F-Distribution

Study guide: Exam MAS-I

The F-Distribution F(n, m) with n numerator and m denominator degrees of freedom is the distribution of the ratio of two independent chi-squared variables, each divided by its degrees of freedom. It is the sampling distribution for comparing two variances and for testing groups of coefficients in linear models.

F = \frac{U/n}{V/m} \sim F(n, m), \quad U \sim \chi^2(n),\ V \sim \chi^2(m)

\begin{aligned} f(x) &= \frac{\Gamma\!\left(\frac{n + m}{2}\right)}{\Gamma\!\left(\frac{n}{2}\right)\Gamma\!\left(\frac{m}{2}\right)}\,\frac{n}{m} \\ &\quad \times \frac{\left(\frac{n}{m}x\right)^{n/2 - 1}}{\left(1 + \frac{n}{m}x\right)^{(n + m)/2}}, \quad x > 0 \end{aligned}

E[F] = \frac{m}{m - 2}, \quad m > 2

\text{Var}(F) = 2\left(\frac{m}{m - 2}\right)^2 \frac{n + m - 2}{n(m - 4)}, \quad m > 4

• U and V are independent. The mean depends only on the denominator degrees of freedom and is close to 1 when m is large.
• Reciprocal: 1/F \sim F(m, n). A lower-tail critical value is therefore the reciprocal of an upper-tail one with the degrees of freedom swapped: the value below which F(n, m) falls with probability \alpha is 1/c, where c is the upper-\alpha point of F(m, n).
• Link to t: if T \sim t(m) then T^2 \sim F(1, m) — see Student's t-Distribution.
• Noncentral F: if U is instead noncentral chi-squared, the ratio has the noncentral F-distribution, whose mean is larger than that of the central F(n, m).
• Two normal variances: for independent normal samples of sizes n_X and n_Y, \dfrac{S_X^2/\sigma_X^2}{S_Y^2/\sigma_Y^2} \sim F(n_X - 1,\ n_Y - 1), using the sample variances.
• Nested linear models: dropping q of the p predictors from a linear regression fitted to n observations raises the residual sum of squares from \text{RSS} to \text{RSS}_0; with normal errors and the q coefficients truly zero, \dfrac{(\text{RSS}_0 - \text{RSS})/q}{\text{RSS}/(n - p - 1)} \sim F(q,\ n - p - 1). With q = p this is the overall F-statistic, and ANOVA tables test group means the same way.
• GLMs: to test k added parameters, \dfrac{D_S - D_B}{k\,\hat{\phi}_B} is compared with F(k,\ n - p_B), where D_S and D_B are the unscaled deviances of the small and big models, \hat{\phi}_B is the big model's estimated dispersion parameter and p_B its number of parameters.
• The CAS MAS-I table gives upper-tail critical values for areas 0.20,\ 0.10,\ 0.05,\ 0.02,\ 0.01, numerator df 1–12, 14, 16, 20, 24, 30, 40, 50, 75, 100, 200, 500, \infty, and denominator df 1–26. Find the denominator row, the upper-tail area, then the numerator column.

Can Two Predictors Be Dropped? A linear model for claim severity with p = 5 predictors is fitted to n = 30 observations, with \text{RSS} = 120. Refitting without two of the predictors gives \text{RSS}_0 = 150. Test at the 5% level whether both dropped coefficients are zero.

Answer

\begin{align*}
F &= \frac{(150 - 120)/2}{120/(30 - 5 - 1)} \\
&= \frac{15}{5} \\
&= 3.0
\end{align*}

Compare with F(2, 24). In the table's denominator-24 rows, numerator column 2, the upper 5% point is 3.403 and the upper 10% point is 2.538. Since 2.538 < 3.0 < 3.403, the p-value is between 0.05 and 0.10: do not reject at 5%. The data do not show that the two predictors are needed.

Comparing the Volatility of Two Territories Normal claim amounts are sampled from two territories: territory A has n = 11 and s_A^2 = 2.4; territory B has n = 9 and s_B^2 = 0.8. Test H_0: \sigma_A^2 = \sigma_B^2 against a two-sided alternative at the 10% level.

Answer Under H_0, s_A^2/s_B^2 \sim F(10, 8), and the observed ratio is 2.4/0.8 = 3.0. A two-sided 10% test puts 5% in each tail.

• Upper 5% point of F(10, 8), from the denominator-8 rows, numerator column 10: 3.347.
• Lower 5% point: the reciprocal of the upper 5% point of F(8, 10), which is 3.072 (denominator-10 rows, numerator column 8), so 1/3.072 = 0.326.

Since 0.326 < 3.0 < 3.347, do not reject H_0 at 10%. The ratio is large, but with samples this small an F-ratio of 3 is not unusual.