diff --git a/NEWS.md b/NEWS.md
index 14be5a6..b425c09 100644
--- a/NEWS.md
+++ b/NEWS.md
@@ -2,6 +2,8 @@
 
 Items for next release go here
 
+Extended `loo_pit()` for discrete data.
+
 # rstantools 2.4.0
 
 * Update to match CRAN's patched version by @jgabry in #114
diff --git a/R/loo-functions.R b/R/loo-functions.R
index 056ea54..184dd51 100644
--- a/R/loo-functions.R
+++ b/R/loo-functions.R
@@ -10,12 +10,15 @@
 #' @return `loo_predict()`, `loo_linpred()`, and `loo_pit()`
 #'   (probability integral transform) methods should return a vector with length
 #'   equal to the number of observations in the data.
+#'   For discrete observations, probability integral transform is randomised to
+#'   ensure theoretical uniformity. Fix random seed for reproducible results 
+#'   with discrete data. For more details, see Czado et al. (2009).
 #'   `loo_predictive_interval()` methods should return a two-column matrix
 #'   formatted in the same way as for [predictive_interval()].
 #'
 #' @template seealso-rstanarm-pkg
 #' @template seealso-vignettes
-#'
+#' @template reference-randomised-pit
 
 #' @rdname loo-prediction
 #' @export
@@ -60,11 +63,32 @@ loo_pit.default <- function(object, y, lw, ...) {
 
 # internal ----------------------------------------------------------------
 .loo_pit <- function(y, yrep, lw) {
-  vapply(seq_len(ncol(yrep)), function(j) {
-    sel <- yrep[, j] <= y[j]
-    .exp_log_sum_exp(lw[sel, j])
+  if (is.null(lw) || !all(is.finite(lw))) {
+    stop("lw needs to be not null and finite.")
+  }
+  pits <- vapply(seq_len(ncol(yrep)), function(j) {
+    sel_min <- yrep[, j] < y[j]
+    pit <- .exp_log_sum_exp(lw[sel_min, j])
+    sel_sup <- yrep[, j] == y[j]
+    if (any(sel_sup)) {
+      # randomized PIT for discrete y (see, e.g., Czado, C., Gneiting, T.,
+      # Held, L.: Predictive model assessment for count data.
+      # Biometrics 65(4), 1254–1261 (2009).)
+      pit_sup <- pit + .exp_log_sum_exp(lw[sel_sup, j])
+      pit <- stats::runif(1, pit, pit_sup)
+    }
+    pit
   }, FUN.VALUE = 1)
+  if (any(pits > 1)) {
+    warning(cat(
+      "Some PIT values larger than 1! Largest: ",
+      max(pits),
+      "\nRounding PIT > 1 to 1."
+    ))
+  }
+  pmin(1, pits)
 }
+
 .exp_log_sum_exp <- function(x) {
   m <- suppressWarnings(max(x))
   exp(m + log(sum(exp(x - m))))
diff --git a/man-roxygen/details-license.R b/man-roxygen/details-license.R
index e118853..009b90f 100644
--- a/man-roxygen/details-license.R
+++ b/man-roxygen/details-license.R
@@ -1,6 +1,6 @@
 #' @details In order to enable Stan functionality, \pkg{\link{rstantools}}
-#'   copies some files to your package.  Since these files are licensed as GPL
-#'   >= 3, the same license applies to your package should you choose to
+#'   copies some files to your package.  Since these files are licensed as 
+#'   GPL >= 3, the same license applies to your package should you choose to
 #'   distribute it.  Even if you don't use \pkg{\link{rstantools}} to create
 #'   your package, it is likely that you will be linking to \pkg{\link{Rcpp}} to
 #'   export the Stan C++ `stanmodel` objects to \R.  Since
diff --git a/man-roxygen/reference-randomised-pit.R b/man-roxygen/reference-randomised-pit.R
new file mode 100644
index 0000000..8e9a48a
--- /dev/null
+++ b/man-roxygen/reference-randomised-pit.R
@@ -0,0 +1,5 @@
+#' @references Czado, C., Gneiting, T., and Held, L. (2009).
+#'   Predictive Model Assessment for Count Data.
+#'   *Biometrics*. 65(4), 1254-1261.
+#'   doi:10.1111/j.1541-0420.2009.01191.x.
+#'   Journal version: <https://doi.org/10.1111/j.1541-0420.2009.01191.x>
diff --git a/man/loo-prediction.Rd b/man/loo-prediction.Rd
index 9655a59..dda7866 100644
--- a/man/loo-prediction.Rd
+++ b/man/loo-prediction.Rd
@@ -35,12 +35,22 @@ the same length as the number of columns in the matrix used as \code{object}.}
 \code{loo_predict()}, \code{loo_linpred()}, and \code{loo_pit()}
 (probability integral transform) methods should return a vector with length
 equal to the number of observations in the data.
+For discrete observations, probability integral transform is randomised to
+ensure theoretical uniformity. Fix random seed for reproducible results
+with discrete data. For more details, see Czado et al. (2009).
 \code{loo_predictive_interval()} methods should return a two-column matrix
 formatted in the same way as for \code{\link[=predictive_interval]{predictive_interval()}}.
 }
 \description{
 See the methods in the \pkg{rstanarm} package for examples.
 }
+\references{
+Czado, C., Gneiting, T., and Held, L. (2009).
+Predictive Model Assessment for Count Data.
+\emph{Biometrics}. 65(4), 1254-1261.
+doi:10.1111/j.1541-0420.2009.01191.x.
+Journal version: \url{https://doi.org/10.1111/j.1541-0420.2009.01191.x}
+}
 \seealso{
 \itemize{
 \item The \pkg{rstanarm} package (\href{https://mc-stan.org/rstanarm/}{mc-stan.org/rstanarm})
diff --git a/man/rstan_create_package.Rd b/man/rstan_create_package.Rd
index c4184ad..c8b048e 100644
--- a/man/rstan_create_package.Rd
+++ b/man/rstan_create_package.Rd
@@ -114,7 +114,13 @@ must be manually configured by running \code{\link[=rstan_config]{rstan_config()
 \code{stanmodel} files in \code{inst/stan} are added, removed, or modified.
 
 In order to enable Stan functionality, \pkg{\link{rstantools}}
-copies some files to your package.  Since these files are licensed as GPL= 3, the same license applies to your package should you choose todistribute it.  Even if you don't use \pkg{\link{rstantools}} to createyour package, it is likely that you will be linking to \pkg{\link{Rcpp}} toexport the Stan C++ stanmodel objects to \R.  Since\pkg{\link{Rcpp}} is released under GPL >= 2, the same license would applyto your package upon distribution.
+copies some files to your package.  Since these files are licensed as
+GPL >= 3, the same license applies to your package should you choose to
+distribute it.  Even if you don't use \pkg{\link{rstantools}} to create
+your package, it is likely that you will be linking to \pkg{\link{Rcpp}} to
+export the Stan C++ \code{stanmodel} objects to \R.  Since
+\pkg{\link{Rcpp}} is released under GPL >= 2, the same license would apply
+to your package upon distribution.
 
 Authors willing to license their Stan programs of general interest
 under the GPL are invited to contribute their \code{.stan} files and
diff --git a/tests/testthat/loo_pit.RDS b/tests/testthat/loo_pit.RDS
index b135a07..2b9c1fc 100644
Binary files a/tests/testthat/loo_pit.RDS and b/tests/testthat/loo_pit.RDS differ
diff --git a/tests/testthat/loo_pit_discrete.RDS b/tests/testthat/loo_pit_discrete.RDS
new file mode 100644
index 0000000..7eb82bf
Binary files /dev/null and b/tests/testthat/loo_pit_discrete.RDS differ
diff --git a/tests/testthat/test-default-methods.R b/tests/testthat/test-default-methods.R
index d4ed2ca..0548b34 100644
--- a/tests/testthat/test-default-methods.R
+++ b/tests/testthat/test-default-methods.R
@@ -2,6 +2,13 @@ context("default methods")
 set.seed(1111)
 x <- matrix(rnorm(150), 50, 3)
 y <- rnorm(ncol(x))
+lw <- matrix(rnorm(150), 50, 3)
+lw <- sweep(
+  lw,
+  MARGIN = 2,
+  STATS = apply(lw, 2, \(col) log(sum(exp(col)))),
+  check.margin = FALSE
+)
 
 test_that("posterior_interval.default hasn't changed", {
   expect_equal_to_reference(
@@ -28,12 +35,18 @@ test_that("prior_summary.default works", {
   expect_null(prior_summary(list(abc = "prior_info")))
 })
 test_that("loo_pit.default works", {
-  lw <- matrix(rnorm(150), 50, 3)
   expect_equal_to_reference(
     loo_pit(x, y, lw),
     "loo_pit.RDS"
   )
 })
+test_that("loo_pit-default works for discrete data", {
+  set.seed(1111)
+  expect_equal_to_reference(
+    loo_pit(round(x), round(y), lw),
+    "loo_pit_discrete.RDS"
+  )
+})
 test_that("bayes_R2.default hasn't changed", {
   expect_equal_to_reference(
     bayes_R2(x, y),
@@ -82,4 +95,3 @@ test_that(".pred_errors throws errors", {
                fixed = TRUE)
 })
 
-