jackknifeVar: Estimated variance for the estimator GREG using the jackknife...

In stalynGuerrero/multisae: Small Area Estimation using plaucible values

Description

The function makes the estimation of variances for the Horvitz-Thompson estimator (TH) and the generalized regression estimator (GREG) when plausible values are employed. The estimation of the variance for a domain is given by the equation

EE(\hat{η}) = √{\frac{1}{5}∑_{i=1}^{5}Var_m(\hat{η}_i)+ ≤ft(1 + \frac{1}{5}\right)\frac{1}{4}∑_{i=1}^{5}(\hat{η}_i - \hat{η})^2}

where Var_m(\hat{η}_i) is the variance of the estimator calculated from the observations and the sampling design. The second term of the equation is estimated by using the jackknife method. In the case of the generalized regression estimator, it is necessary to adjust a lineal model, where the plausible value \hat{η}_{ij} included as the response variable, as follows:

\hat{η}_{ij}=β X_i+e_{ij}

The GREG variance is estimated for each doman with X_i as auxiliary information, employing the model’s estimated residuals using the expression

\widehat{Var}(\hat{η}_{GREG})=∑∑_s\frac{Δ_{jl}}{π_{jl}}g_{js} \frac{\hat{e}_{js}}{π_j}g_{ls}\frac{\hat{e}_{ls}}{π_l}

donde

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g_{js} = 1+≤ft( \bar{X}-\hat{\bar{X}}\right)^t ≤ft(∑_m \frac{w_j}{σ^2_l}x_jx_j^t\right)^{-1}x_j

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\hat{e}_{ij} = \hat{η}_{ij} - \hat{β} X_i

-

w_j = \frac{1}{π_j}

Usage

jackknifeVar(x, yhat, weight, total, Xk, stratum = NULL, ...)

Arguments

x	data.frame which contains the information necessary to make the estimations
yhat	Name of the column which contains the estimated values of the variable
weight	Expansion factor of each observation
total	Total for the auxiliary variable
Xk	Vector which contains the names of the covariantes.
stratum	Indicator of the domains in the sample, by default stratum = NULL
...	Additional arguments employed by the calib function.

Value

Returns a data.frame with the results obtained for the Horvitz-Thompson estimator and the GREG as well as their corresponding variances for the indicated groups and subgroups

References

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Shao, J., & Tu, D. (2012). The jackknife and bootstrap. Springer Science & Business Media.

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Rubin, D. B. (2004). Multiple imputation for nonresponse in surveys (Vol. 81). John Wiley & Sons.

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Von Davier, M., Gonzalez, E., & Mislevy, R. (2009). What are plausible values and why are they useful. IERI monograph series, 2, 9-36.

See Also

calib, E.GREG

Examples

data('ResultStudent')
attach(ResultStudent)
sampling <- ResultStudent[["student"]]
#Add dummys
sampling <- data.frame(sampling[,c("schooltype",'weight','prop','ses')],
                      Domains(sampling[["urbanicity"]]),
                      Domains(sampling[["schooltype"]]))

# Covariantes
Xk <- c('Rural', 'Urbana','No.Oficial', 'Oficial', 'ses')

jackknifeVar(x=sampling,yhat = "prop",weight = 'weight',
            Xk=c("ses", "Rural","Urbana"),
            total = total, method='linear')

jackknifeVar(sampling,weight = 'weight',yhat = "prop",
         Xk = 'ses',total = total, method='linear')
jackknifeVar(sampling,yhat = "prop",weight = 'weight',
           Xk = 'ses',total = total, method='linear',
           stratum = 'schooltype')

stalynGuerrero/multisae documentation built on May 30, 2019, 8:44 a.m.