Imbens-Manski interval from replicated estimates of the two bounds

A thin front end over morie_bnd_imbens_manski, which is where the Imbens-Manski critical value actually lives; this function only reduces two samples of bound estimates to the arguments that construction needs. Aliasing rather than re-deriving matters here: a second copy of the critical value would agree with the first at 1e-9 forever and still be a second copy.

Usage

Bndfre(lower, upper, alpha = 0.05)

Arguments

lower, upper

Replicated estimates of the two bounds, same length.

alpha

Miss probability, default 0.05.

Value

List with lower, upper, width, c, z_one_sided, z_two_sided, delta, n.

Details

Formula: [l - c s_l / sqrt(n), u + c s_u / sqrt(n)] with c solving Phi(c + sqrt(n) Delta / max(s_l, s_u)) - Phi(-c) = 1 - alpha.

References

Imbens, G. W. and Manski, C. F. (2004). Confidence intervals for partially identified parameters. Econometrica 72(6), 1845-1857, equation (6). doi:10.1111/j.1468-0262.2004.00555.x .

Examples

V <- c(1, 2, 3, 4, 5, 6, 7, 8)
Bndfre(V, V)
#> $lower
#> [1] 2.802621
#> 
#> $upper
#> [1] 6.197379
#> 
#> $width
#> [1] 3.394757
#> 
#> $c
#> [1] 1.959964
#> 
#> $z_one_sided
#> [1] 1.644854
#> 
#> $z_two_sided
#> [1] 1.959964
#> 
#> $delta
#> [1] 0
#> 
#> $n
#> [1] 8
#> 
#> $method
#> [1] "Frequentist bound with valid coverage"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"