TeachingSampling (version 4.1.1)

p.WR: Generalization of every with replacement sampling design

Description

Computes the selection probability (sampling design) of each with replacement sample

Usage

p.WR(N, m, pk)

Arguments

N

Population size

m

Sample size

pk

A vector containing selection probabilities for each unit in the population

Value

The function returns a vector of selection probabilities for every with-replacement sample.

Details

Every with replacement sampling design is a particular case of a multinomial distribution. $$p(\mathbf{S}=\mathbf{s})=\frac{m!}{n_1!n_2!\cdots n_N!}\prod_{i=1}^N p_k^{n_k}$$ where \(n_k\) is the number of times that the \(k\)-th unit is selected in a sample.

References

Sarndal, C-E. and Swensson, B. and Wretman, J. (1992), Model Assisted Survey Sampling. Springer. Gutierrez, H. A. (2009), Estrategias de muestreo: Diseno de encuestas y estimacion de parametros. Editorial Universidad Santo Tomas.

Examples

# NOT RUN {
############
## Example 1
############
# With replacement simple random sampling
# Vector U contains the label of a population of size N=5
U <- c("Yves", "Ken", "Erik", "Sharon", "Leslie")
# Vector pk is the sel?ection probability of the units in the finite population
pk <- c(0.2, 0.2, 0.2, 0.2, 0.2)
sum(pk)
N <- length(pk)
m <- 3
# The smapling design
p <- p.WR(N, m, pk)
p
sum(p)

############
## Example 2
############
# With replacement PPS random sampling
# Vector U contains the label of a population of size N=5
U <- c("Yves", "Ken", "Erik", "Sharon", "Leslie")
# Vector x is the auxiliary information and y is the variables of interest
x<-c(32, 34, 46, 89, 35)
y<-c(52, 60, 75, 100, 50)
# Vector pk is the sel?ection probability of the units in the finite population
pk <- x/sum(x)
sum(pk)
N <- length(pk)
m <- 3
# The smapling design
p <- p.WR(N, m, pk)
p
sum(p)
# }