diff --git a/tests/middleware/test_middleware.py b/tests/middleware/test_middleware.py
index f4d7a32f0..938250f08 100644
--- a/tests/middleware/test_middleware.py
+++ b/tests/middleware/test_middleware.py
@@ -1,8 +1,10 @@
 from starlette.middleware import Middleware
+from starlette.types import ASGIApp
 
 
 class CustomMiddleware:
-    pass
+    def __init__(self, app: ASGIApp) -> None:
+        self.app = app  # pragma: no cover
 
 
 def test_middleware_repr():
diff --git a/tests/test_authentication.py b/tests/test_authentication.py
index af0beafd0..150482a1b 100644
--- a/tests/test_authentication.py
+++ b/tests/test_authentication.py
@@ -15,7 +15,7 @@
 from starlette.endpoints import HTTPEndpoint
 from starlette.middleware import Middleware
 from starlette.middleware.authentication import AuthenticationMiddleware
-from starlette.requests import Request
+from starlette.requests import HTTPConnection
 from starlette.responses import JSONResponse
 from starlette.routing import Route, WebSocketRoute
 from starlette.websockets import WebSocketDisconnect
@@ -327,7 +327,7 @@ def test_authentication_redirect(test_client_factory):
         assert response.json() == {"authenticated": True, "user": "tomchristie"}
 
 
-def on_auth_error(request: Request, exc: Exception):
+def on_auth_error(request: HTTPConnection, exc: AuthenticationError):
     return JSONResponse({"error": str(exc)}, status_code=401)