Est.EmpDistFunc.Hajek {samplingVarEst} | R Documentation |
The Hajek estimator for the empirical cumulative distribution function
Description
Computes the Hajek (1971) estimator for the empirical cumulative distribution function (ECDF).
Usage
Est.EmpDistFunc.Hajek(VecY.s, VecPk.s, t)
Arguments
VecY.s |
vector of the variable of interest; its length is equal to |
VecPk.s |
vector of the first-order inclusion probabilities; its length is equal to |
t |
value to be evaluated for the empirical cumulative distribution function. It must be an integer or a double-precision scalar. |
Details
For the population empirical cumulative distribution function (ECDF) of the variable y
at the value t
:
Fn(t) = \frac{\#(k\in U:y_k \leq t)}{N} = \frac{1}{N} \sum_{k\in U} I(y_k \leq t)
the approximately unbiased Hajek (1971) estimator of Fn(t)
(implemented by the current function) is given by:
\hat{F}n_{Hajek}(t) = \frac{\sum_{k\in s} w_k I(y_k \leq t)}{\sum_{k\in s} w_k}
where I(y_k \leq t)
denotes the indicator function that takes the value 1
if y_k \leq t
and the value 0
otherwise, and where w_k=1/\pi_k
and \pi_k
denotes the inclusion probability of the k
-th element in the sample s
.
Value
The function returns a value for the empirical cumulative distribution function evaluated at t
.
Author(s)
Emilio Lopez Escobar [aut, cre], Juan Francisco Munoz Rosas [ctb].
References
Hajek, J. (1971) Comment on An essay on the logical foundations of survey sampling by Basu, D. in Foundations of Statistical Inference (Godambe, V.P. and Sprott, D.A. eds.), p. 236. Holt, Rinehart and Winston.
See Also
Examples
data(oaxaca) #Loads Oaxaca municipalities dataset
pik.U <- Pk.PropNorm.U(373, oaxaca$HOMES00) #Reconstructs the inclusion probs.
s <- oaxaca$sHOMES00 #Defines the sample to be used
y1 <- oaxaca$POP10 #Defines the variable of interest y1
Est.EmpDistFunc.Hajek(y1[s==1], pik.U[s==1], 950) #Hajek est. of ECDF for y1 at t=950