Cross-validated Dantzig estimator with missing covariates

Description

Performs K-fold cross-validation for the Dantzig selector in linear regression models with missing covariates. The method optionally incorporates unlabelled covariate data to improve estimation of second-moment matrices. This function is based on Section 3 of Risebrow and Berrett (2026).

Usage

cv_dantzig_missing(
  X, y, X_unlabeled = NULL,
  lambdas = NULL, nlambda = 30, lambda_min_ratio = 1e-3,
  K = 5, standardise = TRUE, gurobi = FALSE,
  seed = 123, fold_ids = NULL, verbose = TRUE,
  plot_path = TRUE
)

Arguments

Details

For each candidate value of the regularisation parameter, the Dantzig selector is fitted using moment estimates computed from the training folds. Prediction performance is assessed on held-out folds via the maximum absolute moment mismatch. The tuning parameter is selected using both the minimum mean cross-validation score and the one-standard-error (1-SE) rule.

Value

A named list with the following components:

lambdas

Numeric vector of tuning parameters used.

cv_scores_matrix

Numeric matrix of cross-validation scores (folds × lambdas).

mean_scores

Mean CV score for each lambda.

se_scores

Standard error of CV scores for each lambda.

lambda_min_mean

Lambda minimising mean CV score.

lambda_1se

Lambda chosen by the 1-SE rule.

beta_path

Optional coefficient path matrix (present if plot_path=TRUE).

design_colnames

Optional design column names (matching beta_path rows).

beta_est

Optional saved coefficient vector from full-data path.

intercept_est

Optional saved intercept corresponding to beta_est.

References

Risebrow BM, Berrett TB (2026). “Semi-supervised linear regression with missing covariates.” arXiv:2602.13729.

Examples

set.seed(1)
n <- 50; p <- 5
X <- matrix(rnorm(n * p), n, p)
y <- X[, 1] + 0.5 * X[, 2] + rnorm(n)
X_unlabeled <- matrix(rnorm(100 * p), 100, p)

cv_fit <- cv_dantzig_missing(
  X = X,
  y = y,
  X_unlabeled = X_unlabeled,
  K = 5,
  nlambda = 20
)

cv_fit$lambda_1se

Dantzig estimator with missing covariates

Description

High-dimensional linear regression estimator based on the Dantzig selector that accommodates missing covariates and optionally leverages unlabelled covariate data. This function is a user-facing wrapper that dispatches to either a standardised or unstandardised implementation depending on the value of standardise. This function is based on Section 3 of Risebrow and Berrett (2026).

Usage

dantzig_missing(
  X_labeled, y, X_unlabeled = NULL, lambda,
  gurobi = FALSE, standardise = TRUE
)

Arguments

Details

Categorical covariates are internally dummy-encoded, with missing values preserved. When standardise = TRUE, covariates are centred and scaled using empirical means and standard deviations computed from the combined labelled and unlabelled samples.

Value

A list with at least the following component:

beta_hat

Numeric vector of estimated regression coefficients, with names corresponding to the encoded design matrix columns.

If standardise = TRUE, the list also contains:

intercept

Numeric scalar giving the estimated intercept term.

References

Risebrow BM, Berrett TB (2026). “Semi-supervised linear regression with missing covariates.” arXiv:2602.13729.

Examples

set.seed(1)
n <- 50; p <- 5
X_full <- matrix(rnorm(n * p), n, p)
beta_true <- c(1, 0.5, rep(0, p - 2))
y <- X_full[, 1] * beta_true[1] + X_full[, 2] * beta_true[2] + rnorm(n)

# introduce missingness into covariates
X_miss <- X_full
X_miss[sample(length(X_miss), size = 0.1 * length(X_miss))] <- NA

# fit Dantzig estimator (example lambda; tune in practice)
fit <- dantzig_missing(
  X_labeled = X_miss,
  y = y,
  lambda = 0.1,
  standardise = TRUE
)
fit$beta_hat

Covariance estimator with missing data

Description

Estimates the covariance matrix of a design matrix in the presence of missing values. Each covariance entry is computed using all observations for which the corresponding pair of covariates is jointly observed.

Usage

estimate_cov_raw(X)

Arguments

Details

Let X_{ij} denote the j-th covariate for observation i. For each pair of variables (j, k), the covariance estimate is

\hat{\Sigma}_{jk} = \frac{1}{n_{jk}} \sum_{i : X_{ij}, X_{ik} \ \mathrm{observed}} X_{ij} X_{ik},

where n_{jk} is the number of observations for which both entries are observed. If no such observations exist, the corresponding covariance entry is set to NA.

This estimator is symmetric by construction and reduces to the usual sample second-moment matrix when the data contain no missing values.

Value

A numeric p x p matrix containing the estimated covariance matrix, where p = ncol(X). Entries corresponding to variable pairs that are never jointly observed are NA.

Examples

set.seed(1)
X <- matrix(rnorm(25), 25, 5)
X[sample(length(X), 10)] <- NA
Sigma_hat <- estimate_cov_raw(X)
Sigma_hat

Linear regression with missing data

Description

Fits a linear regression model in the presence of missing covariates and/or missing responses using the OSS (Ordinary Semi-Supervised) estimator. This corresponds to Section 2 of Risebrow and Berrett (2026). The method exploits partially observed covariates and optionally unlabelled observations to improve estimation efficiency. If sufficient complete cases are present, the weights are estimated from them, otherwise the weights are estimated using an initial consistent estimate. If sufficient unlabelled data is present the covariance matrix is estimated exclusively from them, otherwise the covariance is estimated elementwise.

Usage

oss_estimator(formula, data,
              all_weights_one = FALSE,
              crossfitting = FALSE)

Arguments

Value

An invisible list with components:

coef

Numeric vector of estimated regression coefficients.

sigma2_hat

Estimated noise variance, or NA if not computed.

weights

Named vector of weights associated with each missingness pattern.

groups

Data.frame mapping labelled observations to missingness patterns.

beta_cc

Complete-case coefficient estimates if used, otherwise NULL.

References

Risebrow BM, Berrett TB (2026). “Semi-supervised linear regression with missing covariates.” arXiv:2602.13729.

Examples

dat <- data.frame(
  y = c(1.0, NA, 2.3, 0.5),
  x1 = rnorm(4),
  x2 = rnorm(4)
)

## Without cross-fitting
res <- oss_estimator(y ~ x1 + x2, dat)

## With cross-fitting
res_cf <- oss_estimator(y ~ x1 + x2, dat, crossfitting = TRUE)