Nonparametric Statistics for Exam MAS-I

Why Nonparametric Methods Matter

Nonparametric statistical tests make minimal assumptions about the underlying distribution of the data. Unlike parametric tests (such as the t-test or F-test), which assume normality or other specific distributional forms, nonparametric tests rely only on the ranks of observations or the signs of differences. This makes them robust to outliers, applicable to ordinal data, and valid even when the sample size is too small to verify normality assumptions.

In actuarial work, claim data is often heavily skewed and may contain extreme outliers. A single catastrophic claim can dramatically affect the sample mean and invalidate t-test assumptions. Nonparametric methods provide reliable inference in these situations because they are not influenced by the magnitude of extreme observations, only by their relative ranking.

The Sign Test

The sign test is the simplest nonparametric test for paired data or one-sample location problems. For paired observations (X_i, Y_i), compute the differences D_i = X_i - Y_i and count the number of positive differences (S+). Under the null hypothesis that the median difference is zero, S+ follows a Binomial(n, 0.5) distribution, where n is the number of non-zero differences. Ties (zero differences) are excluded from the analysis.

For example, suppose you compare claim processing times before and after a system upgrade for 12 adjusters. If 9 of the 12 show improvement (positive difference), S+ = 9. Under H0, S+ ~ Binomial(12, 0.5), and P(S+ >= 9) = P(9) + P(10) + P(11) + P(12) = 220/4096 + 66/4096 + 12/4096 + 1/4096 = 299/4096 = 0.073. For a one-sided test at the 5% level, you would not reject H0, though the evidence is suggestive.

The Wilcoxon Signed-Rank Test

The Wilcoxon signed-rank test improves on the sign test by incorporating the magnitudes of differences, not just their signs. For paired data, compute |D_i|, rank the absolute differences from smallest to largest, and assign each rank the sign of the original difference. The test statistic W+ is the sum of ranks with positive signs. Under H0 (symmetric distribution of differences centered at zero), E[W+] = n(n+1)/4 and Var(W+) = n(n+1)(2n+1)/24.

Continuing the claim processing example, suppose the absolute differences sorted are 2, 3, 5, 7, 8, 10, 12, 14, 15, 18, 22, 25, with the first three having negative signs (times increased) and the rest positive. The negative ranks sum to 1 + 2 + 3 = 6, and the positive ranks sum to 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 = 72. With n = 12, E[W+] = 12*13/4 = 39. The large value of W+ = 72 relative to 39 suggests a real improvement.

The Mann-Whitney U Test

The Mann-Whitney U test (also called the Wilcoxon rank-sum test) compares two independent samples. Combine all n_1 + n_2 observations, rank them from 1 to n_1 + n_2, and compute the rank sum R_1 for the first group. The U statistic is U_1 = R_1 - n_1*(n_1+1)/2. Under the null hypothesis that the two populations have the same distribution, E[U_1] = n_1*n_2/2 and Var(U_1) = n_1*n_2*(n_1+n_2+1)/12.

For large samples, (U_1 - E[U_1]) / sqrt(Var(U_1)) is approximately standard normal. For small samples, exact tables or software provide critical values. The Mann-Whitney test is especially useful when comparing claim severities between two groups (such as two regions or two policy types) without assuming that claim sizes follow a specific distribution.

The Kruskal-Wallis Test

The Kruskal-Wallis test generalizes the Mann-Whitney test to k > 2 independent groups, serving as the nonparametric analog of one-way ANOVA. Rank all N = n_1 + n_2 + ... + n_k observations together, and compute the rank sum R_i for each group. The test statistic is H = (12 / (N*(N+1))) * sum(R_i^2 / n_i) - 3*(N+1). Under H0 (all groups have the same distribution), H follows approximately a chi-squared distribution with k-1 degrees of freedom for large samples.

The Kruskal-Wallis test is useful in actuarial contexts when comparing claim experience across multiple territories, agent groups, or coverage types without assuming normality. If the test rejects H0, follow-up pairwise Mann-Whitney tests (with a multiple comparison correction) identify which groups differ.

Spearman Rank Correlation and the Runs Test

The Spearman rank correlation coefficient r_s measures the strength of monotonic association between two variables. Replace each observation with its rank and compute the Pearson correlation of the ranks. Equivalently, r_s = 1 - 6*sum(d_i^2) / (n*(n^2 - 1)), where d_i is the difference between the ranks of the i-th pair. Values near +1 indicate strong positive monotonic association, values near -1 indicate strong negative monotonic association, and values near 0 indicate no monotonic relationship.

The runs test checks whether a sequence of binary outcomes is random. A "run" is a consecutive sequence of the same symbol. Too few runs suggests positive autocorrelation (clustering), while too many runs suggests negative autocorrelation (alternation). For n_1 symbols of type A and n_2 symbols of type B, the expected number of runs under randomness is E[R] = 2*n_1*n_2/(n_1+n_2) + 1.

Common Pitfalls

A frequent mistake is applying nonparametric tests when parametric assumptions are met and sample sizes are large. In such cases, parametric tests are more powerful (more likely to detect a real difference). Nonparametric tests trade power for generality, so they are most valuable when distributional assumptions are questionable.

Another error involves handling ties in rank-based tests. When multiple observations have the same value, assign each the average of the ranks they would occupy. For example, if two observations tie for ranks 3 and 4, both receive rank 3.5. Tied ranks require a correction to the variance formula for the test statistic; ignoring this correction can affect p-values.

Study Tips

Exam MAS-I tests when to use nonparametric methods (small samples, non-normality, ordinal data), how to compute test statistics, and interpretation of results. Practice ranking data sets and computing W+, U, and H by hand. Know which test corresponds to which parametric counterpart: sign test replaces one-sample t-test, Wilcoxon signed-rank replaces paired t-test, Mann-Whitney replaces two-sample t-test, and Kruskal-Wallis replaces one-way ANOVA. Be prepared to explain the advantages and limitations of nonparametric methods relative to their parametric counterparts.