PEL {Frames2} | R Documentation |
Pseudo empirical likelihood estimator
Description
Produces estimates for population totals using the pseudo empirical likelihood estimator from survey data obtained from a dual frame sampling design. Confidence intervals for the population total are also computed, if required.
Usage
PEL(ysA, ysB, pi_A, pi_B, domains_A, domains_B, N_A = NULL, N_B = NULL,
N_ab = NULL, xsAFrameA = NULL, xsBFrameA = NULL, xsAFrameB = NULL, xsBFrameB = NULL,
XA = NULL, XB = NULL, 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 |
N_A |
(Optional) A numeric value indicating the size of frame A. |
N_B |
(Optional) A numeric value indicating the size of frame B. |
N_ab |
(Optional) A numeric value indicating the size of the overlap domain. |
xsAFrameA |
(Optional) A numeric vector of length |
xsBFrameA |
(Optional) A numeric vector of length |
xsAFrameB |
(Optional) A numeric vector of length |
xsBFrameB |
(Optional) A numeric vector of length |
XA |
(Optional) A numeric value or vector of length |
XB |
(Optional) A numeric value or vector of length |
conf_level |
(Optional) A numeric value indicating the confidence level for the confidence intervals, if desired. |
Details
Pseudo empirical likelihood estimator for the population mean is computed as
\hat{\bar{Y}}_{PEL} = \frac{N_a}{N}\hat{\bar{Y}}_a + \frac{\eta N_{ab}}{N}\hat{\bar{Y}}_{ab}^A + \frac{(1 - \eta) N_{ab}}{N}\hat{\bar{Y}}_{ab}^B + \frac{N_b}{N}\hat{\bar{Y}}_b
where \hat{\bar{Y}}_a = \sum_{k \in s_a}\hat{p}_{ak}y_k, \hat{\bar{Y}}_{ab} = \sum_{k \in s_{ab}^A}\hat{p}_{abk}^Ay_k, \hat{\bar{Y}}_{ab}^B = \sum_{k \in s_{ab}^B}\hat{p}_{abk}^By_k
and \hat{\bar{Y}}_b = \sum_{k \in s_b}\hat{p}_{bk}y_k
with \hat{p}_{ak}, \hat{p}_{abk}^A, \hat{p}_{abk}^B
and \hat{p}_{bk}
the weights resulting of applying the pseudo empirical likelihood procedure to a determined function under a determined set of constraints, depending on the case.
Furthermore, \eta \in (0,1)
. In this case, N_A, N_B
and N_{ab}
have been supposed known and no additional auxiliary variables have been considered. This is not happening in some cases.
Function covers following scenarios:
There is not any additional auxiliary variable
-
N_A, N_B
andN_{ab}
unknown -
N_A
andN_B
known andN_{ab}
unknown -
N_A, N_B
andN_{ab}
known
-
At least, one additional auxiliary variable is available
-
N_A
andN_B
known andN_{ab}
unknown -
N_A, N_B
andN_{ab}
known
-
Explicit variance of this estimator is not easy to obtain. Instead, confidence intervals can be computed through the bi-section method. This method constructs intervals in the form \{\theta|r_{ns}(\theta) < \chi_1^2(\alpha)\}
,
where \chi_1^2(\alpha)
is the 1 - \alpha
quantile from a \chi^2
distribution with one degree of freedom and r_{ns}(\theta)
represents the so called pseudo empirical log likelihood ratio statistic,
which can be obtained as a difference of two pseudo empirical likelihood functions.
Value
PEL
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
Rao, J. N. K. and Wu, C. (2010) Pseudo Empirical Likelihood Inference for Multiple Frame Surveys. Journal of the American Statistical Association, 105, 1494 - 1503.
Wu, C. (2005) Algorithms and R codes for the pseudo empirical likelihood methods in survey sampling. Survey Methodology, Vol. 31, 2, pp. 239 - 243.
See Also
Examples
data(DatA)
data(DatB)
data(PiklA)
data(PiklB)
#Let calculate pseudo empirical likelihood estimator for variable Feeding, without
#considering any auxiliary information
PEL(DatA$Feed, DatB$Feed, PiklA, PiklB, DatA$Domain, DatB$Domain)
#Now, let calculate pseudo empirical estimator for variable Clothing when the frame
#sizes and the overlap domain size are known
PEL(DatA$Clo, DatB$Clo, PiklA, PiklB, DatA$Domain, DatB$Domain,
N_A = 1735, N_B = 1191, N_ab = 601)
#Finally, let calculate pseudo empirical likelihood estimator and a 90% confidence interval
#for population total for variable Feeding, considering Income and Metres2 as auxiliary
#variables and with frame sizes and overlap domain size known.
PEL(DatA$Feed, DatB$Feed, PiklA, PiklB, DatA$Domain, DatB$Domain,
N_A = 1735, N_B = 1191, N_ab = 601, xsAFrameA = DatA$Inc, xsBFrameA = DatB$Inc,
xsAFrameB = DatA$M2, xsBFrameB = DatB$M2, XA = 4300260, XB = 176553,
conf_level = 0.90)