{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T08:08:23Z","timestamp":1776845303138,"version":"3.51.2"},"reference-count":40,"publisher":"American Mathematical Society (AMS)","issue":"261","license":[{"start":{"date-parts":[[2008,9,12]],"date-time":"2008-09-12T00:00:00Z","timestamp":1221177600000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    When testing that a sample of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n\">\n                        <mml:semantics>\n                          <mml:mi>n<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    points in the unit hypercube\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-bracket 0 comma 1 right-bracket Superscript d\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mrow>\n                              <mml:mo>[<\/mml:mo>\n                              <mml:mn>0<\/mml:mn>\n                              <mml:mo>,<\/mml:mo>\n                              <mml:mn>1<\/mml:mn>\n                              <mml:mo>]<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi>d<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">\\left [0,1\\right ]^{d}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    comes from a uniform distribution, the Kolmogorov\u2013Smirnov and the Cram\u00e9r\u2013von Mises statistics are simple and well-known procedures. To encompass these measures of uniformity, Hickernell introduced the so-called\n                    <italic>generalized<\/italic>\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script upper L Superscript p\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">L<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi>p<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathcal {L}^{p}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -\n                    <italic>discrepancies<\/italic>\n                    . These discrepancies can be used in numerical integration through Monte Carlo and quasi\u2013Monte Carlo methods, design of experiments, uniformity testing and goodness-of-fit tests. The aim of this paper is to derive the statistical asymptotic properties of these statistics under Monte Carlo sampling. In particular, we show that, under the hypothesis of uniformity of the sample of points, the asymptotic distribution is a complex stochastic integral with respect to a pinned Brownian sheet. On the other hand, if the points are not uniformly distributed, then the asymptotic distribution is Gaussian.\n                  <\/p>","DOI":"10.1090\/s0025-5718-07-01839-x","type":"journal-article","created":{"date-parts":[[2007,10,29]],"date-time":"2007-10-29T06:30:33Z","timestamp":1193639433000},"page":"421-446","source":"Crossref","is-referenced-by-count":3,"title":["Statistical properties of generalized discrepancies"],"prefix":"10.1090","volume":"77","author":[{"given":"Christine","family":"Choirat","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Raffaello","family":"Seri","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2007,9,12]]},"reference":[{"issue":"21","key":"1","doi-asserted-by":"publisher","first-page":"2477","DOI":"10.1080\/03610928308828614","article-title":"On the asymptotic distribution of Cram\u00e9r-von Mises one-sample test statistics under an alternative","volume":"12","author":"Angus, John E.","year":"1983","journal-title":"Comm. 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