# VarSYGHT: Two different varaince estimators for the Horvitz-Thompson... In TeachingSampling: Selection of Samples and Parameter Estimation in Finite Population

## 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

 1 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.

## 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,π}) = ∑_{k \in U}∑_{l\in U}\frac{Δ_{kl}}{π_{kl}}\frac{y_k}{π_k}\frac{y_l}{π_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,π}) = -\frac{1}{2}∑_{k \in U}∑_{l\in U}\frac{Δ_{kl}}{π_{kl}}(\frac{y_k}{π_k} - \frac{y_l}{π_l})^2

## Value

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

## Author(s)

Hugo Andres Gutierrez Rojas <hagutierrezro at gmail.com>

## 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

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 # 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) 

TeachingSampling documentation built on April 22, 2020, 1:05 a.m.