Horvitz-Thompson variance of the estimated total

The diagonal term uses pi_ii = pi_i, so it contributes (1 - pi_i) y_i^2 / pi_i^2. For simple random sampling without replacement the whole expression collapses to N^2 (1 - n/N) S^2 / n, the closed form the tests check against.

Usage

Htvar1(y, pi, pi_ij)

Arguments

y

Observed values for the sampled units.

pi

First-order inclusion probabilities in (0, 1].

pi_ij

Matrix of joint inclusion probabilities.

Value

List with estimate, variance, total, se, n, method.

Details

Formula: V = sum_i sum_j (pi_ij - pi_i pi_j)(y_i/pi_i)(y_j/pi_j)/pi_ij.

References

Horvitz and Thompson (1952), A generalization of sampling without replacement from a finite universe, JASA 47(260):663-685. doi:10.1080/01621459.1952.10483446

Examples

y <- c(10, 20, 15)
pi <- c(0.5, 0.4, 0.6)
pi_ij <- matrix(c(0.5, 0.2, 0.3, 0.2, 0.4, 0.25, 0.3, 0.25, 0.6), 3, 3)
Htvar1(y, pi, pi_ij)
#> $estimate
#> [1] 2050
#> 
#> $variance
#> [1] 2050
#> 
#> $total
#> [1] 95
#> 
#> $se
#> [1] 45.27693
#> 
#> $n
#> [1] 3
#> 
#> $method
#> [1] "sum_ij (pi_ij - pi_i pi_j)(y_i/pi_i)(y_j/pi_j)/pi_ij, Horvitz & Thompson (1952)"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"