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If we define
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If
we define
to
be a new estimator of 
, then the bias of 
is
which
is less than the bias of 
since it
eliminates the term of order 1/n. In practice the estimator 
is computed as
where
i = 1, ... n. This
is the first order jackknife estimator. It is
useful in that it is a less biased estimator although being somewhat more variable than
the un-jackknifed estimator, but this increased variability is at maximum
.
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<p class="MsoPlainText"><span style="font-size: 12pt; font-family: Arial;">If we define<o:p></o:p></span></p> <p class="MsoPlainText"><span style="font-size: 12pt; font-family: Arial;"><!--[if !supportEmptyParas]--> <!--[endif]--><o:p></o:p></span></p> <p class="MsoPlainText"><span style="font-size: 12pt; font-family: Arial;"><span style=""> </span><sub><!--[if gte vml 1]><v:shape id="_x0000_i1070" type="#_x0000_t75" style='width:155.25pt;height:20.25pt' o:ole="" fillcolor="window"> <v:imagedata src="./BOOTS_files/image060.wmz" o:title=""/></v:shape><![endif]--><!--[if !vml]--><img src="BOOTS_files/image061.gif" v:shapes="_x0000_i1070" height="27" width="207"><!--[endif]--></sub><!--[if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.3" ShapeID="_x0000_i1070" DrawAspect="Content" ObjectID="_1031637828"> </o:OLEObject></xml><![endif]--><o:p></o:p></span></p> <p class="MsoPlainText"><span style="font-size: 12pt; font-family: Arial;"><!--[if !supportEmptyParas]--> <!--[endif]--><o:p></o:p></span></p> <p class="MsoPlainText"><span style="font-size: 12pt; font-family: Arial;">to be a new estimator of <sub><!--[if gte vml 1]><v:shape id="_x0000_i1071" type="#_x0000_t75" style='width:9.75pt;height:14.25pt' o:ole="" fillcolor="window"> <v:imagedata src="./BOOTS_files/image047.wmz" o:title=""/></v:shape><![endif]--><!--[if !vml]--><img src="BOOTS_files/image062.gif" v:shapes="_x0000_i1071" height="19" width="13"><!--[endif]--></sub><!--[if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.3" ShapeID="_x0000_i1071" DrawAspect="Content" ObjectID="_1031637829"> </o:OLEObject></xml><![endif]-->, then the bias of <sub><!--[if gte vml 1]><v:shape id="_x0000_i1072" type="#_x0000_t75" style='width:15.75pt;height:17.25pt' o:ole="" fillcolor="window"> <v:imagedata src="./BOOTS_files/image063.wmz" o:title=""/></v:shape><![endif]--><!--[if !vml]--><img src="BOOTS_files/image064.gif" v:shapes="_x0000_i1072" height="23" width="21"><!--[endif]--></sub><!--[if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.3" ShapeID="_x0000_i1072" DrawAspect="Content" ObjectID="_1031637830"> </o:OLEObject></xml><![endif]--><span style=""> </span>is<o:p></o:p></span></p> <p class="MsoPlainText"><span style="font-size: 12pt; font-family: Arial;"><!--[if !supportEmptyParas]--> <!--[endif]--><o:p></o:p></span></p> <p class="MsoPlainText"><span style="font-size: 12pt; font-family: Arial;"><span style=""> </span><sub><!--[if gte vml 1]><v:shape id="_x0000_i1073" type="#_x0000_t75" style='width:158.25pt;height:32.25pt' o:ole="" fillcolor="window"> <v:imagedata src="./BOOTS_files/image065.wmz" o:title=""/></v:shape><![endif]--><!--[if !vml]--><img src="BOOTS_files/image066.gif" v:shapes="_x0000_i1073" height="43" width="211"><!--[endif]--></sub><!--[if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.3" ShapeID="_x0000_i1073" DrawAspect="Content" ObjectID="_1031637832"> </o:OLEObject></xml><![endif]--><o:p></o:p></span></p> <p class="MsoPlainText"><span style="font-size: 12pt; font-family: Arial;"><!--[if !supportEmptyParas]--> <!--[endif]--><o:p></o:p></span></p> <p class="MsoPlainText"><span style="font-size: 12pt; font-family: Arial;">which is less than the bias of <sub><!--[if gte vml 1]><v:shape id="_x0000_i1074" type="#_x0000_t75" style='width:9.75pt;height:17.25pt' o:ole="" fillcolor="window"> <v:imagedata src="./BOOTS_files/image067.wmz" o:title=""/></v:shape><![endif]--><!--[if !vml]--><img src="BOOTS_files/image068.gif" v:shapes="_x0000_i1074" height="23" width="13"><!--[endif]--></sub><!--[if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.3" ShapeID="_x0000_i1074" DrawAspect="Content" ObjectID="_1031637833"> </o:OLEObject></xml><![endif]--><span style=""> </span>since it eliminates the term of order 1/<i style="">n</i>.<span style=""> </span>In practice the estimator <sub><!--[if gte vml 1]><v:shape id="_x0000_i1075" type="#_x0000_t75" style='width:15.75pt;height:17.25pt' o:ole="" fillcolor="window"> <v:imagedata src="./BOOTS_files/image069.wmz" o:title=""/></v:shape><![endif]--><!--[if !vml]--><img src="BOOTS_files/image070.gif" v:shapes="_x0000_i1075" height="23" width="21"><!--[endif]--></sub><!--[if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.3" ShapeID="_x0000_i1075" DrawAspect="Content" ObjectID="_1031637834"> </o:OLEObject></xml><![endif]--><span style=""> </span>is computed as<o:p></o:p></span></p> <p class="MsoPlainText"><span style="font-size: 12pt; font-family: Arial;"><!--[if !supportEmptyParas]--> <!--[endif]--><o:p></o:p></span></p> <p class="MsoPlainText"><span style="font-size: 12pt; font-family: Arial;"><span style=""> </span><sub><!--[if gte vml 1]><v:shape id="_x0000_i1076" type="#_x0000_t75" style='width:176.25pt;height:33pt' o:ole="" fillcolor="window"> <v:imagedata src="./BOOTS_files/image071.wmz" o:title=""/></v:shape><![endif]--><!--[if !vml]--><img src="BOOTS_files/image072.gif" v:shapes="_x0000_i1076" height="44" width="235"><!--[endif]--></sub><!--[if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.3" ShapeID="_x0000_i1076" DrawAspect="Content" ObjectID="_1031637835"> </o:OLEObject></xml><![endif]--><o:p></o:p></span></p> <p class="MsoPlainText"><span style="font-size: 12pt; font-family: Arial;"><!--[if !supportEmptyParas]--> <!--[endif]--><o:p></o:p></span></p> <p class="MsoPlainText"><span style="font-size: 12pt; font-family: Arial;">where <i style="">i</i> = 1, ... <i style="">n</i>.<span style=""> </span>This is the first order jackknife estimator.<span style=""> </span>It is useful in that it is a less biased estimator although being somewhat more variable than the un-jackknifed estimator, but this increased variability is at maximum<o:p></o:p></span></p> <p class="MsoPlainText"><span style="font-size: 12pt; font-family: Arial;"><!--[if !supportEmptyParas]--> <!--[endif]--><o:p></o:p></span></p> <p class="MsoPlainText" style="margin-left: 0.5in; text-indent: 0.5in;"><span style="font-size: 12pt; font-family: Arial;"><sub><!--[if gte vml 1]><v:shape id="_x0000_i1077" type="#_x0000_t75" style='width:117.75pt;height:32.25pt' o:ole="" fillcolor="window"> <v:imagedata src="./BOOTS_files/image073.wmz" o:title=""/></v:shape><![endif]--><!--[if !vml]--><img src="BOOTS_files/image074.gif" v:shapes="_x0000_i1077" height="43" width="157"><!--[endif]--></sub><!--[if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.3" ShapeID="_x0000_i1077" DrawAspect="Content" ObjectID="_1031637836"> </o:OLEObject></xml><![endif]-->.<o:p></o:p></span></p> <p class="MsoPlainText"><span style="font-size: 12pt; font-family: Arial;"><!--[if !supportEmptyParas]--> </span></p> |
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