# FB: Fuller-Burmeister estimator In Frames2: Estimation in Dual Frame Surveys

## Description

Produces estimates for population totals and means using the Fuller - Burmeister estimator from survey data obtained from a dual frame sampling desing. Confidence intervals are also computed, if required.

## Usage

 1 FB(ysA, ysB, pi_A, pi_B, domains_A, domains_B, conf_level = NULL) 

## Arguments

 ysA A numeric vector of length n_A or a numeric matrix or data frame of dimensions n_A x c containing information about variable of interest from s_A. ysB A numeric vector of length n_B or a numeric matrix or data frame of dimensions n_B x c containing information about variable of interest from s_B. pi_A A numeric vector of length n_A or a square numeric matrix of dimension n_A containing first order or first and second order inclusion probabilities for units included in s_A. pi_B A numeric vector of length n_B or a square numeric matrix of dimension n_B containing first order or first and second order inclusion probabilities for units included in s_B. domains_A A character vector of size n_A indicating the domain each unit from s_A belongs to. Possible values are "a" and "ab". domains_B A character vector of size n_B indicating the domain each unit from s_B belongs to. Possible values are "b" and "ba". conf_level (Optional) A numeric value indicating the confidence level for the confidence intervals.

## Details

Fuller-Burmeister estimator of population total is given by

\hat{Y}_{FB} = \hat{Y}_a^A + \hat{β_1}\hat{Y}_{ab}^A + (1 - \hat{β_1})\hat{Y}_{ab}^B + \hat{Y}_b^B + \hat{β_2}(\hat{N}_{ab}^A - \hat{N}_{ab}^B)

where optimal values for \hat{β} to minimize variance of the estimator are:

≤ft( \begin{array}{c} \hat{β}_1\\ \hat{β}_2 \end{array} \right) = - ≤ft( \begin{array}{cc} \hat{V}(\hat{Y}_{ab}^A - \hat{Y}_{ab}^B) & \widehat{Cov}(\hat{Y}_{ab}^A - \hat{Y}_{ab}^B, \hat{N}_{ab}^A - \hat{N}_{ab}^B)\\ \widehat{Cov}(\hat{Y}_{ab}^A - \hat{Y}_{ab}^B, \hat{N}_{ab}^A - \hat{N}_{ab}^B) & \hat{V}(\hat{N}_{ab}^A - \hat{N}_{ab}^B) \end{array} \right)^{-1} \times

≤ft( \begin{array}{c} \widehat{Cov}(\hat{Y}_a^A + \hat{Y}_b^B + \hat{Y}_{ab}^B, \hat{Y}_{ab}^A - \hat{Y}_{ab}^B)\\ \widehat{Cov}(\hat{Y}_a^A + \hat{Y}_b^B + \hat{Y}_{ab}^B, \hat{N}_{ab}^A - \hat{N}_{ab}^B) \end{array} \right)

Due to Fuller-Burmeister estimator is not defined for estimating population sizes, estimation of the mean is computed as \hat{Y}_{FB} / \hat{N}_H, where \hat{N}_H is the estimation of the population size using Hartley estimator. Estimated variance for the Fuller-Burmeister estimator can be obtained through expression

\hat{V}(\hat{Y}_{FB}) = \hat{V}(\hat{Y}_a^A) + \hat{V}(\hat{Y}^B) + \hat{β}_1[\widehat{Cov}(\hat{Y}_a^A, \hat{Y}_{ab}^A) - \widehat{Cov}(\hat{Y}^B, \hat{Y}_{ab}^B)]

+ \hat{β}_2[\widehat{Cov}(\hat{Y}_a^A, \hat{N}_{ab}^A) - \widehat{Cov}(\hat{Y}^B, \hat{N}_{ab}^B)]

If both first and second order probabilities are known, variances and covariances involved in calculation of \hat{β} and \hat{V}(\hat{Y}_{FB}) are estimated using functions VarHT and CovHT, respectively. If only first order probabilities are known, variances are estimated using Deville's method and covariances are estimated using following expression

\widehat{Cov}(\hat{X}, \hat{Y}) = \frac{\hat{V}(X + Y) - \hat{V}(X) - \hat{V}(Y)}{2}

## Value

FB returns an object of class "EstimatorDF" which is a list with, at least, the following components:

 Call the matched call. Est total and mean estimation for main variable(s). VarEst variance estimation for main variable(s).

If parameter conf_level is different from NULL, object includes component

 ConfInt total and mean estimation and confidence intervals for main variables(s).

In addition, components TotDomEst and MeanDomEst are available when estimator is based on estimators of the domains. Component Param shows value of parameters involded in calculation of the estimator (if any). By default, only Est component (or ConfInt component, if parameter conf_level is different from NULL) is shown. It is possible to access to all the components of the objects by using function summary.

## References

Fuller, W.A. and Burmeister, L.F. (1972). Estimation for Samples Selected From Two Overlapping Frames ASA Proceedings of the Social Statistics Sections, 245 - 249.

Hartley JackFB
  1 2 3 4 5 6 7 8 9 10 11 12 data(DatA) data(DatB) data(PiklA) data(PiklB) #Let calculate Fuller-Burmeister estimator for variable Clothing FB(DatA$Clo, DatB$Clo, PiklA, PiklB, DatA$Domain, DatB$Domain) #Now, let calculate Fuller-Burmeister estimator and a 90% confidence interval #for variable Leisure, considering only first order inclusion probabilities FB(DatA$Lei, DatB$Lei, DatA$ProbA, DatB$ProbB, DatA$Domain, DatB$Domain, 0.90)