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The receiver operating characteristic (ROC) curve is a valuable statistical tool in medical research. It assesses a biomarker’s ability to distinguish between diseased and healthy individuals. The area under the ROC curve (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>U</mi><mi>C</mi></mrow></semantics></math></inline-formula>) and the Youden index (<i>J</i>) are common summary indices used to evaluate a biomarker’s diagnostic accuracy. Simultaneously examining <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>U</mi><mi>C</mi></mrow></semantics></math></inline-formula> and <i>J</i> offers a more comprehensive understanding of the ROC curve’s characteristics. In this paper, we utilize a semiparametric density ratio model to link the distributions of a biomarker for healthy and diseased individuals. Under this model, we establish the joint asymptotic normality of the maximum empirical likelihood estimator of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>A</mi><mi>U</mi><mi>C</mi><mo>,</mo><mi>J</mi><mo>)</mo></mrow></semantics></math></inline-formula> and construct an asymptotically valid confidence region for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>A</mi><mi>U</mi><mi>C</mi><mo>,</mo><mi>J</mi><mo>)</mo></mrow></semantics></math></inline-formula>. Furthermore, we propose a new test to determine whether a biomarker simultaneously exceeds prespecified target values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>U</mi><msub><mi>C</mi><mn>0</mn></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>J</mi><mn>0</mn></msub></semantics></math></inline-formula> with the null hypothesis <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mn>0</mn></msub><mo>:</mo><mi>A</mi><mi>U</mi><mi>C</mi><mo>≤</mo><mi>A</mi><mi>U</mi><msub><mi>C</mi><mn>0</mn></msub></mrow></semantics></math></inline-formula> or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>J</mi><mo>≤</mo><msub><mi>J</mi><mn>0</mn></msub></mrow></semantics></math></inline-formula> against the alternative hypothesis <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mi>a</mi></msub><mo>:</mo><mi>A</mi><mi>U</mi><mi>C</mi><mo>></mo><mi>A</mi><mi>U</mi><msub><mi>C</mi><mn>0</mn></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>J</mi><mo>></mo><msub><mi>J</mi><mn>0</mn></msub></mrow></semantics></math></inline-formula>. Simulation studies and a real data example on Duchenne Muscular Dystrophy are used to demonstrate the effectiveness of our proposed method and highlight its advantages over existing methods.