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1 理論 (Journal of Economcs and Management 5 Vol. No 參數量羅 * 數率數羅說例 ( 暴露量暴露罹率 ( 路 (3 (4 立行量 (5 臨數數數年車率數量羅數數聯不率來參數參數數數不數量羅數參數 JEL 類 C5 羅來立數率量羅 (unvarate logstc regresson 梁 (3 羅 4 年年年 7 * 精理精 ( FCU-RD-89-5

2 4 路路梁 (3 兩了類羅益 ( 羅車金數行立 (997 利羅度 (cross-valdaton methodology (tenure (ncome 率 (994 羅臨 (997 利量數 975 量數李 98 年度數羅 (tenure (ncome 兩數 (bnary resonse varable 率利 ( 年 ( 兩數度兩來數量數 (bvarate bnary resonse varable 數 (ared bnary varable Dagenas (975 車行參數立數 (ndeendent varable 數數數 (deendent varable 數 ( 997 年 45 年 ( ( ( ( 數 (Dagenas 975 類參數量羅

3 參數量羅 4 Lstz et al. (99 療率羅數 (ared logt functon 異類來數 (elanatory varable 數異數兩數 (beteen-grou varable 年 8 說率數 (thn-grou varable 率易不量羅數數不參數 (threshold arameter 什量異什量參數量羅 (bvarate logstc regresson 參數 (mamum lkelhood estmate 流利數 (arametrc bootstrang methodology 論利行論論. 參數量羅令 Y ( Y Y... n 數 Y 數 (bnary varable 不 3 ( 理數 Y Y 說年 8 3 ( 數錄了數

4 4 類數年度狀類 Z ( q 來 Y 錄來數量... 數說 k k... q k 數數數率說年車率率 ( 數率 (success robablty logt (... ( I ( I (... 參數... 數論 ( 裡數 (ndcator functon I 數數數 I 數數數 ( 數率 ( 羅數 (logt dfference functon 異 logt ( q q... ( λ I q q ( λ I λ q λ q (... q 參數 λ... q 數了 ( ( 數 Y Y 羅

5 參數量羅 43 數說 logt( logt( logt logt logt (3 數 Y Y 率數 Y (3 羅數數 Y (3 羅數更 (3 率 (margnal robablty 數數 e( logt e( logt 率數數 e( logt logt. e( logt logt Y Y 聯 (assocaton 率 (odds rato (relatve rsk 數 (correlaton coeffcent 聯率 (ont robablty 來量度度讀不參 Lstz et al. (99 令 Y Y 聯率 : P( Y P( Y Y Y Z Z P( Y P( Y Y Y Z Z. Bahadur (96 量數率數

6 44 率數率 n 率參數數數 (log lkelhood functon l( λ y Z ;... n n ( y logt( ylogt( y y (... (... q (... λ ( λ λ... λq (... n y ( y y Y ( Y Y 參數數數數 (3 不數 (smle algorthm. 利參數不數 Nelder and Mead (965 利 Ulm (99 量羅量狀 logt ( I logt ( λ I λ. 說兩數數數聯不率來

7 參數量羅 45 數數說 ( ( m mn( ma( Ulm (99 切 m 切 ( ( ( m m ( (... m m 理切 m 切 ( k ( m ( m ( ( k k... m ( ( m mn( λ ma( m m m 令 ( λ k ( k 參數 ( k ( ( ( k Z ( ( I ( ( I ( k 數數數 - 數數 ( k ( k ( k ( k ( k ( k ( k 令 ϑ ( ( λ k ( 參數 m k m 六 m m 數數 ( k 參數 arg ma ( k ( ( k l ( ϑ λ ( k ( k ( ( k l l( ϑ λ

8 46.3 利不利 (Fsher nformaton matr 參數數參數 ( ( 兩量率利例說說 (3 ( ( e( ( ( e( ( e( ( e( λ λ λ λ I I I I I I λ λ 數 Y Y 率說不 ^ (hat Lstz et al. (99 率兩量率 ( Y Y P ( ( ( [ ] V ( [ ] ( 4 v v 狀率 ( Y P Y ( Y P Y

9 參數量羅 47 ( Y P Y 了量數率 (multnomal dstrbuton 數率利參數 ( λ b ( ϑb ϑ b ϑb ϑ3b ϑ4b ϑ5b ϑ6b ϑ7b 說 BOOT (BOOT 數 BOOT BOOT ( b BOOT ϑ kb ϑ k BOOT ϑk b ϑkb / BOOT k... 7 k 數參數 3 利參數 (reflecton (eanson (contracton 數例了數 3. Sendley et al. (96Nelder and Mead (965 數 K 數數數 y f (z z K 數量

10 48 z 數 f (z 令 P P... PK z K (smle 不 z f (z 令 y k f ( P k y h ma k ( yk y l mnk ( yk k... K K P k k h Pk /( K ( 令 Ph P k f ( P h yh P h P P ( A P A P h A 數 f ( P y 兩 (a 若 y < yl 行 (b 若 y > y h y > yh 若立行若 y < yh P P h 行 ( P P B Ph ( B P B 數 f ( P y 兩 (a 若 y < yh P P h (b 若 y < yh 不立了 P l ( f ( P l yl P k ( P k Pl / ( P P P C P ( C P C 數 f ( P y 兩 (a 若 y < yl P P h

11 參數量羅 49 (b 若 y < yl 不立 P P h K 令 y k yk /( K ε K 數 k ( yk y / K ε 度 P l 流讀參 (3 錄 3 讀 (Nelder and Mead 了論率 logt ( ( ( I I logt. 讀. 兩羅數不不數 ( 量度 69 ( 不率說.45 女.55 數利亂數說不不狀狀數 7 9 ( 69

12 5 Z ( Z ( Z3 ( Z 4 ( Z 5 ( ( ( Z 量 Ulm (99 3 數度 ( mg/m 暴露 ( years/ 不行度. 數數 ε. ( 數了理來數數 3.3 不數 (A 數 (B 數 (C 參數參數 A.5 來 A.75 降利數數數數數降 5 6 兩 6 都讀數不數 BOOT

13 參數量羅 5 BOOT BOOT 數不 BOOT 4 數數數來數不不理論數讀行料數數數力說路路兩 (smultaneously 行 Lstz et al. (99 量羅

14 5 不數 (A 數 (B 數 (C A B C 數數數 3 數

15 參數量羅 53 利數參數 (A.75 B.5 C.5 BOOT 參 (975 讀立論李 (98 女度理 33-7 (3 參數羅數羅數量羅精論梁 (3 路羅理 (997 羅度 ( 羅行 5-9 (994 臨 Bahadur R. R. (96 A Reresentaton of the Jont Dstrbuton of Resonses to N Dchotomous Items n Studes n Item Analyss and Predcton Solomon H. (ed. Stanford Unversty Press Stanford Calf Dagenas M. G. (969 A Threshold Regresson Model Econometrca Dagenas M. G. (975 Alcaton of a Threshold Regresson Model to Household Purchases of Automobles The Reve of Economcs and Statstcs

16 54 De Janos P. E. (956 Factors Influencng the Demand for Ne Automobles: A Cross-Secton Analyss Ph.D. dssertaton Unversty of Mchgan Ann Arbor. Goldberger A. S. and M. L. Lee (96 Toard a Mcroanalytc Model of the Household Sector Amercan Economc Reve Lstz S. R. N. M. Lard and D. P. Harrngton (99 Mamum Lkelhood Regresson Methods for Pared Bnary Data Statstcs n Medcne Nelder J. A. and R. Mead (965 A Smle Method for Functon Mnmzaton Comuter Journal Sendley W. G. R. Het and F. R. Hmsorth (96 Sequental Alcaton of Smle Desgns n Otmsaton and Evolutonary Oeraton Technometrcs Ulm K. (99 Statstcal Method for Assessng a Threshold n Edemologcal Studes Statstcs n Medcne

17 參數量羅 55 Mamum Lkelhood Estmaton of the Bvarate Logstc Regresson th Threshold Parameters Jungn Wu Deartment of Statstcs Feng Cha Unversty Chung Wen Chang Deartment of Statstcs Feng Cha Unversty In dscussng the relatonsh beteen the successful robablty of a bnary random varable and a certan set of elanatory varables the logstc regresson s oular n varous felds esecally n medcne researches. For eamle ( usng remedy methods to elan the robablty of recovery from some dsease; ( buldng a redcton model for the e-commence customer buyng behavor; (3 buldng a model to decde the cut-ont of the lfe nsurance sales hrng; (4 buldng a credt score functon; or (5 decdng the threshold arameter for the etreme human nure. Sometmes a varable must eceed a certan value to have nfluence on the resonse varable. For eamle the ncome of a household must eceed some value to nfluence the successful robablty. We call ths value threshold. On the other hand t s ossble to observe ared bnary varables n the busness study. In ths artcle e roose a bvarate logstc regresson model and also consder varables that may have threshold arameter. The assocaton beteen the ar bnary varables s consdered as a constant odds rato for demonstraton. We used mamum lkelhood methodology to fnd arameter estmates va a modfed smle algorthm and suggested a arametrc bootstrang method to estmate the standard devaton of the mamum lkelhood estmator. Keyords: bvarate logstc regresson ared bnary varable threshold arameter smle algorthm JEL Classfcaton: C5

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