{"id":323,"date":"2017-05-09T10:32:27","date_gmt":"2017-05-09T10:32:27","guid":{"rendered":"http:\/\/jsr.isrt.ac.bd\/?post_type=article&p=323"},"modified":"2017-05-09T10:33:09","modified_gmt":"2017-05-09T10:33:09","slug":"note-weighted-bootstrap-approximation-bickel-rosenblatt-statistic","status":"publish","type":"article","link":"http:\/\/jsr.isrt.ac.bd\/article\/note-weighted-bootstrap-approximation-bickel-rosenblatt-statistic\/","title":{"rendered":"A note on the weighted bootstrap approximation of the Bickel-Rosenblatt Statistic"},"content":{"rendered":"

In this article we propose a weighted bootstrap approximation to the distribution
\nof the supremum (over compact sets) of the classical Bickel-Rosenblatt statistic
\n\"|f_n(t) − f(t)|/\sqrt{f(t)}\" as well as its \u201cStudentized\u201d version \"|f_n(t) − f(t)|/\sqrt{f_n(t)}\",
\nwhere \"f_n\" is the usual kernel density estimator of the true density \"f\". Follow-
\ning Horvath et al. (2000), we showed that the proposed weighted bootstrap
\nmethod is consistent (in capturing the true limiting distributions derived by Bickel
\nand Rosenblatt (1973)). Furthermore, simulation results show that the proposed
\nweighted bootstrap has a much better finite-sample performance than the results
\nbased on asymptotic theory. For comparison purposes, we also consider Efron\u2019s
\n(1979) original bootstrap.<\/p>\n

 <\/p>\n

44n2_2<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

In this article we propose a weighted bootstrap approximation to the distribution of the supremum (over compact sets) of the classical Bickel-Rosenblatt statistic as well as its \u201cStudentized\u201d version , where is the usual kernel density estimator of the true density . Follow- ing Horvath et al. (2000), we showed that the proposed weighted bootstrap […]<\/p>\n","protected":false},"author":2,"featured_media":0,"menu_order":0,"comment_status":"closed","ping_status":"open","template":"","format":"standard","meta":{"_mi_skip_tracking":false,"_exactmetrics_sitenote_active":false,"_exactmetrics_sitenote_note":"","_exactmetrics_sitenote_category":0,"footnotes":""},"issuem_issue":[10],"issuem_issue_categories":[],"issuem_issue_tags":[],"yoast_head":"\nA note on the weighted bootstrap approximation of the Bickel-Rosenblatt Statistic - JSR<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/jsr.isrt.ac.bd\/article\/note-weighted-bootstrap-approximation-bickel-rosenblatt-statistic\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"A note on the weighted bootstrap approximation of the Bickel-Rosenblatt Statistic - JSR\" \/>\n<meta property=\"og:description\" content=\"In this article we propose a weighted bootstrap approximation to the distribution of the supremum (over compact sets) of the classical Bickel-Rosenblatt statistic as well as its \u201cStudentized\u201d version , where is the usual kernel density estimator of the true density . Follow- ing Horvath et al. (2000), we showed that the proposed weighted bootstrap […]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/jsr.isrt.ac.bd\/article\/note-weighted-bootstrap-approximation-bickel-rosenblatt-statistic\/\" \/>\n<meta property=\"og:site_name\" content=\"JSR\" \/>\n<meta property=\"article:modified_time\" content=\"2017-05-09T10:33:09+00:00\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data1\" content=\"1 minute\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\/\/jsr.isrt.ac.bd\/article\/note-weighted-bootstrap-approximation-bickel-rosenblatt-statistic\/\",\"url\":\"https:\/\/jsr.isrt.ac.bd\/article\/note-weighted-bootstrap-approximation-bickel-rosenblatt-statistic\/\",\"name\":\"A note on the weighted bootstrap approximation of the Bickel-Rosenblatt Statistic - JSR\",\"isPartOf\":{\"@id\":\"https:\/\/jsr.isrt.ac.bd\/#website\"},\"datePublished\":\"2017-05-09T10:32:27+00:00\",\"dateModified\":\"2017-05-09T10:33:09+00:00\",\"breadcrumb\":{\"@id\":\"https:\/\/jsr.isrt.ac.bd\/article\/note-weighted-bootstrap-approximation-bickel-rosenblatt-statistic\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/jsr.isrt.ac.bd\/article\/note-weighted-bootstrap-approximation-bickel-rosenblatt-statistic\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/jsr.isrt.ac.bd\/article\/note-weighted-bootstrap-approximation-bickel-rosenblatt-statistic\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/jsr.isrt.ac.bd\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Articles\",\"item\":\"https:\/\/jsr.isrt.ac.bd\/article\/\"},{\"@type\":\"ListItem\",\"position\":3,\"name\":\"A note on the weighted bootstrap approximation of the Bickel-Rosenblatt Statistic\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/jsr.isrt.ac.bd\/#website\",\"url\":\"https:\/\/jsr.isrt.ac.bd\/\",\"name\":\"JSR\",\"description\":\"Journal of Statistical Research\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/jsr.isrt.ac.bd\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"en-US\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"A note on the weighted bootstrap approximation of the Bickel-Rosenblatt Statistic - JSR","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/jsr.isrt.ac.bd\/article\/note-weighted-bootstrap-approximation-bickel-rosenblatt-statistic\/","og_locale":"en_US","og_type":"article","og_title":"A note on the weighted bootstrap approximation of the Bickel-Rosenblatt Statistic - JSR","og_description":"In this article we propose a weighted bootstrap approximation to the distribution of the supremum (over compact sets) of the classical Bickel-Rosenblatt statistic as well as its \u201cStudentized\u201d version , where is the usual kernel density estimator of the true density . Follow- ing Horvath et al. (2000), we showed that the proposed weighted bootstrap […]","og_url":"https:\/\/jsr.isrt.ac.bd\/article\/note-weighted-bootstrap-approximation-bickel-rosenblatt-statistic\/","og_site_name":"JSR","article_modified_time":"2017-05-09T10:33:09+00:00","twitter_card":"summary_large_image","twitter_misc":{"Est. reading time":"1 minute"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/jsr.isrt.ac.bd\/article\/note-weighted-bootstrap-approximation-bickel-rosenblatt-statistic\/","url":"https:\/\/jsr.isrt.ac.bd\/article\/note-weighted-bootstrap-approximation-bickel-rosenblatt-statistic\/","name":"A note on the weighted bootstrap approximation of the Bickel-Rosenblatt Statistic - JSR","isPartOf":{"@id":"https:\/\/jsr.isrt.ac.bd\/#website"},"datePublished":"2017-05-09T10:32:27+00:00","dateModified":"2017-05-09T10:33:09+00:00","breadcrumb":{"@id":"https:\/\/jsr.isrt.ac.bd\/article\/note-weighted-bootstrap-approximation-bickel-rosenblatt-statistic\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/jsr.isrt.ac.bd\/article\/note-weighted-bootstrap-approximation-bickel-rosenblatt-statistic\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/jsr.isrt.ac.bd\/article\/note-weighted-bootstrap-approximation-bickel-rosenblatt-statistic\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/jsr.isrt.ac.bd\/"},{"@type":"ListItem","position":2,"name":"Articles","item":"https:\/\/jsr.isrt.ac.bd\/article\/"},{"@type":"ListItem","position":3,"name":"A note on the weighted bootstrap approximation of the Bickel-Rosenblatt Statistic"}]},{"@type":"WebSite","@id":"https:\/\/jsr.isrt.ac.bd\/#website","url":"https:\/\/jsr.isrt.ac.bd\/","name":"JSR","description":"Journal of Statistical Research","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/jsr.isrt.ac.bd\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-US"}]}},"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"http:\/\/jsr.isrt.ac.bd\/wp-json\/wp\/v2\/article\/323"}],"collection":[{"href":"http:\/\/jsr.isrt.ac.bd\/wp-json\/wp\/v2\/article"}],"about":[{"href":"http:\/\/jsr.isrt.ac.bd\/wp-json\/wp\/v2\/types\/article"}],"author":[{"embeddable":true,"href":"http:\/\/jsr.isrt.ac.bd\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"http:\/\/jsr.isrt.ac.bd\/wp-json\/wp\/v2\/comments?post=323"}],"version-history":[{"count":0,"href":"http:\/\/jsr.isrt.ac.bd\/wp-json\/wp\/v2\/article\/323\/revisions"}],"wp:attachment":[{"href":"http:\/\/jsr.isrt.ac.bd\/wp-json\/wp\/v2\/media?parent=323"}],"wp:term":[{"taxonomy":"issuem_issue","embeddable":true,"href":"http:\/\/jsr.isrt.ac.bd\/wp-json\/wp\/v2\/issuem_issue?post=323"},{"taxonomy":"issuem_issue_categories","embeddable":true,"href":"http:\/\/jsr.isrt.ac.bd\/wp-json\/wp\/v2\/issuem_issue_categories?post=323"},{"taxonomy":"issuem_issue_tags","embeddable":true,"href":"http:\/\/jsr.isrt.ac.bd\/wp-json\/wp\/v2\/issuem_issue_tags?post=323"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}