Zero-inflated Poisson regression with right-censored data

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Zero-inflated Poisson regression with right-censored data

van Trinh Nguyen, Jean-François Dupuy

To cite this version:

van Trinh Nguyen, Jean-François Dupuy. Zero-inflated Poisson regression with right-censored data.

2018. �hal-01811949�

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Zero-inated Poisson regression with right-censored data

Nguyen Van Trinh^a, Jean-François Dupuy^a

aUniv Rennes, INSA Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France

Abstract

The zero-inated Poisson regression model is often used to analyse count data with an excess of zeros. This paper extends the model to randomly right-censored count data. Right- censoring occurs when one only knows that the true count value is higher than the observed one. In this setting, maximum likelihood estimators (MLE) are constructed and their properties are investigated. In particular, MLE are shown to be consistent and asymptotically normal. A simulation study is conducted to assess nite-sample behaviour of the MLE.

Finally, an application in health economics is described.

Keywords: Count data, excess of zeros, health-care utilization, large-sample properties, simulations

1. Introduction

Statistical modeling of count data is an important issue in various elds, including agricul- ture, econometrics, epidemiology, industrial applications, public health. . . Generalized linear models (McCullagh and Nelder, 1989) provide a powerful framework for analysing such data.

In many applications however, count data show an excess of zeros, that is, a number of zeros that cannot be explained by models based on standard distributional assumptions. A large number of statistical tools have been developed to tackle this issue, such as zero-inated regression models which mix a degenerate distribution with point mass of one at zero with a standard count regression model.

For example, zero-inated Poisson (ZIP) regression model was proposed by Lambert (1992) and further developed by Dietz and Böhning (2000), Lim et al. (2014) and Monod (2014), among many others. Recent variants of ZIP regression include random-eects ZIP models (Hall, 2000; Min and Agresti, 2005) and semiparametric ZIP models (Lam et al., 2006;

Feng and Zhu, 2011). Zero-inated negative binomial (ZINB) regression model was proposed by Ridout et al. (2001), see also Moghimbeigi et al. (2008) and Mwalili et al. (2008). When counts have an upper bound, ZIP and ZINB regression models are no longer appropriate. Hall (2000) thus introduced the zero-inated binomial (ZIB) model, see also Hall and Berenhaut (2002), Diop et al. (2011), Diop et al. (2016) and Diallo et al. (2017). Statistical modeling of bounded count data containing both extra zeros and extra right-endpoints has recently

Email addresses: van-trinh.nguyen@insa-rennes.fr (Nguyen Van Trinh),

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attracted attention. Deng and Zhang (2015) proposed a zero-one inated binomial regression model for such data, see also Tian et al. (2015) and Dupuy (2017). In Diallo et al. (2018), authors propose a zero-inated regression model for multinomial counts with joint zero- ination.

In this paper, we investigate estimation in zero-inated Poisson regression when the count response is subject to right-censoring. Right-censoring occurs when only a lower bound of the count of interest is observed, or, equivalently, when we know that the true count value is higher than the observed one. For example, in a study investigating demand for medical care, patients number of visits to a physician is right-censored at C if we only know that the true number of visits is greater than C. Ignoring censoring can yield biased estimates and incorrect statistical inference. Terza (1985) considers estimation in Poisson regression when the response is right-censored at a constant threshold. Famoye and Wang (2004) and Karlis et al. (2016) investigate random right-censoring in generalized Poisson regression model and in mixtures of Poisson regressions respectively. Estimation in zero- inated Poisson regression model with censoring is less documented. Saari and Adnan (2001) propose maximum likelihood estimation (MLE) in a ZIP model with censoring at a constant threshold. However, no asymptotic results are proved for the MLE (in particular, distributional properties of the MLE are not discussed). In the current paper, we undertake a comprehensive theoretical and numerical analysis of MLE in ZIP regression with random censoring.

The remainder of the paper is organized as follows. In Section 2, we recall the denition of ZIP regression model, we describe maximum likelihood estimation under random right- censoring and we introduce some useful notations. In Section 3, we establish consistency and asymptotic normality of the MLE. Section 4 reports results of a comprehensive simulation study. Saari and Adnan (2001) already conducted a simulation study in censored ZIP model. But censoring was xed at a constant threshold and their model had only one predictor for zero ination and two predictors for Poisson mean. In our paper, we consider random censoring and a larger number of predictors. Moreover, Saari and Adnan (2001) did not investigate the eect of the proportion of zero on the performance of the MLE, nor the nite-sample distribution of the MLE. These issues are discussed in our simulation study.

An application to a health-care utilization dataset is described in Section 5. A discussion and some perspectives are provided in Section 6.

2. Notations and likelihood calculation

In this section, we briey recall the denition of the ZIP model and we describe maximum likelihood estimation when the count response is randomly right censored.

2.1. Maximum likelihood estimation in ZIP regression with right-censoring

The ZIP model assumes that the response variableZ_i (where the lower indicei indicates the individual) is such that

Zi ∼

0 with probabilityωi,

P(λ_i) with probability1−ω_i, (2.1)

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where P(λ_i)denotes Poisson distribution with parameter λ_i >0. Obviously, the ZIP model reduces to a standard Poisson distribution ifω_i = 0. In ZIP regression, the mixing probability ω_i and parameterλ_i are usually modeled by logistic and log-linear models respectively, that is:

logit(ω_i(γ)) =γ^>W_i, (2.2)

and

log(λ_i(β)) = β^>X_i, (2.3)

whereX_i = (1, X_i2, . . . , X_ip)^>and W_i = (1, W_i2, . . . , W_iq)^>are random vectors of predictors or covariates (both categorical and continuous covariates are allowed), β ∈ R^p and γ ∈ R^q are unknown parameters and > denotes the transpose operator.

Assume that we observe n independent vectors(Z₁,X₁,W₁), . . . ,(Z_n,X_n,W_n) from the model (2.1)-(2.2)-(2.3), all dened on the probability space (Ω,C,P). The log-likelihood of (β, γ)based on these observations is (see Dupuy, 2018):

n

X

i=1

n

1{Zi=0}log

e^γ^>^Wⁱ+e⁻^exp(β^>^Xⁱ⁾

+ 1{Zi>0}

Ziβ^>Xi−e^β^>^Xⁱ −log(Zi!)

−log

1 +e^γ^>^Wⁱ o

. The maximum likelihood estimator of (β, γ) is obtained by maximizing this function. This

estimator is consistent and asymptotically normally distributed (see Czado et al., 2007).

Assume now that the count response Z_i can be right-censored, that is, for some individ- uals, we only observe a lower bound on Z_i. This can be modeled by introducing a censoring random variable Ci and by dening the observation for the i-th individual as the vector (Z_i^∗, δ_i,X_i,W_i), where Z_i^∗ = min(Z_i, C_i) and δ_i = 1_{Z_i_<C_i_} (if Z_i = C_i, we let Z_i^∗ = C_i and δ_i = 0). Let J_i = 1{Z_i^∗=0}. The likelihood of ψ := (β^>, γ^>)^> based on observations (Z_i^∗, δi,Xi,Wi),i= 1, . . . , n is calculated as:

L_n(ψ) =

n

Y

i=1

P(Z_i =Z_i^∗|X_i,W_i)^δⁱP(Z_i ≥Z_i^∗|X_i,W_i)^1−δⁱ,

=

n

Y

i=1

P(Z_i =Z_i^∗|X_i,W_i)^1−JⁱP(Z_i = 0|X_i,W_i)^Jⁱδi

P(Z_i ≥Z_i^∗|X_i,W_i)^(1−δⁱ^)(1−Jⁱ⁾,

=

n

Y

i=1



 e^−λⁱλ^Z_iⁱ^∗

Z_i^∗!(1−ωi)

!^1−Ji

ωi+ (1−ωi)e^−λⁱJi





δi

×



1−

Z^∗_i−1

X

k=0

e^−λⁱλ^k_i

k!(1−ω_i)−ω_i





(1−δi)(1−Ji)

,

from which we easily obtain the loglikelihood . If and are given by

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(2.2) and (2.3), straightforward algebra yields:

`_n(ψ) =

n

X

i=1

n δ_ih

J_ilog

e^γ^>^Wⁱ+e⁻^exp(β^>^Xⁱ⁾

+ (1−J_i)

Z_i^∗β^>X_i−e^β^>^Xⁱ−log(Z_i^∗!)i

+(1−δ_i)(1−J_i) ln



1−

Z_i^∗−1

X

k=0

e⁻^exp(β^>^Xⁱ^)+kβ^>^Xⁱ k!



−log

1 +e^γ^>^Wⁱ





 . Note that `_n(ψ) reduces to the log-likelihood given above when there is no censoring (that is, whenδ_i = 1 for all i= 1, . . . , n).

The maximum likelihood estimator ψˆ_n := ( ˆβ_n^>,γˆ_n^>)^> ofψ is solution of the k-dimensional score equation

∂`n(ψ)

∂ψ = 0, (2.4)

where k = p+q. In the next section, we establish existence, consistency and asymptotic normality of ψˆ_n. First, we need to introduce some further notations.

2.2. Some additional notations

In what follows, we note ki(γ) = e^γ^>^Wⁱ and Li(β) = e⁻^exp(β^>^Xⁱ⁾, i = 1, . . . , n. Let also S_λ_i_(β)(u) = P(P(λ_i(β)) ≥ u), u = 0,1, . . . denote the survival function of P(λ_i(β)) distribution. We have:

∂`_n(ψ)

∂β_` =

n

X

i=1

X_i`

−δ_iJ_i λ_i(β)L_i(β)

k_i(γ) +L_i(β) +δ_i(1−J_i) (Z_i^∗−λ_i(β)) (2.5)

−(1−δ_i)(1−J_i)

Z_i^∗−1

X

k=0

L_i(β)λ^k_i(β)(k−λ_i(β)) k!Sλi(β)(Z_i^∗)



, ` = 1, . . . , p, and

∂`_n(ψ)

∂γ_` =

n

X

i=1

W_i`

δ_iJ_ik_i(γ)

k_i(γ) +L_i(β) − k_i(γ) k_i(γ) + 1

, ` = 1, . . . , q. (2.6) Let

u_i(ψ) = λi(β)Li(β)

(k_i(γ) +L_i(β))²[k_i(γ) +L_i(β)−λ_i(β)k_i(γ)], i= 1, . . . , n, and for Z_i^∗ ≥1, let

v_i(ψ) =

Z_i^∗−1

X

k=0

L_i(β)λ^k_i(β) k!S_λ²

i(β)(Z_i^∗)

S_λ_i_(β)(Z_i^∗) (λ_i(β)−k)²−λ_i(β)

−λ_i(β)(k−λ_i(β))P(P(λ_i(β)) =Z_i^∗−1)}, i= 1, . . . , n.

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Then, some tedious albeit not dicult algebra shows that

∂²`n(ψ)

∂β_`∂β_m =

n

X

i=1

X_i`X_im{−δ_iJ_iu_i(ψ)−δ_i(1−J_i)λ_i(β)−(1−δ_i)(1−J_i)v_i(ψ)}, `, m= 1, . . . , p

∂²`n(ψ)

∂β_`∂γ_m =

n

X

i=1

Xi`Wim

δiJiki(γ)λi(β)Li(β)

(k_i(γ) +L_i(β))² , `= 1, . . . , p and m= 1, . . . , q

∂²`_n(ψ)

∂γ_`∂γ_m =

n

X

i=1

Wi`Wimki(γ)

δ_iJ_iL_i(β)

(k_i(γ) +L_i(β))² − 1 (k_i(γ) + 1)²

, `, m= 1, . . . , q.

We note S_n(ψ) = ∂`_n(ψ)/∂ψ, H_n(ψ) = −∂²`_n(ψ)/∂ψ∂ψ^>, F_n(ψ) = E(H_n(ψ)) and I_k the identity matrix of order k. H_n(ψ) is assumed positive denite.

3. Asymptotic results

In this section, we establish consistency and asymptotic normality ofψˆ_n. In what follows, the spaceR^kofk-dimensional vectors is provided with the Euclidean normk·k₂ and the space of (k×k) real matrices is provided with the norm |||A|||₂ := sup_kxk₂₌₁kAxk₂ (for notations simplicity, we use k · k for both norms). Recall that for a symmetric real (k×k)-matrix A with eigenvalues λ₁, . . . , λ_k, kAk= max_i|λ_i|(from now on, λ_min(A)and λ_max(A) will denote the smallest and largest eigenvalues of A respectively).

We rst state some regularity conditions:

C1 Covariates are bounded, that is, there exist compact sets X ⊂ R^p and W ⊂ R^q such that X_i ∈ X and W_i ∈ W for every i= 1,2, . . .

C2 The true parameter value ψ0 = (β₀^>, γ₀^>)^> lies in the interior of some known compact and convex set C =B × G ⊂R^k (where B ⊂ R^p and G ⊂R^q are the parameter spaces of β and γ respectively).

C3 There exists a positive constantc1 such thatn/λmin(Fn(ψ0))≤c1 for every n= 1,2, . . . C4 Censoring random variables C_i, i = 1,2, . . . are strictly positive and bounded by some constant M <∞ (for example, M can be the end of the study period, at which every individual still under study is censored).

Conditions C1-C3 are classical in generalized linear regression and zero-inated regression models (see Fahrmeir and Kaufmann, 1985; Czado et al., 2007). Condition C4 is required in the censored setting.

For each n = 1,2, . . . and ε > 0, dene the neighbourhood N_n(ε) = {ψ ∈ C : (ψ − ψ₀)^>F_n(ψ−ψ₀)≤ε²} of ψ₀, where F_n is a short notation forF_n(ψ₀). Our rst result states that the solution of (2.4) exists, lies in the neighbourhoodN_n(ε)ofψ₀ when n is suciently large and is consistent for ψ₀.

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Theorem 3.1 (Existence and consistency). Assume conditions C1-C4 hold. Then the probability that ψˆ_n exists and lies in N_n(ε) for some ε tends to 1 as n → ∞. Furthermore, ψˆ_n converges in probability to ψ₀ as n→ ∞.

Proof of Theorem 3.1. Our proof follows the lines of Fahrmeir and Kaufmann (1985) but technical details are dierent. Moreover, we rely on dierent arguments in several parts, leading to more direct proofs. A technical lemma is proved in an Appendix.

i) We rst prove asymptotic existence of ψˆn. We show that for every η > 0, there exists ε >0and n₁ ∈N such that

P(`_n(ψ)−`_n(ψ₀)<0 for all ψ ∈∂N_n(ε))≥1−η, for n ≥n₁, (3.7) where∂N_n(ε)is the boundary{ψ ∈ C : (ψ−ψ₀)^>F_n(ψ−ψ₀) = ε²}ofN_n(ε). This will imply the existence of a local maximum of `_n in N_n(ε). Positive-deniteness of H_n and convexity of C will ensure that this maximum is global and unique.

In fact, equivalently to (3.7), we show that for every η >0, there existsε >0and n₁ ∈N such that

P(`n(ψ)−`n(ψ0)≥0for some ψ ∈∂Nn(ε))≤η, for n≥n₁. To see this, we use Taylor's expansion to write

`n(ψ)−`n(ψ0) = (ψ −ψ0)^TSn(ψ0)− 1

2(ψ−ψ0)^THn( ˜ψ)(ψ−ψ0), := (ψ −ψ₀)^TS_n(ψ₀)−Q_n(ψ),

where ψ˜=aψ+ (1−a)ψ₀ (for some 0≤a ≤1) lies between ψ and ψ₀. Let 0< c < ¹₂ and write f.s. for "for some". Then we have:

P(`_n(ψ)−`_n(ψ₀)≥0, f.s. ψ ∈∂N_n(ε))

=P (ψ−ψ₀)^TS_n(ψ₀)≥Q_n(ψ) and Q_n(ψ)> cε², f.s. ψ ∈∂N_n(ε)

+P (ψ−ψ₀)^TS_n(ψ₀)≥Q_n(ψ) and Q_n(ψ)≤cε², f.s. ψ ∈∂N_n(ε) ,

≤P(A) +P(B),

where A and B denote events A = {(ψ −ψ₀)^TS_n(ψ₀) > cε², f.s. ψ ∈ ∂N_n(ε)} and B = {Q_n(ψ)≤cε², f.s. ψ ∈∂N_n(ε)} respectively. Letu_n(ψ) = ¹_εF

1

n2(ψ −ψ₀). Then A = {u_n(ψ)^>F⁻

1

n 2S_n(ψ₀)> cε, f.s. ψ ∈∂N_n(ε)},

⊆ { sup

ψ∈∂Nn(ε)

|u_n(ψ)^>F⁻

1

n 2S_n(ψ₀)|> cε},

⊆ { sup

kun(ψ)k=1

|u_n(ψ)^>F⁻

1

n 2S_n(ψ₀)|> cε},

= {kF⁻

1

n 2S_n(ψ₀)k> cε}.

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where the second to third line comes from the fact that ψ ∈ ∂N_n(ε) implies ku_n(ψ)k = 1. It follows that P(A) ≤ P(kF⁻

1

n 2S_n(ψ₀)k > cε). By Theorem 1.5 of Seber and Lee (2012), EkF⁻

1

n 2Sn(ψ0)k² =k and Chebyshev's inequality implies P(A)≤ k

c²ε². Finally, letting ε=q

2k

ηc² implies that P(A)≤η/2. Now,

B =

1

2(ψ−ψ₀)^TH_n( ˜ψ)(ψ −ψ₀)≤cε², f.s. ψ ∈∂N_n(ε)

,

= 1

2u_n(ψ)^>F⁻

1

n 2H_n( ˜ψ)F⁻

1

n 2u_n(ψ)≤c, f.s. ψ ∈∂N_n(ε)

,

⊆ 1

2λ_min F⁻

1

n 2H_n( ˜ψ)F⁻

1

n 2

u_n(ψ)^>u_n(ψ)≤c, f.s. ψ ∈∂N_n(ε)

,

= 1

2λ_min F⁻

1

n 2H_n( ˜ψ)F⁻

1

n 2

≤c, f.s. ψ ∈∂N_n(ε)

. Thus,P(B)≤P(there exists ψ ∈∂N_n(ε)such that λ_min(F⁻

1

n 2H_n( ˜ψ)F⁻

1

n 2)≤ 2c). By Lemma 6.1 in Appendix, F⁻

1

n 2H_n(ψ)F⁻

1

n 2 converges in probability to I_k uniformly in ψ ∈ N_n(ε), as n → ∞. Thus, by Maller (2003), λ_min(F⁻

1

n 2H_n(ψ)F⁻

1

n 2) converges in probability to 1 uniformly inψ ∈N_n(ε), as n → ∞.

If ψ˜=aψ+ (1−a)ψ0 for some 0≤a≤1and ψ ∈Nn(ε), then kF

1

n2( ˜ψ−ψ₀)k = kF

1

n2(aψ+ (1−a)ψ₀−ψ₀)k,

= akF

1

n2(ψ−ψ₀)k,

≤ kF

1

n2(ψ−ψ₀)k,

≤ , and thusψ˜∈Nn(ε). If follows thatλmin(F⁻

1

n 2Hn( ˜ψ)F⁻

1

n 2)converges in probability to1asn→

∞, since|λ_min(F⁻

1

n 2H_n( ˜ψ)F⁻

1

n 2)−1| ≤sup_ψ∈N_n_(ε)|λ_min(F⁻

1

n 2H_n(ψ)F⁻

1

n 2)−1|. Therefore, forn suciently large (say,n ≥n₁),P(there exists ψ ∈∂N_n(ε) such that λ_min(F⁻

1

n 2H_n( ˜ψ)F⁻

1

n 2)≤ 2c)≤η/2, since 2c < 1. This implies that P(B)≤η/2. Finally,

P(`n(ψ)−`n(ψ0)≥0, f.s. ψ ∈∂Nn(ε))≤P(A) +P(B)≤η,

which proves (3.7) and in turn, the existence of a unique global maximum of `_n on N_n(ε), which coincides withψˆ_n.

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ii) We turn to the consistency of ψˆ_n. We have:

λ_min(F_n)kψˆ_n−ψ₀k² = ( ˆψ_n−ψ₀)^>λ_min(F_n)I_k( ˆψ_n−ψ₀),

≤ ( ˆψ_n−ψ₀)^>F_n( ˆψ_n−ψ₀),

= kFn¹²( ˆψ_n−ψ₀)k²,

≤ ε²,

with probability tending to 1 as n → ∞, by i). By condition C3, λ_min(F_n) tends to ∞ as n→ ∞. Therefore kψˆ_n−ψ₀k converges to 0 with probability tending to 1 asn→ ∞, which

concludes the proof.

Our second result is:

Theorem 3.2 (Asymptotic normality). Assume conditions C1-C4 hold. Then F

1

n2( ˆψ_n− ψ0) converges in distribution to the Gaussian vector N(0, Ik), as n → ∞.

Proof of Theorem 3.2. Our proof proceeds along the same lines as the proof of asymptotic normality of MLE in uncensored zero-inated generalized Poisson regression (Czado and Min, 2005). However, Czado and Min (2005) use a central limit theorem with Lyapunov condition.

Here, we rely on the weaker Lindeberg condition, which yields a much shorter proof.

We rst prove asymptotic normality of the normalized score vectorF⁻

1

n 2S_n, whereS_nis a short notation for S_n(ψ₀). Let u be any vector in R^k. We show thatu^>F⁻

1

n 2S_n converges in distribution toN(0, u^>u)(without loss of generality, we set kuk= 1). From (2.5) and (2.6), we remark that Sn can be written as a sum Sn = Pn

i=1Sn,i of independent k-dimensional random vectors S_n,i = (S_n,i,1, . . . , S_n,i,k)^>. It is not dicult to see that under conditions C1, C2 and C4, components of S_n,i are bounded by some nite positive constant c₂ that is,

|Sn,i,`|< c2, `= 1, . . . , k. Therefore, kSn,ik² < c3 :=kc²₂. Let

u^>F⁻

1

n 2S_n =u^>F⁻

1

n 2

n

X

i=1

S_n,i:=

n

X

i=1

S_n,i^∗ . Then E(S_n,i^∗ ) = 0 and var(Pn

i=1S_n,i^∗ ) = 1. We now verify Lindeberg condition, namely:

for every ε >0,

n

X

i=1

E

S_n,i^∗21{|S_n,i^∗ |>ε}

→0 asn → ∞.

Letε >0. We have:

n

X

i=1

E

S_n,i^∗21_{|S^∗

n,i|>ε}

≤

n

X

i=1

E

kuk²kFn⁻¹²k²kS_n,ik²1_{|S^∗

n,i|>ε}

,

≤ c₁c₃ n

n

X

i=1

E(1{|S_n,i^∗ |>ε}),

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by condition C3. Now, {|S_n,i^∗ | > ε} implies that {λ_min(F_n) < c₃/ε²}, therefore, 1_{|S^∗

n,i|>ε} ≤ 1_{λ_min_(F_n_)<c₃_/ε²_} and thus,

n

X

i=1

E

S_n,i^∗21{|S^∗_n,i|>ε}

≤ c₁c₃ n

n

X

i=1

1{λ_min(Fn)<c3/ε²} =c₁c₃1{λ_min(Fn)<c3/ε²}. Under C3, λ_min(F_n) → ∞ as n → ∞. Therefore, Pn

i=1E(S_n,i^∗21{|S^∗_n,i|>ε}) → 0 as n → ∞. It follows that for every u ∈ R^k, u^>F⁻

1

n 2Sn converges in distribution to N(0,1) and by Cramer-Wold device, F⁻

1

n 2S_n converges in distribution to N(0, I_k). Weak convergence of F

1

n2( ˆψ_n−ψ₀) is now obtained as usual, by expanding S_n:=S_n(ψ₀) about ψˆ_n. The rest of the proof is similar to proof of Theorem 3 of Fahrmeir and Kaufmann

(1985) and is thus omitted.

In order to construct asymptotic condence intervals and tests of hypothesis for the components ofψ, one needs to estimateFn^1/2 (by Fn^1/2( ˆψ_n) for example). Asymptotic normality of Fn^1/2( ˆψ_n)( ˆψ_n−ψ₀) still holds if one assumes that there exists a nite constant c₄ such that λmax(Fn)/λmin(Fn)≤c4 forn suciently large. This can be proved as in Fahrmeir and Kaufmann (1985) and is omitted.

4. Simulation study

In this section, we investigate nite-samples properties of the MLE under various scenarios obtained by varying the censoring and zero-ination proportions and the sample size.

4.1. Simulation design

We simulate the data according to the ZIP model (2.1)-(2.2)-(2.3) dened by:

log(λ_i(β)) =β₁X_i1+β₂X_i2+β₃X_i3+β₄X_i4+β₅X_i5+β₆X_i6, and

logit(ω_i(γ)) =γ₁W_i1+γ₂W_i2 +γ₃W_i3+γ₄W_i4,+γ₅W_i5,

where X_i1 = W_i1 = 1 and the X_i2, . . . , X_i6, W_i4, W_i5 are independently drawn from normal N(0,1), Bernoulli B(0.3), normal N(1,2.25), exponentialE(1), uniform U(2,5), normal N(−1,1) and Bernoulli B(0.5) distributions respectively. Linear predictors in log(λ_i(β)) and logit(ω_i(γ)) are allowed to share common terms by letting W_i2 = X_i2 and W_i3 = X_i3. We consider the following sample sizes: n = 500,1000,2500. The regression parameter β is chosen as β = (0.7,0.1,0.4,0.85,−0.5,0)^>. The regression parameterγ is chosen as:

case 1: γ = (−0.9,−0.65,−0.2,0.65,0)^>,

case 2: γ = (0.25,−0.7,−0.2,0.65,0)^>.

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Using these values, in case 1 (respectively case 2), the average percentage of zero-ination in the simulated data sets is 20% (respectively 40%). Censoring values are simulated from a zero-truncated Poisson model with parameter µ, where µis chosen to yield various average censoring proportions c in the simulated samples, namely c = 0.1,0.2,0.4. For purpose of comparison, we also provide results that would be obtained if there were no censoring (that is, when c= 0) since these results will constitute a benchmark for assessing performance of the MLE when censoring is present.

For each combination of the simulation design parameters (sample size, proportions of censoring and zero-ination), we simulate N = 1000 samples and we calculate the MLE ψˆn. Simulations are carried out using the statistical software R. To solve the likelihood equation, we use the package maxLik (Henningsen and Toomet, 2011) which implements various Newton-Raphson-like algorithms. We obtain starting values by estimating a ZIP model without taking censoring into account (this step is carried out using the function zeroinfl of the R package pscl, see Zeileis et al. (2008) and Jackman (2017)).

4.2. Results

For each conguration [sample size×censoring proportion×zero-inflation proportion] of the simulation parameters, we calculate the average bias and average relative bias (expressed as a percentage) of the estimatesβˆ_j,n and ˆγ_k,n over theN simulated samples.

For example, the relative bias of βˆ_j,n is obtained as 1

N

X

t=1

βˆ_j,n^(t)−β_j

β_j ×100,

where βˆ_j,n^(t) denotes the MLE ofβ_j in the t-th simulated sample. We also obtain the average standard error (SE), empirical standard deviation (SD) and root mean square error (RMSE) for each βˆ_j,n (j = 1, . . . ,6) and γˆ_k,n (k = 1, . . . ,5). Finally, we provide the empirical coverage probability (CP) and average length of 95%-level condence intervals for the β_j and γ_k. Results are given in Table 1 (case 1, n = 500), Table 2 (case 2, n = 500), Table 3 (case 1, n= 1000), Table 4 (case 2, n= 1000), Table 5 (case 1,n = 2500) and Table 6 (case 2, n= 2500).

From these results, we observe, as expected, that accuracy of MLEs of both β_j and γ_k decreases as sample size decreases. Accuracy of β_js estimates also decreases as censoring increases (note that the relative bias stays moderate though, even when censoring is high).

On the contrary, estimates of the γ_k are rather insensitive to censoring, which can be explained by the fact that censoring does not aect zero counts. For bothβ_j and γ_k, empirical coverage probabilities are close to the nominal condence level in every case. As may also be expected, for a given censoring proportion, we observe that MLEs of theβ_j (respectively γ_k) perform better when the zero-ination proportion decreases (respectively increases).

Finally, in order to assess quality of the Gaussian approximation stated in Theorem 3.2, we obtain normal Q-Q plots of the estimates and histograms of the normalized estimates (βb_j,n−β_j)/standard error(βb_j,n),j = 1, . . . ,6and(bγ_k,n−γ_k)/standard error(bγ_k,n),j = 1, . . . ,5.

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We provide these graphs for n = 500 with a proportion of zero-ination equal to 0.4 and 40% of censoring (see gures 1 to 4). Plots for the other (and more favorable) simulated scenarios yield similar observations and are thus not given. From these gures, it appears that the Gaussian approximation of the distribution of the MLE is reasonably satised, even when the sample size is moderate and the proportions of zero-ination and censoring are as high as 0.4.

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βbn bγn

average proportion

of censoring βb_1,n βb_2,n βb_3,n βb_4,n βb_5,n βb_6,n bγ_1,n bγ_2,n γb_3,n bγ_4,n bγ_5,n 0 bias 0.0029 -0.0013 0.0008 0.0005 -0.0011 -0.0014 -0.0241 -0.0191 0.0060 0.0180 -0.0077

rel. bias 0.4189 -1.3293 0.2094 0.0598 0.2221 - 2.6778 2.9395 -3.0042 2.7756 - SD 0.0921 0.0196 0.0376 0.0136 0.0277 0.0215 0.2707 0.1615 0.3426 0.1586 0.3005 SE 0.0878 0.0189 0.0374 0.0132 0.0276 0.0211 0.2571 0.1613 0.3258 0.1609 0.2973 RMSE 0.1273 0.0273 0.0530 0.0189 0.0391 0.0301 0.3740 0.2290 0.4727 0.2266 0.4227 CP 0.9500 0.9430 0.9480 0.9380 0.9510 0.9540 0.9400 0.9440 0.9400 0.9520 0.9540

` 0.3433 0.0740 0.1463 0.0512 0.1076 0.0824 1.0048 0.6298 1.2729 0.6285 1.1633 0.1 bias -0.0049 -0.0009 0.0025 0.0041 -0.0015 -0.0009 -0.0278 -0.0198 0.0069 0.0192 -0.0073

rel. bias -0.7045 -0.9439 0.6349 0.4850 0.2904 - 3.0882 3.0520 -3.4358 2.9562 - SD 0.1263 0.0273 0.0548 0.0266 0.0347 0.0297 0.2725 0.1621 0.3448 0.1587 0.3014 SE 0.1207 0.0266 0.0549 0.0273 0.0352 0.0296 0.2585 0.1619 0.3272 0.1613 0.2979 RMSE 0.1747 0.0381 0.0776 0.0384 0.0494 0.0419 0.3765 0.2300 0.4753 0.2271 0.4237 CP 0.9430 0.9390 0.9440 0.9590 0.9490 0.9530 0.9400 0.9490 0.9380 0.9540 0.9530

` 0.4725 0.1042 0.2149 0.1070 0.1377 0.1157 1.0101 0.6323 1.2781 0.6299 1.1654 0.2 bias -0.0098 -0.0008 0.0024 0.0069 -0.0031 0.0002 -0.0292 -0.0203 0.0069 0.0198 -0.0077

rel. bias -1.4064 -0.8357 0.5971 0.8076 0.6201 - 3.2453 3.1206 -3.4564 3.0470 - SD 0.1500 0.0330 0.0697 0.0349 0.0419 0.0361 0.2731 0.1631 0.3475 0.1587 0.3013 SE 0.1439 0.0327 0.0683 0.0355 0.0413 0.0361 0.2589 0.1624 0.3281 0.1614 0.2981 RMSE 0.2081 0.0464 0.0976 0.0502 0.0589 0.0511 0.3773 0.2310 0.4778 0.2272 0.4238 CP 0.9440 0.9420 0.9540 0.9460 0.9570 0.9520 0.9400 0.9500 0.9380 0.9540 0.9540

` 0.5635 0.1279 0.2676 0.1390 0.1615 0.1415 1.0118 0.6341 1.2815 0.6303 1.1661 0.4 bias 0.0016 0.0005 0.0050 0.0143 -0.0053 -0.0032 -0.0310 -0.0201 0.0062 0.0205 -0.0071

rel. bias 0.2347 0.4716 1.2491 1.6786 1.0591 - 3.4448 3.0875 -3.0980 3.1517 - SD 0.2018 0.0503 0.0997 0.0534 0.0570 0.0522 0.2741 0.1652 0.3503 0.1593 0.3020 SE 0.2063 0.0491 0.1042 0.0535 0.0567 0.0537 0.2601 0.1638 0.3312 0.1617 0.2984 RMSE 0.2885 0.0703 0.1443 0.0769 0.0805 0.0749 0.3790 0.2335 0.4820 0.2278 0.4245 CP 0.9520 0.9540 0.9610 0.9430 0.9540 0.9490 0.9450 0.9460 0.9370 0.9530 0.9500

` 0.8074 0.1922 0.4079 0.2092 0.2216 0.2101 1.0163 0.6395 1.2935 0.6312 1.1673

Table 1: Simulation results (n= 500, ZI proportion = 20%). SD: empirical standard deviation. SE: average standard error. RMSE: empirical

12

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βbn bγn

average proportion

of censoring βb_1,n βb_2,n βb_3,n βb_4,n βb_5,n βb_6,n bγ_1,n bγ_2,n γb_3,n bγ_4,n bγ_5,n 0 bias 0.0021 -0.0006 0.0014 0.0001 0.0009 -0.0012 0.0051 -0.0166 -0.0089 0.0118 -0.0033

rel. bias 0.3032 -0.5554 0.3613 0.0162 -0.1714 - 2.0303 2.3779 4.4416 1.8185 - SD 0.1045 0.0231 0.0442 0.0156 0.0334 0.0246 0.2234 0.1358 0.2535 0.1306 0.2442 SE 0.1038 0.0226 0.0437 0.0158 0.0323 0.0247 0.2155 0.1315 0.2546 0.1294 0.2349 RMSE 0.1473 0.0323 0.0621 0.0222 0.0465 0.0349 0.3103 0.1897 0.3593 0.1841 0.3388 CP 0.9530 0.9500 0.9480 0.9530 0.9390 0.9520 0.9460 0.9530 0.9540 0.9460 0.9450

` 0.4056 0.0883 0.1708 0.0614 0.1260 0.0967 0.8438 0.5144 0.9973 0.5061 0.9205 0.1 bias -0.0039 -0.0002 0.0014 0.0049 -0.0012 -0.0008 0.0030 -0.0172 -0.0094 0.0125 -0.0034

rel. bias -0.5502 -0.1754 0.3451 0.5726 0.2389 - 1.1946 2.4539 4.6880 1.9284 - SD 0.1496 0.0340 0.0699 0.0348 0.0458 0.0370 0.2237 0.1361 0.2540 0.1308 0.2443 SE 0.1506 0.0341 0.0689 0.0357 0.0438 0.0370 0.2164 0.1320 0.2556 0.1296 0.2352 RMSE 0.2123 0.0482 0.0981 0.0501 0.0634 0.0523 0.3112 0.1904 0.3604 0.1845 0.3391 CP 0.9440 0.9490 0.9430 0.9560 0.9440 0.9440 0.9470 0.9540 0.9520 0.9470 0.9470

` 0.5894 0.1334 0.2698 0.1399 0.1709 0.1450 0.8475 0.5165 1.0012 0.5069 0.9216 0.2 bias -0.0068 0.0016 0.0071 0.0095 -0.0041 -0.0004 0.0015 -0.0169 -0.0079 0.0130 -0.0036

rel. bias -0.9652 1.6303 1.7772 1.1197 0.8294 - 0.5822 2.4105 3.9740 1.9938 - SD 0.1915 0.0452 0.0887 0.0485 0.0556 0.0484 0.2240 0.1378 0.2549 0.1309 0.2443 SE 0.1906 0.0448 0.0915 0.0489 0.0542 0.0483 0.2170 0.1326 0.2565 0.1297 0.2354 RMSE 0.2702 0.0636 0.1276 0.0695 0.0777 0.0683 0.3118 0.1919 0.3616 0.1847 0.3392 CP 0.9470 0.9490 0.9600 0.9530 0.9460 0.9480 0.9430 0.9550 0.9550 0.9460 0.9480

` 0.7457 0.1750 0.3579 0.1912 0.2117 0.1889 0.8496 0.5186 1.0047 0.5074 0.9222 0.4 bias -0.0110 0.0078 0.0254 0.0305 -0.0154 0.0015 -0.0028 -0.0160 -0.0059 0.0147 -0.0039

rel. bias -1.5772 7.8277 6.3481 3.5830 3.0722 - -1.1201 2.2840 2.9409 2.2563 - SD 0.3577 0.0937 0.1880 0.0926 0.0944 0.0944 0.2264 0.1408 0.2632 0.1316 0.2452 SE 0.3550 0.0887 0.1850 0.0912 0.0924 0.0932 0.2191 0.1361 0.2627 0.1302 0.2359 RMSE 0.5040 0.1293 0.2649 0.1335 0.1330 0.1327 0.3149 0.1964 0.3719 0.1857 0.3402 CP 0.9520 0.9410 0.9490 0.9430 0.9480 0.9450 0.9490 0.9490 0.9530 0.9510 0.9470

` 1.3864 0.3456 0.7212 0.3555 0.3601 0.3643 0.8578 0.5321 1.0287 0.5092 0.9241

13

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βbn bγn

average proportion

of censoring βb_1,n βb_2,n βb_3,n βb_4,n βb_5,n βb_6,n bγ_1,n bγ_2,n γb_3,n bγ_4,n γb_5,n 0 bias 0.0008 0.0005 0.0006 0.0001 -0.0001 -0.0002 -0.0135 -0.0059 -0.0003 0.0032 0.0041

rel. bias 0.1106 0.4539 0.1535 0.0141 0.0298 - 1.4961 0.9068 0.1458 0.4936 - SD 0.0605 0.0134 0.0253 0.0090 0.0198 0.0145 0.1719 0.1124 0.2299 0.1127 0.2031 SE 0.0612 0.0131 0.0260 0.0090 0.0192 0.0147 0.1786 0.1116 0.2257 0.1118 0.2065 RMSE 0.0860 0.0187 0.0362 0.0128 0.0276 0.0206 0.2482 0.1584 0.3221 0.1587 0.2896 CP 0.9590 0.9470 0.9550 0.9540 0.9400 0.9540 0.9650 0.9500 0.9480 0.9510 0.9510

` 0.2394 0.0512 0.1018 0.0352 0.0752 0.0575 0.6994 0.4365 0.8834 0.4376 0.8090 0.1 bias -0.0006 0.0005 0.0021 0.0012 -0.0003 -0.0004 -0.0149 -0.0063 0.0004 0.0036 0.0043

rel. bias -0.0901 0.4610 0.5329 0.1424 0.0544 - 1.6521 0.9691 -0.2008 0.5582 - SD 0.0854 0.0189 0.0374 0.0191 0.0245 0.0207 0.1721 0.1128 0.2295 0.1129 0.2032 SE 0.0846 0.0186 0.0384 0.0191 0.0246 0.0207 0.1794 0.1119 0.2263 0.1120 0.2067 RMSE 0.1202 0.0266 0.0536 0.0270 0.0347 0.0293 0.2490 0.1589 0.3222 0.1590 0.2898 CP 0.9450 0.9420 0.9560 0.9550 0.9470 0.9570 0.9670 0.9510 0.9530 0.9490 0.9500

` 0.3315 0.0730 0.1504 0.0750 0.0965 0.0812 0.7024 0.4379 0.8859 0.4382 0.8096 0.2 bias -0.0007 0.0006 0.0045 0.0025 -0.0016 -0.0006 -0.0158 -0.0064 0.0014 0.0038 0.0041

` 0.3947 0.0895 0.1871 0.0971 0.1131 0.0991 0.7035 0.4389 0.8878 0.4383 0.8099 0.4 bias 0.0004 0.0014 0.0074 0.0067 -0.0042 -0.0007 -0.0171 -0.0063 0.0020 0.0041 0.0041

rel. bias 0.0544 1.4001 1.8431 0.7844 0.8493 - 1.8998 0.9736 -0.9913 0.6250 - SD 0.1442 0.0339 0.0721 0.0370 0.0398 0.0371 0.1724 0.1144 0.2307 0.1132 0.2034 SE 0.1444 0.0342 0.0726 0.0372 0.0396 0.0375 0.1804 0.1131 0.2286 0.1121 0.2068 RMSE 0.2040 .0482 0.1026 0.0529 0.0563 0.0528 0.2501 0.1609 0.3247 0.1594 0.2901 CP 0.9580 0.9600 0.9570 0.9610 0.9560 0.9560 0.9690 0.9490 0.9540 0.9490 0.9490

` 0.5655 0.1338 0.2845 0.1458 0.1549 0.1469 0.7062 0.4424 0.8947 0.4387 0.8102

14

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βbn bγn

average proportion

of censoring βb_1,n βb_2,n βb_3,n βb_4,n βb_5,n βb_6,n bγ_1,n bγ_2,n bγ_3,n bγ_4,n γb_5,n 0 bias -0.0011 0.0002 -0.0002 0.0003 0.0000 -0.0001 0.0087 -0.0069 -0.0010 0.0116 -0.0016

rel. bias -0.1638 0.2100 -0.0606 0.0405 0.0012 - 3.4745 0.9848 0.5148 1.7888 - SD 0.0702 0.0159 0.0310 0.0106 0.0226 0.0167 0.1518 0.0938 0.1808 0.0900 0.1625 SE 0.0712 0.0155 0.0301 0.0106 0.0224 0.0170 0.1514 0.0916 0.1781 0.0906 0.1647 RMSE 0.1000 0.0222 0.0432 0.0150 0.0318 0.0238 0.2145 0.1312 0.2538 0.1282 0.2313 CP 0.9530 0.9400 0.9480 0.9460 0.9510 0.9540 0.9590 0.9480 0.9550 0.9550 0.9500

` 0.2785 0.0607 0.1177 0.0412 0.0874 0.0667 0.5932 0.3586 0.6980 0.3547 0.6456 0.1 bias -0.0031 0.0011 -0.0001 0.0028 -0.0025 -0.0001 0.0072 -0.0071 -0.0011 0.0122 -0.0015

rel. bias -0.4410 1.1269 -0.0181 0.3284 0.4976 - 2.8914 1.0083 0.5361 1.8699 - SD 0.1063 0.0245 0.0476 0.0254 0.0317 0.0258 0.1528 0.0942 0.1811 0.0904 0.1626 SE 0.1054 0.0237 0.0480 0.0250 0.0306 0.0259 0.1521 0.0919 0.1788 0.0907 0.1649 RMSE 0.1496 0.0341 0.0676 0.0357 0.0441 0.0366 0.2156 0.1318 0.2544 0.1286 0.2315 CP 0.9520 0.9470 0.9520 0.9540 0.9430 0.9510 0.9580 0.9500 0.9540 0.9560 0.9500

` 0.4128 0.0927 0.1881 0.0980 0.1198 0.1015 0.5957 0.3599 0.7004 0.3552 0.6461 0.2 bias -0.0041 0.0009 0.0034 0.0050 -0.0038 -0.0001 0.0065 -0.0073 -0.0002 0.0124 -0.0018

rel. bias -0.5830 0.8778 0.8493 0.5876 0.7598 - 2.5978 1.0495 0.0864 1.9000 - SD 0.1364 0.0318 0.0616 0.0336 0.0389 0.0338 0.1530 0.0946 0.1822 0.0904 0.1626 SE 0.1329 0.0310 0.0636 0.0341 0.0379 0.0337 0.1524 0.0922 0.1793 0.0908 0.1649 RMSE 0.1905 0.0444 0.0886 0.0481 0.0544 0.0477 0.2159 0.1323 0.2556 0.1287 0.2316 CP 0.9480 0.9440 0.9560 0.9580 0.9460 0.9550 0.9610 0.9490 0.9510 0.9580 0.9490

` 0.5205 0.1212 0.2493 0.1334 0.1482 0.1319 0.5969 0.3611 0.7027 0.3554 0.6462 0.4 bias 0.0012 0.0026 0.0103 0.0139 -0.0106 -0.0003 0.0048 -0.0075 0.0005 0.0129 -0.0018

rel. bias 0.1656 2.6138 2.5854 1.6307 2.1250 - 1.9197 1.0707 -0.2254 1.9869 - SD 0.2476 0.0638 0.1291 0.0613 0.0660 0.0646 0.1544 0.0976 0.1857 0.0905 0.1631 SE 0.2439 0.0603 0.1266 0.0625 0.0637 0.0641 0.1537 0.0943 0.1832 0.0909 0.1651 RMSE 0.3475 0.0878 0.1810 0.0886 0.0923 0.0910 0.2178 0.1359 0.2608 0.1289 0.2320 CP 0.9390 0.9350 0.9480 0.9530 0.9390 0.9470 0.9560 0.9480 0.9500 0.9570 0.9490

` 0.9544 0.2358 0.4951 0.2443 0.2488 0.2508 0.6021 0.3692 0.7178 0.3561 0.6468

15

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βbn bγn

average proportion

of censoring βb_1,n βb_2,n βb_3,n βb_4,n βb_5,n βb_6,n bγ_1,n bγ_2,n γb_3,n bγ_4,n γb_5,n 0 bias -0.0007 0.0001 0.0007 0.0000 0.0000 0.0000 -0.0034 -0.0050 0.0011 0.0032 0.0005

rel. bias -0.0961 0.0764 0.1761 -0.0012 -0.0030 - 0.3731 0.7663 -0.5449 0.4927 - SD 0.0386 0.0081 0.0165 0.0055 0.0118 0.0093 0.1116 0.0687 0.1454 0.0688 0.1311 SE 0.0381 0.0082 0.0162 0.0055 0.0120 0.0092 0.1120 0.0698 0.1415 0.0700 0.1296 RMSE 0.0542 0.0115 0.0232 0.0078 0.0168 0.0130 0.1581 0.0981 0.2028 0.0982 0.1843 CP 0.9490 0.9430 0.9440 0.9460 0.9540 0.9520 0.9550 0.9510 0.9440 0.9560 0.9490

` 0.1493 0.0320 0.0636 0.0216 0.0469 0.0359 0.4389 0.2735 0.5544 0.2742 0.5080 0.1 bias -0.0020 0.0002 0.0010 0.0003 0.0002 0.0002 -0.0037 -0.0049 0.0012 0.0033 0.0006

rel. bias -0.2881 0.2330 0.2396 0.0314 -0.0352 - 0.4082 0.7615 -0.5971 0.5113 - SD 0.0518 0.0116 0.0242 0.0120 0.0158 0.0130 0.1120 0.0689 0.1457 0.0688 0.1312 SE 0.0532 0.0117 0.0241 0.0120 0.0155 0.0130 0.1125 0.0700 0.1418 0.0700 0.1297 RMSE 0.0743 0.0165 0.0342 0.0170 0.0221 0.0184 0.1587 0.0983 0.2033 0.0982 0.1844 CP 0.9570 0.9510 0.9470 0.9530 0.9390 0.9580 0.9570 0.9520 0.9470 0.9550 0.9500

` 0.2086 0.0459 0.0946 0.0472 0.0606 0.0511 0.4406 0.2741 0.5557 0.2744 0.5081 0.2 bias -0.0028 0.0003 0.0012 0.0010 -0.0002 0.0002 -0.0041 -0.0050 0.0012 0.0035 0.0006

` 0.2482 0.0562 0.1175 0.0610 0.0710 0.0623 0.4412 0.2747 0.5570 0.2745 0.5081 0.4 bias -0.0035 0.0005 0.0036 0.0025 -0.0010 0.0004 -0.0048 -0.0051 0.0021 0.0035 0.0005

` 0.3548 0.0838 0.1784 0.0916 0.0972 0.0922 0.4428 0.2768 0.5611 0.2747 0.5082

16

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βbn bγn

average proportion

of censoring βb_1,n βb_2,n βb_3,n βb_4,n βb_5,n βb_6,n bγ_1,n bγ_2,n γb_3,n bγ_4,n bγ_5,n 0 bias -0.0025 0.0002 -0.0006 0.0004 0.0003 0.0003 0.0033 -0.0049 0.0031 0.0050 -0.0019

rel. bias -0.3542 0.2225 -0.1422 0.0482 -0.0621 - 1.3244 0.7032 -1.5480 0.7756 - SD 0.0449 0.0098 0.0188 0.0066 0.0140 0.0110 0.0964 0.0558 0.1125 0.0555 0.1024 SE 0.0444 0.0096 0.0187 0.0065 0.0139 0.0106 0.0952 0.0576 0.1121 0.0569 0.1037 RMSE 0.0631 0.0137 0.0266 0.0092 0.0197 0.0152 0.1355 0.0803 0.1588 0.0797 0.1457 CP 0.9440 0.9430 0.9450 0.9500 0.9540 0.9360 0.9480 0.9580 0.9500 0.9620 0.9630

` 0.1738 0.0376 0.0733 0.0253 0.0545 0.0415 0.3732 0.2256 0.4395 0.2230 0.4064 0.1 bias -0.0085 0.0010 0.0011 0.0024 -0.0003 0.0011 0.0021 -0.0049 0.0037 0.0054 -0.0018

` 0.2592 0.0581 0.1182 0.0617 0.0749 0.0637 0.3747 0.2264 0.4408 0.2232 0.4066 0.2 bias -0.0070 0.0015 0.0023 0.0031 -0.0008 0.0007 0.0019 -0.0048 0.0039 0.0054 -0.0019

rel. bias -1.0047 1.5341 0.5652 0.3633 0.1684 - 0.7405 0.6910 -1.9621 0.8347

SD 0.0839 0.0192 0.0395 0.0219 0.0239 0.0212 0.0973 0.0562 0.1135 0.0557 0.1025 SE 0.0833 0.0194 0.0399 0.0214 0.0236 0.0211 0.0958 0.0580 0.1128 0.0570 0.1038 RMSE 0.1184 0.0273 0.0562 0.0307 0.0336 0.0299 0.1365 0.0808 0.1600 0.0798 0.1458 CP 0.9420 0.9490 0.9520 0.9370 0.9570 0.9370 0.9490 0.9600 0.9480 0.9600 0.9620

` 0.3264 0.0759 0.1564 0.0837 0.0924 0.0827 0.3755 0.2271 0.4422 0.2233 0.4067 0.4 bias -0.0077 0.0006 0.0033 0.0076 -0.0027 0.0013 0.0014 -0.0054 0.0034 0.0056 -0.0017

` 0.5953 0.1467 0.3080 0.1519 0.1544 0.1564 0.3785 0.2319 0.4512 0.2236 0.4068

17