From 6d61eb6a887d55d2f7043ac525fdca12a3aaaadc Mon Sep 17 00:00:00 2001
From: Stefan Jansen <stefan@applied-ai.com>
Date: Wed, 22 Jul 2026 08:14:33 -0400
Subject: [PATCH 1/3] fix: include finite-sample offset in MinTRL

---
 docs/user-guide/statistical-tests.md          | 25 ++++++++----
 .../evaluation/stats/minimum_track_record.py  | 15 +++++---
 .../backtest/statistical_validity.py          | 38 +++++++++++--------
 .../test_evaluation/fixtures/dsr_reference.py |  4 +-
 tests/test_evaluation/test_dsr_validation.py  | 24 ++++++------
 .../test_minimum_track_record.py              | 33 ++++++++++++++++
 tests/test_visualization/test_backtest.py     |  9 +++++
 7 files changed, 107 insertions(+), 41 deletions(-)
 create mode 100644 tests/test_evaluation/test_minimum_track_record.py

diff --git a/docs/user-guide/statistical-tests.md b/docs/user-guide/statistical-tests.md
index 585c477..6869bfa 100644
--- a/docs/user-guide/statistical-tests.md
+++ b/docs/user-guide/statistical-tests.md
@@ -89,18 +89,24 @@ print(f"Complexity penalty: {result.complexity:.4f}")
 Calculate how long a track record must be for statistical significance:
 
 ```python
+import numpy as np
+
 from ml4t.diagnostic.evaluation.stats import compute_min_trl
 
 result = compute_min_trl(
-    sharpe_ratio=1.5,
-    target_pvalue=0.05,
-    frequency='daily'
+    observed_sharpe=1.5 / np.sqrt(252),
+    target_sharpe=0.5 / np.sqrt(252),
+    confidence_level=0.95,
+    frequency="daily",
 )
 
-print(f"Minimum observations: {result.min_observations}")
-print(f"Minimum years: {result.min_years:.1f}")
+print(f"Minimum observations: {result.min_trl:.0f}")
+print(f"Minimum years: {result.min_trl_years:.1f}")
 ```
 
+Sharpe inputs use the return series' native frequency. The example converts
+annualized Sharpe ratios to daily values before requesting a daily MinTRL.
+
 ### MinTRL with Multiple Testing
 
 For FWER-controlled significance across multiple strategies:
@@ -109,9 +115,12 @@ For FWER-controlled significance across multiple strategies:
 from ml4t.diagnostic.evaluation.stats import min_trl_fwer
 
 result = min_trl_fwer(
-    sharpe_ratio=1.5,
-    num_trials=50,
-    alpha=0.05
+    observed_sharpe=1.5 / np.sqrt(252),
+    n_trials=50,
+    variance_trials=0.04 / 252,
+    target_sharpe=0.5 / np.sqrt(252),
+    confidence_level=0.95,
+    frequency="daily",
 )
 ```
 
diff --git a/src/ml4t/diagnostic/evaluation/stats/minimum_track_record.py b/src/ml4t/diagnostic/evaluation/stats/minimum_track_record.py
index 648db2f..31fd3b2 100644
--- a/src/ml4t/diagnostic/evaluation/stats/minimum_track_record.py
+++ b/src/ml4t/diagnostic/evaluation/stats/minimum_track_record.py
@@ -1,13 +1,17 @@
 """Minimum Track Record Length (MinTRL) calculation.
 
 MinTRL is the minimum number of observations required to reject the null
-hypothesis (SR ≤ target) at the specified confidence level.
+hypothesis (SR ≤ target) at the specified confidence level. The implementation
+uses the finite-sample form, with the leading one-observation offset.
 
 References
 ----------
+Bailey, D. H., & López de Prado, M. (2012).
+"The Sharpe Ratio Efficient Frontier." Journal of Risk, 15(2), 3-44.
+
 López de Prado, M., Lipton, A., & Zoonekynd, V. (2025).
 "How to Use the Sharpe Ratio." ADIA Lab Research Paper Series, No. 19.
-Equation 11, page 9.
+Equation 11 supplies the serial-correlation variance adjustment.
 """
 
 from __future__ import annotations
@@ -150,7 +154,8 @@ def _compute_min_trl_core(
     -------
     float
         Minimum number of observations. Returns math.inf if
-        observed SR <= target SR.
+        observed SR <= target SR. The finite-sample result is rounded up
+        after adding the leading one-observation offset.
     """
     rho = autocorrelation
     sr_diff = observed_sharpe - target_sharpe
@@ -178,9 +183,9 @@ def _compute_min_trl_core(
     # Variance term (without 1/T factor)
     var_term = a - b * skewness * target_sharpe + c * (kurtosis - 1) / 4 * target_sharpe**2
 
-    # MinTRL formula (Equation 11)
+    # Finite-sample MinTRL adds one observation to the asymptotic variance term.
     try:
-        min_trl = var_term * (z_alpha / sr_diff) ** 2
+        min_trl = 1.0 + var_term * (z_alpha / sr_diff) ** 2
         if np.isinf(min_trl):
             return float("inf")
         return float(np.ceil(max(min_trl, 1)))
diff --git a/src/ml4t/diagnostic/visualization/backtest/statistical_validity.py b/src/ml4t/diagnostic/visualization/backtest/statistical_validity.py
index 4a715d6..59bdf5c 100644
--- a/src/ml4t/diagnostic/visualization/backtest/statistical_validity.py
+++ b/src/ml4t/diagnostic/visualization/backtest/statistical_validity.py
@@ -17,6 +17,7 @@
 import numpy as np
 import plotly.graph_objects as go
 
+from ml4t.diagnostic.evaluation.stats import compute_min_trl
 from ml4t.diagnostic.visualization._colors import COLORS as _ML4T_COLORS
 from ml4t.diagnostic.visualization.core import get_theme_config
 
@@ -598,26 +599,30 @@ def plot_minimum_track_record(
 
     Notes
     -----
-    The minimum track record length formula is:
-    MinTRL = 1 + (1 - γ₃*SR + γ₄*SR²/4) * (z_α / SR)²
+    The finite-sample minimum track record length formula is:
+    MinTRL = 1 + (1 - γ₃*SR₀ + (γ₄-1)*SR₀²/4) * (z_α / (SR-SR₀))²
 
-    where γ₃ is skewness, γ₄ is excess kurtosis, and z_α is the
-    critical value for confidence level α.
+    where γ₃ is skewness, γ₄ is Pearson kurtosis, and z_α is the
+    critical value for confidence level α. This visualization assumes
+    normally distributed, serially uncorrelated returns.
     """
     from scipy import stats
 
     theme_config = get_theme_config(theme)
     colors = theme_config["colorway"]
 
-    # Calculate MinTRL (simplified, assuming normal returns)
+    # Calculate finite-sample MinTRL under the plot's normal-return assumption.
     z_alpha = stats.norm.ppf(confidence)
-    sharpe_diff = observed_sharpe - sr_benchmark
-
-    if sharpe_diff <= 0:
-        min_trl = float("inf")
-    else:
-        # Simplified MinTRL (assuming γ₃=0, γ₄=3)
-        min_trl = (z_alpha / sharpe_diff) ** 2
+    annualization_factor = np.sqrt(periods_per_year)
+    observed_sharpe_period = observed_sharpe / annualization_factor
+    benchmark_sharpe_period = sr_benchmark / annualization_factor
+    min_trl_result = compute_min_trl(
+        observed_sharpe=observed_sharpe_period,
+        target_sharpe=benchmark_sharpe_period,
+        confidence_level=confidence,
+        periods_per_year=periods_per_year,
+    )
+    min_trl = min_trl_result.min_trl
 
     # Convert to years
     min_trl_years = min_trl / periods_per_year if min_trl != float("inf") else float("inf")
@@ -633,9 +638,12 @@ def plot_minimum_track_record(
     # Generate data for the required SR curve at different track record lengths
     periods_range = np.linspace(10, max_periods, 100)
 
-    # Required SR to achieve significance at each track record length
-    # SR_required = z_alpha / sqrt(T)
-    required_sr = z_alpha / np.sqrt(periods_range) + sr_benchmark
+    # Invert finite-sample MinTRL and return the required annualized Sharpe.
+    variance_adjustment = 1.0 + 0.5 * benchmark_sharpe_period**2
+    required_sr_period = benchmark_sharpe_period + z_alpha * np.sqrt(
+        variance_adjustment / (periods_range - 1.0)
+    )
+    required_sr = required_sr_period * annualization_factor
 
     fig = go.Figure()
 
diff --git a/tests/test_evaluation/fixtures/dsr_reference.py b/tests/test_evaluation/fixtures/dsr_reference.py
index 5f3ac54..575b9ba 100644
--- a/tests/test_evaluation/fixtures/dsr_reference.py
+++ b/tests/test_evaluation/fixtures/dsr_reference.py
@@ -247,7 +247,7 @@ class PowerTestCase(NamedTuple):
     tolerance=0.05,  # Our calculation gives 27.063, paper says 27.109 (0.17% diff)
 )
 
-# Normal returns case - corrected expected value computed via minimum_track_record_length()
+# Normal returns case - continuous asymptotic reference before finite-sample adjustment
 MINTRL_NORMAL = MinTRLTestCase(
     name="MinTRL with normal returns",
     sharpe_ratio=1.0,
@@ -260,7 +260,7 @@ class PowerTestCase(NamedTuple):
     tolerance=0.02,
 )
 
-# High target SR - corrected expected value computed via minimum_track_record_length()
+# High target SR - continuous asymptotic reference before finite-sample adjustment
 MINTRL_HIGH_TARGET = MinTRLTestCase(
     name="MinTRL with high target SR",
     sharpe_ratio=1.5,
diff --git a/tests/test_evaluation/test_dsr_validation.py b/tests/test_evaluation/test_dsr_validation.py
index a56706c..3f47dbf 100644
--- a/tests/test_evaluation/test_dsr_validation.py
+++ b/tests/test_evaluation/test_dsr_validation.py
@@ -548,16 +548,18 @@ def test_psr_sample_size_effect(self):
 # =============================================================================
 
 
-def minimum_track_record_length(
+def asymptotic_minimum_track_record_length(
     sharpe_ratio: float,
     sharpe_star: float,
     skewness: float,
     excess_kurtosis: float,
     alpha: float = 0.05,
 ) -> float:
-    """Calculate Minimum Track Record Length (MinTRL).
+    """Reproduce the paper's continuous asymptotic MinTRL quantity.
 
     Implementation based on equation (11) from López de Prado et al. (2025).
+    The production implementation adds the finite-sample leading offset and
+    rounds up to an integer observation count.
 
     Args:
         sharpe_ratio: Observed Sharpe ratio (SR_hat*)
@@ -582,7 +584,7 @@ def minimum_track_record_length(
     # So (γ₄-1)/4 = (excess_kurtosis + 2)/4
     variance_adjustment = 1 - skewness * sharpe_star + (excess_kurtosis + 2) / 4 * sharpe_star**2
 
-    # MinTRL = variance_adjustment * (z_alpha / (SR - SR_0))²
+    # Continuous asymptotic quantity before the finite-sample offset and rounding.
     mintrl = variance_adjustment * (z_alpha / (sharpe_ratio - sharpe_star)) ** 2
 
     return mintrl
@@ -600,7 +602,7 @@ def test_mintrl_reference(self, case):
         Paper cases (SR_0=0 and SR_0=0.1) match within <0.2%.
         Other cases use computed expected values for consistency validation.
         """
-        result = minimum_track_record_length(
+        result = asymptotic_minimum_track_record_length(
             sharpe_ratio=case.sharpe_ratio,
             sharpe_star=case.sharpe_star,
             skewness=case.skewness,
@@ -620,7 +622,7 @@ def test_mintrl_reference(self, case):
 
     def test_mintrl_zero_target(self):
         """Edge case: MinTRL with SR_0 = 0."""
-        mintrl = minimum_track_record_length(
+        mintrl = asymptotic_minimum_track_record_length(
             sharpe_ratio=1.0,
             sharpe_star=0.0,
             skewness=0.0,
@@ -634,7 +636,7 @@ def test_mintrl_zero_target(self):
 
     def test_mintrl_equal_sharpes(self):
         """Edge case: MinTRL when SR = SR_0."""
-        mintrl = minimum_track_record_length(
+        mintrl = asymptotic_minimum_track_record_length(
             sharpe_ratio=1.0,
             sharpe_star=1.0,
             skewness=0.0,
@@ -647,7 +649,7 @@ def test_mintrl_equal_sharpes(self):
 
     def test_mintrl_below_target(self):
         """Edge case: MinTRL when SR < SR_0."""
-        mintrl = minimum_track_record_length(
+        mintrl = asymptotic_minimum_track_record_length(
             sharpe_ratio=0.5,
             sharpe_star=1.0,
             skewness=0.0,
@@ -663,7 +665,7 @@ def test_mintrl_alpha_sensitivity(self):
         sharpe_ratio = 1.0
         sharpe_star = 0.5
 
-        mintrl_lenient = minimum_track_record_length(
+        mintrl_lenient = asymptotic_minimum_track_record_length(
             sharpe_ratio=sharpe_ratio,
             sharpe_star=sharpe_star,
             skewness=0.0,
@@ -671,7 +673,7 @@ def test_mintrl_alpha_sensitivity(self):
             alpha=0.10,  # Lenient
         )
 
-        mintrl_strict = minimum_track_record_length(
+        mintrl_strict = asymptotic_minimum_track_record_length(
             sharpe_ratio=sharpe_ratio,
             sharpe_star=sharpe_star,
             skewness=0.0,
@@ -1380,7 +1382,7 @@ def test_non_normality_consistent_across_functions(self):
         )
 
         # Calculate MinTRL
-        mintrl = minimum_track_record_length(
+        mintrl = asymptotic_minimum_track_record_length(
             sharpe_ratio=sharpe_ratio,
             sharpe_star=sharpe_star,
             skewness=skewness,
@@ -1400,7 +1402,7 @@ def test_non_normality_consistent_across_functions(self):
         assert psr < psr_normal
 
         # MinTRL should be higher with fat tails
-        mintrl_normal = minimum_track_record_length(
+        mintrl_normal = asymptotic_minimum_track_record_length(
             sharpe_ratio=sharpe_ratio,
             sharpe_star=0.0,
             skewness=0.0,
diff --git a/tests/test_evaluation/test_minimum_track_record.py b/tests/test_evaluation/test_minimum_track_record.py
new file mode 100644
index 0000000..dad067d
--- /dev/null
+++ b/tests/test_evaluation/test_minimum_track_record.py
@@ -0,0 +1,33 @@
+"""Regression tests for Minimum Track Record Length calculations."""
+
+import numpy as np
+
+from ml4t.diagnostic.evaluation.stats import compute_min_trl
+
+
+def test_min_trl_includes_finite_sample_offset_for_strategy_14() -> None:
+    """Match the seeded Chapter 16 strategy-selection example."""
+    rng = np.random.default_rng(123)
+    n_strategies = 30
+    n_days = 504
+    true_sharpes = rng.permutation([0.8] * 5 + [0.0] * (n_strategies - 5))
+
+    strategy_returns: dict[str, np.ndarray] = {}
+    observed_sharpes: dict[str, float] = {}
+    for index, true_sharpe in enumerate(true_sharpes, start=1):
+        daily_volatility = 0.15 / np.sqrt(252)
+        daily_mean = true_sharpe * 0.15 / 252
+        returns = rng.normal(daily_mean, daily_volatility, n_days)
+        name = f"Strategy_{index}"
+        strategy_returns[name] = returns
+        observed_sharpes[name] = returns.mean() / returns.std(ddof=1) * np.sqrt(252)
+
+    best_name = max(observed_sharpes, key=observed_sharpes.__getitem__)
+    result = compute_min_trl(
+        returns=strategy_returns[best_name],
+        target_sharpe=0.5 / np.sqrt(252),
+        frequency="daily",
+    )
+
+    assert best_name == "Strategy_14"
+    assert result.min_trl == 213.0
diff --git a/tests/test_visualization/test_backtest.py b/tests/test_visualization/test_backtest.py
index 477d376..64f5049 100644
--- a/tests/test_visualization/test_backtest.py
+++ b/tests/test_visualization/test_backtest.py
@@ -620,6 +620,7 @@ def test_plot_ras_analysis(self):
 
     def test_plot_minimum_track_record(self):
         """Test MinTRL visualization."""
+        from ml4t.diagnostic.evaluation.stats import compute_min_trl
         from ml4t.diagnostic.visualization.backtest import plot_minimum_track_record
 
         fig = plot_minimum_track_record(
@@ -629,6 +630,14 @@ def test_plot_minimum_track_record(self):
         )
 
         assert isinstance(fig, go.Figure)
+        expected = compute_min_trl(
+            observed_sharpe=1.8 / np.sqrt(252),
+            target_sharpe=0.5 / np.sqrt(252),
+            periods_per_year=252,
+        )
+        min_trl_line = fig.layout.shapes[0]
+        assert min_trl_line.x0 == pytest.approx(expected.min_trl_years)
+        assert min_trl_line.x1 == pytest.approx(expected.min_trl_years)
 
 
 # =============================================================================

From 407f3b6576dba5160871b8ee139395c780fdc405 Mon Sep 17 00:00:00 2001
From: Stefan Jansen <stefan@applied-ai.com>
Date: Wed, 22 Jul 2026 12:53:41 -0400
Subject: [PATCH 2/3] chore: bump version to 0.1.0b24

---
 src/ml4t/diagnostic/__init__.py | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/ml4t/diagnostic/__init__.py b/src/ml4t/diagnostic/__init__.py
index aca0697..9cf3c15 100644
--- a/src/ml4t/diagnostic/__init__.py
+++ b/src/ml4t/diagnostic/__init__.py
@@ -31,7 +31,7 @@
 exported in __all__. Breaking changes will only occur in major version bumps.
 """
 
-__version__ = "0.1.0b23"
+__version__ = "0.1.0b24"
 
 # Sub-modules for advanced usage
 from . import (

From fde968dae26cda7b652dcf707be9896adf2d5c31 Mon Sep 17 00:00:00 2001
From: Stefan Jansen <stefan@applied-ai.com>
Date: Wed, 22 Jul 2026 12:55:00 -0400
Subject: [PATCH 3/3] ci: align Ruff release checks

---
 .pre-commit-config.yaml                         | 2 +-
 src/ml4t/diagnostic/integration/__init__.py     | 1 +
 tests/test_integration/test_backtest_profile.py | 2 +-
 tests/test_integration/test_backtest_result.py  | 4 ++--
 tests/test_visualization/test_backtest.py       | 2 +-
 5 files changed, 6 insertions(+), 5 deletions(-)

diff --git a/.pre-commit-config.yaml b/.pre-commit-config.yaml
index 44c41b2..1cc9a62 100644
--- a/.pre-commit-config.yaml
+++ b/.pre-commit-config.yaml
@@ -1,6 +1,6 @@
 repos:
   - repo: https://github.com/astral-sh/ruff-pre-commit
-    rev: v0.12.10
+    rev: v0.13.2
     hooks:
       - id: ruff-format
         exclude: ^examples/
diff --git a/src/ml4t/diagnostic/integration/__init__.py b/src/ml4t/diagnostic/integration/__init__.py
index 4d62538..b37ea6c 100644
--- a/src/ml4t/diagnostic/integration/__init__.py
+++ b/src/ml4t/diagnostic/integration/__init__.py
@@ -34,6 +34,7 @@
 
 if TYPE_CHECKING:
     from ml4t.backtest import BacktestResult
+
     from ml4t.diagnostic.evaluation import PortfolioAnalysis
 
 
diff --git a/tests/test_integration/test_backtest_profile.py b/tests/test_integration/test_backtest_profile.py
index e525857..b3bc4ce 100644
--- a/tests/test_integration/test_backtest_profile.py
+++ b/tests/test_integration/test_backtest_profile.py
@@ -4,9 +4,9 @@
 
 import polars as pl
 import pytest
-
 from ml4t.backtest import BacktestResult
 from ml4t.backtest.types import Fill, OrderSide, Trade
+
 from ml4t.diagnostic.integration import analyze_backtest_result
 
 
diff --git a/tests/test_integration/test_backtest_result.py b/tests/test_integration/test_backtest_result.py
index 4e9cbb9..e2f7f26 100644
--- a/tests/test_integration/test_backtest_result.py
+++ b/tests/test_integration/test_backtest_result.py
@@ -10,9 +10,10 @@
 import numpy as np
 import polars as pl
 import pytest
-
 from ml4t.backtest import BacktestConfig, BacktestResult
 from ml4t.backtest.types import Trade
+from ml4t.specs import FeedSpec
+
 from ml4t.diagnostic.integration import (
     BacktestReportMetadata,
     compute_metrics_from_result,
@@ -21,7 +22,6 @@
     portfolio_analysis_from_result,
     profile_from_run_artifacts,
 )
-from ml4t.specs import FeedSpec
 
 
 def create_sample_result(n_trades: int = 20) -> BacktestResult:
diff --git a/tests/test_visualization/test_backtest.py b/tests/test_visualization/test_backtest.py
index 64f5049..c387946 100644
--- a/tests/test_visualization/test_backtest.py
+++ b/tests/test_visualization/test_backtest.py
@@ -17,9 +17,9 @@
 import plotly.graph_objects as go
 import polars as pl
 import pytest
-
 from ml4t.backtest import BacktestResult
 from ml4t.backtest.types import Fill, OrderSide, Trade
+
 from ml4t.diagnostic.integration import BacktestReportMetadata
 
 # =============================================================================