Confidence sets for ranks based on multinomial data

Marginal and simultaneous confidence sets for ranks of categories, where categories are ranked by the probabilities of being chosen.

csranks_multinom(
  x,
  coverage = 0.95,
  cstype = "two-sided",
  simul = TRUE,
  multcorr = "Holm",
  indices = NA,
  na.rm = FALSE
)

Arguments

x

vector of counts indicating how often each category was chosen.

coverage

nominal coverage of the confidence set. Default is 0.95.

cstype

type of confidence set (two-sided, upper, lower). Default is two-sided.

simul

logical; if TRUE (default), then simultaneous confidence sets are computed, which jointly cover all populations indicated by indices. Otherwise, for each population indicated in indices a marginal confidence set is computed.

multcorr

multiplicity correction to be used: Holm (default) or Bonferroni. See Details section for more.

indices

vector of indices of x for whose ranks the confidence sets are computed. indices=NA (default) means computation of confidence sets for all populations.

na.rm

logical; if TRUE, then NA's are removed from x and Sigma (if any).

Value

A csranks object, which is a list with three items:

L

Lower bounds of the confidence sets for ranks indicated in indices

rank

Estimated ranks from irank with default parameters

U

Upper bounds of the confidence sets.

Details

This function computes confidence sets for ranks similarly as csranks, but it is tailored to the special case of multinomial data. Suppose there are \(p\) populations (for the case of multinomial data, we will refer to them as "categories") such as political parties, for example, that one wants to rank by the probabilities of them being chosen. For political parties, this would correspond to the share of votes each party obtains. Here, the underlying data are multinomial: each observation corresponds to a choice among the \(p\) categories. The vector x contains the counts of how often each category was chosen in the data.

In this setting, link{csranks} could be applied to compute confidence sets for the ranks of each category, but instead this function implements a different method proposed by Bazylik, Mogstad, Romano, Shaikh, and Wilhelm (2023), which exploits the multinomial structure of the problem and yields confidence sets for the ranks that are valid in finite samples (whereas csranks produces confidence sets that are valid only asymptotically).

The procedure involves testing multiple hypotheses. The \code{multcorr} indicates a method for multiplicity correction. See the paper for details.

References

Bazylik, Mogstad, Romano, Shaikh, and Wilhelm. "Finite-and large-sample inference for ranks using multinomial data with an application to ranking political parties". cemmap working paper

Examples

x <- c(rmultinom(1, 1000, 1:10))
csranks_multinom(x)
#> $L
#>  [1] 8 8 7 6 2 2 2 2 2 1
#> 
#> $rank
#>  [1] 10  9  8  7  6  5  3  3  2  1
#> 
#> $U
#>  [1] 10 10 10  8  7  6  6  6  6  1
#> 
#> attr(,"class")
#> [1] "csranks"