Construct a discrete confidence set for argmax.

Description

This is a wrapper to construct a confidence set for the argmax by negating the input and reusing CS.argmin.

Usage

CS.argmax(data, method = "softmin.LOO", alpha = 0.05, ...)

Arguments

Details

The supported methods include:

Value

A vector of indices (1-based) representing the confidence set for the argmax.

References

Zhang T, Lee H, Lei J (2024). “Winners with confidence: Discrete argmin inference with an application to model selection.” arXiv preprint arXiv:2408.02060.

Gupta SS (1965). “On Some Multiple Decision (Selection and Ranking) Rules.” Technometrics, 7(2), 225–245. doi:10.1080/00401706.1965.10490251.

Futschik A, Pflug G (1995). “Confidence Sets for Discrete Stochastic Optimization.” Annals of Operations Research, 56(1), 95–108. doi:10.1007/BF02031702.

Chernozhukov V, Chetverikov D, Kato K (2013). “Testing many moment inequalities.” RePEc. IDEAS Working Paper Series.

Examples

set.seed(108)
n <- 200
p <- 20
mu <- (1:p)/p
cov <- diag(p)
data <- MASS::mvrnorm(n, mu, cov)

## softmin.LOO (SML)
CS.argmax(data)

## argmin.LOO (HML)
CS.argmax(data, method = "HML")

## nonsplit (NS) - requires lambda
CS.argmax(data, method = "NS", lambda = sqrt(n)/2.5)

## Bonferroni (MT) - t test default
CS.argmax(data, method = "MT", test = "t")

## Gupta (GTA)
CS.argmax(data, method = "GTA")

## Futschik (FCHK) with default alpha.1 and alpha.2
CS.argmax(data, method = "FCHK")

## Futschik (FCHK) with user-specified alpha.1 and alpha.2
alpha.1 <- 0.001
alpha.2 <- 1 - (0.95 / (1 - alpha.1))
CS.argmax(data, method = "FCHK", alpha.1 = alpha.1, alpha.2 = alpha.2)

Construct a discrete confidence set for argmin.

Description

This is a wrapper to construct a discrete confidence set for argmin. Multiple methods are supported.

Usage

CS.argmin(data, method = "softmin.LOO", alpha = 0.05, ...)

Arguments

Details

The supported methods include:

Value

A vector of indices (1-based) representing the (1 - alpha) confidence set.

References

Zhang T, Lee H, Lei J (2024). “Winners with confidence: Discrete argmin inference with an application to model selection.” arXiv preprint arXiv:2408.02060.

Chernozhukov V, Chetverikov D, Kato K (2013). “Testing many moment inequalities.” RePEc. IDEAS Working Paper Series.

Gupta SS (1965). “On Some Multiple Decision (Selection and Ranking) Rules.” Technometrics, 7(2), 225–245. doi:10.1080/00401706.1965.10490251.

Futschik A, Pflug G (1995). “Confidence Sets for Discrete Stochastic Optimization.” Annals of Operations Research, 56(1), 95–108. doi:10.1007/BF02031702.

Examples

r <- 4
n <- 200
mu <- (1:20)/20
cov <- diag(length(mu))
set.seed(108)
data <- MASS::mvrnorm(n, mu, cov)
sample.mean <- colMeans(data)

## softmin.LOO
CS.argmin(data)

## use seed
CS.argmin(data, seed=13)

## argmin.LOO
CS.argmin(data, method='HML')

## nonsplit
CS.argmin(data, method='NS', lambda=sqrt(n)/2.5)

## Bonferroni (choose t test because of normal data)
CS.argmin(data, method='MT', test='t')

## Gupta
CS.argmin(data, method='GTA')

## Futschik two-step method
# default alpha.1, alpha.2
CS.argmin(data, method='FCHK')

alpha.1 <- 0.0005
alpha.2 <- 1 - (0.95/(1 - alpha.1))
CS.argmin(data, method='FCHK', alpha.1=0.0005, alpha.2=alpha.2)

A wrapper to perform argmax hypothesis test.

Description

This function performs a hypothesis test to evaluate whether a given dimension may be the argmax. It internally negates the data and reuses the implementation from argmin.HT.

Usage

argmax.HT(data, r = NULL, method = "softmin.LOO", ...)

Arguments

Details

The supported methods include:

Value

A character string: 'Accept' or 'Reject', indicating whether the dimension could be an argmax, and relevant statistics.

References

Chernozhukov V, Chetverikov D, Kato K (2013). “Testing many moment inequalities.” RePEc. IDEAS Working Paper Series.

Gupta SS (1965). “On Some Multiple Decision (Selection and Ranking) Rules.” Technometrics, 7(2), 225–245. doi:10.1080/00401706.1965.10490251.

Futschik A, Pflug G (1995). “Confidence Sets for Discrete Stochastic Optimization.” Annals of Operations Research, 56(1), 95–108. doi:10.1007/BF02031702.

Examples

set.seed(108)
n <- 200
p <- 20
mu <- (1:p)/p
cov <- diag(p)
data <- MASS::mvrnorm(n, mu, cov)

## Define the dimension of interest
r <- 4

## Construct difference matrix for dimension r
difference.matrix.r <- matrix(rep(data[, r], p - 1), ncol = p - 1, byrow = FALSE) - data[, -r]

## softmin.LOO (SML)
argmax.HT(difference.matrix.r)

## use seed
argmax.HT(difference.matrix.r, seed=19)

## With known true difference
true.mean.diff <- mu[r] - mu[-r]
argmax.HT(difference.matrix.r, true.mean = true.mean.diff)

## Without scaling
argmax.HT(difference.matrix.r, scale.input = FALSE)

## With a user-specified lambda
argmax.HT(difference.matrix.r, lambda = sqrt(n) / 2.5)

## Add a seed for reproducibility
argmax.HT(difference.matrix.r, seed = 17)

## argmin.LOO (HML)
argmax.HT(difference.matrix.r, method = "HML")

## nonsplit method
argmax.HT(difference.matrix.r, method = "NS", lambda = sqrt(n)/2.5)

## Bonferroni method (choose t test for normal data)
argmax.HT(difference.matrix.r, method = "MT", test = "t")

## Gupta method (pass full data matrix)
critical.val <- get.quantile.gupta.selection(p = length(mu))
argmax.HT(data, r, method = "GTA", critical.val = critical.val)

A wrapper to perform argmin hypothesis test.

Description

This is a wrapper to perform hypothesis test to see if a given dimension may be an argmin. Multiple methods are supported.

Usage

argmin.HT(data, r = NULL, method = "softmin.LOO", ...)

Arguments

Details

The supported methods include:

Value

'Accept' or 'Reject'. A string indicating whether the given dimension could be an argmin (Accept) or not (Reject), and relevant statistics.

References

Zhang T, Lee H, Lei J (2024). “Winners with confidence: Discrete argmin inference with an application to model selection.” arXiv preprint arXiv:2408.02060.

Chernozhukov V, Chetverikov D, Kato K (2013). “Testing many moment inequalities.” RePEc. IDEAS Working Paper Series.

Gupta SS (1965). “On Some Multiple Decision (Selection and Ranking) Rules.” Technometrics, 7(2), 225–245. doi:10.1080/00401706.1965.10490251.

Futschik A, Pflug G (1995). “Confidence Sets for Discrete Stochastic Optimization.” Annals of Operations Research, 56(1), 95–108. doi:10.1007/BF02031702.

Examples

r <- 4
n <- 200
p <- 20
mu <- (1:p)/p
cov <- diag(length(mu))
set.seed(108)
data <- MASS::mvrnorm(n, mu, cov)
sample.mean <- colMeans(data)

## softmin.LOO
difference.matrix.r <- matrix(rep(data[,r], p-1), ncol=p-1, byrow=FALSE) - data[,-r]
argmin.HT(difference.matrix.r)

## use seed
argmin.HT(difference.matrix.r, seed=19)

# provide centered test statistic (to simulate asymptotic normality)
true.mean.difference.r <- mu[r] - mu[-r]
argmin.HT(difference.matrix.r, true.mean=true.mean.difference.r)

# keep the data unstandardized
argmin.HT(difference.matrix.r, scale.input=FALSE)

# use an user-specified lambda
argmin.HT(difference.matrix.r, lambda=sqrt(n)/2.5)

# add a seed
argmin.HT(difference.matrix.r, seed=19)

## argmin.LOO/hard min
argmin.HT(difference.matrix.r, method='HML')

## nonsplit
argmin.HT(difference.matrix.r, method='NS', lambda=sqrt(n)/2.5)

## Bonferroni (choose t test because of normal data)
argmin.HT(difference.matrix.r, method='MT', test='t')
## z test
argmin.HT(difference.matrix.r, method='MT', test='z')

## Gupta
critical.val <- get.quantile.gupta.selection(p=length(mu))
argmin.HT(data, r, method='GTA', critical.val=critical.val)

Perform argmin hypothesis test.

Description

Test if a dimension may be argmin, using the LOO (leave-one-out) algorithm in Zhang et al 2024.

Usage

argmin.HT.LOO(
  difference.matrix,
  sample.mean = NULL,
  min.algor = "softmin",
  lambda = NULL,
  const = 2.5,
  enlarge = TRUE,
  alpha = 0.05,
  true.mean.difference = NULL,
  output.weights = FALSE,
  scale.input = TRUE,
  seed = NULL,
  ...
)

Arguments

Value

A list containing:

Perform argmin hypothesis test.

Description

Test if a dimension may be argmin, using multiple testing with Bonferroni's correction.

Usage

argmin.HT.MT(difference.matrix, sample.mean = NULL, test = "z", alpha = 0.05)

Arguments

Value

A list containing:

Perform argmin hypothesis test using Gupta's method.

Description

Test whether a dimension is the argmin, using the method in (Gupta 1965).

Usage

argmin.HT.gupta(
  data,
  r,
  sample.mean = NULL,
  stds = NULL,
  critical.val = NULL,
  alpha = 0.05,
  ...
)

Arguments

Value

A list containing:

Note

This method requires independence among the dimensions.

References

Gupta SS (1965). “On Some Multiple Decision (Selection and Ranking) Rules.” Technometrics, 7(2), 225–245. doi:10.1080/00401706.1965.10490251.

Futschik A, Pflug G (1995). “Confidence Sets for Discrete Stochastic Optimization.” Annals of Operations Research, 56(1), 95–108. doi:10.1007/BF02031702.

Perform argmin hypothesis test.

Description

Test if a dimension may be argmin without any splitting.

Usage

argmin.HT.nonsplit(
  difference.matrix,
  lambda,
  sample.mean = NULL,
  alpha = 0.05,
  scale.input = TRUE
)

Arguments

Details

This method is not recommended, given its poor performance when p is small.

Value

A list containing:

Get the index of the smallest dimension apart from an index

Description

Get the index of the smallest dimension apart from an index

Usage

find.sub.argmin(nums, idx, seed = NULL)

Arguments

Value

The index of the second smallest dimension (as an integer).

Examples

nums <- c(1,3,2)
find.sub.argmin(nums,1)
## return 3

nums <- c(1,1,2)
find.sub.argmin(nums,1)
## return 2

Construct a difference matrix for argmin hypothesis testing

Description

Given a data matrix and a reference column index, construct the difference matrix used in hypothesis testing procedures. Each column represents the difference between the reference dimension and one of the remaining dimensions.

Usage

get.difference.matrix(data, r)

Arguments

Value

A n by (p-1) matrix where each row is the difference between the r-th column and the remaining columns.

Examples

set.seed(1)
data <- matrix(rnorm(50), nrow = 10)
diff.mat <- get.difference.matrix(data, r = 2)

Generate the quantile used for the selection procedure in (Gupta 1965).

Description

Generate the quantile used for the selection procedure in (Gupta 1965) by Monte Carlo estimation.

Usage

get.quantile.gupta.selection(p, alpha = 0.05, N = 1e+05)

Arguments

Value

A list containing:

Note

The quantile is pre-calculated for some common configurations of (p, alpha)

References

Gupta SS (1965). “On Some Multiple Decision (Selection and Ranking) Rules.” Technometrics, 7(2), 225–245. doi:10.1080/00401706.1965.10490251.

Futschik A, Pflug G (1995). “Confidence Sets for Discrete Stochastic Optimization.” Annals of Operations Research, 56(1), 95–108. doi:10.1007/BF02031702.

Examples

get.quantile.gupta.selection(p=10)

get.quantile.gupta.selection(p=100)

Compute sample mean differences for hypothesis testing

Description

Computes the vector of differences between the sample mean at the reference index and the remaining dimensions.

Usage

get.sample.mean.r(sample.mean, r)

Arguments

Value

A vector of length p - 1 giving the differences: sample.mean[r] - sample.mean[-r].

Examples

sample.mean <- 1:5
get.sample.mean.r(sample.mean, r = 3)

Check the feasibility of a tuning parameter \lambda for LOO algorithm.

Description

Check the feasibility of a tuning parameter \lambda for LOO algorithm by examining whether its resulting \nabla_i K_j is less than a threshold value, i.e., the first order stability is likely achieved. For further details, we refer to the paper Zhang et al 2024.

Usage

is.lambda.feasible.LOO(
  lambda,
  scaled.difference.matrix,
  sample.mean = NULL,
  threshold = 0.08,
  n.pairs = 100,
  seed = NULL
)

Arguments

Value

A boolean value indicating if the given \lambda likely gives the first order stability.

Generate a scaled.difference.matrix-driven \lambda for LOO algorithm.

Description

Generate a scaled.difference.matrix-driven \lambda for LOO algorithm motivated by the derivation of the first order stability. For its precise definition, we refer to the paper Zhang et al 2024.

Usage

lambda.adaptive.LOO(
  scaled.difference.matrix,
  sample.mean = NULL,
  const = 2.5,
  seed = NULL
)

Arguments

Value

A scaled.difference.matrix-driven \lambda for LOO algorithm.

Examples

# Simulate data
set.seed(123)
r <- 4
n <- 200
mu <- (1:20)/20
cov <- diag(length(mu))
set.seed(108)
data <- MASS::mvrnorm(n, mu, cov)
sample.mean <- colMeans(data)
diff.mat <- get.difference.matrix(data, r)
sample.mean.r <- get.sample.mean.r(sample.mean, r)
lambda <- lambda.adaptive.LOO(diff.mat, sample.mean=sample.mean.r)

Iteratively enlarge a tuning parameter \lambda in a data-driven way.

Description

Iteratively enlarge a tuning parameter \lambda to enhance the power of hypothesis testing. The iterative algorithm ends when an enlarged \lambda unlikely yields the first order stability.

Usage

lambda.adaptive.enlarge(
  lambda,
  scaled.difference.matrix,
  sample.mean = NULL,
  mult.factor = 2,
  verbose = FALSE,
  seed = NULL,
  ...
)

Arguments

Value

A list containing:

Examples

# Simulate data
set.seed(123)
r <- 4
n <- 200
mu <- (1:20)/20
cov <- diag(length(mu))
set.seed(108)
data <- MASS::mvrnorm(n, mu, cov)
sample.mean <- colMeans(data)
diff.mat <- get.difference.matrix(data, r)
sample.mean.r <- get.sample.mean.r(sample.mean, r)
lambda <- lambda.adaptive.LOO(diff.mat, sample.mean=sample.mean.r)

# Run the enlargement algorithm
res <- lambda.adaptive.enlarge(lambda, diff.mat, sample.mean=sample.mean.r)
res
# with a seed
res <- lambda.adaptive.enlarge(lambda, diff.mat, sample.mean=sample.mean.r, seed=3)
res