How many trades separate edge from noise; false-positive rate of 'edge' tests

"""M14 · How many trades separate edge from noise; false-positive rate of 'edge' tests.

Exercise stubs. See notes.md §14.10. Trades are measured in R-multiples (see M03).
"""

from __future__ import annotations


def t_stat(r) -> float:
    """One-sample t-statistic mean(r) / (s/sqrt(N)) with s the ddof=1 standard deviation."""
    raise NotImplementedError


def trades_needed(expectancy: float, sd: float, t_crit: float = 2.0) -> int:
    """Smallest integer N with expectancy*sqrt(N)/sd >= t_crit: ceil((t_crit*sd/expectancy)^2).
    Guard against float round-up (subtract 1e-12 before ceil). M03 Corollary 3.1.1."""
    raise NotImplementedError


def trades_needed_power(expectancy: float, sd: float, t_crit: float = 2.0, power: float = 0.8) -> int:
    """N giving probability `power` that the t-stat clears t_crit when the true mean is `expectancy`
    (normal approximation): ceil(((t_crit + z_power) * sd / expectancy)^2), z_power = Phi^-1(power).
    M03 Corollary 3.1.2."""
    raise NotImplementedError


def approx_power(expectancy: float, sd: float, n: int, t_crit: float = 2.0) -> float:
    """Normal approximation 1 - Phi(t_crit - expectancy*sqrt(n)/sd)."""
    raise NotImplementedError


def simulate_pass_rate(n_trades: int, n_sims: int, seed: int, true_mean: float = 0.0,
                       t_crit: float = 2.0, kind: str = "normal") -> float:
    """Fraction of n_sims simulated traders (each with n_trades i.i.d. R-multiples) whose
    t-stat >= t_crit. Use np.random.default_rng(seed) and vectorise over a (n_sims, n_trades) array.

    kind='normal':   R ~ N(true_mean, 1).
    kind='breakout': R = +0.65/0.35 with prob 0.35, else -1, then add true_mean
                     (zero-mean two-point distribution typical of breakout systems).
    Raise ValueError for any other kind.
    """
    raise NotImplementedError


def familywise_false_positive(alpha: float, k: int) -> float:
    """P(at least one of k independent null setups passes) = 1 - (1 - alpha)^k."""
    raise NotImplementedError