Semiparametric Mixed Models in Small Area Estimation

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Semiparametric Mixed Models in Small Area
Estimation

• Mark Delorey
• F. Jay Breidt
• Colorado State University
• September 22, 2002

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FUNDING SOURCE

• This presentation was developed under the STAR
  Research Assistance Agreement CR-829095 awarded
  by the U.S. Environmental Protection Agency (EPA)
  to Colorado State University.  This presentation
  has not been formally reviewed by EPA.  The views
  expressed here are solely those of its authors
  and the STARMAP Program. EPA does not endorse any
  products or commercial services mentioned in this
  presentation.

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Outline

• Small area estimation
• Standard parametric small area model
• Semiparametric model/estimation
• Future work

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Small Area Estimation

• Probability samples are not sufficiently dense in
  small watersheds
• Need to incorporate auxiliary information (remote
  sensing, GIS) through model
• Problem High bias if model is misspecified

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Standard Parametric Mixed Model with Site
Specific Auxiliary Data

• Battese, Harter, Fuller (1988)
• where
• i 1,,ng are the sites within small area g
• ?g iid N(0, ?2) and ?gi N(0, ?2)

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Battese, Harter, Fuller (cont.)

• Small area mean is
• Estimate by convex combination of model-based
  prediction and survey regression estimator

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Semiparametric Model with Smooth Function

• Adapting time series case from Zhang, Lin, Raz,
  and Sowers (1988)
• f(t) is a twice differentiable smooth function of
  time
• ?g N(0, ?2)
• ?gi ? N(0, ?2)

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Smooth Function Representation

• Impose the constraint f T? Ba where a is iid
  N(0, ?2I) (Green, 1987)
• T is a matrix consisting of the coordinates of
  observed locations in time (space)
• B is constructed based on relative positions of
  observed locations

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Linear Mixed Model

• Model can then be written as linear mixed model
• a N(0, ?2I)
• ?g N(0, ?2)
• ?gi ? N(0, ?2)

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Prediction with Linear Mixed Model

• Then,
• ?11, ?12, and ?22 are known up to some variance
  components

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Prediction with Linear Mixed Model

• Using the form of composite estimator from BHF,
  we get
• for small ng, where and are the gls
  estimates of ? and ?, respectively.

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Future Work

• Estimate variance components and smoothing
  parameter
• Compare results withwhere U(t) is a
  correlated random process (black-box kriging,
  Barry and Ver Hoef, 1996)

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References

• Barry and Ver Hoef (1996). Blackbox Kriging
  Spatial prediction without specifying variogram
  models. Journal of Agricultural, Biological, and
  Environmental Statistics, 1297-322.
• Battese, Harter, Fuller (1988). An
  error-component model for prediction of county
  crop areas using survey and satellite data. JASA
  8328-36.
• Green (1987). Penalized likelihood for general
  semi-parametric regression models. International
  Statistical Review 55245-260.
• Zhang, Lin, Raz, and Sowers (1998).
  Semiparametric stochastic mixed models for
  longitudinal data. JASA 93710-719.