{"id":3819,"date":"2020-04-22T22:26:27","date_gmt":"2020-04-22T20:26:27","guid":{"rendered":"https:\/\/myoceane.fr\/?p=3819"},"modified":"2022-09-06T11:40:25","modified_gmt":"2022-09-06T09:40:25","slug":"chi-square-test","status":"publish","type":"post","link":"https:\/\/myoceane.fr\/index.php\/chi-square-test\/","title":{"rendered":"[\u7d71\u8a08] Chi-Square Test \u5361\u65b9\u6aa2\u5b9a"},"content":{"rendered":"
<\/div>\n\n

Chi-Square Test \u5361\u65b9\u6aa2\u5b9a\u6216\u7a31\u70ba Chi-Squared Goodness of Fit Test \u4e3b\u8981\u662f\u900f\u904e\u4e00\u500b\u6a23\u672c\u53d6\u6a23\u4f86\u6aa2\u6e2c\u6211\u5011\u662f\u5426\u6709\u8db3\u5920\u7684\u4fe1\u5fc3\u53bb\u4f7f\u7528 Multinomial \u5206\u4f48\u63cf\u8ff0\u9019\u500b\u53d6\u6a23\u76ee\u6a19\u7684\u96a8\u6a5f\u884c\u70ba\uff1f\u672c\u7bc7\u7684\u6578\u5b78\u63a8\u5c0e\u4e3b\u8981\u53c3\u8003 Michael J. Evans and Jeffrey S.Rosenthal \u8457\u4f5c\u7684 Probability and Statistic, The Science of Uncertainty<\/a> \u4e00\u66f8\u3002<\/p>\n\n\n\n\n\u76ee\u6a19\u8b8a\u6578\u5047\u8a2d\u7e3d\u5171\u6709 \"k\" \u500b\u5206\u985e\uff0c\u5047\u8a2d\u505a\u53d6\u6a23\u7684\u7d50\u679c\u5728 \"k\" \u500b\u5206\u985e\u4e2d\u5404\u81ea\u5171\u6709 \"X=(X_1,\cdots,X_k)\" \u7684\u6a23\u672c\u6578\uff0c\u4e14\u7e3d\u5171\u53d6\u6a23\u6a23\u672c\u6578\u70ba \"\sum_kX_k = n\"\uff0c\u5982\u679c \"(X_1,\cdots,X_k) \sim\" Multinomial\"(n, p_1, \cdots, p_k)\" \u7684\u8a71\uff0c\u90a3\u9ebc \"X^2\" \u6703\u5448\u73fe\u81ea\u7531\u5ea6 \"k-1\" \u7684 \"\chi^2\" \u5206\u4f48\u3002\n\n

(1) <\/span>   <\/span>\"\begin{equation*}<\/p>\n\n\n\n

\u5229\u7528\u6578\u5b78\u5f0f (1) \u7684\u7279\u6027\uff0c\u6211\u5011\u5c31\u53ef\u4ee5\u8a08\u7b97\u51fa\u53d6\u6a23\u7684\u6a23\u672c\u4f4d\u65bc\u6574\u500b\u5206\u4f48\u7684\u54ea\u4e00\u500b\u4f4d\u7f6e\uff0c\u4ee5\u6b64\u6211\u5011\u5c31\u53ef\u4ee5\u5229\u7528\u7d71\u8a08\u7684\u7279\u6027\u4f86\u63a8\u7ffb\u6211\u5011\u60f3\u8981\u6aa2\u5b9a\u7684\u5047\u8a2d\u8207\u5426\uff01\u4ee5\u4e0b\u6211\u5011\u60f3\u8981\u7d00\u9304\u6578\u5b78\u7684\u63a8\u5c0e\u904e\u7a0b\uff01\u63a8\u5c0e\u53c3\u8003\u7684\u662f\u7531 A. W. van der Vaart, \u8457\u4f5c\u7684 Asymptotic Statistics \u4e00\u66f8<\/a>\u4e2d\u7684 17.2 \u5b9a\u7406\uff1a<\/p>\n\n\n\n\n\u9996\u5148\u6211\u5011\u5c07 \"X\" \u5411\u91cf\u62c6\u89e3\u6210 \"Y_1, \cdots, Y_n\" \u500b\u55ae\u4f4d\u5411\u91cf\uff0c\u4e14 \"\sum^n_{i=1}Y_i = X\"\uff0c\u5247\u95dc\u65bc \"Y_i\" \u9019\u4e00\u500b\u96a8\u6a5f\u5411\u91cf\u7684\u7d71\u8a08\u7279\u6027\u7684\u63cf\u8ff0\u53ef\u4ee5\u7528\u4ee5\u4e0b\u8868\u793a\uff1a\n

(2) <\/span>   <\/span>\"\begin{equation*}<\/p>\n\u6839\u64da\u591a\u8b8a\u6578\u4e2d\u592e\u6975\u9650\u5b9a\u7406 (multivariate central limit theorem)\uff0c\u5247 \"X_n\" \u53ef\u4ee5\u7531 \"Y_i\" \u7684\u7d71\u8a08\u7279\u6027\u63cf\u8ff0\uff1a\n

(3) <\/span>   <\/span>\"\begin{align*}<\/p>\n\n\n\n

\u5957\u7528\u4ee5\u4e0b\u7684\u5b9a\u7406\u4e00\uff0c\u6211\u5011\u53ef\u4ee5\u63a8\u5c0e<\/p>\n\n\n\n\u4ee3\u5165 \"\Sigma = I-\sqrt{p}\sqrt{p}^T\"\uff0c\u7531\u65bc \"\sum_ip_i = 1\"\uff0c\u6240\u4ee5\u77e9\u9663\u4e2d\u6709\u4e00\u884c\u53ef\u4ee5\u662f\u5176\u4ed6\u7684\u7dda\u6027\u7d44\u5408\uff0c\u6240\u4ee5 rank < k\uff0c\u6240\u4ee5 \"\lambda_i\" \u4e2d\u6709\u4e00\u500b\u662f \"0\" \uff0c\u5176\u4ed6 \"k-1\" \u90fd\u662f \"1\"\uff0c\u6240\u4ee5\n

(4) <\/span>   <\/span>\"\begin{equation*}<\/p>\n\n\n\n

\u5b9a\u7406\u4e00\uff1a<\/h5>\n\n\n\n\u5982\u679c \"X\" \u5411\u91cf\u662f \"N_k(0,\Sigma)\" \u7684\u9ad8\u65af\u5206\u4f48\u7684\u8a71\uff0c\u90a3\u9ebc \"\|X\|^2\" \u7684\u5206\u4f48\u8207 \"\sum^{k}_{i=1}\lambda_i Z_i\" \u4e00\u81f4\uff0c\u5176\u4e2d \"\lambda_1,\cdots,\lambda_k\" \u662f \"\Sigma\" \u77e9\u9663\u7684 eigenvalues\uff0c\u800c \"Z_1, \cdots, Z_k\" \u5247\u662f i.i.d \u7684 \"N(0,1)\" \u7684\u9ad8\u65af\u5206\u4f48\u3002\n\n\n\n
\u5b9a\u7406\u4e00\u8b49\u660e\uff1a<\/h5>\n\n\n\n\n\u5047\u8a2d\u5b58\u5728\u4e00\u500b\u6b63\u4ea4\u7684\u77e9\u9663 \"O\"\uff0c\u4e5f\u5c31\u662f\u8aaa \"O^TO=OO^T=I\"\uff0c\u53ef\u4ee5\u4f7f\u5f97 \"O\SigmaO^T = \text{diag}(\lambda_i)\"\uff0c\u90a3\u9ebc \"OX\" \u5411\u91cf\u5c31\u6703\u5448\u73fe \"N_k(0, \text{diag}(\lambda_i))\" \u9019\u4e00\u500b \"k\" \u7dad\u7684\u9ad8\u65af\u591a\u8b8a\u6578\u5206\u4f48\uff0c\u65bc\u662f\u4e4e \"\|X\|^2=\|OX\|^2=\sum_i\lambda_i Z^2_i\"\u3002\n\n\n\n

\u4ee5\u4e0b\u5206\u4eab\u4e00\u500b Chi-Square Test \u7684\u5be6\u4f5c\u7bc4\u4f8b\uff0c\u53ef\u4ee5\u900f\u904e\u53d6\u6a23\u7684\u6578\u64da\u53bb\u6aa2\u6e2c\u6211\u5011\u7684\u6a23\u672c\u662f\u5426\u53ef\u4ee5\u7528 Multinomial \u7684\u5206\u4f48\u4f86\u63cf\u8ff0\uff1f<\/p>\n\n\n\n

\n