Cross-Sectional Model

SW2023Model

class sw2023.SW2023Model(X, Y, direction='mean', method='HMS', h=None, log_transform=True, standardize=True, bandwidth_method='silverman')[source]

Bases: object

Simar & Wilson (2023) nonparametric multiple-output stochastic frontier model.

Parameters:

X (array-like (n, p) inputs)
Y (array-like (n, q) outputs)
direction ('mean' | 'median' | array (p+q,) direction vector)
method ('SVKZ' | 'HMS' inefficiency estimation method)
h (bandwidth (if None, Silverman rule is applied automatically))

__repr__()[source]: Return repr(self).

fit(verbose=True)[source]

Estimate the SW(2023) nonparametric stochastic frontier model.

Executes Steps 1–10 of Simar & Wilson (2023): data preprocessing, direction-vector rotation, three-moment LLLS regressions with LOO-CV bandwidth selection, sigma_eta estimation (SVKZ or HMS), frontier recovery, and JLMS individual efficiency scoring.

Parameters:

verbose (bool, default True) – If True, print progress messages during estimation.

Returns:

self – Fitted model. Key attributes set after calling fit():

phi_hat_ndarray, shape (n,): Estimated frontier values at each observation.
efficiency_ndarray, shape (n,): JLMS efficiency scores exp(-eta_hat) in (0, 1].
sigma_eta_ndarray, shape (n,): Estimated inefficiency std dev at each observation.
r1_, r2_, r3_ndarray, shape (n,): Estimated first, second, and third conditional moments.
h_ndarray, shape (n,): Hat-matrix leverages for r1 regression (used in CI).

Return type:

SW2023Model

summary()[source]: Print a summary of estimation results.

confint_asymptotic(alpha=0.05)[source]

Asymptotic normal confidence intervals (SW 2023 CLT + delta method).

SW(2023) Eq.(3.8) CLT:: (nh^{d-1})^{1/2} (r̂_j(z) − r_j(z)) →^L N(0, s²_j(z))
Hat-matrix based variance estimation:: AVar(r̂_1(z_i)) ≈ h_ii(r1) × r̂_2(z_i) AVar(r̂_3(z_i)) ≈ h_ii(r3) × V̂ar(ε³ | Z=z_i)
φ̂(z) = r̂_1(z) + ||d|| √(2/π) σ̂_η(z) delta method:: ∂φ/∂r1 = 1 ∂φ/∂r3 = −||d|| √(2/π) / (3 × A3_PLUS × σ̂²_η) [when r3 ≤ 0]
Note: Asymptotic CI is valid when n is sufficiently large.: For small samples, bootstrap_sw() CI is more appropriate.

Parameters:: alpha (significance level (1-alpha confidence interval))
Returns:: phi_hat_ci : (n, 2) r1_ci : (n, 2) r3_ci : (n, 2) se_phi : (n,) standard error se_r1 : (n,) se_r3 : (n,) alpha : float
Return type:: ConfintResult

predict_at(Z_eval, U_eval=None)[source]

Predict at new evaluation points Z_eval using the fitted model.

Used during bootstrap CI computation: fit on bootstrap sample, then evaluate at the original Z points.

Parameters:

Z_eval ((m, d) evaluation points (rotated coordinates, same as training Z))
U_eval ((m,) observed directional distances (if provided, efficiency is also computed))

Returns:

dict

Return type:

phi_hat, sigma_eta, sigma_eps, mu_eta, [efficiency, eta_hat]

results_dataframe()[source]: Return estimation results as a DataFrame.

plot_efficiency(bins=30, figsize=(7, 4), ax=None)[source]

Histogram of efficiency scores with mean and median lines.

Parameters:

bins (int, default 30)
figsize (tuple, default (7, 4))
ax (matplotlib Axes or None) – If provided, plot into this axes; otherwise a new figure is created.

Returns:

fig, ax

Return type:

matplotlib Figure and Axes

plot_frontier(dim=0, figsize=(7, 5), ax=None)[source]

Scatter plot of U vs Z[dim] with the estimated frontier phi_hat(Z).

Points are colour-coded by efficiency score (red = inefficient, green = efficient).

Parameters:

dim (int, default 0) – Index of the Z dimension to use as the x-axis.
figsize (tuple, default (7, 5))
ax (matplotlib Axes or None)

Returns:

fig, ax

Return type:

matplotlib Figure and Axes

plot_diagnostics(figsize=(12, 9))[source]

Comprehensive diagnostic panel (2×2 layout).

Panels

(top-left) Efficiency distribution (top-right) Efficiency ranking (caterpillar) (bottom-left) Residual distribution (eps = U − r̂₁) (bottom-right) Wrong-skewness diagnostic (r̂₃ sign)

param figsize:
type figsize:: tuple, default (12, 9)
returns:: fig
rtype:: matplotlib Figure

bootstrap(B=200, alpha=0.05, bandwidth_method='silverman', seed=None, verbose=True)[source]

Pairs bootstrap confidence intervals for this fitted model.

Resamples (X, Y) with replacement, re-estimates the model using the same fixed direction vector and preprocessing, then evaluates each bootstrap estimate at the original data points. Yields per-observation CIs for phi_hat(z), sigma_eta(z), and individual efficiency, as well as a CI for the mean efficiency.

Parameters:

B (int, default 200) – Number of bootstrap draws.
alpha (float, default 0.05) – Significance level; produces (1 - alpha) confidence intervals.
bandwidth_method ({'silverman', 'loocv'}, default 'silverman') – Bandwidth selection for each bootstrap replicate. ‘silverman’ is strongly recommended here — LOO-CV per replicate is very slow.
seed (int or None) – Random seed for reproducibility.
verbose (bool, default True) – Print progress messages.

Returns:

Object with attributes phi_hat_ci, eff_mean_ci, eff_individual_ci, sigma_eta_ci, and a summary() method.

Return type:

BootstrapResult