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:

- \(y_s\) - a vector containing the observed values;
- \(N\) - 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.

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 disgn-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.

- 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
#> [1] 375.586
ys <- sample(Expend, size = 20, replace = FALSE)
```

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

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$est.beta
#> Beta
#> 1 390.8338
Estimator$Vest.beta
#> V1
#> 1 2524.999
Estimator$est.mean[1,]
#> [1] 390.8338
Estimator$Vest.mean[1:5,1:5]
#> V1 V2 V3 V4 V5
#> 1 102524.999 2524.999 2524.999 2524.999 2524.999
#> 2 2524.999 102524.999 2524.999 2524.999 2524.999
#> 3 2524.999 2524.999 102524.999 2524.999 2524.999
#> 4 2524.999 2524.999 2524.999 102524.999 2524.999
#> 5 2524.999 2524.999 2524.999 2524.999 102524.999
```

- Example from the help page

```
ys <- c(5,6,8)
m <- 6
v <- 5
sigma <- 1
N <- 5
Estimator <- BLE_SRS(ys,N,m,v,sigma)
Estimator
#> $est.beta
#> Beta
#> 1 6.307692
#>
#> $Vest.beta
#> V1
#> 1 0.3076923
#>
#> $est.mean
#> y_nots
#> 1 6.307692
#> 2 6.307692
#>
#> $Vest.mean
#> V1 V2
#> 1 1.3076923 0.3076923
#> 2 0.3076923 1.3076923
#>
#> $est.tot
#> [1] 31.61538
#>
#> $Vest.tot
#> [1] 3.230769
```