Estimation in zero-inflated binomial regression with missing covariates

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Estimation in zero-inflated binomial regression with missing covariates

Alpha Diallo, Aliou Diop, Jean-François Dupuy

To cite this version:

Alpha Diallo, Aliou Diop, Jean-François Dupuy. Estimation in zero-inflated binomial regression with missing covariates. Statistics, Taylor & Francis: STM, Behavioural Science and Public Health Titles, 2019, 53 (4), pp.839-865. �10.1080/02331888.2019.1619741�. �hal-01585220�

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Estimation in zero-inated binomial regression with missing covariates

Alpha Oumar DIALLO^a,b, Aliou DIOP^a, Jean-François DUPUY^b

aLERSTAD, CEA-MITIC, Gaston Berger University, Saint Louis, Senegal.

bIRMAR-INSA, Rennes, France.

Abstract

The zero-inated binomial (ZIB) regression model was recently proposed to account for excess zeros in binomial regression. Since then, the model has been applied in various domains, such as dental epidemiology and health economics. In practice, it often arises that some covariates involved in ZIB regression have missing values. Assuming that the missingness probability can be estimated parametrically, we propose an inverse-probability-weighted estimator of the parameters of a ZIB model with missing-at-random covariates. Consistency and asymptotic normality of the proposed estimator are established. A consistent estimator of the asymptotic variance-covariance matrix is also provided. The nite-sample behavior of the estimator is assessed via simulations.

Keywords: Asymptotics, count data, excess of zeros, inverse-probability-weighting 1. Introduction

Count data with excess zeros arise in many disciplines, such as agriculture, economics, epidemiology, industry, insurance, terrorism study, trac safety research. . . Excess of zeros refers to the situation where the number of observed zeros is larger than predicted by standard models for count data. Zero-inated regression models, which are obtained by mixing a degenerate distribution at zero with a standard count regression model (such as Poisson, negative binomial or binomial) have been developed to analyze such data. For example, the 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), semi-varying coecient ZIP models (Zhao et al., 2015) and semi- parametric ZIP models (Lam et al., 2006; Feng and Zhu, 2011). The zero-inated negative binomial (ZINB) regression model was proposed by Ridout et al. (2001), see also Moghim- beigi et al. (2008), Mwalili et al. (2008), Garay et al. (2011). When counts have an upper bound, ZIP and ZINB regression models are no longer appropriate. Hall (2000) and Vieira et

Email addresses: Alpha-Oumar.Diallo1@insa-rennes.fr (Alpha Oumar DIALLO),

aliou.diop@ugb.edu.sn (Aliou DIOP), Jean-Francois.Dupuy@insa-rennes.fr (Jean-François DUPUY)

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al. (2000) thus introduced the zero-inated binomial (ZIB) regression model, see also Diop et al. (2016). ZIB regression was recently used in dental caries epidemiology (Gilthorpe et al., 2009; Matranga et al., 2013) and in health economics (Diallo et al., 2017a). A zero-inated model for multinomial counts (or ZIM model) was recently proposed by Diallo et al. (2017b).

In addition, missing data arise in a wide variety of disciplines. In the past decades, there has been an enormous literature on estimation in regression models with missing covariates, including missing covariates in linear models, generalized linear models, generalized linear mixed models, survival models. . . Despite this interest, only a few papers have focused on missing covariates in zero-inated regression models. Chen and Fu (2011) develop a model selection criterion for zero-inated regression models with missing covariates. Lukusa et al.

(2016) consider estimation in ZIP regression with missing-at-random covariates. Motivated by this work, we investigate estimation in ZIB regression when some covariate values are missing for some of the sample individuals.

One simple approach to estimation with missing data is the so-called complete-case method, which consists in removing all incomplete cases from the statistical analysis. How- ever, this method usually produces asymptotically biased estimators (unless data are missing completely at random, which is rarely the case in practice). Therefore, in this paper, we propose to rely on the alternative and more sophisticated inverse-probability-weighted approach, which has not been investigated yet in ZIB regression with missing covariates.

Inverse-probability-weighting (IPW) is a general estimation method under missing data. It was originally proposed by Horvitz and Thompson (1952) and further developed by Zhao and Lipsitz (1992). The basic idea of IPW is to correct for missing data by giving extra weight to subjects with fully observed data. This idea has already proved useful in a variety of models, such as the logistic regression model (Hsieh et al., 2010), proportional hazards regression model (Qi et al., 2005) and single-index model (Li and Hu, 2016), for example.

The rest of the paper is organized as follows. In Section 2, we provide a brief review of ZIB regression, including model formulation and maximum likelihood estimation without missing data. Then, we introduce a IPW estimator of the parameters of a ZIB regression model with missing-at-random covariates. In Section 3, we establish consistency and asymptotic normality of the proposed estimator. A consistent estimator of its asymptotic variance- covariance matrix is provided. Section 4 reports results of a simulation study. A discussion and some perspectives are provided in Section 5.

2. ZIB regression with missing covariates

We rst provide a brief review of ZIB regression and maximum likelihood estimation in ZIB model with complete data.

2.1. A brief review of ZIB regression

Let Z_i denote the random count of interest for individual i,i= 1, . . . , n. Individuals are assumed to be independent. The ZIB distribution is a mixture of a degenerate distribution

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at zero and a binomial distribution. It is given as follows:

Z_i ∼

0 with probability p_i,

B(m_i, π_i) with probability 1−p_i, (2.1) where p_i is a mixing probability for the accommodation of extra zeros and B(m, π)denotes the binomial distribution with sizem and event probabilityπ. The ZIB distribution reduces to a standard binomial distribution when p_i = 0.

In ZIB regression, the mixing probabilities p_i and event probabilities π_i are usually modeled via logistic regressions: logit(pi) = γ^>Wi and logit(πi) = β^>Xi, where Xi = (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) and > denotes the transpose operator. Vectors Xi and Wi may either have some common components or be distinct (note that some caution is required in the special case where m_i = 1 for all i= 1, . . . , n, see Diop et al., 2011). Here, β and γ are respectively p and q-dimensional vectors of unknown regression parameters to be estimated.

Let {(Z_i,X_i,W_i), i = 1, . . . , n} be a sample of independent observations and ψ = (β^>, γ^>)^> denote the whole unknown k-dimensional (k := p +q) parameter. The log- likelihood function ``_n(ψ)based on the observed sample is:

``n(ψ) =

n

X

i=1

n Jilog

e^γ^>^Wⁱ + (1 +e^β^>^Xⁱ)^−mⁱ

−log

1 +e^γ^>^Wⁱ

+(1−J_i)h

Z_iβ^>X_i−m_ilog

1 +e^β^>^Xⁱio , :=

n

X

i=1

`_i(ψ),

where Ji := 1{Zi=0} (see Hall, 2000). The maximum likelihood estimator (MLE) ψˆn :=

( ˆβ_n^>,γˆ_n^>)^> of ψ is obtained by solving the score equation U_n(ψ) = 0, where U_n(ψ) = 1

√n

∂``_n(ψ)

∂ψ = 1

√n

n

X

i=1

∂`_i(ψ)

∂ψ = 1

√n

n

X

i=1

`˙_i(ψ). (2.2) This estimating equation can be solved by using the expectation-maximization (EM) algo- rithm (Hall, 2000) or by a direct maximization of``_n(ψ)(Diallo et al., 2017a). The MLE ψˆ_n is a consistent and asymptotically normal estimator of the true ψ, see Diallo et al. (2017a).

The next section describes the problem and the proposed estimator.

2.2. ZIB regression with missing covariates: the proposed estimator

In this work, we assume that some components ofX_imay be missing for some individuals.

Decompose X_i as X_i = (X^(obs),>_i ,X(miss),>

i )^>, where X^(obs)_i and X^(miss)_i contain the observed and missing components ofX_i respectively (we assume that the same components ofX_i may be missing for all individuals). Let δ_i be a dummy variable indicating whether X_i is fully

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observed (δ_i = 1) or not (δ_i = 0). Finally, letS_i := (Z_i,X^(obs),>_i ,W^>_i )^> denote the vector of variables that are always observed on each individual. Then {Z_i,X_i,W_i} = {S_i,X^(miss)_i }. Letd denote the dimension of Si.

We assume that X^(miss)_i is missing at random (MAR, see Rubin, 1976): the probability that some components of X_i are missing depends only on the observed variables. The MAR assumption can be expressed in terms of the missingness (or selection) probability P(δ_i = 1|S_i,X^(miss)_i ), as:

P(δi = 1|Si,X^(miss)_i ) = P(δi = 1|Si).

Under missing data, we propose to estimateψ in ZIB model (2.1) by using the IPW method.

Originally proposed by Horvitz and Thompson (1952), IPW has recently been used for estimating various regression models with missing or mismeasured covariates. Basic idea is to inversely weight the observed data by the selection probability P(δ_i = 1|S_i), so as to reduce the bias due to incomplete cases deletion.

Recall that without missing data, ψ in model (2.1) can be estimated by solving the score equation (2.2). Under missing data, we propose to estimate ψ by solving the following estimating equation, derived from (2.2) by weighting individuals with fully observed data by the inverse of their selection probability:

√1 n

n

X

i=1

δ_i P(δi = 1|Si)

`˙_i(ψ) = 0. (2.3)

In practice, selection probabilities P(δi = 1|Si) are usually unknown and need to be estimated. Several estimation procedures can be used. In this work, we consider the case where the P(δ_i = 1|S_i), i = 1, . . . , n, can be estimated parametrically. In this case, logistic regression is the most frequently used option. Let ri(α) :=P(δi = 1|Si)be dened as:

r_i(α) = exp(α^>S_i)

1 + exp(α^>S_i), (2.4)

where α is a d-dimensional vector of unknown regression parameters. We need to estimate α before solving the weighted score equation (2.3). Maximum likelihood estimation can be used for that purpose. The MLE αˆ_n = arg max_αQn

i=1{r_i(α)^δⁱ(1−r_i(α))^1−δⁱ}in model (2.4) is known to be consistent and asymptotically Gaussian (Gouriéroux and Monfort, 1981).

Once αˆ_n is available, one can estimate ψ by solving the estimated weighted score equation U_w,n(ψ,αˆ_n) = 0, where

U_w,n(ψ,αˆ_n) = 1

√n

n

X

i=1

δ_i r_i( ˆα_n)

`˙_i(ψ) = 0. (2.5)

In what follows, the resulting estimator of ψ will be denoted byψˆ_n. Asymptotic properties of this estimator are established in Section 3. First, we need to introduce some further notations.

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2.3. Some further notations

Dene rst the (k×d),(k×k) and (d×d) matrices

B(ψ, α) = lim

n→∞ −1

n

X

i=1

E

δ_i1−ri(α) r_i(α)

`˙_i(ψ)S^>_i !

,

J(ψ, α) = lim

n→∞

1 n

n

X

i=1

E

"

δi

`˙_i(ψ) ˙`_i(ψ)^>

r²_i(α)

#!

,

and

Σ(α) = E

S_iS^>_i r_i(α)(1−r_i(α))

. (2.6)

For every i = 1, . . . , n, let U_i = (U_i1, . . . , U_ik)^> denote the k-dimensional column vector U_i :=B(ψ₀, α₀)Σ(α₀)⁻¹S_i. Then, dene the (p×n) , (q×n) and (k×n) matrices

X=







1 1 · · · 1 X12 X22 · · · Xn2

... ... ... ...

X_1p X_2p · · · X_np







, W=







1 1 · · · 1 W12 W22 · · · Wn2

... ... ... ...

W_1q W_2q · · · W_nq





 ,

and

U=







U₁₁ U₂₁ · · · U_n1 U₁₂ U₂₂ · · · U_n2 ... ... ... ...

U_1k U_2k · · · U_nk







= U1

U2

,

where U1 is the (p×n) sub-matrix of U consisting of the rst p rows of U and U2 is the (q×n)sub-matrix ofUconsisting of the lastq rows ofU. LetVbe the(k×3n)block-matrix dened as

V=

X 0_p,n U1

0_q,n W U2

,

and C(ψ, α) = (C_j(ψ, α))1≤j≤3n be the 3n-dimensional column vector dened by C(ψ, α) =

δ₁

r₁(α)A1(ψ), . . . , δ_n

r_n(α)An(ψ), δ₁

r₁(α)B1(ψ), . . . , δ_n

r_n(α)Bn(ψ), δ1−r1(α), . . . , δn−rn(α) >

, where 0_a,b denotes the (a×b) matrix whose components are all equal to zero and for every

i= 1, . . . , n,

A_i(ψ) = −J_i m_ie^β^>^Xⁱ

e^γ^>^Wⁱ(h_i(β))^mⁱ⁺¹+h_i(β) + (1−J_i) Z_i− m_ie^β^>^Xⁱ hi(β)

!

(2.7)

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and

B_i(ψ) = J_ie^γ^>^Wⁱ(h_i(β))^mⁱ

e^γ^>^Wⁱ(h_i(β))^mⁱ+ 1 − e^γ^>^Wⁱ

1 +e^γ^>^Wⁱ, (2.8) with h_i(β) := 1 +e^β^>^Xⁱ.

If A= (A_ij)1≤i≤a,1≤j≤b is a (a×b) matrix, A_•j will denote its j-th column (j = 1, . . . , b) that is, A•j = (A_1j, . . . , A_aj)^>.

Finally, under model (2.4), it is known that the MLE αˆ_n veries

√n( ˆα_n−α₀) = Σ(α₀)⁻¹M_n(α₀) +o_P(1), (2.9) where Σ(α₀) is given by (2.6) and M_n(α) =n^−1/2Pn

i=1S_i(δ_i−r_i(α)).

In the next section, we establish rigorously the consistency and asymptotic normality of the proposed IPW-MLE ψˆ_n.

3. Asymptotic results

We rst state some regularity conditions that will be needed for proving our asymptotic results.

3.1. Regularity conditions and consistency

C1 Covariates are bounded, that is, there exists a nite positive constant c₁ such that

|X_ij| ≤ c₁ and |W_i`| ≤ c₁ for every i = 1, . . . , n, j = 2, . . . , p and ` = 2, . . . , q. For every i = 1, . . . , n, j = 2, . . . , p and ` = 2, . . . , q, var[X_ij] > 0 and var[W_i`] > 0. For every i = 1, . . . , n, the X_ij (j = 1, . . . , p) are linearly independent and the W_i`

(`= 1, . . . , q) are linearly independent.

C2 The true parameter value ψ0 := (β₀^>, γ₀^>)^> lies in the interior of some known compact set ofR^p×R^q. The true α₀ belongs to the interior of some known compact set of R^d. C3 As n→ ∞, n⁻¹Pn

i=1E h δi

ri(α0)

∂²`i(ψ)

∂ψ∂ψ^>

i converges to some invertible matrix A(ψ, α₀) and the smallest eigenvalueλ_n of VV^> tends to +∞.

C4 For every i= 1, . . . , n, we have m_i ∈ {2, . . . , M} for some nite integer value M. In what follows, the space R^k of k-dimensional (column) vectors will be provided with the Euclidean norm k · k₂ and the space of (k×k) real matrices will be provided with the norm

|||A|||₂ := maxkxk₂=1kAxk₂ (for notations simplicity, we will use k · k for both norms).

We rst prove consistency of ψˆ_n:

Theorem 3.1. Assume that conditions C1-C4 hold. Then, as n → ∞, ψˆ_n converges in probability to ψ₀.

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Proof of Theorem 3.1. To prove consistency of ψˆ_n, we verify the conditions of the inverse function theorem of Foutz (1977). These conditions are proved in a series of technical lemmas.

Lemma 3.2. ∂U_w,n(ψ,αˆ_n)/∂ψ^> exists and is continuous in an open neighborhood of ψ₀. Proof of Lemma 3.2. The `i(ψ), i = 1, . . . , n are twice dierentiable with respect to ψ. Continuity of ^∂_∂ψ∂ψ²^`ⁱ^(ψ)^> is straightforward and is omitted.

Lemma 3.3. As n → ∞, n^−1/2U_w,n(ψ₀,αˆ_n) converges in probability to 0.

Proof of Lemma 3.3. Decompose n^−1/2Uw,n(ψ0,αˆn) as:

n^−1/2U_w,n(ψ₀,αˆ_n) = n^−1/2U_w,n(ψ₀,αˆ_n)−n^−1/2U_w,n(ψ₀, α₀)

+n^−1/2U_w,n(ψ₀, α₀), (3.10) and consider the rst term on the right-hand side of this decomposition. We have:

n^−1/2U_w,n(ψ₀,αˆ_n)−n^−1/2U_w,n(ψ₀, α₀) = 1 n

n

X

i=1

δ_i 1

r_i( ˆα_n) − 1 r_i(α₀)

`˙_i(ψ₀),

= 1 n

n

X

i=1

δ_i 1

e^α^ˆ^>ⁿ^Sⁱ − 1 e^α^>⁰^Sⁱ

`˙_i(ψ₀).

By a Taylor expansion of 1/e^α^ˆ^>ⁿ^Sⁱ aroundα₀, n^−1/2U_w,n(ψ₀,αˆ_n)−n^−1/2U_w,n(ψ₀, α₀) = 1

n

X

i=1

δ_i(α₀−αˆ_n)^>S_i 1 e^α^>^∗^Sⁱ

`˙_i(ψ₀),

where α∗ is on the line segment between αˆ_n and α₀. Then, we have:

kn^−1/2U_w,n(ψ₀,αˆ_n)−n^−1/2U_w,n(ψ₀, α₀)k ≤ 1 n

n

X

i=1

`˙_i(ψ₀) ,

≤ 1 n

n

X

i=1

`˙_i(ψ₀) ,

≤ 1 n

n

X

i=1

kα₀−αˆ_nk kS_ik 1 e^α^>^∗^Sⁱ

`˙_i(ψ₀) , where the second to third line comes from Cauchy-Schwarz inequality. Now, straightforward calculations show that

`˙_i(ψ₀) = (X^>_i ,0^>_q)^>·A_i(ψ₀) + (0^>_p,W_i^>)^>·B_i(ψ₀),

where0_p := 0_p,1 is the p-dimensional column vector having all its components equal to 0 and A_i(ψ₀), B_i(ψ₀) are given by (2.7) and (2.8) respectively. Thus we have:

k`˙_i(ψ₀)k ≤ k(X^>_i ,0^>_q)^>k · |A_i(ψ₀)|+k(0^>_p,W^>_i )^>k · |B_i(ψ₀)|.

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Under conditions C1, C2 and C4, it is easy to see that |A_i(ψ₀)| and |B_i(ψ₀)| are bounded above. Thus, there exists a nite constant c₂ such that n⁻¹Pn

i=1k`˙_i(ψ₀)k ≤ c₂. Note that kS_ik and 1/e^α^>^∗^Sⁱ are also bounded, by conditions C1 and C2. Therefore, there exists some nite constantc3 such thatkn^−1/2Uw,n(ψ0,αˆn)−n^−1/2Uw,n(ψ0, α0)k ≤c3kα₀−αˆnk. Finally, the convergence of αˆ_n toα₀ imply thatkn^−1/2U_w,n(ψ₀,αˆ_n)−n^−1/2U_w,n(ψ₀, α₀)kconverges in probability to 0 asn → ∞.

Next, consider the term n^−1/2Uw,n(ψ0, α0) in decomposition (3.10). Some simple algebra yields:

n^−1/2U_w,n(ψ₀, α₀) =







1 n

Pn i=1

δi

ri(α0)X_i1A_i(ψ₀) ...

1 n

Pn i=1

δi

ri(α0)X_ipA_i(ψ₀)

1 n

Pn i=1

δi

ri(α0)Wi1Bi(ψ0) ...

1 n

Pn i=1

δi

ri(α0)WiqBi(ψ0)





 .

We prove that n^−1/2U_w,n(ψ₀, α₀) converges in probability to 0 as n → ∞. To see this, note rst that for everyi= 1, . . . , nand `= 1, . . . , q:

E δ_i

ri(α0)W_i`B_i(ψ₀)

= E

E

δ_i

S_i

,

= E

1

ri(α0)W_i`E[δ_iB_i(ψ₀)|S_i]

.

Given S_i, B_i(ψ₀) is a function of X^(miss)_i only. Thus, by the MAR assumption, B_i(ψ₀) and δ_i are independent givenS_i. It follows that:

E δ_i

= E

1

ri(α0)W_i`E[δ_i|S_i]E[B_i(ψ₀)|S_i]

,

= E[W_i`E[B_i(ψ₀)|S_i]],

= E[W_i`B_i(ψ₀)].

Diallo et al. (2017a) proved that E[W_i`B_i(ψ₀)] = 0 for every i = 1, . . . , n and ` = 1, . . . , q. Thus, E

h δi

ri(α0)W_i`B_i(ψ₀)i

= 0 for every i = 1, . . . , n and ` = 1, . . . , q. Similarly, for every i= 1, . . . , n and j = 1, . . . , p, we have:

E δ_i

r_i(α₀)X_ijA_i(ψ₀)

=E

E δ_i

S_i

.

Two cases should be considered, namely: i) X_ij is a component of X^(miss)_i and ii) X_ij is a component of X^(obs)_i . In case i), we have:

E

E δi

r_i(α₀)XijAi(ψ0)

Si

=E 1

r_i(α₀)E[δiXijAi(ψ0)|Si]

.

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Given S_i, X_ijA_i(ψ₀) is a function of X^(miss)_i only. Thus, by the MAR assumption, E

1

r_i(α₀)E[δ_iX_ijA_i(ψ₀)|S_i]

= E 1

r_i(α₀)E[δ_i|S_i]E[X_ijA_i(ψ₀)|S_i]

,

= E[X_ijA_i(ψ₀)].

Diallo et al. (2017a) proved that E[X_ijA_i(ψ₀)] = 0 for every i = 1, . . . , n and j = 1, . . . , p. Therefore,E

h δi

ri(α0)X_ijA_i(ψ₀)i

= 0. In case ii), E

E

δ_i

S_i

= E 1

r_i(α₀)X_ijE[δ_iA_i(ψ₀)|S_i]

,

= E 1

r_i(α₀)X_ijE[δ_i|S_i]E[A_i(ψ₀)|S_i]

,

= E[X_ijA_i(ψ₀)],

= 0,

where the rst to second line comes from the fact that under MAR, A_i(ψ₀) and δ_i are independent given Si. Finally, in caseii), we also have: E

h δi

ri(α0)XijAi(ψ0) i

= 0. Now, for every i= 1, . . . , n and ` = 1, . . . , q, we have:

var δ_i

r_i(α₀)W_i`B_i(ψ₀)

≤E δ_i

r_i²(α₀)W_i`²B_i²(ψ₀)

.

By C1, C2, C4, there exists nite constantsc₄andc₅such that1/r²_i(α₀)≤c₄andB_i²(ψ₀)≤c₅ for every i= 1, . . . , n. Therefore,

var δ_i

≤c₆ :=c²₁c₄c₅. It follows that

∞

X

i=1

var

δi

i² ≤c₆

∞

X

i=1

1 i² <∞.

One can easily show that a similar result holds for var

δi

ri(α0)X_ijA_i(ψ₀)

. By Kolmogorov's law of large numbers (see for example Jiang (2010), Theorem 6.7),

1 n

n

X

i=1

δ_i

r_i(α₀)Wi`Bi(ψ0)−E δ_i

r_i(α₀)Wi`Bi(ψ0)

= 1 n

n

X

i=1

δ_i

r_i(α₀)Wi`Bi(ψ0), ` = 1, . . . , q and

1 n

n

X

i=1

δ_i

ri(α0)X_ijA_i(ψ₀)−E δ_i

ri(α0)X_ijA_i(ψ₀)

= 1 n

n

X

i=1

δ_i

ri(α0)X_ijA_i(ψ₀), j = 1, . . . , p converge in probability to 0 as n → ∞ and thus, n^−1/2U_w,n(ψ₀, α₀) converges in probability to 0. This implies that n^−1/2U_w,n(ψ₀,αˆ_n) converges to 0, which concludes the proof.

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Lemma 3.4. Asn→ ∞,n^−1/2∂U_w,n(ψ,αˆ_n)/∂ψ^>converges in probability to a xed function A(ψ, α₀), uniformly in an open neighborhood of ψ₀.

Proof of Lemma 3.4. Let Ue_w,n(ψ, α) := n^−1/2∂U_w,n(ψ, α)/∂ψ^> and V_ψ₀ be an open neighborhood of ψ₀. Let ψ ∈ V_ψ₀. Using similar arguments as in proof of Lemma 3.3, we have:

kUe_w,n(ψ,αˆ_n)−Ue_w,n(ψ, α₀)k ≤ 1 n

n

X

i=1

kα₀−αˆ_nk kS_ik 1 e^α^>^∗^Sⁱ

`¨_i(ψ) ,

where α∗ is on the line segment between αˆ_n and α₀. From this, one easily proves that kUew,n(ψ,αˆn)−Uew,n(ψ, α0)k converges in probability to 0 as n → ∞. Details are omitted.

Now, consider the(`, j)-th element of Uew,n(ψ, α0), namely:

Ue_w,n(ψ, α₀)

(`,j)

= 1 n

n

X

i=1

δ_i r_i(α₀)

∂²`_i(ψ)

∂ψ_`∂ψ_j

. We have:

Ue_w,n(ψ, α₀)

(`,j) = 1 n

n

X

i=1

δi

r_i(α₀)

∂²`i(ψ)

∂ψ_`∂ψ_j −E δi

r_i(α₀)

∂²`i(ψ)

∂ψ_`∂ψ_j

+ 1 n

n

X

i=1

E δi

r_i(α₀)

∂²`i(ψ)

∂ψ_`∂ψ_j

.

Now,

var δ_i

r_i(α₀)

∂²`_i(ψ)

∂ψ_`∂ψ_j

≤ E δ_i r²_i(α₀)

∂²`_i(ψ)

∂ψ_`∂ψ_j 2!

,

≤ c₄E

∂²`_i(ψ)

∂ψ`∂ψj

2! . We prove that var

δi

ri(α0)

∂²`i(ψ)

∂ψ`∂ψj

is bounded. Some tedious albeit easy calculations show that ^∂_∂ψ²^`_`ⁱ_∂ψ^(ψ)_j is the (`, j)-th element of the (k ×k) matrix (−ViUi(ψ)V_i^>), where Vi is the (k×2) matrix dened as

V_i =

X_i 0_p 0_q W_i

and

U_i(ψ) =

U_i,1(ψ) U_i,2(ψ) U_i,2(ψ) U_i,3(ψ)

is the (2×2) symmetric matrix dened by U_i,1(ψ) = J_im_ie^β^>^Xⁱ

(k_i(ψ))²

k_i(ψ)−e^β^>^Xⁱh

e^γ^>^Wⁱ(m_i+ 1)(h_i(β))^mⁱ+ 1i

+m_i(1−J_i)e^β^>^Xⁱ (h_i(β))² ,

U_i,2(ψ) = −J_im_ie^β^>^Xⁱ^+γ^>^Wⁱ(h_i(β))^mⁱ⁺¹ (ki(ψ))² ,

U_i,3(ψ) = J_ie^γ^>^Wⁱ(h_i(β))^mⁱ⁺¹ (k_i(ψ))²

e^γ^>^Wⁱ(h_i(β))^mⁱ⁺¹−k_i(ψ)

+ e^γ^>^Wⁱ 1 +e^γ^>^Wⁱ2,

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with k_i(ψ) :=e^γ^>^Wⁱ(h_i(β))^mⁱ⁺¹+h_i(β), i= 1, . . . , n. Using these notations, it is easy to see that

∂²`i(ψ)

∂ψ_`∂ψ_j = − V_i,(`,1)U_i,1(ψ) +V_i,(`,2)U_i,2(ψ) V_i,(j,1)

− V_i,(`,1)U_i,2(ψ) +V_i,(`,2)U_i,3(ψ)

V_i,(j,2), (3.11)

where Vi,(a,b) denotes the (a, b)-th element of matrix Vi. For a given row ` (` = 1, . . . , k), exactly one ofV_i,(`,1) and V_i,(`,2) must be equal to 0 (this is straightforward from the expres- sion ofV_i). Suppose for example thatV_i,(`,1) = 0andV_i,(j,2) = 0(other combinations of null and non-null values among (Vi,(`,1),Vi,(`,2)) and (Vi,(j,1),Vi,(j,2)) can be treated similarly).

Then (3.11) reduces to:

∂²`_i(ψ)

∂ψ`∂ψj

=−V_i,(`,2)U_i,2(ψ)V_i,(j,1).

Let MX := maxβ,Xe^β^>^X and MW := maxγ,We^γ^>^W. Under conditions C1, C2 and C4, we have:

|Ui,2(ψ)| ≤M^∗ :=M ·MX·MW·(1 +MX)^M⁺¹<∞, which implies

E

∂²`_i(ψ)

∂ψ_`∂ψ_j 2!

≤c⁴₁M^∗2,

and nally,

var δ_i

r_i(α₀)

∂²`_i(ψ)

∂ψ_`∂ψ_j

≤c₄c⁴₁M^∗2 <∞.

It follows that

∞

X

i=1

var

δi

ri(α0)

∂²`i(ψ)

∂ψ`∂ψj

i² ≤c₄c⁴₁M^∗2

∞

X

i=1

1 i² <∞.

Therefore, Kolmogorov's law of large numbers implies that 1

n

X

i=1

δ_i ri(α0)

∂²`_i(ψ)

∂ψ`∂ψj

−E δ_i

ri(α0)

∂²`_i(ψ)

∂ψ`∂ψj

converges in probability to 0 as n → ∞ and by condition C3, (Ue_w,n(ψ, α₀))_(`,j) converges in probability to the(`, j)-th element of the matrix A(ψ, α₀). Finally,Ue_w,n(ψ,αˆ_n)converges in probability toA(ψ, α₀). Under conditions C1, C2 and C4, the derivative ofUe_w,n(ψ,αˆ_n)with respect toψ is bounded, for every n. Therefore, the sequence (Ue_w,n(ψ,αˆ_n))_n is equicontinu- ous. It follows that the convergence of Ue_w,n(ψ,αˆ_n)to A(ψ, α₀) is uniform on V_ψ₀. Having now veried the conditions of Foutz (1977) inverse function theorem, we conclude

that ψˆ_n converges in probability to ψ₀.

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3.2. Asymptotic normality

Our second main result asserts that the IPW-MLE ψˆ_n is asymptotically Gaussian.

Theorem 3.5. Assume that conditions C1-C4 hold. Then √

n( ˆψ_n−ψ₀) is asymptotically normally distributed with mean zero and covariance matrix ∆, where

∆ := A(ψ₀, α₀)⁻¹{J(ψ₀, α₀)−B(ψ₀, α₀)Σ(α₀)⁻¹B(ψ₀, α₀)^>}

A(ψ₀, α₀)⁻¹^>

.

Proof of Theorem 3.5. A Taylor series expansion of Uw,n( ˆψn,αˆn)at (ψ0, α0) yields 0 =U_w,n( ˆψ_n,αˆ_n) =U_w,n(ψ₀, α₀) + ∂U_w,n(ψ₀, α₀)

∂ψ^> ( ˆψ_n−ψ₀) + ∂U_w,n(ψ₀, α₀)

∂α^> ( ˆα_n−α₀) +o_P(1).

LetUˇ_w,n(ψ, α) :=n^−1/2∂U_w,n(ψ, α)/∂α^>. Then we have:

0 =U_w,n(ψ₀, α₀) +Ue_w,n(ψ₀, α₀)√

n( ˆψ_n−ψ₀) + ˇU_w,n(ψ₀, α₀)√

n( ˆα_n−α₀) +o_P(1). (3.12) Now, straightforward calculations yield

Uˇ_w,n(ψ, α) =−1 n

n

X

i=1

δ_i1−ri(α) r_i(α)

`˙_i(ψ)S^>_i ,

and it can be proved that Uˇ_w,n(ψ₀, α₀) converges in probability toB(ψ₀, α₀) (arguments are similar to those in proof of Lemma 3.4 and are thus omitted). Combining this with (2.9), we can re-express (3.12) as:

0 = U_w,n(ψ₀, α₀) +Ue_w,n(ψ₀, α₀)√

n( ˆψ_n−ψ₀) +B(ψ₀, α₀)Σ(α₀)⁻¹M_n(α₀) +o_P(1), and it follows that:

√n( ˆψ_n−ψ₀) =−Ue_w,n(ψ₀, α₀)⁻¹ U_w,n(ψ₀, α₀) +B(ψ₀, α₀)Σ(α₀)⁻¹M_n(α₀)

+o_P(1).

Using notations introduced in Section 2.3, we nally obtain:

√n( ˆψ_n−ψ₀) = −Ue_w,n(ψ₀, α₀)⁻¹n^−1/2VC(ψ₀, α₀) +o_P(1),

= −Ue_w,n(ψ₀, α₀)⁻¹

3n

X

j=1

V^•jC_j,n(ψ₀, α₀) +o_P(1),

whereC_j,n(ψ₀, α₀) =n^−1/2C_j(ψ₀, α₀). LetC²n =var(U_w,n(ψ₀, α₀) +B(ψ₀, α₀)Σ(α₀)⁻¹M_n(α₀)). Then, by Eicker (1966), the random linear form C⁻¹n

P3n

j=1V^•jC_j,n(ψ₀, α₀) converges in distribution to the k-dimensional standard Gaussian distribution if the following conditions are satised: a) max

1≤j≤3nV^>•j(VV^>)⁻¹V^•j →0asn → ∞, b) sup

1≤j≤3nE[C_j,n² (ψ₀, α₀)1_{|C_j,n_(ψ₀_,α₀_)|>c}]→ 0 asc→ ∞, c) inf

1≤j≤3nE[C_j,n² (ψ0, α0)]>0. Note rst that 0< max

1≤j≤3nV^>•j(VV^>)⁻¹V^•j ≤ max

1≤j≤3nkV^•jk²k(VV^>)⁻¹k= max

1≤j≤3nkV^•jk²/λn.

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Since kV^•jk is bounded, condition C3 implies that condition a) is satised. Condition b) follows by noting thatC_j,n(ψ₀, α₀)(forj = 1, . . . ,3n) are bounded under C1, C2, C4. Finally, under C1, C2 and C4, we have E[C_j,n² (ψ₀, α₀)]>0 for every j = 1, . . . ,3n.

Moreover,

C²n =var(U_w,n(ψ₀, α₀)) +var B(ψ₀, α₀)Σ(α₀)⁻¹M_n(α₀)

+2cov U_w,n(ψ₀, α₀), B(ψ₀, α₀)Σ(α₀)⁻¹M_n(α₀) . Straightforward calculations yield: var(U_w,n(ψ₀, α₀)) = _n¹Pn

i=1E h

δ_i^`^˙ⁱ^(ψ_r⁰2^{) ˙}^`ⁱ^(ψ⁰⁾^>

i(α0)

i, var(M_n(α₀)) = Σ(α₀)and

cov U_w,n(ψ₀, α₀), B(ψ₀, α₀)Σ(α₀)⁻¹M_n(α₀)

= 1 n

n

X

i=1

E

`˙_i(ψ₀)δ_i(1−r_i(α₀)) ri(α0) S^>_i

Σ(α₀)⁻¹B(ψ₀, α₀)^>.

Hence,C²nconverges toJ(ψ0, α0)−B(ψ0, α0)Σ(α0)⁻¹B(ψ0, α0)^>. It follows thatP3n

j=1V^•jCj,n(ψ0, α0) converges in distribution to a k-dimensional Gaussian vector with mean zero and variance

J(ψ₀, α₀)−B(ψ₀, α₀)Σ(α₀)⁻¹B(ψ₀, α₀)^>. Finally, using Lemma 3.4, condition C3 and Slut- sky's theorem, √

n( ˆψn−ψ0) converges in distribution to a mean-zero Gaussian vector with

variance ∆, where ∆is dened in Theorem 3.5.

Remark. A consistent estimator of ∆is given by

∆ˆ_n:=A_n( ˆψ_n,αˆ_n)⁻¹{J_n( ˆψ_n,αˆ_n)−B_n( ˆψ_n,αˆ_n)Σ_n( ˆα_n)⁻¹B_n( ˆψ_n,αˆ_n)^>}h

A_n( ˆψ_n,αˆ_n)⁻¹i>

(3.13) where

An(ψ, α) = 1 n

n

X

i=1

δ_i r_i(α)

∂²`_i(ψ)

∂ψ∂ψ^>,

B_n(ψ, α) = −1 n

n

X

i=1

δ_i1−r_i(α) r_i(α)

`˙_i(ψ)S^>_i ,

J_n(ψ, α) = 1 n

n

X

i=1

δ_i

`˙_i(ψ) ˙`_i(ψ)^>

r²_i(α) ,

Σ_n(α) = 1 n

n

X

i=1

S_iS^>_i r_i(α)(1−r_i(α)).

The proof proceeds along the same lines as proof of Lemma 3.4 and is therefore omitted.

4. Simulation study

In this section, we investigate the nite-sample performances of the IPW estimator under various conditions.

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4.1. Simulation design

The following ZIB regression model is used to simulate data:

logit(π_i) =β₁X_i1+β₂X_i2+β₃X_i3+β₄X_i4+β₅X_i5+β₆X_i6 and

logit(p_i) =γ₁W_i1+γ₂W_i2+γ₃W_i3+γ₄W_i4,

where X_i1 = W_i1 = 1 and the X_i2, . . . , X_i6 and W_i4 are independently drawn from normal N(0,1), uniform U(2,5), normal N(1,1.5), exponentialE(1), binomialB(1,0.3) and normal N(−1,1) distributions respectively. Linear predictors in logit(π_i) and logit(p_i) are allowed to share common terms by lettingW_i2 =X_i2 and W_i3 =X_i6. The regression parameter β is chosen as β = (−0.3,1.2,0.5,−0.75,−1,0.8)^>. The regression parameter γ is chosen as:

case 1: γ = (−0.55,−0.7,−1,0.45)^>

case 2: γ = (0.25,−0.4,0.8,0.45)^>

We consider the following sample sizes, n= 500,1000. The numbersmi are allowed to vary across subjects, with m_i ∈ {4,8,10,15}. Let n_j = card{i : m_i = j}, for j = 4,8,10,15. For n = 500, we let (n₄, n₈, n₁₀, n₁₅) = (125,125,125,125) and for n = 1000, we let (n4, n8, n10, n15) = (250,250,250,250).

Using these values, in case 1 (respectively case 2), the average percentage of zero-ination in the simulated data sets is 25% (respectively 50%). Missingness indicators δ_i are simulated from a logistic regression model with selection probability ri(α) := P(δi = 1|Si) = logit⁻¹(α^>S_i), with S_i := (1, Z_i, X_i2, W_i4). The regression parameter α is chosen to yield average missingness proportions in the simulated samples successively equal to 0.2 and 0.4.

Finally, for each combination of the simulation design parameters (sample size, proportions of zero-ination and missing data), we simulateN = 1000samples and we calculate the IPW estimate ψˆ_n. Simulations are carried out using the statistical software R. We use the package maxLik (Henningsen and Toomet, 2011) to solve the estimating equation (2.5).

4.2. Results

For each conguration [sample size×zero-inflation proportion×proportion of missing data] of the simulation parameters, we calculate the average absolute relative bias (as a percentage) of the estimates βˆ_j,n and γˆ_k,n over the N simulated samples (for example, the absolute relative bias of βˆ_j,n is obtained as N⁻¹PN

t=1|( ˆβ_j,n^(t) −β_j)/β_j| ×100, where βˆ_j,n^(t) denotes the IPW estimate of β_j in the t-th simulated sample). We also obtain the average standard error SE (calculated from (3.13)), empirical standard deviation (SD) and root mean square error (RMSE) for each estimator βˆ_j,n (j = 1, . . . ,6)and γˆ_k,n (k = 1, . . . ,4). Finally, we provide the empirical coverage probability (CP) of 95%-level condence intervals for the βj and γk. Results are given in Table 1 (case 1, n = 500), Table 2 (case 1, n = 1000), Table 3 (case 2, n = 500), Table 4 (case 2, n = 1000). For purpose of comparison, we also