In pedrosfig/BayesSampling: Bayes Linear Estimators for Finite Population

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)

library(BayesSampling)

Application of the BLE to the Simple Random Sample design

(From Section 2.3.1 of the "Gonçalves, Moura and Migon: Bayes linear estimation for finite population with emphasis on categorical data")

In a simple model, where there is no auxiliary variable, and a Simple Random Sample was taken from the population, we can calculate the Bayes Linear Estimator for the individuals of the population with the BLE_SRS() function, which receives the following parameters:

• $y_s$ - either a vector containing the observed values or just the value for the sample mean ($\sigma$ and $n$ parameters will be required in this case);
• $N$ - total size of the population;
• $m$ - prior mean. If NULL, sample mean will be used (non-informative prior);
• $v$ - prior variance of an element from the population ($> \sigma^2$). If NULL, it will tend to infinity (non-informative prior);
• $\sigma$ - prior estimate of variability (standard deviation) within the population. If NULL, sample variance will be used;
• $n$ - sample size. Necessary only if $y_s$ represent sample mean (will not be used otherwise).

Vague Prior Distribution

Letting $v \to \infty$ and keeping $\sigma^2$ fixed, that is, assuming prior ignorance, the resulting estimator will be the same as the one seen in the design-based context for the simple random sampling case.\

This can be achieved using the BLE_SRS() function by omitting either the prior mean and/or the prior variance, that is:

• $m =$ NULL - the sample mean will be used
• $v =$ NULL - prior variance will tend to infinity

Examples

1. We will use the TeachingSampling's BigCity dataset for this example (actually we have to take a sample of size $10000$ from this dataset so that R can perform the calculations). Imagine that we want to estimate the mean or the total Expenditure of this population, after taking a simple random sample of only 20 individuals, but applying a prior information (taken from a previous study or an expert's judgment) about the mean expenditure (a priori mean = $300$).

data(BigCity)
set.seed(1)
Expend <- sample(BigCity$Expenditure,10000)
mean(Expend)          #Real mean expenditure value, goal of the estimation
ys <- sample(Expend, size = 20, replace = FALSE)

Our design-based estimator for the mean will be the sample mean:

mean(ys)

Applying the prior information about the population we can get a better estimate, especially in cases when only a small sample is available:

Estimator <- BLE_SRS(ys, N = 10000, m=300, v=10.1^5, sigma = sqrt(10^5))

Estimator$est.beta
Estimator$Vest.beta
Estimator$est.mean[1,]
Estimator$Vest.mean[1:5,1:5]

1. Example from the help page

ys <- c(5,6,8)
N <- 5
m <- 6
v <- 5
sigma <- 1

Estimator <- BLE_SRS(ys, N, m, v, sigma)
Estimator

1. Example from the help page, but informing sample mean and sample size instead of sample observations

ys <- mean(c(5,6,8))
n <- 3
N <- 5
m <- 6
v <- 5
sigma <- 1

Estimator <- BLE_SRS(ys, N, m, v, sigma, n)
Estimator

pedrosfig/BayesSampling documentation built on May 5, 2021, 1:02 p.m.