Adjust some of the [rand.dist] critical values that are too strict

- Most critical values are determined empirically by running each test 51 times with a different PRNG seed and finding the smallest symmetric interval around the median that contains 90% of the sample means, variances, etc. - For the Kolmogorov-Smirnov tests, the alpha=0.1 critical value for large N is 1.224/sqrt(N). - For normally distributed variates, the sample kurtosis is distributed as Normal(0, 24/N). For N=1e5, this gives a 90% confidence interval of 0+/-0.0255. For Binomial(40, 0.25), which is approximately normal, the kurtosis is -0.0167, so the relative 90% CI is large, on the order of 0.0255/0.0167 = 153%. In most cases the distribution of the sample kurtosis isn't known analytically, but similarly large relative tolerances can be expected if the kurtosis is near zero.
llvm · Apr 27, 2024 · 788fc37 · 788fc37
1 parent 5927492
commit 788fc37
Show file tree

Hide file tree

Showing 28 changed files with 134 additions and 134 deletions.
diff --git a/...xx/test/std/numerics/rand/rand.dist/rand.dist.bern/rand.dist.bern.bernoulli/eval.pass.cpp b/...xx/test/std/numerics/rand/rand.dist/rand.dist.bern/rand.dist.bern.bernoulli/eval.pass.cpp
@@ -63,8 +63,8 @@ int main(int, char**)
         double x_kurtosis = (6 * sqr(d.p()) - 6 * d.p() + 1)/x_var;
         assert(std::abs((mean - x_mean) / x_mean) < 0.01);
         assert(std::abs((var - x_var) / x_var) < 0.01);
-        assert(std::abs((skew - x_skew) / x_skew) < 0.01);
-        assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.02);
+        assert(std::abs((skew - x_skew) / x_skew) < 0.02);
+        assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.05);
     }
     {
         typedef std::bernoulli_distribution D;
@@ -99,8 +99,8 @@ int main(int, char**)
         double x_kurtosis = (6 * sqr(d.p()) - 6 * d.p() + 1)/x_var;
         assert(std::abs((mean - x_mean) / x_mean) < 0.01);
         assert(std::abs((var - x_var) / x_var) < 0.01);
-        assert(std::abs((skew - x_skew) / x_skew) < 0.01);
-        assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.02);
+        assert(std::abs((skew - x_skew) / x_skew) < 0.02);
+        assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.05);
     }
 
   return 0;

diff --git a/...t/std/numerics/rand/rand.dist/rand.dist.bern/rand.dist.bern.bernoulli/eval_param.pass.cpp b/...t/std/numerics/rand/rand.dist/rand.dist.bern/rand.dist.bern.bernoulli/eval_param.pass.cpp
@@ -65,8 +65,8 @@ int main(int, char**)
         double x_kurtosis = (6 * sqr(p.p()) - 6 * p.p() + 1)/x_var;
         assert(std::abs((mean - x_mean) / x_mean) < 0.01);
         assert(std::abs((var - x_var) / x_var) < 0.01);
-        assert(std::abs((skew - x_skew) / x_skew) < 0.01);
-        assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.02);
+        assert(std::abs((skew - x_skew) / x_skew) < 0.02);
+        assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.05);
     }
     {
         typedef std::bernoulli_distribution D;
@@ -103,8 +103,8 @@ int main(int, char**)
         double x_kurtosis = (6 * sqr(p.p()) - 6 * p.p() + 1)/x_var;
         assert(std::abs((mean - x_mean) / x_mean) < 0.01);
         assert(std::abs((var - x_var) / x_var) < 0.01);
-        assert(std::abs((skew - x_skew) / x_skew) < 0.01);
-        assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.02);
+        assert(std::abs((skew - x_skew) / x_skew) < 0.02);
+        assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.05);
     }
 
   return 0;

diff --git a/.../test/std/numerics/rand/rand.dist/rand.dist.bern/rand.dist.bern.bin/eval.PR44847.pass.cpp b/.../test/std/numerics/rand/rand.dist/rand.dist.bern/rand.dist.bern.bin/eval.PR44847.pass.cpp
@@ -163,7 +163,7 @@ int main(int, char**) {
     assert(std::abs((mean - x_mean) / x_mean) < 0.01);
     assert(std::abs((var - x_var) / x_var) < 0.01);
     assert(std::abs((skew - x_skew) / x_skew) < 0.01);
-    assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.04);
+    assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.06);
 
     return 0;
 }
diff --git a/libcxx/test/std/numerics/rand/rand.dist/rand.dist.bern/rand.dist.bern.bin/eval.pass.cpp b/libcxx/test/std/numerics/rand/rand.dist/rand.dist.bern/rand.dist.bern.bin/eval.pass.cpp
@@ -68,7 +68,7 @@ void test1() {
     assert(std::abs((mean - x_mean) / x_mean) < 0.01);
     assert(std::abs((var - x_var) / x_var) < 0.01);
     assert(std::abs((skew - x_skew) / x_skew) < 0.01);
-    assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.04);
+    assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.08);
 }
 
 template <class T>
@@ -109,8 +109,8 @@ void test2() {
     double x_kurtosis = (1-6*d.p()*(1-d.p())) / x_var;
     assert(std::abs((mean - x_mean) / x_mean) < 0.01);
     assert(std::abs((var - x_var) / x_var) < 0.01);
-    assert(std::abs((skew - x_skew) / x_skew) < 0.01);
-    assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.01);
+    assert(std::abs((skew - x_skew) / x_skew) < 0.02);
+    assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.08);
 }
 
 template <class T>
@@ -151,8 +151,8 @@ void test3() {
     double x_kurtosis = (1-6*d.p()*(1-d.p())) / x_var;
     assert(std::abs((mean - x_mean) / x_mean) < 0.01);
     assert(std::abs((var - x_var) / x_var) < 0.01);
-    assert(std::abs((skew - x_skew) / x_skew) < 0.03);
-    assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.3);
+    assert(std::abs((skew - x_skew) / x_skew) < 0.07);
+    assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 2.0);
 }
 
 template <class T>
@@ -292,7 +292,7 @@ void test6() {
     assert(std::abs((mean - x_mean) / x_mean) < 0.01);
     assert(std::abs((var - x_var) / x_var) < 0.01);
     assert(std::abs(skew - x_skew) < 0.02);
-    assert(std::abs(kurtosis - x_kurtosis) < 0.01);
+    assert(std::abs(kurtosis - x_kurtosis) < 0.03);
 }
 
 template <class T>

diff --git a/...xx/test/std/numerics/rand/rand.dist/rand.dist.bern/rand.dist.bern.bin/eval_param.pass.cpp b/...xx/test/std/numerics/rand/rand.dist/rand.dist.bern/rand.dist.bern.bin/eval_param.pass.cpp
@@ -72,7 +72,7 @@ int main(int, char**)
         assert(std::abs((mean - x_mean) / x_mean) < 0.01);
         assert(std::abs((var - x_var) / x_var) < 0.01);
         assert(std::abs((skew - x_skew) / x_skew) < 0.01);
-        assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.04);
+        assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.08);
     }
     {
         typedef std::binomial_distribution<> D;
@@ -113,8 +113,8 @@ int main(int, char**)
         double x_kurtosis = (1-6*p.p()*(1-p.p())) / x_var;
         assert(std::abs((mean - x_mean) / x_mean) < 0.01);
         assert(std::abs((var - x_var) / x_var) < 0.01);
-        assert(std::abs((skew - x_skew) / x_skew) < 0.01);
-        assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.01);
+        assert(std::abs((skew - x_skew) / x_skew) < 0.02);
+        assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.08);
     }
     {
         typedef std::binomial_distribution<> D;
@@ -155,8 +155,8 @@ int main(int, char**)
         double x_kurtosis = (1-6*p.p()*(1-p.p())) / x_var;
         assert(std::abs((mean - x_mean) / x_mean) < 0.01);
         assert(std::abs((var - x_var) / x_var) < 0.01);
-        assert(std::abs((skew - x_skew) / x_skew) < 0.04);
-        assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.3);
+        assert(std::abs((skew - x_skew) / x_skew) < 0.07);
+        assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 2.0);
     }
 
   return 0;

diff --git a/libcxx/test/std/numerics/rand/rand.dist/rand.dist.bern/rand.dist.bern.geo/eval.pass.cpp b/libcxx/test/std/numerics/rand/rand.dist/rand.dist.bern/rand.dist.bern.geo/eval.pass.cpp
@@ -77,7 +77,7 @@ void test1() {
     assert(std::abs((mean - x_mean) / x_mean) < 0.01);
     assert(std::abs((var - x_var) / x_var) < 0.01);
     assert(std::abs((skew - x_skew) / x_skew) < 0.01);
-    assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.01);
+    assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.03);
 }
 
 template <class T>
@@ -161,7 +161,7 @@ void test3() {
     assert(std::abs((mean - x_mean) / x_mean) < 0.01);
     assert(std::abs((var - x_var) / x_var) < 0.01);
     assert(std::abs((skew - x_skew) / x_skew) < 0.01);
-    assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.02);
+    assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.03);
 }
 
 template <class T>
@@ -203,7 +203,7 @@ void test4() {
     assert(std::abs((mean - x_mean) / x_mean) < 0.01);
     assert(std::abs((var - x_var) / x_var) < 0.01);
     assert(std::abs((skew - x_skew) / x_skew) < 0.01);
-    assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.02);
+    assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.03);
 }
 
 template <class T>
@@ -245,7 +245,7 @@ void test5() {
     assert(std::abs((mean - x_mean) / x_mean) < 0.01);
     assert(std::abs((var - x_var) / x_var) < 0.01);
     assert(std::abs((skew - x_skew) / x_skew) < 0.01);
-    assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.02);
+    assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.03);
 }
 
 template <class T>
@@ -284,8 +284,8 @@ void test6() {
     double x_var = x_mean / d.p();
     double x_skew = (2 - d.p()) / std::sqrt((1 - d.p()));
     double x_kurtosis = 6 + sqr(d.p()) / (1 - d.p());
-    assert(std::abs((mean - x_mean) / x_mean) < 0.01);
-    assert(std::abs((var - x_var) / x_var) < 0.01);
+    assert(std::abs((mean - x_mean) / x_mean) < 0.02);
+    assert(std::abs((var - x_var) / x_var) < 0.02);
     assert(std::abs((skew - x_skew) / x_skew) < 0.01);
     assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.02);
 }

diff --git a/...xx/test/std/numerics/rand/rand.dist/rand.dist.bern/rand.dist.bern.geo/eval_param.pass.cpp b/...xx/test/std/numerics/rand/rand.dist/rand.dist.bern/rand.dist.bern.geo/eval_param.pass.cpp
@@ -72,7 +72,7 @@ int main(int, char**)
         assert(std::abs((mean - x_mean) / x_mean) < 0.01);
         assert(std::abs((var - x_var) / x_var) < 0.01);
         assert(std::abs((skew - x_skew) / x_skew) < 0.01);
-        assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.01);
+        assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.03);
     }
     {
         typedef std::geometric_distribution<> D;
@@ -156,7 +156,7 @@ int main(int, char**)
         assert(std::abs((mean - x_mean) / x_mean) < 0.01);
         assert(std::abs((var - x_var) / x_var) < 0.01);
         assert(std::abs((skew - x_skew) / x_skew) < 0.01);
-        assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.02);
+        assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.03);
     }
 
   return 0;

diff --git a/libcxx/test/std/numerics/rand/rand.dist/rand.dist.bern/rand.dist.bern.negbin/eval.pass.cpp b/libcxx/test/std/numerics/rand/rand.dist/rand.dist.bern/rand.dist.bern.negbin/eval.pass.cpp
@@ -74,7 +74,7 @@ void test1() {
     assert(std::abs((mean - x_mean) / x_mean) < 0.01);
     assert(std::abs((var - x_var) / x_var) < 0.01);
     assert(std::abs((skew - x_skew) / x_skew) < 0.01);
-    assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.02);
+    assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.03);
 }
 
 template <class T>
@@ -115,8 +115,8 @@ void test2() {
     double x_kurtosis = 6. / d.k() + sqr(d.p()) / (d.k() * (1 - d.p()));
     assert(std::abs((mean - x_mean) / x_mean) < 0.01);
     assert(std::abs((var - x_var) / x_var) < 0.01);
-    assert(std::abs((skew - x_skew) / x_skew) < 0.01);
-    assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.01);
+    assert(std::abs((skew - x_skew) / x_skew) < 0.02);
+    assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.1);
 }
 
 template <class T>
@@ -157,8 +157,8 @@ void test3() {
     double x_kurtosis = 6. / d.k() + sqr(d.p()) / (d.k() * (1 - d.p()));
     assert(std::abs((mean - x_mean) / x_mean) < 0.01);
     assert(std::abs((var - x_var) / x_var) < 0.01);
-    assert(std::abs((skew - x_skew) / x_skew) < 0.01);
-    assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.03);
+    assert(std::abs((skew - x_skew) / x_skew) < 0.02);
+    assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.08);
 }
 
 template <class T>
@@ -243,8 +243,8 @@ void test5() {
     double x_kurtosis = 6. / d.k() + sqr(d.p()) / (d.k() * (1 - d.p()));
     assert(std::abs((mean - x_mean) / x_mean) < 0.01);
     assert(std::abs((var - x_var) / x_var) < 0.01);
-    assert(std::abs((skew - x_skew) / x_skew) < 0.04);
-    assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.05);
+    assert(std::abs((skew - x_skew) / x_skew) < 0.02);
+    assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.3);
 }
 
 template <class T>

diff --git a/...test/std/numerics/rand/rand.dist/rand.dist.bern/rand.dist.bern.negbin/eval_param.pass.cpp b/...test/std/numerics/rand/rand.dist/rand.dist.bern/rand.dist.bern.negbin/eval_param.pass.cpp
@@ -72,7 +72,7 @@ int main(int, char**)
         assert(std::abs((mean - x_mean) / x_mean) < 0.01);
         assert(std::abs((var - x_var) / x_var) < 0.01);
         assert(std::abs((skew - x_skew) / x_skew) < 0.01);
-        assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.01);
+        assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.03);
     }
     {
         typedef std::negative_binomial_distribution<> D;
@@ -113,8 +113,8 @@ int main(int, char**)
         double x_kurtosis = 6. / p.k() + sqr(p.p()) / (p.k() * (1 - p.p()));
         assert(std::abs((mean - x_mean) / x_mean) < 0.01);
         assert(std::abs((var - x_var) / x_var) < 0.01);
-        assert(std::abs((skew - x_skew) / x_skew) < 0.01);
-        assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.01);
+        assert(std::abs((skew - x_skew) / x_skew) < 0.02);
+        assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.1);
     }
     {
         typedef std::negative_binomial_distribution<> D;
@@ -155,8 +155,8 @@ int main(int, char**)
         double x_kurtosis = 6. / p.k() + sqr(p.p()) / (p.k() * (1 - p.p()));
         assert(std::abs((mean - x_mean) / x_mean) < 0.01);
         assert(std::abs((var - x_var) / x_var) < 0.01);
-        assert(std::abs((skew - x_skew) / x_skew) < 0.01);
-        assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.03);
+        assert(std::abs((skew - x_skew) / x_skew) < 0.02);
+        assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.08);
     }
 
   return 0;

diff --git a/libcxx/test/std/numerics/rand/rand.dist/rand.dist.norm/rand.dist.norm.cauchy/eval.pass.cpp b/libcxx/test/std/numerics/rand/rand.dist/rand.dist.norm/rand.dist.norm.cauchy/eval.pass.cpp
@@ -45,7 +45,7 @@ int main(int, char**)
             u.push_back(d(g));
         std::sort(u.begin(), u.end());
         for (int i = 0; i < N; ++i)
-            assert(std::abs(f(u[i], a, b) - double(i)/N) < .001);
+            assert(std::abs(f(u[i], a, b) - double(i)/N) < .0013);
     }
     {
         typedef std::cauchy_distribution<> D;
@@ -60,7 +60,7 @@ int main(int, char**)
             u.push_back(d(g));
         std::sort(u.begin(), u.end());
         for (int i = 0; i < N; ++i)
-            assert(std::abs(f(u[i], a, b) - double(i)/N) < .001);
+            assert(std::abs(f(u[i], a, b) - double(i)/N) < .0013);
     }
     {
         typedef std::cauchy_distribution<> D;
@@ -75,7 +75,7 @@ int main(int, char**)
             u.push_back(d(g));
         std::sort(u.begin(), u.end());
         for (int i = 0; i < N; ++i)
-            assert(std::abs(f(u[i], a, b) - double(i)/N) < .001);
+            assert(std::abs(f(u[i], a, b) - double(i)/N) < .0013);
     }
 
   return 0;

diff --git a/libcxx/test/std/numerics/rand/rand.dist/rand.dist.norm/rand.dist.norm.chisq/eval.pass.cpp b/libcxx/test/std/numerics/rand/rand.dist/rand.dist.norm/rand.dist.norm.chisq/eval.pass.cpp
@@ -70,7 +70,7 @@ int main(int, char**)
         assert(std::abs((mean - x_mean) / x_mean) < 0.01);
         assert(std::abs((var - x_var) / x_var) < 0.01);
         assert(std::abs((skew - x_skew) / x_skew) < 0.01);
-        assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.01);
+        assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.04);
     }
     {
         typedef std::chi_squared_distribution<> D;
@@ -109,7 +109,7 @@ int main(int, char**)
         assert(std::abs((mean - x_mean) / x_mean) < 0.01);
         assert(std::abs((var - x_var) / x_var) < 0.01);
         assert(std::abs((skew - x_skew) / x_skew) < 0.01);
-        assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.01);
+        assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.03);
     }
     {
         typedef std::chi_squared_distribution<> D;
@@ -148,7 +148,7 @@ int main(int, char**)
         assert(std::abs((mean - x_mean) / x_mean) < 0.01);
         assert(std::abs((var - x_var) / x_var) < 0.01);
         assert(std::abs((skew - x_skew) / x_skew) < 0.01);
-        assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.01);
+        assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.03);
     }
 
   return 0;

diff --git a/.../test/std/numerics/rand/rand.dist/rand.dist.norm/rand.dist.norm.chisq/eval_param.pass.cpp b/.../test/std/numerics/rand/rand.dist/rand.dist.norm/rand.dist.norm.chisq/eval_param.pass.cpp
@@ -72,7 +72,7 @@ int main(int, char**)
         assert(std::abs((mean - x_mean) / x_mean) < 0.01);
         assert(std::abs((var - x_var) / x_var) < 0.01);
         assert(std::abs((skew - x_skew) / x_skew) < 0.01);
-        assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.01);
+        assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.03);
     }
     {
         typedef std::chi_squared_distribution<> D;
@@ -113,7 +113,7 @@ int main(int, char**)
         assert(std::abs((mean - x_mean) / x_mean) < 0.01);
         assert(std::abs((var - x_var) / x_var) < 0.01);
         assert(std::abs((skew - x_skew) / x_skew) < 0.01);
-        assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.01);
+        assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.03);
     }
     {
         typedef std::chi_squared_distribution<> D;
@@ -154,7 +154,7 @@ int main(int, char**)
         assert(std::abs((mean - x_mean) / x_mean) < 0.01);
         assert(std::abs((var - x_var) / x_var) < 0.01);
         assert(std::abs((skew - x_skew) / x_skew) < 0.01);
-        assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.01);
+        assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.04);
     }
 
   return 0;

diff --git a/...xx/test/std/numerics/rand/rand.dist/rand.dist.norm/rand.dist.norm.lognormal/eval.pass.cpp b/...xx/test/std/numerics/rand/rand.dist/rand.dist.norm/rand.dist.norm.lognormal/eval.pass.cpp
@@ -70,8 +70,8 @@ test1()
                       3*std::exp(2*sqr(d.s())) - 6;
     assert(std::abs((mean - x_mean) / x_mean) < 0.01);
     assert(std::abs((var - x_var) / x_var) < 0.01);
-    assert(std::abs((skew - x_skew) / x_skew) < 0.05);
-    assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.25);
+    assert(std::abs((skew - x_skew) / x_skew) < 0.1);
+    assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 1.9);
 }
 
 void
@@ -115,7 +115,7 @@ test2()
     assert(std::abs((mean - x_mean) / x_mean) < 0.01);
     assert(std::abs((var - x_var) / x_var) < 0.01);
     assert(std::abs((skew - x_skew) / x_skew) < 0.01);
-    assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.03);
+    assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.04);
 }
 
 void
@@ -159,7 +159,7 @@ test3()
     assert(std::abs((mean - x_mean) / x_mean) < 0.01);
     assert(std::abs((var - x_var) / x_var) < 0.01);
     assert(std::abs((skew - x_skew) / x_skew) < 0.02);
-    assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.05);
+    assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.06);
 }
 
 void
@@ -202,8 +202,8 @@ test4()
                       3*std::exp(2*sqr(d.s())) - 6;
     assert(std::abs((mean - x_mean) / x_mean) < 0.01);
     assert(std::abs((var - x_var) / x_var) < 0.02);
-    assert(std::abs((skew - x_skew) / x_skew) < 0.08);
-    assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.4);
+    assert(std::abs((skew - x_skew) / x_skew) < 0.1);
+    assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.5);
 }
 
 void
@@ -245,9 +245,9 @@ test5()
     double x_kurtosis = std::exp(4*sqr(d.s())) + 2*std::exp(3*sqr(d.s())) +
                       3*std::exp(2*sqr(d.s())) - 6;
     assert(std::abs((mean - x_mean) / x_mean) < 0.01);
-    assert(std::abs((var - x_var) / x_var) < 0.04);
-    assert(std::abs((skew - x_skew) / x_skew) < 0.2);
-    assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 0.7);
+    assert(std::abs((var - x_var) / x_var) < 0.05);
+    assert(std::abs((skew - x_skew) / x_skew) < 0.3);
+    assert(std::abs((kurtosis - x_kurtosis) / x_kurtosis) < 1.0);
 }
 
 int main(int, char**)