diff --git a/zerver/lib/request.pyi b/zerver/lib/request.pyi
index a31d1f9394..835d9d2214 100644
--- a/zerver/lib/request.pyi
+++ b/zerver/lib/request.pyi
@@ -1,14 +1,31 @@
 # This mypy stubs file ensures that mypy can correctly analyze REQ.
-from typing import Any, Callable, Text, TypeVar
+#
+# Note that here REQ is claimed to be a function, with a return type to match
+# that of the parameter of which it is the default value, allowing type
+# checking. However, in request.py, REQ is a class to enable the decorator to
+# scan the parameter list for REQ objects and patch the parameters as the true
+# types.
+
+from typing import Any, Callable, Text, TypeVar, Optional, Union
 from django.http import HttpResponse
 
 from zerver.lib.exceptions import JsonableError as JsonableError
 
+Validator = Callable[[str, Any], Optional[str]]
+
+ResultT = TypeVar('ResultT')
 ViewFuncT = TypeVar('ViewFuncT', bound=Callable[..., HttpResponse])
 
 class RequestVariableMissingError(JsonableError): ...
 class RequestVariableConversionError(JsonableError): ...
 
-def REQ(*args: Any, **kwargs: Any) -> Any: ...
+class _NotSpecified: ...
+NotSpecified = _NotSpecified()
+
+def REQ(whence: Optional[str] = None,
+        converter: Optional[Callable[[str], ResultT]] = None,
+        default: Union[_NotSpecified, ResultT] = NotSpecified,
+        validator: Optional[Validator] = None,
+        argument_type: Optional[str] = None) -> ResultT: ...
 
 def has_request_variables(view_func: ViewFuncT) -> ViewFuncT: ...
diff --git a/zerver/tests/test_decorators.py b/zerver/tests/test_decorators.py
index 36d54a3654..c43f96698f 100644
--- a/zerver/tests/test_decorators.py
+++ b/zerver/tests/test_decorators.py
@@ -102,13 +102,13 @@ class DecoratorTestCase(TestCase):
         # type: () -> None
 
         def my_converter(data):
-            # type: (str) -> List[str]
+            # type: (str) -> List[int]
             lst = ujson.loads(data)
             if not isinstance(lst, list):
                 raise ValueError('not a list')
             if 13 in lst:
                 raise JsonableError('13 is an unlucky number!')
-            return lst
+            return [int(elem) for elem in lst]
 
         @has_request_variables
         def get_total(request, numbers=REQ(converter=my_converter)):
@@ -148,7 +148,7 @@ class DecoratorTestCase(TestCase):
         with self.assertRaisesRegex(AssertionError, "converter and validator are mutually exclusive"):
             @has_request_variables
             def get_total(request, numbers=REQ(validator=check_list(check_int),
-                                               converter=lambda: None)):
+                                               converter=lambda x: [])):
                 # type: (HttpRequest, Iterable[int]) -> int
                 return sum(numbers)  # nocoverage -- isn't intended to be run
 
diff --git a/zerver/views/report.py b/zerver/views/report.py
index e38b53cfb3..082394ab45 100644
--- a/zerver/views/report.py
+++ b/zerver/views/report.py
@@ -1,6 +1,6 @@
 # System documented in https://zulip.readthedocs.io/en/latest/logging.html
 
-from typing import Any, Dict, Optional, Text
+from typing import Any, Dict, Optional, Text, Union
 
 from django.conf import settings
 from django.http import HttpRequest, HttpResponse
@@ -33,20 +33,27 @@ def get_js_source_map() -> Optional[SourceMap]:
 
 @human_users_only
 @has_request_variables
-def report_send_times(request, user_profile,
-                      time=REQ(converter=to_non_negative_int),
-                      received=REQ(converter=to_non_negative_int, default="(unknown)"),
-                      displayed=REQ(converter=to_non_negative_int, default="(unknown)"),
-                      locally_echoed=REQ(validator=check_bool, default=False),
-                      rendered_content_disparity=REQ(validator=check_bool, default=False)):
-    # type: (HttpRequest, UserProfile, int, int, int, bool, bool) -> HttpResponse
+def report_send_times(request: HttpRequest, user_profile: UserProfile,
+                      time: int=REQ(converter=to_non_negative_int),
+                      received: int=REQ(converter=to_non_negative_int, default=-1),
+                      displayed: int=REQ(converter=to_non_negative_int, default=-1),
+                      locally_echoed: bool=REQ(validator=check_bool, default=False),
+                      rendered_content_disparity: bool=REQ(validator=check_bool, default=False)) -> HttpResponse:
+    received_str = "(unknown)"
+    if received > 0:
+        received_str = str(received)
+    displayed_str = "(unknown)"
+    if displayed > 0:
+        displayed_str = str(displayed)
+
     request._log_data["extra"] = "[%sms/%sms/%sms/echo:%s/diff:%s]" \
-        % (time, received, displayed, locally_echoed, rendered_content_disparity)
+        % (time, received_str, displayed_str, locally_echoed, rendered_content_disparity)
+
     base_key = statsd_key(user_profile.realm.string_id, clean_periods=True)
     statsd.timing("endtoend.send_time.%s" % (base_key,), time)
-    if received != "(unknown)":
+    if received > 0:
         statsd.timing("endtoend.receive_time.%s" % (base_key,), received)
-    if displayed != "(unknown)":
+    if displayed > 0:
         statsd.timing("endtoend.displayed_time.%s" % (base_key,), displayed)
     if locally_echoed:
         statsd.incr('locally_echoed')
diff --git a/zerver/webhooks/travis/view.py b/zerver/webhooks/travis/view.py
index 050e5742b0..8f27d9370c 100644
--- a/zerver/webhooks/travis/view.py
+++ b/zerver/webhooks/travis/view.py
@@ -32,7 +32,7 @@ def api_travis_webhook(request, user_profile,
                            ('status_message', check_string),
                            ('compare_url', check_string),
                        ]))):
-    # type: (HttpRequest, UserProfile, str, str, str, Dict[str, str]) -> HttpResponse
+    # type: (HttpRequest, UserProfile, str, str, bool, Dict[str, str]) -> HttpResponse
 
     message_status = message['status_message']
     if ignore_pull_requests and message['type'] == 'pull_request':