Efficient estimation in missing data and survey sampling problems

Transcription

1 Graduate Theses and Dssertatons Iowa State Unversty Capstones, Theses and Dssertatons 2012 Effcent estmaton n mssng data and survey samplng problems Sxa Chen Iowa State Unversty Follow ths and addtonal works at: Part of the Statstcs and Probablty Commons Recommended Ctaton Chen, Sxa, "Effcent estmaton n mssng data and survey samplng problems" (2012). Graduate Theses and Dssertatons Ths Dssertaton s brought to you for free and open access by the Iowa State Unversty Capstones, Theses and Dssertatons at Iowa State Unversty Dgtal Repostory. It has been accepted for ncluson n Graduate Theses and Dssertatons by an authorzed admnstrator of Iowa State Unversty Dgtal Repostory. For more nformaton, please contact dgrep@astate.edu.

2 Effcent estmaton n mssng data and survey samplng problems by Sxa Chen A dssertaton submtted to the graduate faculty n partal fulfllment of the requrements for the degree of DOCTOR OF PHILOSOPHY Major: Statstcs Program of Study Commttee: Jae Kwang Km, Co-major Professor Wayne A. Fuller, Co-major Professor Zhengyuan Zhu Nordman Dan Cndy L. Yu Iowa State Unversty Ames, Iowa 2012 Copyrght c Sxa Chen, All rghts reserved.

3 DEDICATION I would lke to dedcate ths thess to my parents and my wfe wthout whose support I would not have been able to complete ths work. I would also lke to thank my frends and famly for ther lovng gudance and fnancal assstance durng the wrtng of ths work.

4 TABLE OF CONTENTS LIST OF TABLES v LIST OF FIGURES v ACKNOWLEDGEMENTS ABSTRACT v x CHAPTER 1. GENERAL INTRODUCTION CHAPTER 2. SEMI-PARAMETRIC INFERENCE WITH A FUNCTIONAL- FORM EMPIRICAL LIKELIHOOD Introducton Man Results Extenson Computatonal Aspects Smulaton Study Conclusons CHAPTER 3. A UNIFIED THEORY ON EMPIRICAL LIKELIHOOD METH- ODS WITH MISSING DATA AND SURVEY SAMPLING Introducton Basc setup Estmaton wth known response probablty Estmaton wth unknown response probablty Nonparametrc estmaton of the response mechansm Extenson to two-phase samplng

5 v 3.7 Smulaton Study Smulaton One Smulaton Two CHAPTER 4. POPULATION EMPIRICAL LIKELIHOOD FOR NON- PARAMETRIC INFERENCE IN SURVEY SAMPLING Introducton Populaton emprcal lkelhood Man results Extenson to rejectve Posson samplng Combnng nformaton from two ndependent surveys Smulaton Study Smulaton One Smulaton Two Concludng remarks CHAPTER 5. TWO-PHASE SAMPLING FOR PROPENSITY SCORE ES- TIMATION IN VOLUNTARY SAMPLES Introducton Basc Setup Man Results Extenson to non-nested two-phase samplng Smulaton Study Smulaton One Smulaton Two Emprcal Study Concludng Remarks CHAPTER 6. FUTURE RESEARCH TOPICS Jackknfe emprcal lkelhood for nference wth mputed data Nonparametrc propensty score estmaton

6 v 6.3 Inference wth parametrc fractonal mputaton APPENDIX A. PROOFS FOR CHAPTER APPENDIX B. PROOFS FOR CHAPTER APPENDIX C. PROOFS FOR CHAPTER APPENDIX D. PROOFS FOR CHAPTER BIBLIOGRAPHY

7 v LIST OF TABLES 2.1 Monte Carlo relatve effcency of the pont estmators Power comparsons for testng H 0 : ρ = Data structure for two-phase samplng Bases, Varances and Mean squared errors (MSE) of the estmators under four dfferent scenaros n smulaton one The Monte Carlo bases, varances, and the mean squared errors (MSE) of the pont estmators n smulaton two Monte Carlo bases, varances, and mean squared errors of the pont estmators Coverage rate and average length comparson for Wald s and Wlk s type 95% confdence ntervals of proposed POEL2 method The Monte Carlo bases, varances, and the mean squared errors (MSE) of the pont estmators n Smulaton Two Smulaton results of the pont estmators for θ 1 and θ 2 n Smulaton One Smulaton results of the pont estmators for θ n Smulaton Two Estmated coeffcents n the propensty model Estmated parameters (s.e.) for 2012 Iowa Caucus Survey Results... 89

8 v LIST OF FIGURES 2.1 Parameter estmatons versus penalty parameter Sample structure of 2012 Iowa Caucus Survey

9 v ACKNOWLEDGEMENTS I would lke to take ths opportunty to express my thanks to those who helped me wth varous aspects of conductng research and the wrtng of ths thess. Frst and foremost, Dr. Jae Kwang Km for hs gudance, patence and support throughout ths research and the wrtng of ths thess. Hs nsghts and words of encouragement have often nspred me and renewed my hopes for completng my graduate educaton. I am partcularly grateful to Dr. Wayne A. Fuller for hs helpful comments and suggestons. I would also lke to thank my commttee members for ther efforts and contrbutons to ths work.

10 x ABSTRACT The thess conssts of four research papers. The frst paper deals wth general theory for emprcal lkelhood under the standard setup. Instead of maxmzng the emprcal lkelhood functon, a functonal-form approach s proposed to generalze the theory of emprcal lkelhood and to acheve computatonal effcency. The second paper deals wth an emprcal lkelhood approach for mssng data. The proposed method uses a partal lkelhood for the respondents and theores are developed for both a parametrc response model and a nonparametrc response model. Also, the proposed method s extended to two-phase samplng where the frst-phase sample s obtaned by complex survey samplng. The thrd paper deals wth emprcal lkelhood n the survey samplng setup. In the proposed method, called the populaton emprcal lkelhood method, the emprcal lkelhood functon s defned for the fnte populaton and the samplng desgn s ncorporated nto one of the constrants n the optmzaton problem. The proposed method s qute useful when combnng nformaton from several ndependent surveys. The fourth paper proposes a novel applcaton of the capture-recapture experment to estmate the propensty score for nongnorable nonresponse. The proposed method can be used to reduce the selecton bas assocated wth voluntary samplng.

11 1 CHAPTER 1. GENERAL INTRODUCTION Hartley and Rao (1968) ntroduced the emprcal lkelhood (EL) approach under the name of scale load. Owen (1988,1990) brought the EL method to standard statstcal problems. For a comprehensve overvew of EL method, see Owen (2001). Chen and Hall (1993) extended the EL method to nference for quantles. Qn and Lawless (1994) extended the EL method to nference for parameters defned by some general estmatng equatons. DCcco et al. (1991) and Chen and Cu (2006) used bartlett correcton technques to mprove the convergence rate of emprcal lkelhood rato. The applcaton of EL method n tme seres has been consdered by Ktamura (1997), Nordman et al. (2007) and others. Recently, Hjort, McKeague and Van Kelogom (2009), Chen, Peng and Qn (2009) and Tang and Leng (2010) showed that the EL method contnues to work when data dmensonalty s growng. Newey and Smth (2004) proposed generalzed emprcal lkelhood (GEL) whch extended the scope of the tradtonal EL method. In chapter 2, we propose a dfferent extenson by usng the functonal-form emprcal lkelhood (FEL) method. The basc dea s to generalze the form of the EL weght or of the objectve functon. We prove the frst order equvalence between our proposed estmator and the tradtonal EL estmator. The proposed estmator has certan advantages n terms of computaton and choce of weghts. Mssng data happens frequently n observatonal studes. If the mssng mechansm s completely mssng at random (CMAR) n the sense of Rubn (1976), we can safely remove the mssng part of the data. However, f the response mechansm s mssng at random (MAR) or not mssng at random (NMAR), we may not gnore the mssng data n order to produce effcent and consstent estmates. There are two man approaches for nference wth mssng data: Imputaton and Propensty Score Weghtng. Wang and Rao (2002) and Wang and Chen (2009) consdered combnng EL and mputaton methods for nference wth data mssng

12 2 at random. Alternatvely, Qn, Leung and Shao (2002) proposed the EL method to deal wth nongnorable mssng data by usng the propensty score method. Qn and Zhang (2007) appled the EL method n mssng response problems. Chen, Leung and Qn (2008) proposed constructng two dfferent emprcal lkelhood method wth data MAR. Most recently, Qn et al. (2009) provded the complete EL method for mssng covarate problem. The lterature s somewhat sparse for modelng the response mechansm nonparametrcally. Cheng (1994) dscussed some asymptotc propertes of the mean estmator based on the kernel regresson method under gnorable mssng data. Recently, Km and Yu (2011) extended the approach of Cheng (1994) to handle nongnorable nonresponse. Xue (2009) dscussed an emprcal lkelhood method for lnear models usng the weghts computed from a nonparametrc model where the kernel regresson method s used to estmate the response model. Da Slva and Opsomer (2009) consdered another type of nonparametrc response probablty estmator usng local polynomal regresson. Hrano et al (2003) and Cattaneo (2010) dscussed semparametrc effcency of the nonparametrc response propensty estmators n the context of estmatng average treatment effect n econometrcs. In chapter 3, we propose a response EL method whch can be used to handle both survey samplng and mssng data problems. Specfcally, we propose estmatng the propensty score nonparametrcally n the EL method. By dong ths, the sem-parametrc lower bound can be acheved automatcally. The use of the EL method for a fnte populaton parameter was frst consdered by Chen and Qn (1993), but ther method s only applcable under smple random samplng (SRS). Chen and Stter (1999) proposed pseduo emprcal lkelhood (PEL) whch can be used to deal wth complex survey data. Wu and Rao (2006) constructed a lkelhood rato-based confdence nterval for the populaton mean by usng PEL. For the most recent development of PEL, see Rao and Wu (2009). The lkelhood rato property s the most attractve property of the EL method. The correspondng confdence regon has several advantages compared to the normal approxmaton (NA) confdence regon. These nclude better coverage rate, shape respectng, and transformaton nvarance. However, the PEL rato converges to a scaled chsqured dstrbuton nstead of the standard ch-squred dstrbuton. The scale factor needs to be estmated and t often depends on the complex samplng desgn. In addton, the PEL

13 3 estmator s not equvalent to the desgn optmal estmator. To avod those drawbacks, we propose usng the populaton emprcal lkelhood (POEL) estmator n chapter 4. The POEL lkelhood rato converges to the standard ch-squred dstrbuton; the proposed estmator s equvalent to the desgn optmal estmator and the POEL method can combne several sources of auxlary nformaton. A voluntary sample s a self-selected sample whose frst order ncluson probabltes are unknown. The most popular method for the nference for a voluntary sample s propensty score weghtng. Rosenbaum and Rubn (1983) and Rosenbaum (1987) proposed usng propensty scores to estmate treatment effects n observatonal studes. Duncan and Stasny (2001) used the propensty score method to control coverage bas n telephone surveys. Lee (2006) appled the propensty score method to a volunteer panel web survey. Lee and Vallant (2009) and Vallant and Dever (2011) consdered the propensty score method for a web-based voluntary sample. All of these studes assumed an gnorable selecton mechansm. However, we often confront the case where the selecton mechansm does depend on the study varable tself. In chapter 5, we propose a novel two-phase approach for estmators wth a voluntary sample. The proposed method can be extended to handle a non-nested two-phase voluntary sample. The auxlary nformaton can be ncorporated va the generalzed method of moment (GMM). We organze the thess as followngs. In chapter 2, we present the new functonal form emprcal lkelhood (EL) method; We proposed a unfed theory of usng the EL method n mssng data problems n chapter 3; In chapter 4, we propose usng the populaton emprcal lkelhood (POEL) method for nference wth survey data; In chapter 5, a novel approach s proposed for nference n the voluntary sample problem. Future works are presented n chapter 6. Techncal detals are presented n the appendxes.

14 4 CHAPTER 2. SEMI-PARAMETRIC INFERENCE WITH A FUNCTIONAL-FORM EMPIRICAL LIKELIHOOD A paper submtted to the Journal of the Korean Statstcal Socety Sxa Chen and Jae Kwang Km Abstract A functonal-form emprcal lkelhood method s proposed as an alternatve for the emprcal lkelhood method. The proposed method has the same asymptotc propertes as the emprcal lkelhood method but has more flexblty n choosng the weght constructon. Also, some computatonal effcency can be ganed. Because t enjoys the lkelhood-based nterpretaton, the profle lkelhood rato test has a ch-square lmtng dstrbuton. Some computatonal detals are also dscussed, and results from lmted smulaton studes are presented. Key Words: Exponental tltng, Generalzed method of moments, Nonparametrc maxmum lkelhood method, Profle lkelhood rato test. 2.1 Introducton The emprcal lkelhood method, proposed by Owen (1988, 1990), provdes a useful tool for obtanng nonparametrc confdence regons for statstcal functonals. Even though the emprcal lkelhood method s a nonparametrc approach n the sense that t does not requre a parametrc model for the underlyng dstrbuton of the sample observaton, the emprcal lkelhood method enjoys some of the desrable propertes of the lkelhood-based method. Usng a nonparametrc lkelhood functon, the emprcal lkelhood method can easly ncorporate

15 5 known constrants on parameters and also ncorporate pror nformaton on parameters. For example, Chen and Qn (1993) and Qn (2000) dscuss combnng nformaton usng the emprcal lkelhood. A comprehensve overvew of the emprcal lkelhood method s provded by Owen (2001). We consder an extenson of the emprcal lkelhood method by provdng a class of nonparametrc estmators that have the same asymptotc propertes as the emprcal lkelhood method. In partcular, nstead of assumng a nonparametrc lkelhood, we consder a generalzaton of the emprcal lkelhood that uses a functonal-form lkelhood functon n the lkelhood maxmzaton. The class of functonal-form lkelhood functon contans the emprcal lkelhood functon as a specal case. The functonal-form lkelhood approach provdes several useful alternatves to the classcal emprcal lkelhood method n the sense that some of the computatonal dffculty of the emprcal lkelhood method can be avoded, and more clear nsghts can be obtaned from the emprcal lkelhood method. Let z 1,, z n be n ndependent realzatons of a vector-valued random varable Z wth a dstrbuton functon F (z) that s completely unspecfed. In the emprcal lkelhood approach, we consder a class of dstrbuton functons, F 1 F, that have support on z 1,., z n. Thus, the elements n F 1 can be wrtten as F w (x) = w I(z x) wth n w = 1 and w > 0, where I(z x) takes the value one f z x and takes the value zero otherwse. The parameter w s the amount of pont mass that unt z represents n the populaton. We are nterested n makng an nference about θ 0 that s defned as a unque soluton to E U (Z; θ) = 0, where U (Z; θ) s an r-dmensonal vector of some functon U(Z; θ) known up to θ and the dmenson of θ equals p r. Hansen (1982) and Imbens (1997) consdered ths over-dentfed stuaton n the context of a generalzed method of moments n econometrcs. In ths setup, Qn and Lawless (1994) consdered the emprcal lkelhood estmator of θ 0 that can be obtaned by maxmzng ln(w ) (2.1)

16 6 subject to w 1, U(z ; θ) = (1, 0). (2.2) Note that (2.2) s equal to the condton E U (Z; θ) = 0 for F F 1. Usng the Lagrange multpler method, the emprcal lkelhood estmator can be obtaned by maxmzng l e (θ) = ln w (θ), (2.3) where w (θ) s of the form w (θ) = 1 1 n 1 + ˆλ T θ U(z (2.4) ; θ) and ˆλ θ satsfes the second equaton of (2.2). Qn and Lawless (1994) showed that the emprcal lkelhood estmator satsfes 2 l e (ˆθ) l e (θ 0 ) d χ 2 p (2.5) where ˆθ s the emprcal lkelhood estmator. The result (2.5) s often called the Wlk s theorem for emprcal lkelhood and s qute useful n obtanng confdence regons for θ 0. The weght (2.4) used to compute the emprcal lkelhood estmator can be expressed as w (θ, ˆλ m ˆλT θ U(z ; θ) θ ) = (2.6) n j=1 ˆλT m θ U(z j ; θ), where m(x) = 1/(1 x) and ˆλ θ = ˆλ(θ; z 1,, z n ) satsfes w (θ, ˆλ θ )U(z ; θ) = 0. (2.7) The Lagrange multpler ˆλ θ = ˆλ(θ; z 1,, z n ) s completely determned by (2.7). We assume that, for gven θ, the soluton ˆλ θ to (2.7) s unque. The unque soluton exsts for any gven θ f 0 s nsde the convex hull of the ponts U(z 1 ; θ),, U(z n ; θ). We consder an extenson of the emprcal lkelhood estmator by allowng m(x) n (2.6) to be some smooth functon other than m(x) = 1/(1 x). The proposed estmator can be called the functonal-form emprcal lkelhood (FEL) estmator because t uses a known functon m(x) n computng the weghts n the FEL estmator. For example, the exponental tltng (ET) estmator consdered n Ktamura and Stutzer (1997) and Schennach (2007) s the same form (2.6) wth m(x) = exp(x). Imbens, Spady, and Johnson (1998) advocated usng the ET

17 7 estmator over the emprcal lkelhood (EL) estmator based on Monte Carlo nvestgaton and analytc comparson usng hgher order asymptotc expanson. In ths paper, we dscuss some asymptotc propertes for the FEL estmator. In partcular, asymptotc normalty and a verson of Wlk s theorem for the FEL estmator are establshed. We found that the asymptotc results n Qn and Lawless (1994) are specal cases of the general results n ths paper. The results n ths paper can also be used to make nferences for other types of FEL estmators, ncludng the ET estmator. The man results are presented n Secton 2. Some extensons are ntroduced n Secton 3 to llustrate possble theoretcal results of the proposed FEL estmator. In Secton 4, the underlyng algorthm s dscussed. Results from a lmted smulaton study are presented n Secton 5 and concludng remarks are made n Secton Man Results Based on the functonal form of the FEL weghts n (2.6), we can defne a functonal-form emprcal log-lkelhood functon l(θ) = l(θ, ˆλ θ ) = ln ω (θ, ˆλ θ ) = m (θ, ln ˆλ θ ) n m (θ, ˆλ θ ) (2.8) where m (θ, ˆλ θ ) = mˆλ T θ U(z ; θ) for some functon m( ) and ˆλ θ satsfes (2.7). The loglkelhood functon n (2.8) s a parametrc form n the sense that the lkelhood functon s known except for some unknown parameter (θ, λ). The computaton for optmzaton usng (2.8) s generally smpler than the computaton usng the nonparametrc lkelhood (2.1) snce the parameter space s reduced from n to p + r. The parameter λ s used to facltate the computaton for constraned optmzaton. Furthermore, the log-lkelhood functon (2.8) does not drectly use any dstrbutonal assumptons. Thus, the nature of the maxmum lkelhood estmator usng (2.8) s stll nonparametrc n the sense that t s vald wthout assumng any dstrbutonal assumptons. The only assumpton we use s E U(Z; θ 0 ) = 0. Let ˆθ be the soluton that maxmzes l(θ, ˆλ θ ) n (2.8). Let ˆQ 1 (θ, λ) = n ω (θ, λ) U(z ; θ) and ˆQ 2 (θ, λ) n 1 dl(θ, ˆλ θ )/dθ. The soluton ˆθ and ts correspondng λ-value, denoted by

18 8 ˆλ = ˆλ(ˆθ), satsfes ˆQ 1 (ˆθ, ˆλ) = 0 and ˆQ 2 (ˆθ, ˆλ) = 0. The soluton ˆθ s called the FEL estmator of θ 0. For smplcty of notaton, let γ = (θ, λ) and ˆγ = (ˆθ, ˆλ). Also, let ˆQ(γ) = ( ˆQ 1 (γ), ˆQ 2 (γ)). To dscuss the asymptotc propertes of the FEL estmator, we assume the followng condtons: (C1) The soluton θ 0 to E U(Z; θ) = 0 s unque. (C2) In the weght functon (2.6), the functon m(x) s always postve and has contnuous second-order dervatves at x = 0 wth m(0) = m (0) = 1. (C3) The partal dervatve U (θ) = U(θ)/ θ s a contnuous functon of θ n the compact set A and θ 0 A almost surely. (C4) The random functons ˆQ(γ) converge unformly n probablty to Q(γ) = E ˆQ(γ) n the compact set B and γ 0 B, where γ 0 = (θ 0, 0). The followng theorem provdes the consstency of the FEL estmator. Theorem Assume that condtons (C1)-(C4) hold. Assume that the soluton (ˆθ, ˆλ) to ˆQ 1 (θ, λ) = 0 and ˆQ 2 (θ, λ) = 0 s unquely determned. Then, the soluton (ˆθ, ˆλ) satsfes where θ 0 s a unque soluton to E U(Z; θ) = 0. p lm n (ˆθ, ˆλ) = (θ 0, 0) (2.9) In the specal case of the emprcal lkelhood method, Qn and Lawless (1994) also proved (2.9). The proof of Theorem 2.2.1, whch s dfferent from that of Qn and Lawless (1994), s presented n Secton A of Appendx A. Theorem In addton to the condtons of Theorem 2.2.1, assume that (C5) 2 U(z, θ)/( θ θ T ) s contnuous at θ n the compact set A almost surely. (C6) U(Z; θ) 3, U(Z; θ)/ θ, and 2 U(Z, θ)/( θ θ T ) are bounded by some ntegrable functon G(Z).

19 9 (C7) The r p matrx E U(Z; θ 0 )/ θ has full column rank p. Also, V aru(z; θ) s postve defnte n the compact set A. Then, we have where where and n ˆθ θ 0 ˆλ 0 V = d N(0, V) (2.10) V V 2 V 1 = E( U θ )T (EUU T ) 1 E( U θ ) 1 V 2 = E(UU T ) 1 I E( U θ )V 1E( U θ )T [E(UU T )] 1. The proof of Theorem s presented n Secton B of Appendx A. Usng Theorem 2.2.2, we can construct a Wald-type confdence nterval for θ 0. The asymptotc varance V 1 of n(ˆθ θ 0 ) can be consstently estmated by w U ( z ; ˆθ ) T w U ( z ; ˆθ ) ( U z ; ˆθ ) 1 T w U where w = w (ˆθ, ˆλ) s the fnal FEL weght n (2.6) evaluated at ˆθ and ˆλ. ( ˆθ) 1 z ;, By Theorem 2.2.2, asymptotc varance of the FEL estmator can be derved. For example, f z = (x, y ) T and µ x = E(x) s known, the FEL estmator of θ = E(y) can be obtaned usng ˆθ = n ˆm y / n ˆm wth ˆm = mˆλ(x µ x ) where ˆλ satsfes n ˆm (x µ x ) = 0. The asymptotc varance of ˆθ s equal to n 1 V (y) 1 ρ 2 where ρ s the correlaton coeffcent of x and y n the populaton. Note that the asymptotc varance s equal to the asymptotc varance of the regresson estmator ˆθ reg = ȳ + S yx S 1 xx (µ x x) (2.11) and so the FEL estmator n ths setup s asymptotcally equvalent to the regresson estmator (2.11). The regresson estmator (2.11) s the maxmum lkelhood estmator under the

20 10 bvarate normalty assumpton (Anderson, 1957). The asymptotc varance V 1 s equal to the semparametrc lower bound dscussed n Chamberlan (1987) and so the FEL estmator acheves semparametrc effcency. Theorem The functonal-form emprcal lkelhood rato statstc for testng H 0 : θ = θ 0 s W (θ 0 ) = l(ˆθ) l(θ 0 ) (2.12) where l(θ) s gven by (2.8). Under the assumpton of Theorem 2.2.1,we have that 2W (θ 0 ) d χ 2 p (2.13) as n, when H 0 s true. Theorem 2.2.3, whch can be called the Wlk s theorem for FEL method, shows that the FEL log-lkelhood n (2.8) can be used to construct a confdence nterval based on the lkelhood rato statstcs (2.12) as n the parametrc lkelhood method. In the followng corollary, we show that the FEL method can be used to construct a profle of lkelhood rato confdence ntervals. The proofs of Theorem and Corollary are presented n Sectons C and D of Appendx A, respectvely. Results smlar to Corollary are also presented n Qn and Lawless (1994) n the context of emprcal lkelhood method, but we presents a dfferent proof of the corollary. Corollary Let θ T = (θ 1, θ 2 ) T, where θ 1 and θ 2 are q 1 and (p q) 1 vectors, respectvely. For H 0 : θ 1 = θ1 0, the profle generalzed emprcal lkelhood rato test statstc s defned by W 2 = l(ˆθ 1, ˆθ 2 ) l(θ 0 1, ˆθ 0 2) (2.14) where ˆθ 0 2 maxmzes l(θ0 1, θ 2) wth respect to θ 2. Then, under H 0, we have that 2W 2 d χ 2 q as n.

21 11 Remark The FEL method could be called a generalzed emprcal lkelhood method because t s essentally a generalzaton of the emprcal lkelhood method usng functonal-form weght functon. The term generalzed emprcal lkelhood, however, was already used by Smth (1997) and Newey and Smth (2004) to denote another type of extenson to emprcal lkelhood method n econometrcs usng a saddle pont optmzaton problem. Our method s dfferent from the GEL method because we do not have to specfy the objectve functon for saddle pont computaton and we have only to drectly specfy the functonal-form for the weghts n FEL estmators. 2.3 Extenson The log-lkelhood functon n (2.8) can be vewed as a negatve dvergence functon between 1/n and w. Instead of usng a dvergence functon based on the log-lkelhood (2.8), one can also consder a more general class of dvergence functons. Specfcally, we consder a class of dvergence functons based on power-dvergence statstcs, proposed by Cresse and Read (1984), CR(α) = 2 α(α + 1) ( ) 1/n α 1. (2.15) Note that CR(0) = 2 n log(nω ), whch s the log-lkelhood functon n (2.6) and CR( 1) = 2 n nω log(nω ), whch s often called the Kullback-Lebler dvergence measure. The results n Secton 2 show that the choce of weght functon s not crtcal because the ω resultng estmators are all asymptotcally equvalent. Surprsngly, we show n ths secton that the choce of the objectve functon s not crtcal ether. The results presented here are an extenson of Baggerly (1998) to the case when θ s defned through the soluton to an estmatng equaton. Theorem Let ˆQ 1 (θ, λ) = n ω U(z ; θ) and ˆQ 2 (θ, λ) = n 1 dl 3 (θ, λ)/dθ where ω s defned n (2.6) and 1 l 3 (θ, λ) = α(α + 1) [ ω (θ, λ)n α 1 ]. (2.16)

22 12 Suppose that (ˆθ, ˆλ) s the soluton of ˆQ1 (θ, λ) = 0 and ˆQ 2 (θ, λ) = 0. Then under condtons stated n theorem and theorem 2.2.2, we have ˆθ θ 0 n d N(0, V ) (2.17) ˆλ where V s defned n (2.10). Also, the generalzed emprcal lkelhood rato statstc for testng H 0 : θ = θ 0 satsfes 2 l 3 (ˆθ) l 3 (θ 0 ) d χ 2 p (2.18) where l 3 (θ) s gven by (2.16). Theorem s a general result n the sense that, for the specal case of α = 0 n (2.15), t leads to Theorem and Theorem Also, for the specal case of α = 1, we have the followng result. Its proof s very smlar to that of Theorem and s not presented here. Corollary Let l 2 (θ) = n nω log(nω ) and assume that ˆθ maxmzes l 2 (θ). Then we have and θ 0 s the true value of θ. 2 l 2 (ˆθ) l 2 (θ 0 ) d χ 2 p, 2.4 Computatonal Aspects The FEL estmator that maxmzes the objectve functon (2.8) subject to the constrant (2.7) could be vewed as a standard optmzaton problem n the (θ, λ) space of dmenson p + r. However, as shown n Secton A of Appendx A, the probablty lmt Q 2 (θ, λ) of ˆQ 2 (θ, λ) satsfes Q 2 (θ, 0) = 0 for all θ. Thus, standard approaches to solvng the systems of equatons ˆQ 1 (θ, λ) = 0 and ˆQ 2 (θ, λ) = 0 can have erratc behavor n the neghborhood of λ = 0. To avod ths numercal problem, we consder an approach usng a penalty term used n the rdge regresson method, as was also consdered by Imbens, Spady, and Johnson (1998). The objectve functon wth a penalty term can be expressed as l (θ, λ) = l (θ, λ) 0.5K ˆQ 1 (θ, λ) T W ˆQ 1 (θ, λ), (2.19)

23 13 where l (θ, λ) s the orgnal objectve functon, such as (2.8) or (2.16), and K s a scalar penalty term that makes the optmzaton problem locally convex, and W s some r r postve defnte matrx. Note that ˆQ 2 (θ, λ) = n 1 l (θ, λ) / θ can be wrtten ˆQ 2 (θ, λ) = Q 2 (θ, λ) K n 1 Q 1θ (θ, λ) T W ˆQ 1 (θ, λ), where Q 1θ (θ, λ) = ˆQ 1 (θ, λ)/ θ. Thus, for suffcently large K = O(n), we have Q 2(θ, 0) 0 for θ θ 0 and Q 2(θ 0, 0) = 0, (2.20) where Q 2 (θ, λ) s the probablty lmt of ˆQ 2 (θ, λ). Property (2.20) follows because Q 2(θ, λ) = Q 2 (θ, λ) + C(θ, λ)q 1 (θ, λ) (ˆθ for some matrx C(θ, λ), and Q 1 (θ, λ) satsfes (2.20). Once the soluton, ˆλ ) that maxmzes l (θ, λ) n (2.19) s obtaned, we solve ˆQ (ˆθ ) 1, λ = m λ T U(z ; ˆθ ) U(z ; ˆθ ) = 0 (2.21) for λ to get the fnal soluton. The Newton-type soluton to (2.21) can be computed by ˆλ (t+1) = ˆλ ) 1 ) (t) ṁ (ˆλT (t) U U U T m (ˆλT (t) U U, where U = U(z ; ˆθ ), wth an ntal value ˆλ (0) = 0. To demonstrate the computaton, we use a sample of sze n = 50 generated from a bvarate normal dstrbuton (X, Y ) d N 1, (2.22) In the computaton, we set W = I and let K vary from 10 to We assume that µ x = 1 s known and we are nterested n estmatng µ y. We used the exponental tltng weght of the form ω = exp(λ 1 x + λ 2 y ) n j=1 exp(λ 1x j + λ 2 y j ) From the realzed sample, the estmates of (µ y, λ 1, λ 2 ) that maxmze the penalzed lkelhood (2.19) are computed for each K usng ( ) ˆQ 1 (θ, λ) = ω (x 1), ω (y θ).

24 14 < Fgure 2.1 around here. > Fgure 2.1 presents the plot of the soluton (ˆµ y, ˆλ 1, ˆλ 2 ) aganst the value of the penalty parameter K. The estmates of µ y and λ 1 converge as K gets larger, but the estmate of λ 2 does not converge even for large K. Because the computaton n Fgure 1 s based on a sngle realzaton of the sample, the resultng ˆµ y s not necessarly equal to µ y = 1. The estmate for µ y can be used for fnal computaton but ˆλ = (ˆλ 1, ˆλ 2 ) need to be updated usng (2.21). 2.5 Smulaton Study To check the fnte sample performance of the FEL estmators, we performed two lmted smulaton studes. In the frst smulaton study, we generated two sets of bvarate data (x, y ) from two dfferent samplng dstrbutons: the bvarate normal dstrbuton (2.22) and a bvarate non-normal dstrbuton defned by x χ 2 (1) y = M(x 1) + e, (2.23) where M = 0.5, e exp(1), and e s ndependent of x for = 1, 2,..., n. Note that, n both dstrbutons, E(X) = E(Y ), V (X) = V (Y ), and Corr(X, Y ) = 0.5. For each dstrbuton, we generated B = 2, 000 ndependent Monte Carlo samples of sze n, where we used the three dfferent sample szes: n = 20, 50, and 100. For each sample generated above, we computed three FEL estmators of µ y = E(Y ) under the followng scenaros: (Scenaro 1) We have no extra nformaton. (Scenaro 2) We use µ x = 1 as the constrant. (Scenaro 3) We use µ x = µ y as the constrant. (Scenaro 4) We use µ x = µ y and σ x = σ y as the constrants. In Scenaro 1, we used the sample mean to estmate θ. In Scenaros 2-4, the FEL methods are used to ncorporate the addtonal nformaton. In Scenaro 3, for example, the addtonal

25 15 nformaton can be ncorporated by usng the FEL weghts ω = m λ 1 (x y ) + λ 2 (y θ) n j=1 m λ 1(x j y j ) + λ 2 (y θ) where λ 1 and λ 2 are computed by (2.21) wth U(x, y ; θ) = (x y, y θ) and θ s determned by maxmzng the gven objectve functon. For the choce of m( ) functon n ω, we consdered three dfferent FEL estmators as below: 1. Emprcal lkelhood estmator (EL) usng m(x) = 1/(1 x) wth the objectve functon (2.8). 2. Exponental tltng estmator (ET1) usng m(x) = exp(x) wth the objectve functon l(θ) = n nω log(nω ). 3. Exponental tltng estmator (ET2) wth the objectve functon (2.8). Monte Carlo mean and Monte Carlo varance of the FEL estmators are computed for each scenaro based on the Monte Carlo sample of sze B = 2, 000. All of the FEL estmators are essentally unbased, and the Monte Carlo means are not presented here. Table 2.1 presents the Monte Carlo estmates of the relatve effcency of the FEL estmators. The effcency s computed by the rato of the varance of the sample mean (under Scenaro 1) to the varance of the correspondng FEL estmator. Under the normal dstrbuton, the theoretcal values of the standardzed varance of the FEL estmators are all approxmately equal to 1/(1 ρ 2 ) = for the three scenaros, whch s consstent wth the smulaton results n Table 2.1. The smulaton results n Table 2.1 show that all of the FEL estmators show smlar effcency for large sample sze (n = 100) but the ET estmators are slghtly more effcent than the EL estmator for small sample sze (n = 20, 50). In the second smulaton study, we compared the statstcal power of test statstcs derved from the FEL methods. In ths smulaton study, we frst generated 6 dfferent samples from (X, Y ) d N 1, 1 ρ. 1 ρ 1

26 16 wth 6 dfferent values of ρ, varyng from 0 to 0.5. In addton to the normal model, we also generated samples from the non-normal model (2.23) where M s chosen to make ρ = (0, 0.1, 0.2, 0.3, 0.4, 0.5). In the second study we consdered the same three FEl estmators. We used θ = (µ x, µ y, σx, 2 σy, 2 ρ) and U(x, y; θ) s a 5-dmensonal vector of unbased estmatng functon for θ. For each FEL method, the profle lkelhood test s constructed by computng the full maxmum lkelhood estmator ˆθ and the profle maxmum lkelhood estmator ˆθ 0 that s computed under the null hypothess H 0 : ρ = 0. H 0 : ρ = 0 f The profle lkelhood test wth level α rejects the null hypothess 2 l (ˆθ1, ˆθ ) ( 2 l 0, ˆθ ) 2 0 χ 2 1(1 α) where θ 1 = ρ, θ 2 = (µ x, µ y, σx, 2 σy) 2 and χ 2 1 (1 α) s the 1 α quantle of the ch-square dstrbuton wth 1 degrees of freedom. In addtonal to the FEL method, we also computed the normal-based Pearson test for comparson. The Monte Carlo power of the level α = 0.05 test statstc was computed by the relatve frequency of rejectng the null hypothess H 0 : ρ = 0. Table 2.2 presents the Monte Carlo power of the test statstcs obtaned from three FEL methods for each sample. For ρ = 0, the power s the sze of the test and t converges to α = 0.05 as n gets larger. In the normal sample, the power of the test based on ET method s hgher than that for EL method when n = 100. The ET1 method shows smaller type-1 error than the ET2 method when the sample sze s small. In the non-normal sample, the EL method seems to have better statstcal powers than the ET methods. Overall, the three FEL methods show smlar performances n most cases, whch s consstent wth our theory. 2.6 Conclusons Emprcal lkelhood method s useful n ncorporatng the known constrants of parameters and also n combnng nformaton from dfferent sources. The functonal-form emprcal lkelhood method proposed n ths paper provdes a unfed approach of handlng such constrants wthout usng dstrbutonal assumptons on the sample observaton. FEL methods allow us

27 17 to set a more flexble objectve functon as well as a flexble weght functon. Thus, computatonal effcency can be acheved by fndng a smple weght functon n the FEL method. For example, n the smulaton study, the computng tme for the ET method s much shorter than the computng tme for the EL method. The FEL method can be used to provde a lkelhood rato test wth a ch-square lmtng dstrbuton. Also, a profle lkelhood rato test can be derved usng the orthogonalty of the log-lkelhood functons. To mprove the coverage propertes of the FEL n the small sample szes, some cuttng-edge technques such as bootstrap calbraton (Hall and Horowtz, 1996) or the Bartlett correcton (Chen and Cu, 2006) can be used. Further nvestgaton n ths drecton, ncludng the Hgher order expanson as n Lu and Chen (2010), s not dscussed here and wll be a topc of future research. muy lambda1 lambda2 emuy elambda elambda2 4e 08 2e 08 0e+00 2e 08 4e K K K Fgure 2.1 Parameter estmatons versus penalty parameter.

28 18 Table 2.1 Monte Carlo relatve effcency of the pont estmators. Model Stuaton Sample sze(n) EL ET1 ET2 n = S1 n = n = n = S2 n = Normal n = n = S3 n = n = n = S4 n = n = n = S1 n = n = n = S2 n = Non-normal n = n = S3 n = n = n = S4 n = n =

29 19 Table 2.2 Power comparsons for testng H 0 : ρ = 0 ρ Model Method Sample sze n = Pearson n = n = n = EL n = Normal n = n = ET1 n = n = n = ET2 n = n = n = Pearson n = n = n = EL n = Non n = normal n = ET1 n = n = n = ET2 n = n =

30 20 CHAPTER 3. A UNIFIED THEORY ON EMPIRICAL LIKELIHOOD METHODS WITH MISSING DATA AND SURVEY SAMPLING A paper submtted to the Australan and New Zealand Journal of Statstcs(revson nvted) Sxa Chen and Jae Kwang Km Abstract Effcent estmaton wth mssng data s an mportant practcal problem wth many applcaton areas. Survey samplng can be treated as a mssng data problem where the sample s treated as a realzaton of a known response mechansm. Parameter estmaton under nonresponse s consdered when the parameter s defned as a soluton to an estmatng equaton. Usng a response probablty model, a complete-response emprcal lkelhood method can be constructed and the nonparametrc maxmum lkelhood estmator can be obtaned by solvng the weghted estmatng equaton where the weghts are computed by maxmzng the complete-response emprcal lkelhood subject to the constrants that ncorporate the auxlary nformaton obtaned from the full sample. Often the constrants are constructed from the workng outcome regresson model for the condtonal dstrbuton of the estmatng functon gven the observaton. The proposed method acheves the sem-parametrc lower bound when we correctly specfy the condtonal expectaton of the estmatng functon, regardless of whether the response probablty s known or estmated. When the response probablty s estmated nonparametrcally, the resultng emprcal lkelhood method automatcally acheves the sem-parametrc lower bound wthout specfyng the condtonal dstrbuton of the estmatng functon. The proposed method s also applcable to two-phase samplng. Asymptotc theores

31 21 are derved and smulaton studes are also presented. Key Words: Mssng at random; Nonparametrc estmaton; Response mechansm; Propensty score. 3.1 Introducton The emprcal lkelhood (EL) method, proposed by Owen (1988, 1990), has become a very powerful tool for nonparametrc nference n statstcs. It uses a lkelhood-based approach wthout havng to make a parametrc dstrbutonal assumpton about the data observaton. Thus, the EL method often leads to effcent estmaton and enables lkelhood-rato type nference. Qn and Lawless (1994) consdered the stuaton when the parameter of nterest s the soluton to a system of estmatng equatons. Owen (2001) provdes a comprehensve overvew of the EL method. Under exstence of mssng data or survey data, however, the EL method s not drectly applcable and some adjustment needs to be made. Qn (1993) addressed ths problem usng a based samplng argument of Vard (1985). Wang and Rao (2002) used regresson-type mputaton approaches to emprcal lkelhood nference. Wang and Chen (2009) used a nonparametrc regresson mputaton approach to handle mssng data n the emprcal lkelhood nference. The mputaton approach uses some assumptons about the mssng data gven the observed data and usually assumes that the response mechansm s gnorable n the sense of Rubn (1976). Under an gnorable mssng mechansm, the explct modelng of the response model s avoded. In the case of survey samplng, Chen and Stter (1999) consdered the pseudo emprcal lkelhood estmator that uses the samplng weght n the emprcal log-lkelhood functon. Km (2009) consdered an alternatve emprcal lkelhood functon based on the based samplng lkelhood of Vard (1985) and Qn (1993). Wu and Rao (2006) dscussed nterval estmaton usng the pseudo emprcal lkelhood. Note that the survey samplng can be treated as a specal case of mssng data problem, where the sample s obtaned by a planned mssng mechansm and the frst-order sample ncluson probablty corresponds to the response probablty n the usual mssng data problem. The man dfference s that the sample ncluson probabltes are

32 22 known n survey samplng, as the mssng mechansm s planned by the samplng desgn. In ths paper, we consder an alternatve approach to handlng mssng data usng a model for response probablty. Use of parametrc response probablty model n the emprcal lkelhood nference has been consdered n Qn and Zhang (2007) and n Chen et al. (2008). Qn et al. (2009) and Tan (2011) consdered usng EL to model the complete lkelhood, where the nonparametrc lkelhood functon s computed for the whole sample ncludng the unts wth mssng data. The use of complete lkelhood attans the full effcency and also provdes a nce theory of the lmtng ch-square dstrbuton n the lkelhood rato test statstcs. However, n some practcal case, the unt-level nformaton for the complete lkelhood s not always avalable and the complete lkelhood cannot be computed. For example, n survey samplng, the ndvdual values of auxlary varable n the non-sampled part are not usually avalable. In ths case, the approach of usng the complete lkelhood for the fnte populaton may not be applcable. If the response mechansm s nonparametrcally modeled, the lterature s somewhat sparse. Cheng (1994) dscussed some asymptotc propertes of the mean estmator usng the kernel regresson method to estmate the condtonal outcome regresson model under an gnorable mssng case. Recently, Km and Yu (2011) extended the approach of Cheng (1994) to handle nongnorable nonresponse. Xue (2009) dscussed an emprcal lkelhood method for lnear models usng the weghts computed from a nonparametrc model where the kernel regresson method s used to estmate the response model. Da Slva and Opsomer (2009) consdered another type of nonparametrc response probablty estmaton usng local polynomal regresson. Hrano et al (2003) and Cattaneo (2010) dscussed semparametrc effcency of the nonparametrc response propensty estmators n the context of estmatng average treatment effect n econometrcs. In ths paper, we propose a unfed approach of the EL method wth mssng data that avods usng the complete lkelhood. Under the setup of estmatng functon n Qn and Lawless (1994), the proposed method can handle the stuaton regardless of whether the response probabltes are known or estmated, parametrcally or even nonparametrcally. When the response probabltes are known, the proposed method can be appled to survey weghtng

33 23 problems when the frst-order ncluson probabltes are known. Incorporatng the populaton level auxlary nformaton nto the weghts n the sample s an mportant problem n survey samplng and s often called calbraton weghtng. Calbraton weghtng s consdered n Devlle and Särndal (1992), Fuller (2002), and Km and Park (2010), among others. The proposed method can be drectly applcable to the calbraton weghtng problem. When the response probabltes are estmated from a parametrc model, the proposed method under gnorable response mechansm s smlar to the method of Qn and Zhang (2007). The proposed method s drectly applcable to the problem of the propensty score weghtng method. The propensty score weghtng method can be found, for example, n Durrant and Sknner (2006), Km and Km (2007), and Chang and Kott (2008). We show that employng EL method usng a sutable choce of control varable leads to effcent estmaton n the sense that t acheves the lower bound of the asymptotc varance. Optmal choce of the control varable requres correct specfcaton of the condtonal dstrbuton of the mssng data gven the observaton. Under the nonparametrc propensty score method, whch wll be dscussed n Secton 5, the lower bound of the asymptotc varance can be acheved wthout correctly specfyng the condtonal dstrbuton. In Secton 2, we frst revew the exstng methods of emprcal lkelhood under mssng data and dscuss a unfed approach of the EL method. Asymptotc propertes of the proposed estmator under known response probabltes are dscussed n Secton 3. The proposed EL estmator s dscussed under estmated response probablty n Secton 4. Use of the nonparametrc response model for the EL approach s dscussed n Secton 5. The proposed method s extended to two-phase samplng n Secton 6. Results from two smulaton studes are reported n Secton Basc setup Consder a multvarate random varable (X, Y ) wth dstrbuton functon F (x, y) whch s completely unspecfed except that EU(X, Y ; θ 0 ) = 0 for some θ 0. We are nterested n estmatng the parameter θ 0 from a random sample of the dstrbuton. To avod unnecessary detals, we assume that the soluton to EU(X, Y ; θ) = 0 s unque. For smplcty, we assume

34 24 that the dmenson of U s equal to the dmenson of θ. If (x, y ), = 1, 2,..., n, are n ndependent realzatons of the random varable (X, Y ), a consstent estmator of θ 0 can be obtaned by solvng U(x, y ; θ) = 0. (3.1) In ths paper, we consder the problem of estmatng θ 0 when x s always observed and y s subject to mssngness. Let r = 1 f y s observed and r = 0 otherwse. We consder an approach based on the emprcal lkelhood (EL) method. To explan the dea, frst note that the jont densty of the observed data can be wrtten as p nr (1 p) n nr f(x, y r = 1) f(x r = 0), (3.2) r =1 r =0 where n r s the response sample sze, p = P r(r = 1), f(x, y r) s the condtonal densty of (X, Y ) gven r, and f(x r = 0) = f(x, y r = 0)dy s the margnal densty of X among r = 0. In the emprcal lkelhood approach, the dstrbuton s assumed to have the support on the sample observaton. Let F 1 (x, y) = P r(x x, Y y r = 1) and F 0 (x, y) = P r(x x, Y y r = 0). Under the emprcal lkelhood approach, we can express F 1 (x, y) = r =1 ω I(x x, y y), (3.3) where r =1 ω = 1, ω s the pont mass assgned to (x, y ) n the nonparametrc dstrbuton of F 1 (x, y), and I(B) s an ndcator functon for event B. To express F 0 (x, y) usng ω, note that we can wrte f(x, y r = 0) = f(x, y r = 1) Odd(x, y ) EOdd(x, y ) r = 1, where Odd(x, y) = P r(r = 0 x, y) P r(r = 1 x, y). Thus, we can express F 0 (x, y) = P r(x x, Y y r = 0) by F 0 (x, y) = r =1 ω O I(x x, y y) r =1 ω O, (3.4)

35 25 where O = Odd(x, y ). Note that F 0 (x, y) s completely determned by two factors: ω and O. The factor ω s determned by the dstrbuton F 1 (x, y) and the factor O s determned by the response mechansm. If Odd(x, y) s a known functon of (x, y), then we have only to determne ω. From (3.4), the jont dstrbuton of (x, y) can be wrtten as F w (x, y) = p r ω I(x x, y y) + (1 p) =1 ω O I(x x, y y) r =1 r =1 ω O r = p ω I(x x, y y) + (1/p 1) =1 ω O I(x x, y y) r =1 ω. O Note that (3.3) mples r =1 r =1 Thus, we have r =1 ω O = 1/p 1 and 1 ω (O + 1) = E π(x, Y ) r = 1 1 = f(x, y r = 1)dxdy π(x, y) F w (x, y) = = 1 π(x, y) π(x, y)f(x, y) dxdy = 1/p. p r =1 ω (1 + O )I(x x, y y) r =1 ω. (O + 1) We propose maxmzng the partal lkelhood r =1 f(x, y r = 1) n (3.2) n constructng the emprcal lkelhood. Thus, the proposed emprcal lkelhood approach can be formulated as maxmzng subject to ω = 1, r =1 l e (θ) = r =1 log (ω ), (3.5) ω (1 + O )U(x, y ; θ) = 0. (3.6) r =1 Note that, n constrant (3.6), the observed values of x wth r = 0 are not used. To ncorporate the partal nformaton, we can mpose r =1 ω (1 + O )h(x ; θ) r =1 ω (1 + O ) = n 1 h(x ; θ). (3.7) as an addtonal constrant for some h(x ; θ). The choce of h(x; θ) wll be dscussed later.

36 26 There are several other approaches usng the emprcal lkelhood wth mssng data. Qn et al. (2002) consdered usng emprcal lkelhood for nongnorable nonresponse. Wang and Rao (2002) proposed emprcal lkelhood-based nference under mputaton for mssng response data. Qn and Zhang (2007) proposed an emprcal lkelhood method for estmatng the mean response under gnorable mssng data where the response probablty π = P r(r = 1 X ) s parametrcally modeled by π = π (φ 0 ) for some φ 0. Specfcally, they proposed maxmzng l = r =1 log π ( ˆφ)p /ˆν, subject to p = 1, r =1 p π ( ˆφ) = ˆν, r =1 p h(x ) = n 1 r =1 h(x ), (3.8) where ˆφ s the maxmum lkelhood estmator of φ 0 n the response probablty, h(x ) s an arbtrary varable and ˆν = n 1 n π ( ˆφ). Once the estmated probablty ˆp s computed by the above maxmzaton procedure, the populaton mean can be estmated by ˆθ = r =1 ˆp y. Chen et al. (2008) bult two emprcal lkelhoods for response and non-response varables separately and formulated two estmatng equatons based on these two emprcal lkelhoods. In the context of the current setup, ther proposed method can be descrbed as maxmzng l = r =1 log(p ) + r j =0 log(q j), subject to r =1 p = 1, p 0, r j =0 q j = 1, q j 0, and r =1 p h(x ; θ) µ π ( ˆφ) = 0, r j =0 q j h(x j ; θ) µ 1 π j ( ˆφ) = 0, (3.9) where ˆφ s the maxmum lkelhood estmator. Qn et al. (2009) consdered maxmzng the complete lkelhood l c = n log(ω ) subject to and ω = 1, ω ( r π 1)h (θ) = 0, ω r π U (θ) = 0, (3.10) π (φ)/ φ ω r π (φ) = 0. (3.11) π (φ)1 π (φ) The computaton requres that the ndvdual values of x for r = 0 be avalable, whch s not always possble, as dscussed n Secton 1. For example, n survey samplng problem, we only observe (x, y ) for r = 1 and the aggregate nformaton x n = n 1 n x s avalable. In ths case, the method of Qn et al. (2009) s not applcable.