{'fraction_non_alphanumeric': 0.013793103448275862, 'fraction_numerical': 0.0, 'mean_word_length': 5.177304964539007, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Stratified sampling in the simplex with applications to estimating distributions

Many problems in statistics and operations research reduce to the evaluation of the distribution of a random variable, called the response, known to be a complicated function of a number, d say, of independent uniform variables. Monte Carlo estimation is often used for this purpose if the distribution is analytically intractable. Often the response possesses symmetry properties with respect to its arguments. It is then possible to restrict sampling to simplex regions of the sample space. This can be easily combined with stratified sampling to give variance reduction of order O(d) compared with normal stratified sampling. The theory of such methods is discussed and a simple stratified sampling scheme is applied to two examples giving a two to five fold reduction in the variance.