Estimation and convergence rates in the distributional single index model
IF 1.4
3区 数学
Q2 STATISTICS & PROBABILITY
Fadoua Balabdaoui, Alexander Henzi, Lukas Looser
{"title":"Estimation and convergence rates in the distributional single index model","authors":"Fadoua Balabdaoui, Alexander Henzi, Lukas Looser","doi":"10.1111/stan.12336","DOIUrl":null,"url":null,"abstract":"The distributional single index model is a semiparametric regression model in which the conditional distribution functions <mjx-container aria-label=\"upper P left parenthesis upper Y less than or equals y vertical bar upper X equals x right parenthesis equals upper F 0 left parenthesis theta 0 left parenthesis x right parenthesis comma y right parenthesis\" ctxtmenu_counter=\"0\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow data-semantic-children=\"36,34\" data-semantic-content=\"10\" data-semantic- data-semantic-role=\"equality\" data-semantic-speech=\"upper P left parenthesis upper Y less than or equals y vertical bar upper X equals x right parenthesis equals upper F 0 left parenthesis theta 0 left parenthesis x right parenthesis comma y right parenthesis\" data-semantic-type=\"relseq\"><mjx-mrow data-semantic-children=\"0,27\" data-semantic-content=\"35,0\" data-semantic- data-semantic-parent=\"37\" data-semantic-role=\"simple function\" data-semantic-type=\"appl\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"36\" data-semantic-role=\"simple function\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"36\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\"26\" data-semantic-content=\"1,9\" data-semantic- data-semantic-parent=\"36\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"27\" data-semantic-role=\"open\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\"24,5,25\" data-semantic-content=\"5\" data-semantic- data-semantic-parent=\"27\" data-semantic-role=\"sequence\" data-semantic-type=\"punctuated\"><mjx-mrow data-semantic-children=\"2,4\" data-semantic-content=\"3\" data-semantic- data-semantic-parent=\"26\" data-semantic-role=\"inequality\" data-semantic-type=\"relseq\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"24\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\"relseq,≤\" data-semantic-parent=\"24\" data-semantic-role=\"inequality\" data-semantic-type=\"relation\" rspace=\"5\" space=\"5\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"24\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-mo data-semantic- data-semantic-operator=\"punctuated\" data-semantic-parent=\"26\" data-semantic-role=\"vbar\" data-semantic-type=\"punctuation\" rspace=\"2\" space=\"2\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\"6,8\" data-semantic-content=\"7\" data-semantic- data-semantic-parent=\"26\" data-semantic-role=\"equality\" data-semantic-type=\"relseq\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"25\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\"relseq,=\" data-semantic-parent=\"25\" data-semantic-role=\"equality\" data-semantic-type=\"relation\" rspace=\"5\" space=\"5\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"25\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-mrow><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"27\" data-semantic-role=\"close\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo></mjx-mrow></mjx-mrow><mjx-mo data-semantic- data-semantic-operator=\"relseq,=\" data-semantic-parent=\"37\" data-semantic-role=\"equality\" data-semantic-type=\"relation\" rspace=\"5\" space=\"5\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\"13,32\" data-semantic-content=\"33,11\" data-semantic- data-semantic-parent=\"37\" data-semantic-role=\"simple function\" data-semantic-type=\"appl\"><mjx-msub data-semantic-children=\"11,12\" data-semantic- data-semantic-parent=\"34\" data-semantic-role=\"simple function\" data-semantic-type=\"subscript\"><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"13\" data-semantic-role=\"simple function\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-script style=\"vertical-align: -0.15em; margin-left: -0.106em;\"><mjx-mrow size=\"s\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"13\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn></mjx-mrow></mjx-script></mjx-msub><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"34\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\"31\" data-semantic-content=\"14,23\" data-semantic- data-semantic-parent=\"34\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"32\" data-semantic-role=\"open\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\"30,21,22\" data-semantic-content=\"21\" data-semantic- data-semantic-parent=\"32\" data-semantic-role=\"sequence\" data-semantic-type=\"punctuated\"><mjx-mrow data-semantic-annotation=\"clearspeak:simple\" data-semantic-children=\"17,28\" data-semantic-content=\"29,15\" data-semantic- data-semantic-parent=\"31\" data-semantic-role=\"simple function\" data-semantic-type=\"appl\"><mjx-msub data-semantic-children=\"15,16\" data-semantic- data-semantic-parent=\"30\" data-semantic-role=\"simple function\" data-semantic-type=\"subscript\"><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"17\" data-semantic-role=\"simple function\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-script style=\"vertical-align: -0.15em;\"><mjx-mrow size=\"s\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"17\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn></mjx-mrow></mjx-script></mjx-msub><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"30\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\"19\" data-semantic-content=\"18,20\" data-semantic- data-semantic-parent=\"30\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"28\" data-semantic-role=\"open\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"28\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"28\" data-semantic-role=\"close\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo></mjx-mrow></mjx-mrow><mjx-mo data-semantic- data-semantic-operator=\"punctuated\" data-semantic-parent=\"31\" data-semantic-role=\"comma\" data-semantic-type=\"punctuation\" rspace=\"3\" style=\"margin-left: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"31\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"32\" data-semantic-role=\"close\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo></mjx-mrow></mjx-mrow></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/1db80d93-0b28-4938-bec3-0ca347be8ec3/stan12336-math-0001.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-children=\"36,34\" data-semantic-content=\"10\" data-semantic-role=\"equality\" data-semantic-speech=\"upper P left parenthesis upper Y less than or equals y vertical bar upper X equals x right parenthesis equals upper F 0 left parenthesis theta 0 left parenthesis x right parenthesis comma y right parenthesis\" data-semantic-type=\"relseq\"><mrow data-semantic-=\"\" data-semantic-children=\"0,27\" data-semantic-content=\"35,0\" data-semantic-parent=\"37\" data-semantic-role=\"simple function\" data-semantic-type=\"appl\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-operator=\"appl\" data-semantic-parent=\"36\" data-semantic-role=\"simple function\" data-semantic-type=\"identifier\">P</mi><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"appl\" data-semantic-parent=\"36\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\"></mo><mrow data-semantic-=\"\" data-semantic-children=\"26\" data-semantic-content=\"1,9\" data-semantic-parent=\"36\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"27\" data-semantic-role=\"open\" data-semantic-type=\"fence\" stretchy=\"false\">(</mo><mrow data-semantic-=\"\" data-semantic-children=\"24,5,25\" data-semantic-content=\"5\" data-semantic-parent=\"27\" data-semantic-role=\"sequence\" data-semantic-type=\"punctuated\"><mrow data-semantic-=\"\" data-semantic-children=\"2,4\" data-semantic-content=\"3\" data-semantic-parent=\"26\" data-semantic-role=\"inequality\" data-semantic-type=\"relseq\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"24\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">Y</mi><mo data-semantic-=\"\" data-semantic-operator=\"relseq,≤\" data-semantic-parent=\"24\" data-semantic-role=\"inequality\" data-semantic-type=\"relation\">≤</mo><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"24\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">y</mi></mrow><mo data-semantic-=\"\" data-semantic-operator=\"punctuated\" data-semantic-parent=\"26\" data-semantic-role=\"vbar\" data-semantic-type=\"punctuation\" stretchy=\"false\">|</mo><mrow data-semantic-=\"\" data-semantic-children=\"6,8\" data-semantic-content=\"7\" data-semantic-parent=\"26\" data-semantic-role=\"equality\" data-semantic-type=\"relseq\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"25\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">X</mi><mo data-semantic-=\"\" data-semantic-operator=\"relseq,=\" data-semantic-parent=\"25\" data-semantic-role=\"equality\" data-semantic-type=\"relation\">=</mo><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"25\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">x</mi></mrow></mrow><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"27\" data-semantic-role=\"close\" data-semantic-type=\"fence\" stretchy=\"false\">)</mo></mrow></mrow><mo data-semantic-=\"\" data-semantic-operator=\"relseq,=\" data-semantic-parent=\"37\" data-semantic-role=\"equality\" data-semantic-type=\"relation\">=</mo><mrow data-semantic-=\"\" data-semantic-children=\"13,32\" data-semantic-content=\"33,11\" data-semantic-parent=\"37\" data-semantic-role=\"simple function\" data-semantic-type=\"appl\"><msub data-semantic-=\"\" data-semantic-children=\"11,12\" data-semantic-parent=\"34\" data-semantic-role=\"simple function\" data-semantic-type=\"subscript\"><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-operator=\"appl\" data-semantic-parent=\"13\" data-semantic-role=\"simple function\" data-semantic-type=\"identifier\">F</mi></mrow><mrow><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"13\" data-semantic-role=\"integer\" data-semantic-type=\"number\">0</mn></mrow></msub><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"appl\" data-semantic-parent=\"34\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\"></mo><mrow data-semantic-=\"\" data-semantic-children=\"31\" data-semantic-content=\"14,23\" data-semantic-parent=\"34\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"32\" data-semantic-role=\"open\" data-semantic-type=\"fence\" stretchy=\"false\">(</mo><mrow data-semantic-=\"\" data-semantic-children=\"30,21,22\" data-semantic-content=\"21\" data-semantic-parent=\"32\" data-semantic-role=\"sequence\" data-semantic-type=\"punctuated\"><mrow data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-children=\"17,28\" data-semantic-content=\"29,15\" data-semantic-parent=\"31\" data-semantic-role=\"simple function\" data-semantic-type=\"appl\"><msub data-semantic-=\"\" data-semantic-children=\"15,16\" data-semantic-parent=\"30\" data-semantic-role=\"simple function\" data-semantic-type=\"subscript\"><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-operator=\"appl\" data-semantic-parent=\"17\" data-semantic-role=\"simple function\" data-semantic-type=\"identifier\">θ</mi></mrow><mrow><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"17\" data-semantic-role=\"integer\" data-semantic-type=\"number\">0</mn></mrow></msub><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"appl\" data-semantic-parent=\"30\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\"></mo><mrow data-semantic-=\"\" data-semantic-children=\"19\" data-semantic-content=\"18,20\" data-semantic-parent=\"30\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"28\" data-semantic-role=\"open\" data-semantic-type=\"fence\" stretchy=\"false\">(</mo><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"28\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">x</mi><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"28\" data-semantic-role=\"close\" data-semantic-type=\"fence\" stretchy=\"false\">)</mo></mrow></mrow><mo data-semantic-=\"\" data-semantic-operator=\"punctuated\" data-semantic-parent=\"31\" data-semantic-role=\"comma\" data-semantic-type=\"punctuation\">,</mo><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"31\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">y</mi></mrow><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"32\" data-semantic-role=\"close\" data-semantic-type=\"fence\" stretchy=\"false\">)</mo></mrow></mrow></mrow>$$ P\\left(Y\\le y|X=x\\right)={F}_0\\left({\\theta}_0(x),y\\right) $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> of a real-valued outcome variable <mjx-container aria-label=\"upper Y\" ctxtmenu_counter=\"1\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper Y\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/56a43599-c210-4499-aeea-c63a9e47df43/stan12336-math-0002.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-role=\"latinletter\" data-semantic-speech=\"upper Y\" data-semantic-type=\"identifier\">Y</mi></mrow>$$ Y $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> depend on <mjx-container aria-label=\"d\" ctxtmenu_counter=\"2\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"d\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/c6c44fcd-8d0a-40f6-9b16-5775abbcd573/stan12336-math-0003.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-role=\"latinletter\" data-semantic-speech=\"d\" data-semantic-type=\"identifier\">d</mi></mrow>$$ d $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>-dimensional covariates <mjx-container aria-label=\"upper X\" ctxtmenu_counter=\"3\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper X\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/8c9a209b-7df6-4c89-bd24-630f971e736e/stan12336-math-0004.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-role=\"latinletter\" data-semantic-speech=\"upper X\" data-semantic-type=\"identifier\">X</mi></mrow>$$ X $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> through a univariate, parametric index function <mjx-container aria-label=\"theta 0 left parenthesis x right parenthesis\" ctxtmenu_counter=\"4\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow data-semantic-annotation=\"clearspeak:simple\" data-semantic-children=\"2,6\" data-semantic-content=\"7,0\" data-semantic- data-semantic-role=\"simple function\" data-semantic-speech=\"theta 0 left parenthesis x right parenthesis\" data-semantic-type=\"appl\"><mjx-msub data-semantic-children=\"0,1\" data-semantic- data-semantic-parent=\"8\" data-semantic-role=\"simple function\" data-semantic-type=\"subscript\"><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"2\" data-semantic-role=\"simple function\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-script style=\"vertical-align: -0.15em;\"><mjx-mrow size=\"s\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn></mjx-mrow></mjx-script></mjx-msub><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"8\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\"4\" data-semantic-content=\"3,5\" data-semantic- data-semantic-parent=\"8\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"6\" data-semantic-role=\"open\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"6\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"6\" data-semantic-role=\"close\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo></mjx-mrow></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/a0919671-ab16-495a-8b38-b1a75cf93613/stan12336-math-0005.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-children=\"2,6\" data-semantic-content=\"7,0\" data-semantic-role=\"simple function\" data-semantic-speech=\"theta 0 left parenthesis x right parenthesis\" data-semantic-type=\"appl\"><msub data-semantic-=\"\" data-semantic-children=\"0,1\" data-semantic-parent=\"8\" data-semantic-role=\"simple function\" data-semantic-type=\"subscript\"><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-operator=\"appl\" data-semantic-parent=\"2\" data-semantic-role=\"simple function\" data-semantic-type=\"identifier\">θ</mi></mrow><mrow><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"2\" data-semantic-role=\"integer\" data-semantic-type=\"number\">0</mn></mrow></msub><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"appl\" data-semantic-parent=\"8\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\"></mo><mrow data-semantic-=\"\" data-semantic-children=\"4\" data-semantic-content=\"3,5\" data-semantic-parent=\"8\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"6\" data-semantic-role=\"open\" data-semantic-type=\"fence\" stretchy=\"false\">(</mo><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"6\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">x</mi><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"6\" data-semantic-role=\"close\" data-semantic-type=\"fence\" stretchy=\"false\">)</mo></mrow></mrow>$$ {\\theta}_0(x) $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>, and increase stochastically as <mjx-container aria-label=\"theta 0 left parenthesis x right parenthesis\" ctxtmenu_counter=\"5\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow data-semantic-annotation=\"clearspeak:simple\" data-semantic-children=\"2,6\" data-semantic-content=\"7,0\" data-semantic- data-semantic-role=\"simple function\" data-semantic-speech=\"theta 0 left parenthesis x right parenthesis\" data-semantic-type=\"appl\"><mjx-msub data-semantic-children=\"0,1\" data-semantic- data-semantic-parent=\"8\" data-semantic-role=\"simple function\" data-semantic-type=\"subscript\"><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"2\" data-semantic-role=\"simple function\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-script style=\"vertical-align: -0.15em;\"><mjx-mrow size=\"s\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn></mjx-mrow></mjx-script></mjx-msub><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"8\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\"4\" data-semantic-content=\"3,5\" data-semantic- data-semantic-parent=\"8\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"6\" data-semantic-role=\"open\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"6\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"6\" data-semantic-role=\"close\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo></mjx-mrow></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/3b6cb2ed-f227-4cfc-bda9-9a47e3ce4908/stan12336-math-0006.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-children=\"2,6\" data-semantic-content=\"7,0\" data-semantic-role=\"simple function\" data-semantic-speech=\"theta 0 left parenthesis x right parenthesis\" data-semantic-type=\"appl\"><msub data-semantic-=\"\" data-semantic-children=\"0,1\" data-semantic-parent=\"8\" data-semantic-role=\"simple function\" data-semantic-type=\"subscript\"><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-operator=\"appl\" data-semantic-parent=\"2\" data-semantic-role=\"simple function\" data-semantic-type=\"identifier\">θ</mi></mrow><mrow><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"2\" data-semantic-role=\"integer\" data-semantic-type=\"number\">0</mn></mrow></msub><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"appl\" data-semantic-parent=\"8\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\"></mo><mrow data-semantic-=\"\" data-semantic-children=\"4\" data-semantic-content=\"3,5\" data-semantic-parent=\"8\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"6\" data-semantic-role=\"open\" data-semantic-type=\"fence\" stretchy=\"false\">(</mo><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"6\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">x</mi><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"6\" data-semantic-role=\"close\" data-semantic-type=\"fence\" stretchy=\"false\">)</mo></mrow></mrow>$$ {\\theta}_0(x) $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> increases. We propose least squares approaches for the joint estimation of <mjx-container aria-label=\"theta 0\" ctxtmenu_counter=\"6\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow><mjx-msub data-semantic-children=\"0,1\" data-semantic- data-semantic-role=\"greekletter\" data-semantic-speech=\"theta 0\" data-semantic-type=\"subscript\"><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"greekletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-script style=\"vertical-align: -0.15em;\"><mjx-mrow size=\"s\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn></mjx-mrow></mjx-script></mjx-msub></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/1e9bf7c6-9a49-49e0-bd90-a116bb8c68af/stan12336-math-0007.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub data-semantic-=\"\" data-semantic-children=\"0,1\" data-semantic-role=\"greekletter\" data-semantic-speech=\"theta 0\" data-semantic-type=\"subscript\"><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"2\" data-semantic-role=\"greekletter\" data-semantic-type=\"identifier\">θ</mi></mrow><mrow><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"2\" data-semantic-role=\"integer\" data-semantic-type=\"number\">0</mn></mrow></msub></mrow>$$ {\\theta}_0 $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> and <mjx-container aria-label=\"upper F 0\" ctxtmenu_counter=\"7\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow><mjx-msub data-semantic-children=\"0,1\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper F 0\" data-semantic-type=\"subscript\"><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-script style=\"vertical-align: -0.15em; margin-left: -0.106em;\"><mjx-mrow size=\"s\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn></mjx-mrow></mjx-script></mjx-msub></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/cd9155c7-d592-4ff5-9db4-b96619d66811/stan12336-math-0008.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub data-semantic-=\"\" data-semantic-children=\"0,1\" data-semantic-role=\"latinletter\" data-semantic-speech=\"upper F 0\" data-semantic-type=\"subscript\"><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">F</mi></mrow><mrow><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"2\" data-semantic-role=\"integer\" data-semantic-type=\"number\">0</mn></mrow></msub></mrow>$$ {F}_0 $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> in the important case where <mjx-container aria-label=\"theta 0 left parenthesis x right parenthesis equals alpha 0 Superscript down tack Baseline x\" ctxtmenu_counter=\"8\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow data-semantic-children=\"15,17\" data-semantic-content=\"6\" data-semantic- data-semantic-role=\"equality\" data-semantic-speech=\"theta 0 left parenthesis x right parenthesis equals alpha 0 Superscript down tack Baseline x\" data-semantic-type=\"relseq\"><mjx-mrow data-semantic-annotation=\"clearspeak:simple\" data-semantic-children=\"2,13\" data-semantic-content=\"14,0\" data-semantic- data-semantic-parent=\"18\" data-semantic-role=\"simple function\" data-semantic-type=\"appl\"><mjx-msub data-semantic-children=\"0,1\" data-semantic- data-semantic-parent=\"15\" data-semantic-role=\"simple function\" data-semantic-type=\"subscript\"><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"2\" data-semantic-role=\"simple function\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-script style=\"vertical-align: -0.15em;\"><mjx-mrow size=\"s\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn></mjx-mrow></mjx-script></mjx-msub><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"15\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\"4\" data-semantic-content=\"3,5\" data-semantic- data-semantic-parent=\"15\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"13\" data-semantic-role=\"open\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"13\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"13\" data-semantic-role=\"close\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo></mjx-mrow></mjx-mrow><mjx-mo data-semantic- data-semantic-operator=\"relseq,=\" data-semantic-parent=\"18\" data-semantic-role=\"equality\" data-semantic-type=\"relation\" rspace=\"5\" space=\"5\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-annotation=\"clearspeak:unit\" data-semantic-children=\"11,12\" data-semantic-content=\"16\" data-semantic- data-semantic-parent=\"18\" data-semantic-role=\"implicit\" data-semantic-type=\"infixop\"><mjx-msubsup data-semantic-children=\"7,8,9\" data-semantic-collapsed=\"(11 (10 7 8) 9)\" data-semantic- data-semantic-parent=\"17\" data-semantic-role=\"greekletter\" data-semantic-type=\"subsup\"><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"11\" data-semantic-role=\"greekletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-script style=\"vertical-align: -0.288em; margin-left: 0px;\"><mjx-mrow size=\"s\"><mjx-mo data-semantic- data-semantic-parent=\"11\" data-semantic-role=\"addition\" data-semantic-type=\"operator\"><mjx-c></mjx-c></mjx-mo></mjx-mrow><mjx-spacer style=\"margin-top: 0.18em;\"></mjx-spacer><mjx-mrow size=\"s\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"11\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn></mjx-mrow></mjx-script></mjx-msubsup><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"infixop,\" data-semantic-parent=\"17\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"17\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/d4e3680e-e32c-4cbb-a216-046476cf7971/stan12336-math-0009.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-children=\"15,17\" data-semantic-content=\"6\" data-semantic-role=\"equality\" data-semantic-speech=\"theta 0 left parenthesis x right parenthesis equals alpha 0 Superscript down tack Baseline x\" data-semantic-type=\"relseq\"><mrow data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-children=\"2,13\" data-semantic-content=\"14,0\" data-semantic-parent=\"18\" data-semantic-role=\"simple function\" data-semantic-type=\"appl\"><msub data-semantic-=\"\" data-semantic-children=\"0,1\" data-semantic-parent=\"15\" data-semantic-role=\"simple function\" data-semantic-type=\"subscript\"><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-operator=\"appl\" data-semantic-parent=\"2\" data-semantic-role=\"simple function\" data-semantic-type=\"identifier\">θ</mi></mrow><mrow><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"2\" data-semantic-role=\"integer\" data-semantic-type=\"number\">0</mn></mrow></msub><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"appl\" data-semantic-parent=\"15\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\"></mo><mrow data-semantic-=\"\" data-semantic-children=\"4\" data-semantic-content=\"3,5\" data-semantic-parent=\"15\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"13\" data-semantic-role=\"open\" data-semantic-type=\"fence\" stretchy=\"false\">(</mo><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"13\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">x</mi><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"13\" data-semantic-role=\"close\" data-semantic-type=\"fence\" stretchy=\"false\">)</mo></mrow></mrow><mo data-semantic-=\"\" data-semantic-operator=\"relseq,=\" data-semantic-parent=\"18\" data-semantic-role=\"equality\" data-semantic-type=\"relation\">=</mo><mrow data-semantic-=\"\" data-semantic-annotation=\"clearspeak:unit\" data-semantic-children=\"11,12\" data-semantic-content=\"16\" data-semantic-parent=\"18\" data-semantic-role=\"implicit\" data-semantic-type=\"infixop\"><msubsup data-semantic-=\"\" data-semantic-children=\"7,8,9\" data-semantic-collapsed=\"(11 (10 7 8) 9)\" data-semantic-parent=\"17\" data-semantic-role=\"greekletter\" data-semantic-type=\"subsup\"><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"11\" data-semantic-role=\"greekletter\" data-semantic-type=\"identifier\">α</mi></mrow><mrow><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"11\" data-semantic-role=\"integer\" data-semantic-type=\"number\">0</mn></mrow><mrow><mo data-semantic-=\"\" data-semantic-parent=\"11\" data-semantic-role=\"addition\" data-semantic-type=\"operator\">⊤</mo></mrow></msubsup><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"infixop,\" data-semantic-parent=\"17\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\"></mo><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"17\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">x</mi></mrow></mrow>$$ {\\theta}_0(x)={\\alpha}_0^{\\top }x $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> and obtain convergence rates of <mjx-container aria-label=\"n Superscript negative 1 divided by 3\" ctxtmenu_counter=\"9\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow><mjx-msup data-semantic-children=\"0,6\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"n Superscript negative 1 divided by 3\" data-semantic-type=\"superscript\"><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-script style=\"vertical-align: 0.363em;\"><mjx-mrow data-semantic-children=\"5,4\" data-semantic-content=\"3\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"division\" data-semantic-type=\"infixop\" size=\"s\"><mjx-mrow data-semantic-annotation=\"clearspeak:simple\" data-semantic-children=\"2\" data-semantic-content=\"1\" data-semantic- data-semantic-parent=\"6\" data-semantic-role=\"negative\" data-semantic-type=\"prefixop\"><mjx-mo data-semantic- data-semantic-operator=\"prefixop,−\" data-semantic-parent=\"5\" data-semantic-role=\"subtraction\" data-semantic-type=\"operator\" rspace=\"1\"><mjx-c></mjx-c></mjx-mo><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"5\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn></mjx-mrow><mjx-mo data-semantic- data-semantic-operator=\"infixop,/\" data-semantic-parent=\"6\" data-semantic-role=\"division\" data-semantic-type=\"operator\" rspace=\"1\" space=\"1\"><mjx-c></mjx-c></mjx-mo><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"6\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn></mjx-mrow></mjx-script></mjx-msup></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/b6b785d1-d5e5-4cda-b0a0-a64c31bc70c7/stan12336-math-0010.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msup data-semantic-=\"\" data-semantic-children=\"0,6\" data-semantic-role=\"latinletter\" data-semantic-speech=\"n Superscript negative 1 divided by 3\" data-semantic-type=\"superscript\"><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"7\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">n</mi></mrow><mrow data-semantic-=\"\" data-semantic-children=\"5,4\" data-semantic-content=\"3\" data-semantic-parent=\"7\" data-semantic-role=\"division\" data-semantic-type=\"infixop\"><mrow data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-children=\"2\" data-semantic-content=\"1\" data-semantic-parent=\"6\" data-semantic-role=\"negative\" data-semantic-type=\"prefixop\"><mo data-semantic-=\"\" data-semantic-operator=\"prefixop,−\" data-semantic-parent=\"5\" data-semantic-role=\"subtraction\" data-semantic-type=\"operator\" form=\"prefix\">−</mo><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"5\" data-semantic-role=\"integer\" data-semantic-type=\"number\">1</mn></mrow><mo data-semantic-=\"\" data-semantic-operator=\"infixop,/\" data-semantic-parent=\"6\" data-semantic-role=\"division\" data-semantic-type=\"operator\" stretchy=\"false\">/</mo><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"6\" data-semantic-role=\"integer\" data-semantic-type=\"number\">3</mn></mrow></msup></mrow>$$ {n}^{-1/3} $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>, thereby improving an existing result that gives a rate of <mjx-container aria-label=\"n Superscript negative 1 divided by 6\" ctxtmenu_counter=\"10\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow><mjx-msup data-semantic-children=\"0,6\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"n Superscript negative 1 divided by 6\" data-semantic-type=\"superscript\"><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-script style=\"vertical-align: 0.363em;\"><mjx-mrow data-semantic-children=\"5,4\" data-semantic-content=\"3\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"division\" data-semantic-type=\"infixop\" size=\"s\"><mjx-mrow data-semantic-annotation=\"clearspeak:simple\" data-semantic-children=\"2\" data-semantic-content=\"1\" data-semantic- data-semantic-parent=\"6\" data-semantic-role=\"negative\" data-semantic-type=\"prefixop\"><mjx-mo data-semantic- data-semantic-operator=\"prefixop,−\" data-semantic-parent=\"5\" data-semantic-role=\"subtraction\" data-semantic-type=\"operator\" rspace=\"1\"><mjx-c></mjx-c></mjx-mo><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"5\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn></mjx-mrow><mjx-mo data-semantic- data-semantic-operator=\"infixop,/\" data-semantic-parent=\"6\" data-semantic-role=\"division\" data-semantic-type=\"operator\" rspace=\"1\" space=\"1\"><mjx-c></mjx-c></mjx-mo><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"6\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn></mjx-mrow></mjx-script></mjx-msup></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/87b91976-86b5-4efc-a4cc-bd81187c82e1/stan12336-math-0011.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msup data-semantic-=\"\" data-semantic-children=\"0,6\" data-semantic-role=\"latinletter\" data-semantic-speech=\"n Superscript negative 1 divided by 6\" data-semantic-type=\"superscript\"><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"7\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">n</mi></mrow><mrow data-semantic-=\"\" data-semantic-children=\"5,4\" data-semantic-content=\"3\" data-semantic-parent=\"7\" data-semantic-role=\"division\" data-semantic-type=\"infixop\"><mrow data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-children=\"2\" data-semantic-content=\"1\" data-semantic-parent=\"6\" data-semantic-role=\"negative\" data-semantic-type=\"prefixop\"><mo data-semantic-=\"\" data-semantic-operator=\"prefixop,−\" data-semantic-parent=\"5\" data-semantic-role=\"subtraction\" data-semantic-type=\"operator\" form=\"prefix\">−</mo><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"5\" data-semantic-role=\"integer\" data-semantic-type=\"number\">1</mn></mrow><mo data-semantic-=\"\" data-semantic-operator=\"infixop,/\" data-semantic-parent=\"6\" data-semantic-role=\"division\" data-semantic-type=\"operator\" stretchy=\"false\">/</mo><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"6\" data-semantic-role=\"integer\" data-semantic-type=\"number\">6</mn></mrow></msup></mrow>$$ {n}^{-1/6} $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>. A simulation study indicates that the convergence rate for the estimation of <mjx-container aria-label=\"alpha 0\" ctxtmenu_counter=\"11\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow><mjx-msub data-semantic-children=\"0,1\" data-semantic- data-semantic-role=\"greekletter\" data-semantic-speech=\"alpha 0\" data-semantic-type=\"subscript\"><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"greekletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-script style=\"vertical-align: -0.15em;\"><mjx-mrow size=\"s\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn></mjx-mrow></mjx-script></mjx-msub></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/a392720e-fdcf-4ed1-9cc6-ee90457d08a7/stan12336-math-0012.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub data-semantic-=\"\" data-semantic-children=\"0,1\" data-semantic-role=\"greekletter\" data-semantic-speech=\"alpha 0\" data-semantic-type=\"subscript\"><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"2\" data-semantic-role=\"greekletter\" data-semantic-type=\"identifier\">α</mi></mrow><mrow><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"2\" data-semantic-role=\"integer\" data-semantic-type=\"number\">0</mn></mrow></msub></mrow>$$ {\\alpha}_0 $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> might be faster. Furthermore, we illustrate our methods in an application on house price data that demonstrates the advantages of shape restrictions in single index models.","PeriodicalId":51178,"journal":{"name":"Statistica Neerlandica","volume":"20 1","pages":""},"PeriodicalIF":1.4000,"publicationDate":"2024-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Statistica Neerlandica","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1111/stan.12336","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 0
Abstract
The distributional single index model is a semiparametric regression model in which the conditional distribution functions分布式单一指数模型的估计和收敛率
分布式单一指数模型是一种半参数回归模型，其中条件分布函数 P(Y≤y|X=x)=F0(θ0(x),y)$$ P\left(Y\le y|X=x\right)={F}_0\left({\theta}_0(x)、y\right) $$ 实值结果变量 Y$$ Y$ 通过单变量参数指标函数 θ0(x)$$ {\theta}_0(x) $$ 取决于 d$$ d$ 维协变量 X$$ X$ ，并且随着 θ0(x)$$ {\theta}_0(x) $$ 的增加而随机增加。在 θ0(x)=α0⊤x$$ {\theta}_0(x)={\alpha}_0^{\top }x $$ 的重要情况下，我们提出了联合估计 θ0$$ {\theta}_0 $$ 和 F0$$ {F}_0 $$ 的最小二乘法，并获得了 n-1/3$$ {n}^{-1/3} $$ 的收敛率、从而改进了给出 n-1/6$ {n}^{-1/6} $$ 收敛率的现有结果。模拟研究表明，估计 α0$$ {\alpha}_0 $$ 的收敛速度可能更快。此外，我们还通过对房价数据的应用说明了我们的方法，从而展示了形状限制在单指数模型中的优势。
本文章由计算机程序翻译，如有差异，请以英文原文为准。
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来源期刊
Statistica Neerlandica
数学-统计学与概率论
CiteScore
2.60
自引率
6.70%
发文量
26
审稿时长
>12 weeks
期刊介绍：
Statistica Neerlandica has been the journal of the Netherlands Society for Statistics and Operations Research since 1946. It covers all areas of statistics, from theoretical to applied, with a special emphasis on mathematical statistics, statistics for the behavioural sciences and biostatistics. This wide scope is reflected by the expertise of the journal’s editors representing these areas. The diverse editorial board is committed to a fast and fair reviewing process, and will judge submissions on quality, correctness, relevance and originality. Statistica Neerlandica encourages transparency and reproducibility, and offers online resources to make data, code, simulation results and other additional materials publicly available.
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