Bootstrap the nth percentile

classical_bootstrap_percentile.py

def bootstrap_nth_percentile_and_ci(
    data: List[float], n_iter: int = 1000, alpha: float = 0.95, percentile: float = 95
) -> Tuple[float, float, float, float]:
    """Compute the nth percentile and confidence interval using bootstrapping

    Args:
        data (List[float]): List of data
        n_iter (int, optional): Number of bootstrap samples. Defaults to 1000.
        alpha (float, optional): Confidence level. Defaults to 0.95.
        percentile (float, optional): Percentile to compute. Defaults to 95.

    Returns:
        Tuple[float, float, float, float]: Tuple containing the percentile, mean, 
        lower bound, and upper bound

    """

    n = len(data)

    bootstrap_samples = np.random.choice(data, (n_iter, n), replace=True)
    percentiles = np.percentile(bootstrap_samples, percentile, axis=1)

    mean = percentiles.mean()
    lower = np.percentile(percentiles, (1 - alpha) / 2 * 100)
    upper = np.percentile(percentiles, (1 + alpha) / 2 * 100)

    return percentiles, mean, lower, upper
  

# population specifics
mean_time = 2.0
std_dev = 0.5
processing_time = np.random.normal(loc=mean_time, scale=std_dev, size=1_000_000)

# sample from processing time
sample = np.random.choice(processing_time, 10000) # note we only sample 1/10th of the population

percentiles, mean, lower_bound, upper_bound = bootstrap_nth_percentile_and_ci(
    sample, n_iter=10000, percentile=95
)

The graph is generated using

fig = px.histogram(
    percentiles,
    color_discrete_sequence=["black"],
    labels={"value": "95th Percentile", "count": "Frequency"},
    title="Bootstrap Distribution of the 95th Percentile",
)
fig.add_vline(
    x=np.percentile(processing_time, 95),
    opacity=1,
    line_width=2,
    line_dash="dash",
    line_color="Blue",
    annotation_text="True 95th Percentile",
)
fig.add_vline(
    x=lower_bound,
    opacity=1,
    line_width=2,
    line_dash="dash",
    line_color="Red",
)
fig.add_vline(
    x=upper_bound,
    opacity=1,
    line_width=2,
    line_dash="dash",
    line_color="Red",
)
fig.update_layout(showlegend=False)