Median voter with a standard error that does not assume normality
Source:R/median_frames_dose_native.R
morie_median_voter_ci.RdThe familiar \(1.2533\,s/\sqrt{n}\) is \(\sqrt{\pi/2}\,\sigma/\sqrt{n}\), the asymptotic standard error of the sample median ONLY under normality. The general result is \(1/(2 f(m)\sqrt{n})\), and for a heavy-tailed electorate the normal formula badly overstates the uncertainty because it reads the tails as spread when the median responds only to the density at the centre. Both are returned, with a distribution-free order-statistic interval that assumes neither.
Value
A list with estimate, se (density-based), se_normal,
ci_lower, ci_upper, ci_exact_lower, ci_exact_upper,
exact_coverage, median_interval, unique_winner, warnings.
Details
With an even electorate every point between the two central ideal
points is a Condorcet winner, so the winner is a set; unique_winner
says whether it collapses to a point.
References
Black D (1948) Journal of Political Economy 56(1):23-34, doi:10.1086/256633 .
Examples
V <- c(1, 2, 3, 4, 5, 6, 7, 8)
morie_median_voter_ci(V)
#> $estimate
#> [1] 4.5
#>
#> $mean
#> [1] 4.5
#>
#> $se
#> [1] 1.421369
#>
#> $se_normal
#> [1] 1.085402
#>
#> $density_at_median
#> [1] 0.1243707
#>
#> $ci_lower
#> [1] 1.714168
#>
#> $ci_upper
#> [1] 7.285832
#>
#> $ci_exact_lower
#> [1] 1
#>
#> $ci_exact_upper
#> [1] 8
#>
#> $exact_coverage
#> [1] 0.9921875
#>
#> $median_interval
#> [1] 4 5
#>
#> $unique_winner
#> [1] FALSE
#>
#> $n
#> [1] 8
#>
#> $warnings
#> [1] "With an even electorate (n = 8) every point in [4, 5] is a Condorcet winner. The reported estimate is the midpoint, which is a convention rather than a result."
#> [2] "The normality-based standard error is 0.76 times the density-based one, so this electorate is far from normal and the 1.2533 formula should not be quoted for it."
#>
#> $method
#> [1] "Median voter theorem (Black 1948)"
#>