TeachingSampling (version 4.1.1)

VarSYGHT: Two different varaince estimators for the Horvitz-Thompson estimator

Description

This function estimates the variance of the Horvitz-Thompson estimator. Two different variance estimators are computed: the original one, due to Horvitz-Thompson and the one due to Sen (1953) and Yates, Grundy (1953). The two approaches yield unbiased estimator under fixed-size sampling schemes.

Usage

VarSYGHT(y, N, n, p)

Arguments

y

Vector containing the information of the characteristic of interest for every unit in the population.

N

Population size.

n

Sample size.

p

A vector containing the selection probabilities of a fixed size without replacement sampling design. The sum of the values of this vector must be one.

Value

This function returns a data frame of every possible sample in within a sampling support, with its corresponding variance estimates.

Details

The function returns two variance estimator for every possible sample within a fixed-size sampling support. The first estimator is due to Horvitz-Thompson and is given by the following expression: $$\widehat{Var}_1(\hat{t}_{y,\pi}) = \sum_{k \in U}\sum_{l\in U}\frac{\Delta_{kl}}{\pi_{kl}}\frac{y_k}{\pi_k}\frac{y_l}{\pi_l}$$ The second estimator is due to Sen (1953) and Yates-Grundy (1953). It is given by the following expression: $$\widehat{Var}_2(\hat{t}_{y,\pi}) = -\frac{1}{2}\sum_{k \in U}\sum_{l\in U}\frac{\Delta_{kl}}{\pi_{kl}}(\frac{y_k}{\pi_k} - \frac{y_l}{\pi_l})^2$$

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
# Without replacement sampling
# Vector U contains the label of a population of size N=5
U <- c("Yves", "Ken", "Erik", "Sharon", "Leslie")
# Vector y1 and y2 are the values of the variables of interest
y1<-c(32, 34, 46, 89, 35)
y2<-c(1,1,1,0,0)
# The population size is N=5
N <- length(U)
# The sample size is n=2
n <- 2
# p is the probability of selection of every possible sample
p <- c(0.13, 0.2, 0.15, 0.1, 0.15, 0.04, 0.02, 0.06, 0.07, 0.08)

# Calculates the estimated variance for the HT estimator
VarSYGHT(y1, N, n, p)
VarSYGHT(y2, N, n, p)

# Unbiasedness holds in the estimator of the total
sum(y1)
sum(VarSYGHT(y1, N, n, p)$p * VarSYGHT(y1, N, n, p)$Est.HT)
sum(y2)
sum(VarSYGHT(y2, N, n, p)$p * VarSYGHT(y2, N, n, p)$Est.HT)

# Unbiasedness also holds in the two variances
VarHT(y1, N, n, p)
sum(VarSYGHT(y1, N, n, p)$p * VarSYGHT(y1, N, n, p)$Est.Var1)
sum(VarSYGHT(y1, N, n, p)$p * VarSYGHT(y1, N, n, p)$Est.Var2)

VarHT(y2, N, n, p)
sum(VarSYGHT(y2, N, n, p)$p * VarSYGHT(y2, N, n, p)$Est.Var1)
sum(VarSYGHT(y2, N, n, p)$p * VarSYGHT(y2, N, n, p)$Est.Var2)

# Example 2: negative variance estimates

x = c(2.5, 2.0, 1.1, 0.5)
N = 4
n = 2
p = c(0.31, 0.20, 0.14, 0.03, 0.01, 0.31)

VarSYGHT(x, N, n, p)

# Unbiasedness holds in the estimator of the total
sum(x)
sum(VarSYGHT(x, N, n, p)$p * VarSYGHT(x, N, n, p)$Est.HT)

# Unbiasedness also holds in the two variances
VarHT(x, N, n, p)
sum(VarSYGHT(x, N, n, p)$p * VarSYGHT(x, N, n, p)$Est.Var1)
sum(VarSYGHT(x, N, n, p)$p * VarSYGHT(x, N, n, p)$Est.Var2)
# }