Hartley {Frames2} | R Documentation |
Hartley estimator
Description
Produces estimates for population totals and means using Hartley estimator from survey data obtained from a dual frame sampling design. Confidence intervals are also computed, if required.
Usage
Hartley(ysA, ysB, pi_A, pi_B, domains_A, domains_B, conf_level = NULL)
Arguments
ysA |
A numeric vector of length |
ysB |
A numeric vector of length |
pi_A |
A numeric vector of length |
pi_B |
A numeric vector of length |
domains_A |
A character vector of size |
domains_B |
A character vector of size |
conf_level |
(Optional) A numeric value indicating the confidence level for the confidence intervals. |
Details
Hartley estimator of population total is given by
\hat{Y}_H = \hat{Y}_a^A + \hat{\theta}\hat{Y}_{ab}^A + (1 - \hat{\theta})\hat{Y}_{ab}^B + \hat{Y}_b^B
where \hat{\theta} \in [0, 1]
. Optimum value for \hat{\theta}
to minimize variance of the estimator is
\hat{\theta}_{opt} = \frac{\hat{V}(\hat{Y}_{ab}^B) + \widehat{Cov}(\hat{Y}_b^B, \hat{Y}_{ab}^B) - \widehat{Cov}(\hat{Y}_a^A, \hat{Y}_{ab}^A)}{\hat{V}(\hat{Y}_{ab}^A) + \hat{V}(\hat{Y}_{ab}^B)}
Taking into account the independence between s_A
and s_B
, an estimator for the variance of the Hartley estimator can be obtained as follows:
\hat{V}(\hat{Y}_H) = \hat{V}(\hat{Y}_a^A + \hat{\theta}\hat{Y}_{ab}^A) + \hat{V}((1 - \hat{\theta})\hat{Y}_{ab}^B + \hat{Y}_b^B)
If both first and second order probabilities are known, variances and covariances involved in calculation of \hat{\theta}_{opt}
and \hat{V}(\hat{Y}_H)
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
Hartley
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
Hartley, H. O. (1962) Multiple Frames Surveys. Proceedings of the American Statistical Association, Social Statistics Sections, 203 - 206.
Hartley, H. O. (1974) Multiple frame methodology and selected applications. Sankhya C, Vol. 36, 99 - 118.
See Also
Examples
data(DatA)
data(DatB)
data(PiklA)
data(PiklB)
#Let calculate Hartley estimator for variable Feeding
Hartley(DatA$Feed, DatB$Feed, PiklA, PiklB, DatA$Domain, DatB$Domain)
#Now, let calculate Hartley estimator and a 90% confidence interval
#for variable Leisure, considering only first order inclusion probabilities
Hartley(DatA$Lei, DatB$Lei, DatA$ProbA, DatB$ProbB, DatA$Domain,
DatB$Domain, 0.90)