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  1. 1. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionRobust inference in two phases sampling with an application to unit nonresponse Jean-François Beaumont, Statistics Canada Cyril Favre Martinoz, Crest-Ensai David Haziza, Université de Montréal, Crest-Ensai 30 janvier 2013 1/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  2. 2. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionSommaire 1 Introduction 2 How to measure the influence ? 3 Unit nonresponse 4 Simulation study 5 Conclusion 2/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  3. 3. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionSommaire 1 Introduction 2 How to measure the influence ? 3 Unit nonresponse 4 Simulation study 5 Conclusion 3/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  4. 4. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionInfluential units Unusual observations with possibly large design weights or large value 4/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  5. 5. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionInfluential units Unusual observations with possibly large design weights or large value Many survey statistics are sensitive to the presence of influential units 4/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  6. 6. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionInfluential units Unusual observations with possibly large design weights or large value Many survey statistics are sensitive to the presence of influential units Including or excluding an influential unit in the calculation of these statistics can have a dramatic impact on their magnitude. 4/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  7. 7. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionInfluential units Unusual observations with possibly large design weights or large value Many survey statistics are sensitive to the presence of influential units Including or excluding an influential unit in the calculation of these statistics can have a dramatic impact on their magnitude. The occurrence of outliers is common in business surveys because the distributions of variables (e.g., revenue, sales, etc.) are highly skewed (heavy right tail) 4/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  8. 8. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionInfluential units Unusual observations with possibly large design weights or large value Many survey statistics are sensitive to the presence of influential units Including or excluding an influential unit in the calculation of these statistics can have a dramatic impact on their magnitude. The occurrence of outliers is common in business surveys because the distributions of variables (e.g., revenue, sales, etc.) are highly skewed (heavy right tail) Influential units are legitimate observations 4/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  9. 9. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionInfluential units Unusual observations with possibly large design weights or large value Many survey statistics are sensitive to the presence of influential units Including or excluding an influential unit in the calculation of these statistics can have a dramatic impact on their magnitude. The occurrence of outliers is common in business surveys because the distributions of variables (e.g., revenue, sales, etc.) are highly skewed (heavy right tail) Influential units are legitimate observations The impact of influential units can be minimized by using a good sampling design : for example, stratified sampling with a take-all stratum 4/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  10. 10. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionInfluential units Unusual observations with possibly large design weights or large value Many survey statistics are sensitive to the presence of influential units Including or excluding an influential unit in the calculation of these statistics can have a dramatic impact on their magnitude. The occurrence of outliers is common in business surveys because the distributions of variables (e.g., revenue, sales, etc.) are highly skewed (heavy right tail) Influential units are legitimate observations The impact of influential units can be minimized by using a good sampling design : for example, stratified sampling with a take-all stratum 4/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  11. 11. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionInfluential units Even with a good sampling design, influential units may still be selected in the sample (e.g., stratum jumpers) 5/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  12. 12. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionInfluential units Even with a good sampling design, influential units may still be selected in the sample (e.g., stratum jumpers) In the presence of influential units, survey statistics are (approximately) unbiased but they can have a very large variance. 5/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  13. 13. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionInfluential units Even with a good sampling design, influential units may still be selected in the sample (e.g., stratum jumpers) In the presence of influential units, survey statistics are (approximately) unbiased but they can have a very large variance. Reducing the influence of large values produces stable but biased estimators 5/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  14. 14. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionInfluential units Even with a good sampling design, influential units may still be selected in the sample (e.g., stratum jumpers) In the presence of influential units, survey statistics are (approximately) unbiased but they can have a very large variance. Reducing the influence of large values produces stable but biased estimators Treatment of influential units : trade-off between bias and variance 5/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  15. 15. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionInfluential units Even with a good sampling design, influential units may still be selected in the sample (e.g., stratum jumpers) In the presence of influential units, survey statistics are (approximately) unbiased but they can have a very large variance. Reducing the influence of large values produces stable but biased estimators Treatment of influential units : trade-off between bias and variance 5/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  16. 16. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionTwo-phase designs U : finite population of size N s1 : first-phase sample, of size n1 s2 : second-phase sample, of size n2 , selected from s1 I1i : first-phase sample selection indicator for unit i I2i : second-phase sample selection indicator for unit i Vectors of indicators : I1 = (I11 , · · · , I1N ) and I2 = (I21 , · · · , I2N ) First-phase inclusion probability for unit i : π1i = P (I1i = 1) Second-phase inclusion probability for unit i : π2i (I1 ) = P (I2i = 1|I1 ; I1i = 1) 6/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  17. 17. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionTwo-phase designs U(N) I1i  0, I 2i  0 I1i  1, I 2i  1 s2 (n2 ) I1i  1, I 2i  0 s1 (n1 ) 6/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  18. 18. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionPoint estimation Goal : estimate a population total of a variable of interest y, Y = yi i∈U 7/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  19. 19. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionPoint estimation Goal : estimate a population total of a variable of interest y, Y = yi i∈U y-values : available only for i ∈ s2 7/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  20. 20. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionPoint estimation Goal : estimate a population total of a variable of interest y, Y = yi i∈U y-values : available only for i ∈ s2 Complete data estimator : Double expansion estimator ˆ yi yi YDE = = ∗ i∈s2 π1i π2i i∈s πi 2 7/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  21. 21. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionPoint estimation Goal : estimate a population total of a variable of interest y, Y = yi i∈U y-values : available only for i ∈ s2 Complete data estimator : Double expansion estimator ˆ yi yi YDE = = ∗ i∈s2 π1i π2i i∈s πi 2 ˆ YDE is design-unbiased for Y ; that is, ˆ E1 E2 (YDE |I1 ) = Y 7/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  22. 22. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionSommaire 1 Introduction 2 How to measure the influence ? 3 Unit nonresponse 4 Simulation study 5 Conclusion 8/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  23. 23. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionTotal error ˆ The total error of YDE : ˆ YDE − Y How to measure the influence (or impact) of a unit on both errors ? 9/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  24. 24. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionTotal error ˆ The total error of YDE : ˆ YDE − Y How to measure the influence (or impact) of a unit on both errors ? Single phase sampling : the conditional bias ; Moreno-Rebollo, Munoz-Reyez and Munoz-Pichardo (1999), Beaumont, Haziza and Ruiz-Gazen (2013). 9/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  25. 25. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionTotal error ˆ The total error of YDE : ˆ YDE − Y How to measure the influence (or impact) of a unit on both errors ? Single phase sampling : the conditional bias ; Moreno-Rebollo, Munoz-Reyez and Munoz-Pichardo (1999), Beaumont, Haziza and Ruiz-Gazen (2013). How to construct a robust estimator to the presence of influential units ? Single phase designs : Beaumont, Haziza and Ruiz-Gazen (2013). 9/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  26. 26. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionMeasuring the influence : the conditional bias We distinguish between three cases : i ∈ s2 : sampled unit i ∈ s1 − s2 : sampled in first phase but not in the second phase i ∈ U − s1 : nonsampled unit 10/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  27. 27. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionMeasuring the influence : the conditional bias We distinguish between three cases : i ∈ s2 : sampled unit i ∈ s1 − s2 : sampled in first phase but not in the second phase i ∈ U − s1 : nonsampled unit We can only reduce the influence of the sampled units (i.e., the units belonging to s2 ) 10/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  28. 28. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionMeasuring the influence : the conditional bias We distinguish between three cases : i ∈ s2 : sampled unit i ∈ s1 − s2 : sampled in first phase but not in the second phase i ∈ U − s1 : nonsampled unit We can only reduce the influence of the sampled units (i.e., the units belonging to s2 ) Nothing can be done for the other units at the estimation stage 10/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  29. 29. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionMeasuring the influence : the conditional bias We distinguish between three cases : i ∈ s2 : sampled unit i ∈ s1 − s2 : sampled in first phase but not in the second phase i ∈ U − s1 : nonsampled unit We can only reduce the influence of the sampled units (i.e., the units belonging to s2 ) Nothing can be done for the other units at the estimation stage Influence of sampled unit i ∈ s2 : DE Bi (I1i = 1, I2i = 1) ˆ = E1 E2 (YDE − Y |I1 , I1i = 1, I2i = 1) 10/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  30. 30. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionMeasuring the influence : the conditional bias We distinguish between three cases : i ∈ s2 : sampled unit i ∈ s1 − s2 : sampled in first phase but not in the second phase i ∈ U − s1 : nonsampled unit We can only reduce the influence of the sampled units (i.e., the units belonging to s2 ) Nothing can be done for the other units at the estimation stage Influence of sampled unit i ∈ s2 : DE Bi (I1i = 1, I2i = 1) ˆ = E1 E2 (YDE − Y |I1 , I1i = 1, I2i = 1) 10/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  31. 31. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionThe conditional bias in particular designs Arbitrary two-phase design : ∗ DE πij Bi (I1i = 1, I2i = 1) = ∗ ∗ − 1 yj πi πj j∈U ∗ n1 n2 n2 SRSWOR/SRSWOR with size n1 and n2 : πi = N × n1 = N DE N N ¯ Bi (I1i = 1, I2i = 1) = ( − 1)(yi − Y ) (N − 1) n2 11/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  32. 32. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionThe conditional bias in particular designs Arbitrary two-phase design : ∗ DE πij Bi (I1i = 1, I2i = 1) = ∗ ∗ − 1 yj πi πj j∈U ∗ n1 n2 n2 SRSWOR/SRSWOR with size n1 and n2 : πi = N × n1 = N DE N N ¯ Bi (I1i = 1, I2i = 1) = ( − 1)(yi − Y ) (N − 1) n2 Arbitrary design/Poisson sampling : DE π1ij −1 −1 Bi (I1i = 1, I2i = 1) = − 1 yj + π1i π2i − 1 yi π1i π1j j∈U 11/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  33. 33. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionThe conditional bias in particular designs Arbitrary two-phase design : ∗ DE πij Bi (I1i = 1, I2i = 1) = ∗ ∗ − 1 yj πi πj j∈U ∗ n1 n2 n2 SRSWOR/SRSWOR with size n1 and n2 : πi = N × n1 = N DE N N ¯ Bi (I1i = 1, I2i = 1) = ( − 1)(yi − Y ) (N − 1) n2 Arbitrary design/Poisson sampling : DE π1ij −1 −1 Bi (I1i = 1, I2i = 1) = − 1 yj + π1i π2i − 1 yi π1i π1j j∈U 11/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  34. 34. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionProperties of conditional bias can be interpreted as a contribution of each unit (sampled or nonsampled) to the total error 12/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  35. 35. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionProperties of conditional bias can be interpreted as a contribution of each unit (sampled or nonsampled) to the total error take fully account of the sampling design : an unit may be highly influential under a given sampling design but may have little or no influence under another sampling design 12/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  36. 36. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionProperties of conditional bias can be interpreted as a contribution of each unit (sampled or nonsampled) to the total error take fully account of the sampling design : an unit may be highly influential under a given sampling design but may have little or no influence under another sampling design ∗ DE If πi = 1, then Bi (I1i = 1, I2i = 1) = 0 12/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  37. 37. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionProperties of conditional bias can be interpreted as a contribution of each unit (sampled or nonsampled) to the total error take fully account of the sampling design : an unit may be highly influential under a given sampling design but may have little or no influence under another sampling design ∗ DE If πi = 1, then Bi (I1i = 1, I2i = 1) = 0 unknown ⇒ must be estimated 12/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  38. 38. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionProperties of conditional bias can be interpreted as a contribution of each unit (sampled or nonsampled) to the total error take fully account of the sampling design : an unit may be highly influential under a given sampling design but may have little or no influence under another sampling design ∗ DE If πi = 1, then Bi (I1i = 1, I2i = 1) = 0 unknown ⇒ must be estimated 12/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  39. 39. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionA robust version of the double expansion estimator Following Beaumont, Haziza and Ruiz-Gazen (2011), we obtain ˆR ˆ YDE = YDE − ˆ DE Bi (I1i = 1, I2i = 1) + ˆ DE ψ Bi (I1i = 1, I2i = 1) i∈s2 i∈s2 Example of ψ-function :   c if t > c ψ (t) = t if |t| ≤ c −c if t < −c  c : tuning constant 13/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  40. 40. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionA robust version of the double expansion estimator 13/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  41. 41. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionSommaire 1 Introduction 2 How to measure the influence ? 3 Unit nonresponse 4 Simulation study 5 Conclusion 14/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  42. 42. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionNotation s2 : set of respondents n2 : number of responding units (random) I2i : response indicator for unit i π2i : unknown response probability for unit i. We assume sampled units respond independently of one another (similar to Poisson sampling in the second phase) Propensity score adjusted estimator, assuming the π2i ’s are known : ˜ yi YP SA = i∈s π1i π2i 2 Influence of a responding unit : π1ij −1 −1 Bi SA (I1i = 1, I2i = 1) = P − 1 yj + π1i π2i − 1 yi π1i π1j j∈U Influence of unit i on the nonresponse error Influence of unit i on the sampling error 15/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  43. 43. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionNonresponse model In practice, the response probability π2i is unknown 16/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  44. 44. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionNonresponse model In practice, the response probability π2i is unknown Estimated response probability for unit i with a parametric nonresponse model : π2i = m (xi , α) ˆ ˆ 16/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  45. 45. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionNonresponse model In practice, the response probability π2i is unknown Estimated response probability for unit i with a parametric nonresponse model : π2i = m (xi , α) ˆ ˆ 16/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  46. 46. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionMethod of homogeneous response groups (RHG) Step 1 : we rank every element in G groups by a growing estimated response probabilities Step 2 : The estimated probability of reponse in the g − th group is given by : Si i ∈ g −1 i∈srg π1i π2i = ˆ −1 i∈sg π1i Propensity score adjusted estimator : ˆ yi YP SA = i∈s2 π1i π2i ˆ 17/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  47. 47. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionMethod of homogeneous response groups (RHG) Step 1 : we rank every element in G groups by a growing estimated response probabilities Step 2 : The estimated probability of reponse in the g − th group is given by : Si i ∈ g −1 i∈srg π1i π2i = ˆ −1 i∈sg π1i Propensity score adjusted estimator : ˆ yi YP SA = i∈s2 π1i π2i ˆ 17/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  48. 48. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionAsymptotic conditional bias One can show that ˆ ˆ YP SA − YL = Op (n−1 ), ˆ ˆ where YL is the linearized version of YP SA . 18/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  49. 49. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionAsymptotic conditional bias One can show that ˆ ˆ YP SA − YL = Op (n−1 ), ˆ ˆ where YL is the linearized version of YP SA . Asymptotic conditional bias of a responding unit : ˆ Bi SA (I1i = 1, I2i = 1) ≈ E1 E2 (YL − Y |I1 , I1i = 1, I2i = 1) P π1ij Bi SA (I1i = 1, I2i = 1) ≈ P − 1 (yj − yU ) ¯ π1i π1j j∈U −1 −1 + π1i π2i − 1 (yi − yg ) ¯ 18/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  50. 50. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionRobust estimator in case of nonresponse ˆ Robust version of YP SA ˆR YP SA ˆ = YP SA − ˆP Bi SA (I1i = 1, I2i = 1) i∈s2 + ˆP ψ Bi SA (I1i = 1, I2i = 1) i∈s2 Choice of the tuning constant We use a min-max criterion and we obtain : ˆP ˆP mini∈S (Bi SA ) + maxi∈S (Bi SA ) ˆR ˆ YP SA = YP SA − 2 19/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  51. 51. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionSommaire 1 Introduction 2 How to measure the influence ? 3 Unit nonresponse 4 Simulation study 5 Conclusion 20/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  52. 52. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionSampling and generating the nonresponse We generated three populations of size N = 5000 with two variables : y and x. Select P = 5000 samples, of size n1 = 300 or 500, according to simple random sampling without replacement in first phase Generate nonresponse : Bernoulli trials with probability π2i , where exp (xi α) π2i = 1 + exp (xi α) Global response rate : 70% We computed the min-max method to calibrate the tuning constant 21/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  53. 53. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionResults for a total estimation 22/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  54. 54. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionResults for a total estimation 2 1 P ˆR ˆR θP SA − EM C θP SA P p=1 RVM C ˆR ˆ θP SA , θP SA = 2, 1 P ˆ ˆ θP SA − EM C θP SA P p=1 et 2 1 P ˆR θP SA − θ P p=1 REM C ˆR ˆ θP SA , θP SA = 2. 1 P ˆ θP SA − θ P p=1 22/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  55. 55. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionResults for a total estimation Pop Taille n1 ˆRP RBM C θp SA (%) ˆRP ˆP RVM C θp SA , θp SA REM C 300 0.02 1.0 0.98 1 500 0.04 1.0 0.98 300 −0.98 0.52 0.58 2 500 −0.60 0.61 0.64 300 −1.33 0.50 0.52 3 500 −1.16 0.59 0.63 22/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  56. 56. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionSommaire 1 Introduction 2 How to measure the influence ? 3 Unit nonresponse 4 Simulation study 5 Conclusion 23/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  57. 57. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionConcluding remarks Conditional bias : measure of influence that takes account of the sampling design, the parameter to be estimated and the estimator Results can be extended to the case of calibration estimators ⇒ important in the unit nonresponse context since weight adjustment procedures by the inverse of the estimated response probabilities are generally followed by some form of calibration Proof of the design consistency and the normality of the robust estimator 24/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  58. 58. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionBibliographie I J.F. Beaumont, D. Haziza, and A. Ruiz-Gazen. A unified approach to robust estimation in finite population sampling. En révision, 2011. R.L. Chambers. Outlier robust finite population estimation. Journal of the American Statistical Association, pages 1063–1069, 1986. P.N. Kokic and P.A. Bell. Optimal winsorizing cutoffs for a stratified finite population estimator. Journal of official statistics-Stockholm, 10 :419–419, 1994. 25/26 Ensai-Ensae 2013 Robust inference in two-phases sampling
  59. 59. Introduction How to measure the influence ? Unit nonresponse Simulation study ConclusionBibliographie II J.L. Moreno-Rebollo, A. Munoz-Reyes, M.D. Jimenez-Gamero, and J. Munoz-Pichardo. Influence diagnostic in survey sampling : Estimating the conditional bias. Metrika, 55(3) :209–214, 2002. J.L. Moreno-Rebollo, A. Muñoz-Reyes, and J. Muñoz-Pichardo. Miscellanea. influence diagnostic in survey sampling : conditional bias. Biometrika, 86(4) :923–928, 1999. J. Munoz-Pichardo, J. Munoz-Garcia, J.L. Moreno-Rebollo, and R. Pino-Mejias. A new approach to influence analysis in linear models. Sankhy¯ : The Indian Journal of Statistics, Series A, pages 393–409, a 1995. 26/26 Ensai-Ensae 2013 Robust inference in two-phases sampling

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