{"id":39749,"date":"2025-05-25T22:08:20","date_gmt":"2025-05-25T14:08:20","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/39749.html"},"modified":"2025-05-25T22:08:20","modified_gmt":"2025-05-25T14:08:20","slug":"%e6%a6%82%e7%8e%87%e8%ae%ba%e5%9f%ba%e6%9c%ac%e6%a6%82%e5%bf%b5x%e4%bb%80%e4%b9%88%e6%98%af%e7%bb%8f%e9%aa%8c%e5%88%86%e5%b8%83","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/39749.html","title":{"rendered":"[\u6982\u7387\u8bba\u57fa\u672c\u6982\u5ff5x]\u4ec0\u4e48\u662f\u7ecf\u9a8c\u5206\u5e03"},"content":{"rendered":"

\u5173\u952e\u8bcd： empirical distribution<\/p>\n

\u4e00\u3001\u8bf4\u660e<\/h2>\n

\u63cf\u8ff0\u4e00\u4e2a\u6982\u7387\u6a21\u578b，\u6709\u5bc6\u5ea6\u51fd\u6570\u5f88\u597d\u63cf\u8ff0\u3002\u5982\u679c\u5199\u4e0d\u51fa\u5bc6\u5ea6\u51fd\u6570，\u9000\u800c\u7528\u5206\u5e03\u51fd\u6570\u4e5f\u80fd\u5b8c\u6574\u523b\u753b，\u56e0\u6b64，\u5206\u5e03\u51fd\u6570\u8868\u793a\u6bd4\u5bc6\u5ea6\u51fd\u6570\u8868\u793a\u66f4\u52a0\u5bbd\u6cdb\u666e\u9002\u3002\u672c\u7247\u8bb2\u8ff0\u7ecf\u9a8c\u5206\u5e03\u62df\u5408\u5206\u5e03\u51fd\u6570\u7684\u57fa\u7840\u6982\u5ff5\u3002<\/p>\n

\u4e8c\u3001\u7ecf\u9a8c\u5206\u5e03\u76f4\u89c2\u89e3\u91ca<\/h2>\n

\u5728\u7edf\u8ba1\u5b66\u4e2d，\u7ecf\u9a8c\u5206\u5e03\u51fd\u6570（\u53c8\u79f0\u7ecf\u9a8c\u7d2f\u79ef\u5206\u5e03\u51fd\u6570，eCDF ）\u662f\u4e0e\u6837\u672c\u7684\u7ecf\u9a8c\u6d4b\u5ea6\u76f8\u5173\u7684\u5206\u5e03\u51fd\u6570\u3002\u8be5\u7d2f\u79ef\u5206\u5e03\u51fd\u6570\u662f\u4e00\u4e2a\u9636\u8dc3\u51fd\u6570，\u5728n \u4e2a\u6570\u636e\u70b9\u5904，\u5176\u503c\u5728\u6bcf\u4e2a\u6570\u636e\u70b9\u4e0a\u90fd\u4ee51\/ n \u7684\u5e45\u5ea6\u8dc3\u5347\u3002\u5bf9\u4e8e\u4efb\u4f55\u6307\u5b9a\u7684\u6d4b\u91cf\u53d8\u91cf\u503c，\u5176\u503c\u662f\u8be5\u6d4b\u91cf\u53d8\u91cf\u89c2\u6d4b\u503c\u4e2d\u5c0f\u4e8e\u6216\u7b49\u4e8e\u8be5\u6307\u5b9a\u503c\u7684\u6bd4\u7387\u3002 \"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\" \u7eff\u8272\u66f2\u7ebf\u6e10\u8fd1\u5730\u63a5\u8fd10\u548c1\u7684\u9ad8\u5ea6，\u4f46\u672a\u8fbe\u5230\u8fd9\u4e24\u4e2a\u9ad8\u5ea6，\u5b83\u662f\u6807\u51c6\u6b63\u6001\u5206\u5e03\u7684\u771f\u5b9e\u7d2f\u79ef\u5206\u5e03\u51fd\u6570\u3002 \u7070\u8272\u7ebf\u6bb5\u8868\u793a\u4ece\u8be5\u5206\u5e03\u4e2d\u62bd\u53d6\u7684\u7279\u5b9a\u6837\u672c\u7684\u89c2\u6d4b\u503c，\u84dd\u8272\u9636\u8dc3\u51fd\u6570\u7684\u6c34\u5e73\u9636\u8dc3（\u5305\u62ec\u6bcf\u4e00\u6b65\u7684\u6700\u5de6\u4fa7\u70b9，\u4f46\u4e0d\u5305\u62ec\u6700\u53f3\u4fa7\u70b9）\u6784\u6210\u4e86\u8be5\u6837\u672c\u7684\u7ecf\u9a8c\u5206\u5e03\u51fd\u6570\u3002（\u5355\u51fb\u6b64\u5904\u52a0\u8f7d\u65b0\u56fe\u8868\u3002）<\/p>\n

\u7ecf\u9a8c\u5206\u5e03\u51fd\u6570\u662f\u751f\u6210\u6837\u672c\u70b9\u7684\u7d2f\u79ef\u5206\u5e03\u51fd\u6570\u7684\u4f30\u8ba1\u503c\u3002\u6839\u636e\u683c\u5229\u6587\u79d1-\u574e\u6cf0\u5229\u5b9a\u7406，\u5b83\u4ee5\u6982\u7387 1 \u6536\u655b\u5230\u8be5\u57fa\u7840\u5206\u5e03\u3002\u76ee\u524d\u6709\u8bb8\u591a\u7ed3\u679c\u53ef\u4ee5\u91cf\u5316\u7ecf\u9a8c\u5206\u5e03\u51fd\u6570\u5411\u57fa\u7840\u7d2f\u79ef\u5206\u5e03\u51fd\u6570\u7684 \u6536\u655b\u901f\u5ea6\u3002<\/p>\n

\u4e09\u3001\u7ecf\u9a8c\u5206\u5e03\u89e3\u6790\u8868\u8ff0<\/h2>\n

\u8bbe( X 1 , \u2026, X n )\u4e3a\u72ec\u7acb\u540c\u5206\u5e03\u7684\u5b9e\u968f\u673a\u53d8\u91cf，\u5177\u6709\u5171\u540c\u7684\u7d2f\u79ef\u5206\u5e03\u51fd\u6570 F ( t )\u3002\u5219\u7ecf\u9a8c\u5206\u5e03\u51fd\u6570\u5b9a\u4e49\u4e3a[ \"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\" \u5728\u8fd9\u91cc <\/p>\n

1 <\/p>\n

A <\/p>\n

1_A <\/p>\n

<\/span><\/span>1<\/span><\/span>A<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u662f\u4e8b\u4ef6A\u7684\u6307\u6807,\u5bf9\u4e8e\u7ed9\u5b9a\u7684t，\u6307\u6807 <\/p>\n

1 <\/p>\n

X <\/p>\n

i <\/p>\n

\u2264 <\/p>\n

t <\/p>\n

{\\\\displaystyle \\\\mathbf {1} _{X_{i}\\\\leq t}} <\/p>\n

<\/span><\/span>1<\/span><\/span>X<\/span><\/span>i<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2264<\/span>t<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u662f\u53c2\u6570\u4e3a <\/p>\n

p <\/p>\n

= <\/p>\n

F <\/p>\n

( <\/p>\n

t <\/p>\n

) <\/p>\n

p = F ( t ) <\/p>\n

<\/span><\/span>p<\/span><\/span>=<\/span><\/span><\/span><\/span>F<\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span><\/span>\u7684\u4f2f\u52aa\u5229\u968f\u673a\u53d8\u91cf；\u56e0\u6b64 <\/p>\n

n <\/p>\n

F <\/p>\n

^ <\/p>\n

n <\/p>\n

( <\/p>\n

t <\/p>\n

) <\/p>\n

{\\\\displaystyle n{\\\\widehat {F}}_{n}(t)} <\/p>\n

<\/span><\/span>n<\/span><\/span>F<\/span><\/span><\/span> <\/p>\n

<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span>\u662f\u4e00\u4e2a\u4e8c\u9879\u5f0f\u968f\u673a\u53d8\u91cf，\u5747\u503c\u4e3a <\/p>\n

n <\/p>\n

F <\/p>\n

( <\/p>\n

t <\/p>\n

) <\/p>\n

nF ( t ) <\/p>\n

<\/span><\/span>n<\/span>F<\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span><\/span>，\u65b9\u5dee\u4e3a <\/p>\n

n <\/p>\n

F <\/p>\n

( <\/p>\n

t <\/p>\n

) <\/p>\n

( <\/p>\n

1 <\/p>\n

\u2212 <\/p>\n

F <\/p>\n

( <\/p>\n

t <\/p>\n

) <\/p>\n

) <\/p>\n

nF ( t )(1 \u2212 F ( t )) <\/p>\n

<\/span><\/span>n<\/span>F<\/span>(<\/span>t<\/span>)<\/span>(<\/span>1<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>F<\/span>(<\/span>t<\/span>))<\/span><\/span><\/span><\/span><\/span>\u3002\u8fd9\u610f\u5473\u7740 <\/p>\n

F <\/p>\n

^ <\/p>\n

n <\/p>\n

( <\/p>\n

t <\/p>\n

) <\/p>\n

{\\\\displaystyle {\\\\widehat {F}}_{n}(t)} <\/p>\n

<\/span><\/span><\/span>F<\/span><\/span><\/span> <\/p>\n

<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span>\u662fF ( t )\u7684\u65e0\u504f\u4f30\u8ba1\u91cf\u3002<\/p>\n

\u7136\u800c，\u5728\u4e00\u4e9b\u6559\u79d1\u4e66\u4e2d，\u5176\u5b9a\u4e49\u5982\u4e0b \"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\"<\/p>\n

\u56db\u3001\u6e10\u8fdb\u6027\u8d28<\/h2>\n

\u7531\u4e8e\u5f53 <\/p>\n

n <\/p>\n

n <\/p>\n

<\/span><\/span>n<\/span><\/span><\/span><\/span><\/span>\u8d8b\u5411\u65e0\u7a77\u5927\u65f6，\u6bd4\u7387 <\/p>\n

( <\/p>\n

n <\/p>\n

+ <\/p>\n

1 <\/p>\n

) <\/p>\n

\/ <\/p>\n

n <\/p>\n

( n + 1)\/ n <\/p>\n

<\/span><\/span>(<\/span>n<\/span><\/span>+<\/span><\/span><\/span><\/span>1<\/span>)<\/span>\/<\/span>n<\/span><\/span><\/span><\/span><\/span>\u8d8b\u8fd1\u4e8e 1 ，\u56e0\u6b64\u4e0a\u9762\u7ed9\u51fa\u7684\u4e24\u4e2a\u5b9a\u4e49\u7684\u6e10\u8fd1\u6027\u8d28\u662f\u76f8\u540c\u7684\u3002<\/p>\n

\u6839\u636e\u5f3a\u5927\u6570\u5b9a\u5f8b，\u4f30\u8ba1\u91cf <\/p>\n

F <\/p>\n

^ <\/p>\n

n <\/p>\n

( <\/p>\n

t <\/p>\n

) <\/p>\n

{\\\\displaystyle \\\\scriptstyle {\\\\widehat {F}}_{n}(t)} <\/p>\n

<\/span><\/span><\/span>F<\/span><\/span><\/span> <\/p>\n

<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span>\u5bf9\u4e8et\u7684\u6bcf\u4e00\u4e2a\u503c，\u5f53n \u2009\u2192 \u221e\u65f6\u51e0\u4e4e\u80af\u5b9a\u6536\u655b\u4e8eF ( t )： \"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\" \u56e0\u6b64\u4f30\u8ba1\u91cf <\/p>\n

F <\/p>\n

^ <\/p>\n

n <\/p>\n

( <\/p>\n

t <\/p>\n

) <\/p>\n

{\\\\displaystyle \\\\scriptstyle {\\\\widehat {F}}_{n}(t)} <\/p>\n

<\/span><\/span><\/span>F<\/span><\/span><\/span> <\/p>\n

<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span>\u662f\u4e00\u81f4\u7684\u3002\u8be5\u8868\u8fbe\u5f0f\u65ad\u8a00\u4e86\u7ecf\u9a8c\u5206\u5e03\u51fd\u6570\u9010\u70b9\u6536\u655b\u4e8e\u771f\u5b9e\u7684\u7d2f\u79ef\u5206\u5e03\u51fd\u6570\u3002\u6709\u4e00\u4e2a\u66f4\u5f3a\u7684\u7ed3\u8bba，\u79f0\u4e3a\u683c\u5229\u6587\u79d1-\u574e\u6cf0\u5229\u5b9a\u7406，\u5b83\u6307\u51fa\u6536\u655b\u5b9e\u9645\u4e0a\u5728t\u4e0a\u5747\u5300\u53d1\u751f： \"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\" \u8fd9\u4e2a\u8868\u8fbe\u5f0f\u4e2d\u7684 sup-norm \u79f0\u4e3aKolmogorov\u2013Smirnov \u7edf\u8ba1\u91cf，\u7528\u4e8e\u68c0\u9a8c\u7ecf\u9a8c\u5206\u5e03\u4e4b\u95f4\u7684\u62df\u5408\u4f18\u5ea6 <\/p>\n

F <\/p>\n

^ <\/p>\n

n <\/p>\n

( <\/p>\n

t <\/p>\n

) <\/p>\n

{\\\\displaystyle \\\\scriptstyle {\\\\widehat {F}}_{n}(t)} <\/p>\n

<\/span><\/span><\/span>F<\/span><\/span><\/span> <\/p>\n

<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span>\u4ee5\u53ca\u5047\u8bbe\u7684\u771f\u5b9e\u7d2f\u79ef\u5206\u5e03\u51fd\u6570F\u3002\u8fd9\u91cc\u53ef\u4ee5\u5408\u7406\u5730\u4f7f\u7528\u5176\u4ed6\u8303\u6570\u51fd\u6570\u6765\u4ee3\u66ff sup \u8303\u6570\u3002\u4f8b\u5982，L2\u8303\u6570\u53ef\u4ee5\u63a8\u5bfc\u51faCram\u00e9r\u2013von Mises \u7edf\u8ba1\u91cf\u3002<\/p>\n

\u6e10\u8fd1\u5206\u5e03\u53ef\u4ee5\u7528\u51e0\u79cd\u4e0d\u540c\u7684\u65b9\u5f0f\u8fdb\u4e00\u6b65\u523b\u753b\u3002\u9996\u5148， \u4e2d\u5fc3\u6781\u9650\u5b9a\u7406\u6307\u51fa，\u9010\u70b9 <\/p>\n

F <\/p>\n

^ <\/p>\n

n <\/p>\n

( <\/p>\n

t <\/p>\n

) <\/p>\n

{\\\\displaystyle \\\\scriptstyle {\\\\widehat {F}}_{n}(t)} <\/p>\n

<\/span><\/span><\/span>F<\/span><\/span><\/span> <\/p>\n

<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span>\u5177\u6709\u6e10\u8fd1\u6b63\u6001\u5206\u5e03，\u6807\u51c6 <\/p>\n

n <\/p>\n

{\\\\displaystyle {\\\\sqrt {n}}} <\/p>\n

<\/span><\/span><\/span>n<\/span><\/span><\/span><\/span> <\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u6536\u655b\u901f\u5ea6： \"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\" \u8fd9\u4e2a\u7ed3\u679c\u7531Donkser\u5b9a\u7406\u6269\u5c55\u3002\u8be5\u5b9a\u7406\u65ad\u8a00\u7ecf\u9a8c\u8fc7\u7a0b <\/p>\n

n <\/p>\n

( <\/p>\n

F <\/p>\n

^ <\/p>\n

n <\/p>\n

\u2212 <\/p>\n

F <\/p>\n

) <\/p>\n

{\\\\displaystyle \\\\scriptstyle {\\\\sqrt {n}}({\\\\widehat {F}}_{n}-F)} <\/p>\n

<\/span><\/span><\/span>n<\/span><\/span><\/span><\/span> <\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span><\/span>F<\/span><\/span><\/span> <\/p>\n

<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span>F<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span>，\u88ab\u89c6\u4e3a\u4e00\u4e2a\u51fd\u6570，\u7d22\u5f15\u4e3a <\/p>\n

t <\/p>\n

\u2208 <\/p>\n

R <\/p>\n

{\\\\displaystyle \\\\scriptstyle t\\\\in \\\\mathbb {R} } <\/p>\n

<\/span><\/span>t<\/span>\u2208<\/span>R<\/span><\/span><\/span><\/span><\/span><\/span>，\u5728Skorokhod \u7a7a\u95f4\u4e2d\u5206\u5e03\u6536\u655b <\/p>\n

D <\/p>\n

[ <\/p>\n

\u2212 <\/p>\n

\u221e <\/p>\n

, <\/p>\n

+ <\/p>\n

\u221e <\/p>\n

] <\/p>\n

{\\\\displaystyle \\\\scriptstyle D[-\\\\infty ,+\\\\infty ]} <\/p>\n

<\/span><\/span>D<\/span>[<\/span>\u2212<\/span>\u221e<\/span>,<\/span>+<\/span>\u221e<\/span>]<\/span><\/span><\/span><\/span><\/span><\/span>\u5747\u503c\u96f6\u9ad8\u65af\u8fc7\u7a0b\u683c <\/p>\n

G <\/p>\n

F <\/p>\n

= <\/p>\n

B <\/p>\n

\u2218 <\/p>\n

F <\/p>\n

{\\\\displaystyle \\\\scriptstyle G_{F}=B\\\\circ F} <\/p>\n

<\/span><\/span>G<\/span><\/span>F<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span>B<\/span>\u2218<\/span>F<\/span><\/span><\/span><\/span><\/span><\/span>\u5176\u4e2dB\u662f\u6807\u51c6\u5e03\u6717\u6865\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

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