{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,3]],"date-time":"2026-08-03T07:47:20Z","timestamp":1785743240676,"version":"3.56.0"},"reference-count":19,"publisher":"Cambridge University Press (CUP)","license":[{"start":{"date-parts":[[2026,8,3]],"date-time":"2026-08-03T00:00:00Z","timestamp":1785715200000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Econom. Theory"],"abstract":"<jats:p>\n                    This article studies the power properties of confidence intervals (CIs) for a partially-identified parameter of interest with an interval identified set. We assume that the researcher has bounds estimators needed to construct the CIs proposed by Imbens and Manski (2004), Stoye (2009), and Stoye (2020), denoted by\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\" display=\"inline\">\n                          <mml:mi>C<\/mml:mi>\n                          <mml:msubsup>\n                            <mml:mi>I<\/mml:mi>\n                            <mml:mrow>\n                              <mml:mi>\u03b1<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:mrow>\n                          <\/mml:msubsup>\n                        <\/mml:math>\n                        <jats:tex-math>$CI_{\\alpha }^{1}$<\/jats:tex-math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" content-type=\"simple\" xlink:href=\"S0266466626100565_inline1.png\">\n                          <jats:alt-text content-type=\"machine-generated\">upper C upper I Subscript alpha Superscript 1<\/jats:alt-text>\n                        <\/jats:inline-graphic>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    ,\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\" display=\"inline\">\n                          <mml:mi>C<\/mml:mi>\n                          <mml:msubsup>\n                            <mml:mi>I<\/mml:mi>\n                            <mml:mrow>\n                              <mml:mi>\u03b1<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:mrow>\n                          <\/mml:msubsup>\n                        <\/mml:math>\n                        <jats:tex-math>$CI_{\\alpha }^{2}$<\/jats:tex-math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" content-type=\"simple\" xlink:href=\"S0266466626100565_inline2.png\">\n                          <jats:alt-text content-type=\"machine-generated\">upper C upper I Subscript alpha Superscript 2<\/jats:alt-text>\n                        <\/jats:inline-graphic>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    ,\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\" display=\"inline\">\n                          <mml:mi>C<\/mml:mi>\n                          <mml:msubsup>\n                            <mml:mi>I<\/mml:mi>\n                            <mml:mrow>\n                              <mml:mi>\u03b1<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mn>3<\/mml:mn>\n                            <\/mml:mrow>\n                          <\/mml:msubsup>\n                        <\/mml:math>\n                        <jats:tex-math>$CI_{\\alpha }^{3}$<\/jats:tex-math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" content-type=\"simple\" xlink:href=\"S0266466626100565_inline3.png\">\n                          <jats:alt-text content-type=\"machine-generated\">upper C upper I Subscript alpha Superscript 3<\/jats:alt-text>\n                        <\/jats:inline-graphic>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , and\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\" display=\"inline\">\n                          <mml:mi>C<\/mml:mi>\n                          <mml:msubsup>\n                            <mml:mi>I<\/mml:mi>\n                            <mml:mrow>\n                              <mml:mi>\u03b1<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mn>4<\/mml:mn>\n                            <\/mml:mrow>\n                          <\/mml:msubsup>\n                        <\/mml:math>\n                        <jats:tex-math>$CI_{\\alpha }^{4}$<\/jats:tex-math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" content-type=\"simple\" xlink:href=\"S0266466626100565_inline4.png\">\n                          <jats:alt-text content-type=\"machine-generated\">upper C upper I Subscript alpha Superscript 4<\/jats:alt-text>\n                        <\/jats:inline-graphic>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    . We also assume that these bounds estimators are \u201cordered\u201d: the lower bound estimator is less than or equal to the upper bound estimator. This setup arises in economic applications involving missing data and treatment effects. Under these conditions, we establish two results. First, we show that\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\" display=\"inline\">\n                          <mml:mi>C<\/mml:mi>\n                          <mml:msubsup>\n                            <mml:mi>I<\/mml:mi>\n                            <mml:mrow>\n                              <mml:mi>\u03b1<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:mrow>\n                          <\/mml:msubsup>\n                        <\/mml:math>\n                        <jats:tex-math>$CI_{\\alpha }^{1}$<\/jats:tex-math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" content-type=\"simple\" xlink:href=\"S0266466626100565_inline5.png\">\n                          <jats:alt-text content-type=\"machine-generated\">upper C upper I Subscript alpha Superscript 1<\/jats:alt-text>\n                        <\/jats:inline-graphic>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    and\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\" display=\"inline\">\n                          <mml:mi>C<\/mml:mi>\n                          <mml:msubsup>\n                            <mml:mi>I<\/mml:mi>\n                            <mml:mrow>\n                              <mml:mi>\u03b1<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:mrow>\n                          <\/mml:msubsup>\n                        <\/mml:math>\n                        <jats:tex-math>$CI_{\\alpha }^{2}$<\/jats:tex-math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" content-type=\"simple\" xlink:href=\"S0266466626100565_inline6.png\">\n                          <jats:alt-text content-type=\"machine-generated\">upper C upper I Subscript alpha Superscript 2<\/jats:alt-text>\n                        <\/jats:inline-graphic>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    are equally powerful, and both dominate\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\" display=\"inline\">\n                          <mml:mi>C<\/mml:mi>\n                          <mml:msubsup>\n                            <mml:mi>I<\/mml:mi>\n                            <mml:mrow>\n                              <mml:mi>\u03b1<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mn>3<\/mml:mn>\n                            <\/mml:mrow>\n                          <\/mml:msubsup>\n                        <\/mml:math>\n                        <jats:tex-math>$CI_{\\alpha }^{3}$<\/jats:tex-math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" content-type=\"simple\" xlink:href=\"S0266466626100565_inline7.png\">\n                          <jats:alt-text content-type=\"machine-generated\">upper C upper I Subscript alpha Superscript 3<\/jats:alt-text>\n                        <\/jats:inline-graphic>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    and\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\" display=\"inline\">\n                          <mml:mi>C<\/mml:mi>\n                          <mml:msubsup>\n                            <mml:mi>I<\/mml:mi>\n                            <mml:mrow>\n                              <mml:mi>\u03b1<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mn>4<\/mml:mn>\n                            <\/mml:mrow>\n                          <\/mml:msubsup>\n                        <\/mml:math>\n                        <jats:tex-math>$CI_{\\alpha }^{4}$<\/jats:tex-math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" content-type=\"simple\" xlink:href=\"S0266466626100565_inline8.png\">\n                          <jats:alt-text content-type=\"machine-generated\">upper C upper I Subscript alpha Superscript 4<\/jats:alt-text>\n                        <\/jats:inline-graphic>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    . Second, we consider a favorable situation in which there are two possible bounds estimators to construct these CIs, and one is more efficient than the other. One would expect that the more efficient bounds estimator yields more powerful inference. We prove that this desirable result holds for\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\" display=\"inline\">\n                          <mml:mi>C<\/mml:mi>\n                          <mml:msubsup>\n                            <mml:mi>I<\/mml:mi>\n                            <mml:mrow>\n                              <mml:mi>\u03b1<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:mrow>\n                          <\/mml:msubsup>\n                        <\/mml:math>\n                        <jats:tex-math>$CI_{\\alpha }^{1}$<\/jats:tex-math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" content-type=\"simple\" xlink:href=\"S0266466626100565_inline9.png\">\n                          <jats:alt-text content-type=\"machine-generated\">upper C upper I Subscript alpha Superscript 1<\/jats:alt-text>\n                        <\/jats:inline-graphic>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    and\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\" display=\"inline\">\n                          <mml:mi>C<\/mml:mi>\n                          <mml:msubsup>\n                            <mml:mi>I<\/mml:mi>\n                            <mml:mrow>\n                              <mml:mi>\u03b1<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:mrow>\n                          <\/mml:msubsup>\n                        <\/mml:math>\n                        <jats:tex-math>$CI_{\\alpha }^{2}$<\/jats:tex-math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" content-type=\"simple\" xlink:href=\"S0266466626100565_inline10.png\">\n                          <jats:alt-text content-type=\"machine-generated\">upper C upper I Subscript alpha Superscript 2<\/jats:alt-text>\n                        <\/jats:inline-graphic>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , but not necessarily for\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\" display=\"inline\">\n                          <mml:mi>C<\/mml:mi>\n                          <mml:msubsup>\n                            <mml:mi>I<\/mml:mi>\n                            <mml:mrow>\n                              <mml:mi>\u03b1<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mn>3<\/mml:mn>\n                            <\/mml:mrow>\n                          <\/mml:msubsup>\n                        <\/mml:math>\n                        <jats:tex-math>$CI_{\\alpha }^{3}$<\/jats:tex-math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" content-type=\"simple\" xlink:href=\"S0266466626100565_inline11.png\">\n                          <jats:alt-text content-type=\"machine-generated\">upper C upper I Subscript alpha Superscript 3<\/jats:alt-text>\n                        <\/jats:inline-graphic>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    or\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\" display=\"inline\">\n                          <mml:mi>C<\/mml:mi>\n                          <mml:msubsup>\n                            <mml:mi>I<\/mml:mi>\n                            <mml:mrow>\n                              <mml:mi>\u03b1<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mn>4<\/mml:mn>\n                            <\/mml:mrow>\n                          <\/mml:msubsup>\n                        <\/mml:math>\n                        <jats:tex-math>$CI_{\\alpha }^{4}$<\/jats:tex-math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" content-type=\"simple\" xlink:href=\"S0266466626100565_inline12.png\">\n                          <jats:alt-text content-type=\"machine-generated\">upper C upper I Subscript alpha Superscript 4<\/jats:alt-text>\n                        <\/jats:inline-graphic>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    . In summary, within the class of models considered,\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\" display=\"inline\">\n                          <mml:mi>C<\/mml:mi>\n                          <mml:msubsup>\n                            <mml:mi>I<\/mml:mi>\n                            <mml:mrow>\n                              <mml:mi>\u03b1<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:mrow>\n                          <\/mml:msubsup>\n                        <\/mml:math>\n                        <jats:tex-math>$CI_{\\alpha }^{1}$<\/jats:tex-math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" content-type=\"simple\" xlink:href=\"S0266466626100565_inline13.png\">\n                          <jats:alt-text content-type=\"machine-generated\">upper C upper I Subscript alpha Superscript 1<\/jats:alt-text>\n                        <\/jats:inline-graphic>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    and\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\" display=\"inline\">\n                          <mml:mi>C<\/mml:mi>\n                          <mml:msubsup>\n                            <mml:mi>I<\/mml:mi>\n                            <mml:mrow>\n                              <mml:mi>\u03b1<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:mrow>\n                          <\/mml:msubsup>\n                        <\/mml:math>\n                        <jats:tex-math>$CI_{\\alpha }^{2}$<\/jats:tex-math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" content-type=\"simple\" xlink:href=\"S0266466626100565_inline14.png\">\n                          <jats:alt-text content-type=\"machine-generated\">upper C upper I Subscript alpha Superscript 2<\/jats:alt-text>\n                        <\/jats:inline-graphic>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    have identical power properties, and both compare favorably to\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\" display=\"inline\">\n                          <mml:mi>C<\/mml:mi>\n                          <mml:msubsup>\n                            <mml:mi>I<\/mml:mi>\n                            <mml:mrow>\n                              <mml:mi>\u03b1<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mn>3<\/mml:mn>\n                            <\/mml:mrow>\n                          <\/mml:msubsup>\n                        <\/mml:math>\n                        <jats:tex-math>$CI_{\\alpha }^{3}$<\/jats:tex-math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" content-type=\"simple\" xlink:href=\"S0266466626100565_inline15.png\">\n                          <jats:alt-text content-type=\"machine-generated\">upper C upper I Subscript alpha Superscript 3<\/jats:alt-text>\n                        <\/jats:inline-graphic>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    and\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\" display=\"inline\">\n                          <mml:mi>C<\/mml:mi>\n                          <mml:msubsup>\n                            <mml:mi>I<\/mml:mi>\n                            <mml:mrow>\n                              <mml:mi>\u03b1<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mn>4<\/mml:mn>\n                            <\/mml:mrow>\n                          <\/mml:msubsup>\n                        <\/mml:math>\n                        <jats:tex-math>$CI_{\\alpha }^{4}$<\/jats:tex-math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" content-type=\"simple\" xlink:href=\"S0266466626100565_inline16.png\">\n                          <jats:alt-text content-type=\"machine-generated\">upper C upper I Subscript alpha Superscript 4<\/jats:alt-text>\n                        <\/jats:inline-graphic>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    .\n                  <\/jats:p>","DOI":"10.1017\/s0266466626100565","type":"journal-article","created":{"date-parts":[[2026,8,3]],"date-time":"2026-08-03T07:25:36Z","timestamp":1785741936000},"page":"1-43","source":"Crossref","is-referenced-by-count":0,"title":["ON THE POWER PROPERTIES OF INFERENCE FOR PARAMETERS WITH INTERVAL IDENTIFIED SETS"],"prefix":"10.1017","author":[{"given":"Federico","family":"Bugni","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/000e0be47","id-type":"ROR","asserted-by":"publisher"}],"name":"Northwestern 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