Confidence sets for the tau-best

Computation of confidence sets for the identities of populations among the tau best.

cstaubest(
  x,
  Sigma,
  tau = 2,
  coverage = 0.95,
  stepdown = TRUE,
  R = 1000,
  na.rm = FALSE,
  seed = NA
)

cstauworst(
  x,
  Sigma,
  tau = 2,
  coverage = 0.95,
  stepdown = TRUE,
  R = 1000,
  na.rm = FALSE,
  seed = NA
)

Arguments

x

vector of estimates containing estimated features by which the populations are to be ranked.

Sigma

estimated covariance matrix of x.

tau

the confidence set contains indicators for the elements in x whose rank is less than or equal to tau.

coverage

nominal coverage of the confidence set. Default is 0.95.

stepdown

logical; if TRUE (default), stepwise procedure is used, otherwise single step procedure is used. See Details section for more.

R

number of bootstrap replications. Default is 1000.

na.rm

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

seed

seed for bootstrap random variable draws. If set to NA (default), then seed is not set.

Value

logical vector indicating which of the elements of x are in the confidence set for the tau-best.

Functions

• cstauworst(): Confidence sets for the tau-worst

  Equivalent to calling cstaubest with -x.

Details

The function computes a confidence set containing indicators for the elements in x whose rank is less than or equal to tau with probability approximately equal to the nominal coverage (coverage).

The function implements the projection confidence set for the tau-best developed and described in more detail in Mogstad, Romano, Shaikh, and Wilhelm (2023).

References

Mogstad, Romano, Shaikh, and Wilhelm (2023), "Inference for Ranks with Applications to Mobility across Neighborhoods and Academic Achievements across Countries", forthcoming at Review of Economic Studies cemmap working paper, doi:10.1093/restud/rdad006

Examples

# simple simulated example:
n <- 100
p <- 10
X <- matrix(rep(1:p,n)/p, ncol=p, byrow=TRUE) + matrix(rnorm(n*p), 100, 10)
thetahat <- colMeans(X)
Sigmahat <- cov(X) / n

# confidence set for the populations that may be among the top-3 
# (with probability approximately 0.95):
cstaubest(thetahat, Sigmahat, tau=3)
#>  [1] FALSE FALSE FALSE FALSE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE

# confidence set for the populations that may be among the bottom-3 
# (with probability approximately 0.95):
cstauworst(thetahat, Sigmahat, tau=3)
#>  [1]  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE FALSE FALSE FALSE