Bayesian Nonlinear Regression Models with Scale Mixtures of Skew ...

Bayesian Nonlinear Regression Models with ScaleMixtures of Skew Normal Distributions: Estimation andCase Influence DiagnosticsVicente 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 março (ICMC-USP) Normal 2010Distributions:1 / 35

MotivationMotivationThe routine use of the normality in nonlinear models (N-NLM) has beenrecently 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 março (ICMC-USP) Normal 2010Distributions:2 / 35

MotivationMotivationThe routine use of the normality in nonlinear models (N-NLM) has beenrecently 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 considerableflexibility 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 março (ICMC-USP) Normal 2010Distributions:2 / 35

MotivationMotivationThe routine use of the normality in nonlinear models (N-NLM) has beenrecently 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 considerableflexibility 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 ofusing the skew-normal distribution in the context of nonlinear regressionmodels (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 março (ICMC-USP) Normal 2010Distributions:2 / 35

MotivationMotivationThe routine use of the normality in nonlinear models (N-NLM) has beenrecently 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 considerableflexibility 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 ofusing the skew-normal distribution in the context of nonlinear regressionmodels (SN-NLM).We extend the SN-NLM by assuming that the model errors follow scalemixtures 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 março (ICMC-USP) Normal 2010Distributions:2 / 35

SMSN DistributionsThe Skew-Normal distributionZ ∼ SN(µ, σ 2 , λ) , with pdf( ) λ(z − µ)f (z) = 2φ(z; µ, σ 2 )Φ, (Azzalini, 1985)σicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of Skew março (ICMC-USP) Normal 2010Distributions:3 / 35

SMSN DistributionsThe Skew-Normal distributionZ ∼ SN(µ, σ 2 , λ) , with pdf( ) λ(z − µ)f (z) = 2φ(z; µ, σ 2 )Φ, (Azzalini, 1985)σMarginal stochastic representation:where ∆ = σδ, δ =Z = µ + ∆|T 0 | + Γ 1/2 T 1 ,λ √1+λ 2 , Γ = (1 − δ2 )σ 2 , T 0 and T 1 areindependent standard normal random variables.icente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of Skew março (ICMC-USP) Normal 2010Distributions:3 / 35

SMSN DistributionsScale Mixtures of Skew-Normal distributions(SMSN)Y = µ + κ 1/2 (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/uThe pdf of Y is given by:f (y) = 2∫ ∞0φ(y; µ, u −1 σ 2 )ΦNotation: Y ∼ SMSN(µ, σ 2 , λ; H)()u 1/2 λ(y − µ)dH(u; ν),σVicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of Skew março (ICMC-USP) Normal 2010Distributions:4 / 35

SMSN DistributionsExamples of SMSN distributionsThe skew–t: Y ∼ ST (µ, σ 2 , λ; ν), U ∼ Gamma(ν/2, ν/2).f (y) =ν+1Γ(2 ) (Γ( ν 2 )√ 1 + d ) −ν+1(√ )2 v + 1Tπνσ νd + ν A; ν + 1 , y ∈ R,icente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of Skew março (ICMC-USP) Normal 2010Distributions:5 / 35

SMSN DistributionsExamples of SMSN distributionsThe skew–t: Y ∼ ST (µ, σ 2 , λ; ν), U ∼ Gamma(ν/2, ν/2).f (y) =ν+1Γ(2 ) (Γ( ν 2 )√ 1 + d ) −ν+1(√ )2 v + 1Tπνσ νd + ν A; ν + 1 , y ∈ R,The skew–slash:Y ∼ SSL(µ, σ 2 , λ; ν),U ∼ Beta(ν, 1).f (y) = 2ν∫ 10u ν−1 φ(y; µ, u −1 σ 2 )Φ(u 1/2 A)du, y ∈ R,icente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of Skew março (ICMC-USP) Normal 2010Distributions:5 / 35

SMSN DistributionsExamples of SMSN distributionsThe skew–t: Y ∼ ST (µ, σ 2 , λ; ν), U ∼ Gamma(ν/2, ν/2).f (y) =ν+1Γ(2 ) (Γ( ν 2 )√ 1 + d ) −ν+1(√ )2 v + 1Tπνσ νd + ν A; ν + 1 , y ∈ R,The skew–slash:Y ∼ SSL(µ, σ 2 , λ; ν),U ∼ Beta(ν, 1).f (y) = 2ν∫ 10u ν−1 φ(y; µ, u −1 σ 2 )Φ(u 1/2 A)du, y ∈ R,Applications of the skew–slash distribution can be found in Wang &Genton (2006).icente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of Skew março (ICMC-USP) Normal 2010Distributions:5 / 35

SMSN DistributionsExamples of SMSN distributionsThe skew contaminated normal:Y ∼ SCN(µ, σ 2 , λ; ν, γ). The pdfof U is given byh(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 março (ICMC-USP) Normal 2010Distributions:6 / 35

The SMSN nonlinear regression modelThe SMSN nonlinear regression modelY i = η(β, x i ) + ε i , 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 março (ICMC-USP) Normal 2010Distributions:7 / 35

The SMSN nonlinear regression modelThe SMSN nonlinear regression modelY i = η(β, x i ) + ε i , 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 março (ICMC-USP) Normal 2010Distributions:7 / 35

The SMSN nonlinear regression modelThe SMSN nonlinear regression modelY i = η(β, x i ) + ε i , i = 1, . . . , n,η(.) is injective and twice continuously differentiable,√2ε i ∼ SMSN(−π k 1∆, σ 2 , λ; H) where k r = 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 março (ICMC-USP) Normal 2010Distributions:7 / 35

The SMSN nonlinear regression modelThe SMSN nonlinear regression modelY i = η(β, x i ) + ε i , i = 1, . . . , n,η(.) is injective and twice continuously differentiable,√2ε i ∼ SMSN(−π k 1∆, σ 2 , λ; H) where k r = E[U r |y],√Y i ∼ SMSN(η(β, x i ) + b∆, σ 2 2, λ; H), with b = −π k 1,Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of Skew março (ICMC-USP) Normal 2010Distributions:7 / 35

The SMSN nonlinear regression modelThe SMSN nonlinear regression modelY i = η(β, x i ) + ε i , i = 1, . . . , n,η(.) is injective and twice continuously differentiable,√2ε i ∼ SMSN(−π k 1∆, σ 2 , λ; H) where k r = E[U r |y],√Y i ∼ SMSN(η(β, x i ) + b∆, σ 2 2, λ; H), with b = −π k 1,E[Y i ] = η( β, x i ), Var[Y i ] = k 2 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 março (ICMC-USP) Normal 2010Distributions:7 / 35

Bayesian inferenceBayesian inferenceThe hierarchical representation:Y i |T i = t i ∼ N 1 (η(β, x i ) + ∆t i , U −1iΓ)T i |U i ∼ TN 1 (b, u −1i)I (b, ∞)U i ∼ 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 março (ICMC-USP) Normal 2010Distributions:8 / 35

Bayesian inferenceBayesian inferenceLet y = (y 1 , . . . , y n ) ⊤ , x = (x 1 , . . . , x n ) ⊤ , t = (t 1 , . . . , t n ) ⊤ andu = (u 1 , . . . , u n ) ⊤ . It follows that the complete likelihood function of θassociated with (y, x, t, u) is given byL c (θ|y, x, t, u) ∝n∏[φ 1 (y i ; η(β, x i )+∆t i , u −1 τ)φ 1 (t i ; b, u −1 )I (b,∞) h(u i |ν)].i=1ii(1)Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of Skew março (ICMC-USP) Normal 2010Distributions:9 / 35

Bayesian inferencePrior distributionβ j = N(µ βj , σ 2 β j), j = 1, . . . , p,∆ ∼ N 1 (µ ∆ , σ 2 ∆ ) andτ −1 ∼ Gamma( ρ 2 , ϱ 2 ).For the skew-t model: ν ∼ exp( ς 2 )I (2,∞),For the skew-slash model: ν ∼ Gamma(a, b) (b ≪ a),For the skew-contaminated normal model: ν ∼ U(0, 1) andγ 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 março Skew (ICMC-USP) Normal - 2010 Distributions: 10 / 35

Bayesian inferenceThe posterior distributionThe joint posterior density of all unobservable is is given byπ(θ, t, u|y, x)∝ ∏ ni=1 [φ 1(y i ; η(β, x i ) + ∆t i , u −1iτ) (2)φ 1 (t i ; b, u −1i)I (b,∞) h(u i |ν)]π(θ).Distribution (2) is not tractable analytically but MCMC methods such asthe Gibbs sampler, can be used to draw samples, from which features ofmarginal 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 março Skew (ICMC-USP) Normal - 2010 Distributions: 11 / 35

Bayesian inferenceThe full conditional distributionsT i |β, ∆, τ, ν, y, u ∼ TN 1 (µ Ti + b, u −1iMT 2 )I (b, ∞), i = 1, . . . , n;where MT 2 = τ∆ 2 + τ , µ T i= ∆∆ 2 + τ (y i − η(β, x i ) − ∆b),( Bσ2∆|β, τ, ν, y, u, t ∼ N ∆+ τµ ∆1Aσ∆ 2 + τ , Aσ2 ∆ + τ )τσ∆2 ,where A = ∑ ni=1 u it i and B = ∑ ni=1 u it i (y i − η(β, x i )),Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of março Skew (ICMC-USP) Normal - 2010 Distributions: 12 / 35

Bayesian inferenceThe full conditional distributionsτ −1 |β, ∆, ν, y, u, t ∼ Gamma( 1 2 (ρ + n), 1 n∑2 (ϱ + u i (y i − η(β, x i ) − ∆t i ) 2 ));i=1β|∆, τν, y, u, t ∝ φ p (β; µ β , D)n∏i=1φ 1 (η(β, x i ); (y i − ∆t i ), u −1iτ),where µ β = (µ β1 , . . . , µ βp ) ⊤ , D = diagonal(σ 2 β 1, . . . , σ 2 β p).Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of março Skew (ICMC-USP) Normal - 2010 Distributions: 13 / 35

Bayesian inferenceThe full conditional distributionsFor each element of u, the density is:π(u i |β, ∆, τ, ν, y, x, t) ∝ u i exp{− 12τ u i(y i − η(β, x i ) − ∆t i ) 2 (3)− 1 2 u i(t i − b) 2 }h(u i |ν),for i = 1, . . . , n. For ν, the density is:n∏π(ν|β, ∆, τ, y, u, t) ∝ π(ν) h(u i |ν). (4)i=1Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of março Skew (ICMC-USP) Normal - 2010 Distributions: 14 / 35

Bayesian inferenceThe full conditional distributions(i) Skew-t. The density of the conditional posterior distribution in (3)takes the form:u i |θ (−ν) , y, t ∼ Gamma(2; ν/2 + C i /2),where C i = 1 τ (y i − η(β, x i ) − ∆t i ) 2 + (t i − b) 2 and the full conditionalposterior density of ν isπ(ν|θ (−ν) , y, t, u) ∝ π 1 (ν) × Gamma( nν 2 − 1, 1 ∑(ui − log u i ))I2 (2,∞) ,where π 1 (ν) = (2 ν/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 março Skew (ICMC-USP) Normal - 2010 Distributions: 15 / 35

Bayesian inferenceThe full conditional distributions• Skew-slash.In this case, the fully conditional posterior density of each u i is:u i |θ (−ν) , y, t ∼ Gamma((nu + 1)/2; C i /2)I (0,1) .Further, the conditional posterior density of ν isν|θ (−ν) , y, b, u, t ∼ Gamma(n + a, b − ∑ log u i ).Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of março Skew (ICMC-USP) Normal - 2010 Distributions: 16 / 35

Bayesian inferenceThe full conditional distributions• Skew-contaminated normal distributionThe full conditional posterior density of the proportion of outliers ν is:(ν|θ (−ν) , y, b, u, t ∼ Beta a + n − ∑ ∑ )u i1 − γ ; b + ui − nγ.1 − γThe conditional posterior density of γ is:π(γ|θ (−γ) , y, b, u, t) ∝ ν (n − ∑ ∑u i1 − γ ) ui − nγ× (1 − ν) ( )1 − γ .An interesting Metropolis–Hastings method to update from γ is describedin 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 março Skew (ICMC-USP) Normal - 2010 Distributions: 17 / 35

Bayesian case influence diagnosticsBayesian case influence diagnosticsLet K(P, P (−i) ) denote the Kullback-Leibler (K–L) divergence between Pand 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,∫[ ]π(θ|D)K(P, P (−i) ) = π(θ|D) logπ(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 março Skew (ICMC-USP) Normal - 2010 Distributions: 18 / 35

Bayesian case influence diagnosticsBayesian case influence diagnosticsAs pointed by Peng & Dey (1995) and Cho et al. (2009), calibration ofK(P, P (−i) ) can be done by solving for p i the equationK(P, P (−i) ) = K [ B(0.5), B(p i ) ] = − log [ 4p i (1 − p i ) ] /2,where B(p) denotes the Bernoulli distribution with success probability p.After some algebra it can be shown thatp i = 1 { √1 + 1 − exp [ − 2K(P, P2(−i) ) ]} .This equation implies that 0.5 ≤ p i ≤ 1.Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of março Skew (ICMC-USP) Normal - 2010 Distributions: 19 / 35

Bayesian case influence diagnosticsBayesian case influence diagnosticsFor our model in (7) it can be shown that (5) can be expressed as aposterior expectationK(P, P (−i) ) = log E θ|D{[g(yi |θ)] −1} + E θ|D {log [g(y i |θ)]}= − log(CPO i ) + E θ|D {log [g(y i |θ)]} ,(6)where E θ|D (·) denotes the expectation with respect to the joint posteriorπ(θ|D).Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of março Skew (ICMC-USP) Normal - 2010 Distributions: 20 / 35

Bayesian case influence diagnosticsBayesian case influence diagnosticsThus (6) can be computed by sampling from the posterior distribution ofθ via MCMC methods. Let θ 1 , . . . , θ Q be a sample of size Q of π(θ|D).Then, a Monte Carlo estimate of K(P, P (−i) ) is given bŷ K(P, P (−i) ) = − log(ĈPO i ) + 1 QQ∑log [g(y i |θ q )] . (7)q=1Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of março Skew (ICMC-USP) Normal - 2010 Distributions: 21 / 35

ApplicationsSimulated dataInfluence of outlying observationsWe consider the following logistic model:Y i =β 11 + β 2 exp(−β 3 x i ) + ε i, i = 1, · · · , 50, (8)where ε i ∼ SN(− √ 2/π∆, σ 2 , λ), the variable x i ranging from 1 to 50 andwe fixed the parameter values at: β 1 = 30, β 2 = 5, β 3 = .7, σ 2 = 2 andλ = −3.We selected cases 4, 13 and 45 for perturbation. To create influentialobservation in the dataset, we choose one, two or tree of these selectedcases and perturbed the response variable as followsỹ i = y i + 4S y , i = 4, 13 and 45, where S y is the standard deviations of they ′i s.Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of março Skew (ICMC-USP) Normal - 2010 Distributions: 22 / 35

ApplicationsSimulated dataSimulated data1 The following independent priors were adopted in the Bayesiancomputations. β k ∼ log − N 1 (0, 10 3 ), k = 1, 2, 3, ∆ ∼ N 1 (0, 10 3 ),and 1/τ ∼ Gamma(1, 0.01).2 we generated two parallel independent runs of the Gibbs sampler withsize 60000 for each parameter, disregarding the first 10000 iterationsto eliminate the effect of the initial values and, to avoid correlationproblems, we considered a spacing of size 20, obtaining a sample ofsize 2500.Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of março Skew (ICMC-USP) Normal - 2010 Distributions: 23 / 35

ApplicationsSimulated dataInfluence of outlying observationsTable: Simulated SN data. Posterior means and standard deviations of theparameters from fitting a SN-NLM.Data Perturbed β 1 = 30 β 2 = 5 β 3 = 0.7 σ 2 = 2 λ = −3names case Mean SD Mean SD Mean SD Mean SD Mean SDa None 29.86 0.17 5.05 0.74 0.69 0.05 2.59 0.84 -2.81 1.67b 4 30.24 0.35 10.56 6.93 0.99 0.21 10.65 3.07 1.14 1.18c 13 30.48 0.32 5.91 2.74 0.73 0.12 11.04 2.55 3.60 1.11d 45 30.51 0.32 6.13 3.11 0.74 0.12 11.20 2.67 3.59 1.06e {4,45} 30.93 0.43 10.06 7.25 0.94 0.21 19.60 4.33 4.41 1.34f {4,13,45} 31.55 0.50 10.68 8.74 0.93 0.24 30.18 6.86 5.07 1.41Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of março Skew (ICMC-USP) Normal - 2010 Distributions: 24 / 35

ApplicationsSimulated dataSimulated SN data. Index plots of K(P, P (−i) ) from fittinga SN-NLM.(a)(b)(c)K−L divergence0 1 2 3 4 5 6K−L divergence0 1 2 3 4 5 64K−L divergence0 1 2 3 4 5 6130 10 20 30 40 50Index(d)0 10 20 30 40 50Index(e)0 10 20 30 40 50Index(f)K−L divergence0 1 2 3 4 5 645K−L divergence0 1 2 3 4 5 6445K−L divergence0 1 2 3 4 5 6413450 10 20 30 40 50Index0 10 20 30 40 50Index0 10 20 30 40 50IndexVicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of março Skew (ICMC-USP) Normal - 2010 Distributions: 25 / 35

ApplicationsSimulated dataInfluence of outlying observationsTable: Simulated SN data. Posterior means and standard deviations of theparameters from fitting a ST–NLM.Data β 1 β 2 β 3 σ 2 λ νnames Mean SD Mean SD Mean SD Mean SD Mean SD Mean SDa 29.87 0.17 5.03 0.76 0.69 0.05 1.88 0.76 -2.26 1.36 12.31 8.95b 29.89 0.24 5.43 1.41 0.72 0.09 1.07 0.42 -0.55 0.96 2.88 0.79c 29.92 0.23 5.090 0.85 0.69 0.05 0.99 0.43 -0.77 0.98 2.87 0.80d 29.92 0.23 5.06 0.84 0.69 0.05 0.99 0.42 -0.77 0.98 2.87 0.80e 30.05 0.26 5.31 1.29 0.71 0.08 1.01 0.47 0.04 0.78 2.43 0.42f 30.22 0.28 5.19 1.32 0.70 0.08 1.17 0.58 0.47 0.72 2.31 0.31Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of março Skew (ICMC-USP) Normal - 2010 Distributions: 26 / 35

ApplicationsSimulated dataSimulated SN data. Index plots of K(P, P (−i) ) from fittinga ST-NLM.(a)(b)(c)K−L divergence0 1 2 3 4 5 6K−L divergence0 1 2 3 4 5 64K−L divergence0 1 2 3 4 5 6130 10 20 30 40 50Index(d)0 10 20 30 40 50Index(e)0 10 20 30 40 50Index(f)K−L divergence0 1 2 3 4 5 645K−L divergence0 1 2 3 4 5 6445K−L divergence0 1 2 3 4 5 64 13 450 10 20 30 40 50Index0 10 20 30 40 50Index0 10 20 30 40 50IndexVicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of março Skew (ICMC-USP) Normal - 2010 Distributions: 27 / 35

ApplicationsSimulated dataInfluence of outlying observationsTable: Simulated data. Comparison between SN–NLM and ST–NLM fitting byusing different Bayesian criteria.Data SN–NLM ST–NLMnames B DIC EAIC EBIC B DIC EAIC EBICa -76,802 153.338 157.928 167.4881 -77.141 153.949 159.539 171.011b -116.794 218.119 223.519 233.079 -93.142 176.988 182.389 193.861c -112.506 216.516 221.35 230.910 -90.133 176.508 182.578 194.050d -113.861 216.51 221.594 231.154 -90.766 176.834 182.824 194.296e -126.484 240.758 245.334 254.894 -99.465 195.338 201.348 212.820f -134.148 259.939 264.85 274.4101 -106.483 210.077 215.897 227.369∑where B = n log(ĈPO i ).i=1Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of março Skew (ICMC-USP) Normal - 2010 Distributions: 28 / 35

ApplicationsThe oil palm datasetThe oil palm datasetWe consider a Bayesian analysis of the data set presented in Foong (1999)that describe the oil palm yield. Assuming a nonlinear growth-curvemodel, we fit a NLM to the data as specified by Cancho et al. (2009)Y i =β 11 + β 2 exp(−β 3 x i ) + ε i, (9)where ε iiid ∼ SMSN(−√2π k 1∆, σ 2 , λ, H), for i = 1, . . . , 19. In our analysiswe assume SN-NLM, ST-NLM, SCN-NLM and SSL-NLM from the SMSNclass 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 março Skew (ICMC-USP) Normal - 2010 Distributions: 29 / 35

ApplicationsThe oil palm datasetThe oil palm datasetTable: 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-NLMParameter Mean SD Mean SD Mean SD Mean SDβ 1 37.251 0.491 37.565 0.469 37.461 0.538 37.421 0.478β 2 42.025 14.538 40.343 10.739 40.144 12.407 40.328 12.284β 3 0.730 0.065 0.717 0.050 0.712 0.057 0.720 0.058σ 2 6.447 2.890 1.966 1.482 2.674 1.907 2.484 2.022λ -2.603 2.070 -1.823 1.369 -3.423 2.108 -2.241 1.710ν - - 3.415 1.267 1.967 0.539 0.562 0.197γ - - - - - - 0.389 0.114Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of março Skew (ICMC-USP) Normal - 2010 Distributions: 30 / 35

ApplicationsThe oil palm datasetThe oil palm datasetTable: Oil palm yield data set. Comparison between SMSN–NLM by usingdifferent Bayesian criteria.criterion SN–NLM ST–NLM SSL–NLM SCN–NLMB -40.007 -38.949 -39.459 -39.146DIC 75.469 72.626 73.561 73.183EAIC 85.595 83.349 84.561 84.5612EBIC 90.317 88.015 90.227 90.172Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of março Skew (ICMC-USP) Normal - 2010 Distributions: 31 / 35

ApplicationsThe oil palm datasetThe oil palm datasetThe K-L divergence and related calibration are computed. We do not findhighly influential cases. The K(P, P (−i) ) is smaller that 0.71 and thecorresponding calibrations are smaller than 1. However, for SN-NLM,cases 13,18, 10 and 15 had larger K(P, P (−i) ) when compared with theother cases.Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of março Skew (ICMC-USP) Normal - 2010 Distributions: 32 / 35

ApplicationsThe oil palm datasetThe oil palm datasetTable: Oil palm yield data. Case influence diagnostics.SN–NLM ST–NLM SSL-NLM SCN–NLMCase 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 0.91218 0.703 0.934 0.290 0.832 0.307 0.818 0.561 0.91110 0.306 0.838 0.304 0.836 0.297 0.835 0.214 0.79515 0.272 0.823 0.244 0.810 0.258 0.817 0.256 0.816Vicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of março Skew (ICMC-USP) Normal - 2010 Distributions: 33 / 35

ApplicationsThe oil palm dataset(a)(b)K−L divergence0.0 0.5 1.0 1.5 2.010131518K−L divergence0.0 0.5 1.0 1.5 2.0101315 185 10 15Index(c)5 10 15Index(d)K−L divergence0.0 0.5 1.0 1.5 2.0101315 18K−L divergence0.0 0.5 1.0 1.5 2.0101315185 10 15Index5 10 15IndexVicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of março Skew (ICMC-USP) Normal - 2010 Distributions: 34 / 35

ConclusionsConclusions• 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 getBayesian inference for the proposed model• Bayesian case influence diagnostics based on the Kullback-Leibler divergence in order tostudy the sensitivity of the Bayesian estimates under perturbations in the model/data.• Our analysis indicates that a ST-NLMVicente G. Cancho (ICMC-USP), Victor H. Lachos Bayesian (IMECC-UNICAMP) Nonlinear Regression and Models Marinhowith Andrade Scale(ICMC-USP) Mixtures of março Skew (ICMC-USP) Normal - 2010 Distributions: 35 / 35

ReferencesBranco, 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. StatisticalPapers, 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 heteroscedasticsymmetric 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 OilCong. (Agric.), pages 265–287.Peng, F. & Dey, D. (1995). Bayesian analysis of outlier problems using divergence measures. The Canadian Journal ofStatistics/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 andbayesian 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 março Skew (ICMC-USP) Normal - 2010 Distributions: 35 / 35