Weak limit theorems for univariate k-mean clustering under a nonregular condition

A set of n points sampled from a common distribution F, is partitioned into k ≥ 2 groups that maximize the between group sum of squares. The asymptotic normality of the vector of probabilities of lying in each group and the vector of group means is known under the condition that a particular function, depending on F, has a nonsingular Hessian. This condition is not met by the double exponential distribution with k = 2. However, in this case it is shown that limiting distribution for the probability is b sign(W) √|W| and for the two means it is a_i sign(W) √|W|, where W ∼ N(0,1) and b, a₁, and a₂ are constants. The rate of convergence in n ^{1 4} and the joint asymptotic disstribution for the two means is concentrated on the line x = y. A general theory is then developed for distributions with singular Hessians. It is shown that the projection of the probability vector onto some sequence of subspaces will have normal limiting distribution and that the rate of convergence is n ^{1 2}. Further, a sufficient condition is given to assure that the probability vector and vector of group means have limiting distributions, and the possible limiting distributions under this condition are characterized. The convergence is slower than n ^{1 2}.