random variable with cdf H(·; ν) (pdf h(·; ν)) κ(.) is weight function, this paper we restrict κ(u) = 1/u The pdf of Y is given by: Z f (y ) = 2
∞
φ(y ; µ, u
−1 2
σ )Φ
0
u 1/2 λ(y − µ) σ
! dH(u; ν),
Notation: Y ∼ SMSN(µ, σ 2 , λ; H) Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of Skew mar¸ (ICMC-USP) coNormal - 2010Distributions: 4 / 35
SMSN Distributions
Examples of SMSN distributions The skew–t: Y ∼ ST (µ, σ 2 , λ; ν), U ∼ Gamma(ν/2, ν/2). ! r ν+1 ) Γ( ν+1 d − 2 v +1 2 f (y ) = ν √ 1+ A; ν + 1 , y ∈ R, T ν d +ν Γ( 2 ) πνσ
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of Skew mar¸ (ICMC-USP) coNormal - 2010Distributions: 5 / 35
SMSN Distributions
Examples of SMSN distributions The skew–t: Y ∼ ST (µ, σ 2 , λ; ν), U ∼ Gamma(ν/2, ν/2). ! r ν+1 ) Γ( ν+1 d − 2 v +1 2 f (y ) = ν √ 1+ A; ν + 1 , y ∈ R, T ν d +ν Γ( 2 ) πνσ The skew–slash:Y ∼ SSL(µ, σ 2 , λ; ν),U ∼ Beta(ν, 1). Z f (y ) = 2ν
1
u ν−1 φ(y ; µ, u −1 σ 2 )Φ(u 1/2 A)du, y ∈ R,
0
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of Skew mar¸ (ICMC-USP) coNormal - 2010Distributions: 5 / 35
SMSN Distributions
Examples of SMSN distributions The skew–t: Y ∼ ST (µ, σ 2 , λ; ν), U ∼ Gamma(ν/2, ν/2). ! r ν+1 ) Γ( ν+1 d − 2 v +1 2 f (y ) = ν √ 1+ A; ν + 1 , y ∈ R, T ν d +ν Γ( 2 ) πνσ The skew–slash:Y ∼ SSL(µ, σ 2 , λ; ν),U ∼ Beta(ν, 1). Z f (y ) = 2ν
1
u ν−1 φ(y ; µ, u −1 σ 2 )Φ(u 1/2 A)du, y ∈ R,
0
Applications of the skew–slash distribution can be found in Wang & Genton (2006).
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of Skew mar¸ (ICMC-USP) coNormal - 2010Distributions: 5 / 35
SMSN Distributions
Examples of SMSN distributions
The skew contaminated normal:Y ∼ SCN(µ, σ 2 , λ; ν, γ). The pdf of U is given by h(u|ν) = νI(u=γ) + (1 − ν)I(u=1) , 0 < ν < 1, 0 < γ ≤ 1,
f (y ) = 2{νφ(y ; µ, γ −1 σ 2 )Φ(γ 1/2 A) + (1 − ν)φ(y ; µ, σ 2 )Φ(A)}.
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of Skew mar¸ (ICMC-USP) coNormal - 2010Distributions: 6 / 35
The SMSN nonlinear regression model
The SMSN nonlinear regression model
Yi = η(β, xi ) + εi , i = 1, . . . , n,
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of Skew mar¸ (ICMC-USP) coNormal - 2010Distributions: 7 / 35
The SMSN nonlinear regression model
The SMSN nonlinear regression model
Yi = η(β, xi ) + εi , i = 1, . . . , n,
η(.) is injective and twice continuously differentiable,
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of Skew mar¸ (ICMC-USP) coNormal - 2010Distributions: 7 / 35
The SMSN nonlinear regression model
The SMSN nonlinear regression model
Yi = η(β, xi ) + εi , i = 1, . . . , n,
η(.) is injective and twice continuously differentiable, q εi ∼ SMSN(− π2 k1 ∆, σ 2 , λ; H) where kr = E [U r |y ],
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of Skew mar¸ (ICMC-USP) coNormal - 2010Distributions: 7 / 35
The SMSN nonlinear regression model
The SMSN nonlinear regression model
Yi = η(β, xi ) + εi , i = 1, . . . , n,
η(.) is injective and twice continuously differentiable, q εi ∼ SMSN(− π2 k1 ∆, σ 2 , λ; H) where kr = E [U r |y ], q Yi ∼ SMSN(η(β, xi ) + b∆, σ 2 , λ; H), with b = − π2 k1 ,
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of Skew mar¸ (ICMC-USP) coNormal - 2010Distributions: 7 / 35
The SMSN nonlinear regression model
The SMSN nonlinear regression model
Yi = η(β, xi ) + εi , i = 1, . . . , n,
η(.) is injective and twice continuously differentiable, q εi ∼ SMSN(− π2 k1 ∆, σ 2 , λ; H) where kr = E [U r |y ], q Yi ∼ SMSN(η(β, xi ) + b∆, σ 2 , λ; H), with b = − π2 k1 , E [Yi ] = η( β, xi ), Var [Yi ] = k2 σ 2 − b 2 ∆2 .
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of Skew mar¸ (ICMC-USP) coNormal - 2010Distributions: 7 / 35
Bayesian inference
Bayesian inference
The hierarchical representation: Yi |Ti = ti ∼ N1 (η(β, xi ) + ∆ti , Ui−1 Γ) Ti |Ui ∼ TN1 (b, ui−1 )I (b, ∞) Ui ∼ H(.; ν), where i = 1, . . . , n Γ = (1 − δ 2 )σ 2 , and ∆ = σδ.
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of Skew mar¸ (ICMC-USP) coNormal - 2010Distributions: 8 / 35
Bayesian inference
Bayesian inference
Let y = (y1 , . . . , yn )> , x = (x1 , . . . , xn )> , t = (t1 , . . . , tn )> and u = (u1 , . . . , un )> . It follows that the complete likelihood function of θ associated with (y, x, t, u) is given by n Y Lc (θ|y, x, t, u) ∝ [φ1 (yi ; η(β, xi )+∆ti , ui−1 τ )φ1 (ti ; b, ui−1 )I(b,∞) h(ui |ν)]. i=1
(1)
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of Skew mar¸ (ICMC-USP) coNormal - 2010Distributions: 9 / 35
Bayesian inference
Prior distribution
βj = N(µβj , σβ2j ), j = 1, . . . , p, 2 ) and ∆ ∼ N1 (µ∆ , σ∆ ρ % τ −1 ∼ Gamma( , ). 2 2
ς For the skew-t model: ν ∼ exp( )I(2,∞) , 2 For the skew-slash model: ν ∼ Gamma(a, b) (b a), For the skew-contaminated normal model: ν ∼ U(0, 1) and γ Beta(a, b) (ν ⊥ γ).
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 10 / 35
Bayesian inference
The posterior distribution
The joint posterior density of all unobservable is is given by π(θ, t, u|y, x) ∝
Qn
i=1 [φ1 (yi ; η(β, xi )
+ ∆ti , ui−1 τ )
(2)
φ1 (ti ; b, ui−1 )I(b,∞) h(ui |ν)]π(θ). Distribution (2) is not tractable analytically but MCMC methods such as the Gibbs sampler, can be used to draw samples, from which features of marginal posterior distributions of interest can be inferred.
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 11 / 35
Bayesian inference
The full conditional distributions
Ti |β, ∆, τ, ν, y, u ∼ TN1 (µTi + b, ui−1 MT2 )I (b, ∞), i = 1, . . . , n; ∆ τ , µ Ti = 2 (yi − η(β, xi ) − ∆b), +τ ∆ +τ 2 2 +τ Bσ∆ + τ µ∆ Aσ∆ ∆|β, τ, ν, y, u, t ∼ N1 , , 2 +τ 2 Aσ∆ τ σ∆ P P where A = ni=1 ui ti and B = ni=1 ui ti (yi − η(β, xi )),
where MT2 =
∆2
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 12 / 35
Bayesian inference
The full conditional distributions
n
X 1 1 ui (yi − η(β, xi ) − ∆ti )2 )); τ −1 |β, ∆, ν, y, u, t ∼ Gamma( (ρ + n), (% + 2 2 i=1
β|∆, τ ν, y, u, t ∝ φp (β; µβ , D)
n Y
φ1 (η(β, xi ); (yi − ∆ti ), ui−1 τ ),
i=1
where µβ = (µβ1 , . . . , µβp )> , D = diagonal(σβ21 , . . . , σβ2p ).
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 13 / 35
Bayesian inference
The full conditional distributions
For each element of u, the density is:
1 ui (yi − η(β, xi ) − ∆ti )2 (3) 2τ 1 2 − ui (ti − b) h(ui |ν), 2
π(ui |β, ∆, τ, ν, y, x, t) ∝ ui exp −
for i = 1, . . . , n. For ν, the density is: π(ν|β, ∆, τ, y, u, t) ∝ π(ν)
n Y
h(ui |ν).
(4)
i=1
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 14 / 35
Bayesian inference
The full conditional distributions
(i) Skew-t. The density of the conditional posterior distribution in (3) takes the form: ui |θ(−ν) , y, t ∼ Gamma(2; ν/2 + Ci /2), where Ci = τ1 (yi − η(β, xi ) − ∆ti )2 + (ti − b)2 and the full conditional posterior density of ν is π(ν|θ(−ν) , y, t, u) ∝ π1 (ν) × Gamma(
1X nν − 1, (ui − log ui ))I(2,∞) , 2 2
where π1 (ν) = (2ν/2 Γ(ν/2))−n .
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 15 / 35
Bayesian inference
The full conditional distributions
• Skew-slash. In this case, the fully conditional posterior density of each ui is: ui |θ(−ν) , y, t ∼ Gamma((nu + 1)/2; Ci /2)I(0,1) . Further, the conditional posterior density of ν is ν|θ(−ν) , y, b, u, t ∼ Gamma(n + a, b −
X
log ui ).
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 16 / 35
Bayesian inference
The full conditional distributions • Skew-contaminated normal distribution The full conditional posterior density of the proportion of outliers ν is: P P ui − nγ n − ui ν|θ(−ν) , y, b, u, t ∼ Beta a + ;b + . 1−γ 1−γ The conditional posterior density of γ is: P P n − ui ui − nγ ) ) ( 1 − γ 1 −γ π(γ|θ(−γ) , y, b, u, t) ∝ ν × (1 − ν) . (
An interesting Metropolis–Hastings method to update from γ is described in Rosa et al. (2003). Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 17 / 35
Bayesian case influence diagnostics
Bayesian case influence diagnostics
Let K (P, P(−i) ) denote the Kullback-Leibler (K–L) divergence between P and P(−i) , where P denotes the posterior distribution of θ for full data, and P(−i) denotes the posterior distribution of θ without the ith case. Specifically, "
Z K (P, P(−i) ) =
π(θ|D) log
π(θ|D) π(θ|D (−i) )
# dθ.
(5)
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 18 / 35
Bayesian case influence diagnostics
Bayesian case influence diagnostics
As pointed by Peng & Dey (1995) and Cho et al. (2009), calibration of K (P, P(−i) ) can be done by solving for pi the equation K (P, P(−i) ) = K B(0.5), B(pi ) = − log 4pi (1 − pi ) /2, where B(p) denotes the Bernoulli distribution with success probability p. After some algebra it can be shown that q o 1n pi = 1 + 1 − exp − 2K (P, P(−i) ) . 2 This equation implies that 0.5 ≤ pi ≤ 1.
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 19 / 35
Bayesian case influence diagnostics
Bayesian case influence diagnostics
For our model in (7) it can be shown that (5) can be expressed as a posterior expectation K (P, P(−i) ) = log Eθ|D [g (yi |θ)]−1 + Eθ|D {log [g (yi |θ)]}
(6)
= − log(CPOi ) + Eθ|D {log [g (yi |θ)]} , where Eθ|D (·) denotes the expectation with respect to the joint posterior π(θ|D).
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 20 / 35
Bayesian case influence diagnostics
Bayesian case influence diagnostics
Thus (6) can be computed by sampling from the posterior distribution of θ via MCMC methods. Let θ1 , . . . , θQ be a sample of size Q of π(θ|D). Then, a Monte Carlo estimate of K (P, P(−i) ) is given by Q X \ [i) + 1 K (P, P(−i) ) = − log(CPO log [g (yi |θq )] . Q
(7)
q=1
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 21 / 35
Applications
Simulated data
Influence of outlying observations We consider the following logistic model: Yi =
β1 + εi , i = 1, · · · , 50, 1 + β2 exp(−β3 xi )
(8)
p where εi ∼ SN(− 2/π∆, σ 2 , λ), the variable xi ranging from 1 to 50 and we fixed the parameter values at: β1 = 30, β2 = 5, β3 = .7, σ 2 = 2 and λ = −3. We selected cases 4, 13 and 45 for perturbation. To create influential observation in the dataset, we choose one, two or tree of these selected cases and perturbed the response variable as follows y˜i = yi + 4Sy , i = 4, 13 and 45, where Sy is the standard deviations of the yi0 s. Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 22 / 35
Applications
Simulated data
Simulated data
1
The following independent priors were adopted in the Bayesian computations. βk ∼ log − N1 (0, 103 ), k = 1, 2, 3, ∆ ∼ N1 (0, 103 ), and 1/τ ∼ Gamma(1, 0.01).
2
we generated two parallel independent runs of the Gibbs sampler with size 60000 for each parameter, disregarding the first 10000 iterations to eliminate the effect of the initial values and, to avoid correlation problems, we considered a spacing of size 20, obtaining a sample of size 2500.
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 23 / 35
Applications
Simulated data
Influence of outlying observations
Table: Simulated SN data. Posterior means and standard deviations of the parameters from fitting a SN-NLM. Data
Perturbed
β1 = 30
β2 = 5
β3 = 0.7
σ2 = 2
λ = −3
names
case
Mean
SD
Mean
SD
Mean
SD
Mean
SD
Mean
SD
a
None
29.86
0.17
5.05
0.74
0.69
0.05
2.59
0.84
-2.81
1.67 1.18
b
4
30.24
0.35
10.56
6.93
0.99
0.21
10.65
3.07
1.14
c
13
30.48
0.32
5.91
2.74
0.73
0.12
11.04
2.55
3.60
1.11
d
45
30.51
0.32
6.13
3.11
0.74
0.12
11.20
2.67
3.59
1.06
e
{4,45}
30.93
0.43
10.06
7.25
0.94
0.21
19.60
4.33
4.41
1.34
f
{4,13,45}
31.55
0.50
10.68
8.74
0.93
0.24
30.18
6.86
5.07
1.41
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 24 / 35
Applications
Simulated data
Simulated SN data. Index plots of K (P, P(−i) ) from fitting a SN-NLM.
6
(c)
6
(b)
6
(a)
5
5
5
4
30
40
50
4 3 0
1
2
K−L divergence
4 3 1 0
20
0
10
20
30
40
50
0
10
(f)
5 4
45
40
50
1
13
0
1 0 30
Index
50
3
45
2
K−L divergence
5 4 3
4
2
K−L divergence
5 4 3 2 1 0
20
40
6
Index
(e)
4
10
30
Index
(d) 45
0
20
Index
6
10
6
0
K−L divergence
2
K−L divergence
4 3 2 0
1
K−L divergence
13
0
10
20
30
Index
40
50
0
10
20
30
40
50
Index
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 25 / 35
Applications
Simulated data
Influence of outlying observations
Table: Simulated SN data. Posterior means and standard deviations of the parameters from fitting a ST–NLM. Data
β1
β2
σ2
β3
λ
ν
names
Mean
SD
Mean
SD
Mean
SD
Mean
SD
Mean
SD
Mean
SD
a
29.87
0.17
5.03
0.76
0.69
0.05
1.88
0.76
-2.26
1.36
12.31
8.95
b
29.89
0.24
5.43
1.41
0.72
0.09
1.07
0.42
-0.55
0.96
2.88
0.79
c
29.92
0.23
5.090
0.85
0.69
0.05
0.99
0.43
-0.77
0.98
2.87
0.80
d
29.92
0.23
5.06
0.84
0.69
0.05
0.99
0.42
-0.77
0.98
2.87
0.80
e
30.05
0.26
5.31
1.29
0.71
0.08
1.01
0.47
0.04
0.78
2.43
0.42
f
30.22
0.28
5.19
1.32
0.70
0.08
1.17
0.58
0.47
0.72
2.31
0.31
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 26 / 35
Applications
Simulated data
Simulated SN data. Index plots of K (P, P(−i) ) from fitting a ST-NLM.
30
40
50
6 10
20
30
40
50
0
10
20
30
(f)
Index
40
50
5 4 3 1
4
45
13
45
0
4
0 30
50
2
K−L divergence
5 4 3 2 1
K−L divergence
5 4 3 2 1
20
40
6
Index
(e)
6
Index
(d)
0
10
4
5 0
Index
45
0
3 1 0
1 0 20
13
2
K−L divergence
5 4 3 4
2
K−L divergence
5 4 3 2
K−L divergence
1 0
10
6
0
K−L divergence
(c)
6
(b)
6
(a)
0
10
20
30
Index
40
50
0
10
20
30
40
50
Index
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 27 / 35
Applications
Simulated data
Influence of outlying observations
Table: Simulated data. Comparison between SN–NLM and ST–NLM fitting by using different Bayesian criteria. Data
SN–NLM
ST–NLM
names
B
DIC
EAIC
EBIC
B
DIC
EAIC
EBIC
a
-76,802
153.338
157.928
167.4881
-77.141
153.949
159.539
171.011
b
-116.794
218.119
223.519
233.079
-93.142
176.988
182.389
193.861
c
-112.506
216.516
221.35
230.910
-90.133
176.508
182.578
194.050 194.296
d
-113.861
216.51
221.594
231.154
-90.766
176.834
182.824
e
-126.484
240.758
245.334
254.894
-99.465
195.338
201.348
212.820
f
-134.148
259.939
264.85
274.4101
-106.483
210.077
215.897
227.369
where B =
n P
[ i ). log(CPO
i=1
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 28 / 35
Applications
The oil palm dataset
The oil palm dataset
We consider a Bayesian analysis of the data set presented in Foong (1999) that describe the oil palm yield. Assuming a nonlinear growth-curve model, we fit a NLM to the data as specified by Cancho et al. (2009) Yi =
β1 + εi , 1 + β2 exp(−β3 xi )
(9)
q where εi ∼ SMSN(− π2 k1 ∆, σ 2 , λ, H), for i = 1, . . . , 19. In our analysis iid
we assume SN-NLM, ST-NLM, SCN-NLM and SSL-NLM from the SMSN class for comparative purposes.
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 29 / 35
Applications
The oil palm dataset
The oil palm dataset
Table: Oil palm yield data set. Summary results from the posterior distribution, mean and standard deviation (SD) for parameter under SMSN distributions. SN-NLM
ST-NLM
SSL-NLM
SCN-NLM
Parameter
Mean
SD
Mean
SD
Mean
SD
Mean
SD
β1
37.251
0.491
37.565
0.469
37.461
0.538
37.421
0.478
β2
42.025
14.538
40.343
10.739
40.144
12.407
40.328
12.284
β3
0.730
0.065
0.717
0.050
0.712
0.057
0.720
0.058
σ2
6.447
2.890
1.966
1.482
2.674
1.907
2.484
2.022
λ
-2.603
2.070
-1.823
1.369
-3.423
2.108
-2.241
1.710
ν
-
-
3.415
1.267
1.967
0.539
0.562
0.197
γ
-
-
-
-
-
-
0.389
0.114
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 30 / 35
Applications
The oil palm dataset
The oil palm dataset
Table: Oil palm yield data set. Comparison between SMSN–NLM by using different Bayesian criteria.
criterion
SN–NLM
ST–NLM
SSL–NLM
SCN–NLM
B
-40.007
-38.949
-39.459
-39.146
DIC
75.469
72.626
73.561
73.183
EAIC
85.595
83.349
84.561
84.5612
EBIC
90.317
88.015
90.227
90.172
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 31 / 35
Applications
The oil palm dataset
The oil palm dataset
The K-L divergence and related calibration are computed. We do not find highly influential cases. The K (P, P(−i) ) is smaller that 0.71 and the corresponding calibrations are smaller than 1. However, for SN-NLM, cases 13,18, 10 and 15 had larger K (P, P(−i) ) when compared with the other cases.
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 32 / 35
Applications
The oil palm dataset
The oil palm dataset
Table: Oil palm yield data. Case influence diagnostics.
SN–NLM
ST–NLM
SSL-NLM
SCN–NLM
Case
K (P, P(−i) )
Cal.
K (P, P(−i) )
Cal.
K (P, P(−i) )
Cal.
K (P, P(−i) )
Cal.
13
0.709
0.935
0.549
0.908
0.699
0.934
0.571
0.912
18
0.703
0.934
0.290
0.832
0.307
0.818
0.561
0.911
10
0.306
0.838
0.304
0.836
0.297
0.835
0.214
0.795
15
0.272
0.823
0.244
0.810
0.258
0.817
0.256
0.816
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 33 / 35
Applications
The oil palm dataset
5
15
10
18
Index
(c)
(d)
15
1.5 10
15
18
13
0.5 15
18
10
15
10
15
0.0
10
0.0
0.5
13
1.0
1.0
K−L divergence
1.5
2.0
Index
2.0
5
K−L divergence
1.0
1.5 15
13 10
0.0 10
0.0
15
0.5
18
10
K−L divergence
1.0
13
0.5
K−L divergence
1.5
2.0
(b)
2.0
(a)
5
Index
5
Index
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 34 / 35
Conclusions
Conclusions
• We propose a interesting, useful and novel class of asymmetric heavy-tailed NLM.
• We discus the use of Markov Chain Monte Carlo methods as an alternative way to get Bayesian inference for the proposed model
• Bayesian case influence diagnostics based on the Kullback-Leibler divergence in order to study the sensitivity of the Bayesian estimates under perturbations in the model/data.
• Our analysis indicates that a ST-NLM
Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 35 / 35
References Branco, M. D., Dey, D. K., 2001. A general class of multivariate skew-elliptical distributions. Journal of Multivariate Analysis 79, 99–113. Cancho, V. C., Lachos, V. H. & Ortega, E. M. M. (2009). A nonlinear regression model with skew-normal errors. Statistical Papers, doi:10.1007/s00362-008-0139-y. Cho, H., Ibrahim, J. G., Sinha, D. & Zhu, H. (2009). Bayesian case influence diagnostics for survival models. Biometrics, 65, 116–124. Cordeiro, G. M., Cysneiros, A. H. M. A., Cysneiros, F. J. A., 2009. Corrected maximum likelihood estimators im heteroscedastic symmetric nonlinear models. Journal of Statistical Computation and Simulation doi:10.1080/00949650802706420. Cysneiros, F. J. A., Vanegas, L. H., (2008). Residuals and their statistical properties in symmetrical nonlinear models. Statistics & Probability Letters 78, 3269–3273. Foong, F. S. (1999). Impact of mixture on potential evapotranspiration, growth and yield of palm oil. PORIM Interl. Palm Oil Cong. (Agric.), pages 265–287. Peng, F. & Dey, D. (1995). Bayesian analysis of outlier problems using divergence measures. The Canadian Journal of Statistics/La Revue Canadienne de Statistique, 23(2), 199–213. Rosa, G. J. M., Padovani, C. R. & Gianola, D. (2003). Robust linear mixed models with normal/independent distributions and bayesian mcmc implementation. Biometrical Journal, 45, 573–590. Spiegelhalter, D. J., Best, N. G., Carlin, B. P. & van der Linde, A. (2002). Bayesian measures of model complexity and fit. 64(4), 583–639. Wang, J. & Genton, M. G. (2006). The multivariate skew-slash distribution. Journal of Statistical Planning and Inference, 136, 209–220. Xie, F. C., Wei, B. C., Lin, J. G., 2009a. Homogeneity diagnostics for skew-normal nonlinear regression models. Statistics & Probability Letters 79, 821–827. Xie, F. C., Lin, J. G., Wei, B. C., 2009b. Diagnostics for skew-normal nonlinear regression models with ar(1) errors. Computational Statistics & Data Analysis, (in press). Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of mar¸ Skew (ICMC-USP) co Normal - 2010 Distributions: 35 / 35
