torch.distributions.FisherSnedecor

class FisherSnedecor extends Distribution

new FisherSnedecor(df1: number | Tensor, df2: number | Tensor, options?: DistributionOptions)

readonlydf1(Tensor): – First degrees of freedom parameter.
readonlydf2(Tensor): – Second degrees of freedom parameter.
readonlyarg_constraints(unknown)
readonlysupport(unknown)
readonlyhas_rsample(unknown)
readonlymean(Tensor)
readonlymode(Tensor)
readonlyvariance(Tensor)

Fisher-Snedecor (F) distribution: ratio of two independent chi-squared distributions.

Parameterized by two degrees of freedom parameters: df1 (numerator) and df2 (denominator). If U ~ Chi2(df1) and V ~ Chi2(df2) are independent, then F = (U/df1)/(V/df2) ~ F(df1, df2). Fundamental to hypothesis testing, ANOVA, and variance comparison. Essential for:

Analysis of variance (ANOVA) for comparing group means
F-tests for equality of variances across samples
Linear regression hypothesis testing and model comparison
Testing if one variance is significantly larger than another
Quality control and process capability analysis
Comparing nested models (e.g., model reduction testing)
Repeated measures ANOVA and mixed-effects models
Welch's test and other robust variance ratio tests

Geometric Interpretation: F = (variance of numerator model) / (variance of denominator model). Large F values indicate the numerator is more variable than denominator (reject null hypothesis).

Relationship to Chi2: F(df1, df2) = (Chi2(df1)/df1) / (Chi2(df2)/df2). This mixture of two chi-squared distributions (divided by their df) creates the F-distribution.

\begin{aligned} \text{PDF: } f(x) = \frac{\sqrt{\frac{(d_1 x)^{d_1} d_2^{d_2}}{(d_1 x + d_2)^{d_1+d_2}}}}{x \cdot B(d_1/2, d_2/2)} \quad x > 0 \\ \text{Mean: } \mathbb{E}[X] = \frac{d_2}{d_2 - 2} \quad d_2 > 2 \quad \text{(undefined if } d_2 \leq 2\text{)} \\ \text{Mode: } \frac{d_1 - 2}{d_1} \cdot \frac{d_2}{d_2 + 2} \quad d_1 > 2 \\ \text{Variance: } \text{Var}(X) = \frac{2d_2^2(d_1 + d_2 - 2)}{d_1(d_2 - 2)^2(d_2 - 4)} \quad d_2 > 4 \\ \text{Relationship: } F(d_1, d_2) = \frac{\chi^2(d_1)/d_1}{\chi^2(d_2)/d_2} \end{aligned}

Mean undefined if df2 ≤ 2: Only defined for df2 2; df2 ≤ 2 gives undefined mean
Variance undefined if df2 ≤ 4: Variance only exists for df2 4
Always positive support: F-statistic is always positive, ratio of positive values
Right-skewed: Longer right tail; mode mean for reasonable df values
Reciprocal property: If X ~ F(df1, df2), then 1/X ~ F(df2, df1)
Chi2 ratio: F(df1, df2) = (Chi2(df1)/df1) / (Chi2(df2)/df2)
Heavy tail with small df: Smaller degrees of freedom → heavier tails

df1, df2 must be positive: df1 ≤ 0 or df2 ≤ 0 causes errors
Mean undefined for df2 ≤ 2: Distribution has infinite mean in this case
Variance undefined for df2 ≤ 4: Variance doesn't exist unless df2 4
Large F values test model differences: Very large F statistic → strong evidence against null

Examples

// Simple F-test: compare variance of two samples
// Sample 1 has df1=10 degrees of freedom, Sample 2 has df2=15
const f_test = new torch.distributions.FisherSnedecor(10, 15);
const critical_value = f_test.icdf(torch.tensor([0.95]));  // 95% quantile
// If computed F-statistic > critical_value, reject null hypothesis at 5% level
//

// ANOVA: test if group means differ (F-test for model fit)
// Between-groups chi-squared has df1 = num_groups - 1
// Within-groups chi-squared has df2 = num_samples - num_groups
const num_groups = 4;
const num_samples = 50;
const anova_dist = new torch.distributions.FisherSnedecor(
  num_groups - 1,    // df1 = 3
  num_samples - num_groups  // df2 = 46
);
const f_critical = anova_dist.icdf(torch.tensor([0.95]));  // critical F value
// If F_computed > f_critical, reject null (group means differ)
//

// Linear regression: test if adding predictor helps (nested models)
// Full model has more parameters, reduced model has fewer
// F = (SS_reduced - SS_full)/p_diff / (SS_full/(n - p_full))
const p_diff = 3;  // number of parameters added
const n_residual = 100 - 10;  // residual df of full model
const model_test = new torch.distributions.FisherSnedecor(p_diff, n_residual);
const p_value = 1 - model_test.cdf(torch.tensor([f_stat]));  // compute p-value
//

// Batched F-tests with different degrees of freedom
const df1_values = torch.tensor([1, 3, 5, 10]);
const df2_values = torch.tensor([10, 15, 20, 30]);
const dist = new torch.distributions.FisherSnedecor(df1_values, df2_values);
const critical_vals = dist.icdf(torch.tensor([0.95]));  // [4] shaped critical values
const samples = dist.sample();  // [4] shaped F-statistics

// Distribution characteristics: higher df1 → more concentrated; higher df2 → lower mean
const narrow = new torch.distributions.FisherSnedecor(100, 100);  // both large
const narrow_mean = narrow.mean;  // ≈ 1.04 (very concentrated)
const wide = new torch.distributions.FisherSnedecor(1, 1);  // both small
// wide distribution has very heavy tail, mean undefined

// Reciprocal property: F(df1, df2) reciprocal ~ F(df2, df1)
const f1 = new torch.distributions.FisherSnedecor(5, 10);
const f_reciprocal = f1.icdf(torch.tensor([0.95]));
// Should match approximately with F(10, 5) at 5% quantile