\relax \ifx\hyper@anchor\@undefined \global \let \oldcontentsline\contentsline \gdef \contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}} \global \let \oldnewlabel\newlabel \gdef \newlabel#1#2{\newlabelxx{#1}#2} \gdef \newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}} \AtEndDocument{\let \contentsline\oldcontentsline \let \newlabel\oldnewlabel} \else \global \let \hyper@last\relax \fi \@writefile{toc}{\contentsline {begintocitem}{}{}{}} \@writefile{toc}{\contentsline {jobname}{Tamhane}{}{}} \@writefile{toc}{\contentsline {doi}{0016/0000.0000}{}{}} \@writefile{toc}{\contentsline {arxiv}{math.PR/0000016}{}{}} \gdef\hy@title{Parametric Mixture Models for Estimating the Proportion of True Null Hypotheses and Adaptive Control of FDR} \@writefile{toc}{\contentsline {title}{Parametric Mixture Models for Estimating the Proportion of True Null Hypotheses and Adaptive Control of FDR}{305}{}} \thanksnewlabel{e1@email}{{}{305}} \thanksnewlabel{e2@email}{{}{305}} \thanksnewlabel{a1thanks}{{1}{305}} \thanksnewlabel{a2thanks}{{2}{305}} \@writefile{idx}{\indexentry {Tamhane, A.C., \textit {Northwestern University}}{}} \@writefile{idx}{\indexentry {Shi, J., \textit {Northwestern University}}{}} \gdef\hy@author{Ajit C. Tamhane and Jiaxiao Shi} \gdef\hy@subject{IMS Collections 2009 Vol. 0 305--326} \gdef\hy@keywords{62F10, 62F12, Beta model; Bias-correction; EM method; Mixture model; False discovery rate (FDR); Least squares method; Maximum likelihood method; Normal model; $p$-values., } \@writefile{toc}{\contentsline {author}{Ajit C. Tamhane and Jiaxiao Shi}{305}{}} \@writefile{toc}{\contentsline {endtocitem}{}{}{}} \gdef\author@num{2} \@writefile{toc}{\contentsline {section}{\numberline {1}Introduction}{306}{section.1}} \newlabel{sec:intro}{{1}{306}{Introduction\relax }{section.1}{}} \newlabel{eq:expna0}{{1.1}{306}{Introduction\relax }{equation.1.1}{}} \newlabel{eq:storeyest}{{1.2}{306}{Introduction\relax }{equation.1.2}{}} \newlabel{sec:normalmodel}{{2}{308}{\label {sec:normalmodel} Normal Model\relax }{section.2}{}} \@writefile{toc}{\contentsline {section}{\numberline {2} Normal Model}{308}{section.2}} \newlabel{eq:mixnorm}{{2.1}{308}{\label {sec:normalmodel} Normal Model\relax }{equation.2.1}{}} \newlabel{sec:hcknormal}{{2.1}{308}{\label {sec:hcknormal} Hsueh, Chen, and Kodell (HCK) Method\relax }{subsection.2.1}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {2.1} Hsueh, Chen, and Kodell (HCK) Method}{308}{subsection.2.1}} \newlabel{eq:betanorm}{{2.2}{308}{\label {sec:hcknormal} Hsueh, Chen, and Kodell (HCK) Method\relax }{equation.2.2}{}} \newlabel{eq:expnr}{{2.3}{308}{\label {sec:hcknormal} Hsueh, Chen, and Kodell (HCK) Method\relax }{equation.2.3}{}} \newlabel{eq:xnormhck}{{2.4}{308}{\label {sec:hcknormal} Hsueh, Chen, and Kodell (HCK) Method\relax }{equation.2.4}{}} \newlabel{eq:ynormhck}{{2.5}{308}{\label {sec:hcknormal} Hsueh, Chen, and Kodell (HCK) Method\relax }{equation.2.5}{}} \newlabel{eq:hskest}{{2.6}{308}{\label {sec:hcknormal} Hsueh, Chen, and Kodell (HCK) Method\relax }{equation.2.6}{}} \newlabel{eq:startvaluesnorm}{{2.7}{309}{\label {sec:hcknormal} Hsueh, Chen, and Kodell (HCK) Method\relax }{equation.2.7}{}} \newlabel{sec:tsnormal}{{2.2}{309}{\label {sec:tsnormal} Test Statistics (TS) Method\relax }{subsection.2.2}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {2.2} Test Statistics (TS) Method}{309}{subsection.2.2}} \newlabel{lem:expnorm}{{1}{309}{\label {sec:tsnormal} Test Statistics (TS) Method\relax }{lemma.1}{}} \newlabel{lem:exbarnorm}{{2}{310}{\label {sec:tsnormal} Test Statistics (TS) Method\relax }{lemma.2}{}} \newlabel{eq:gnorm}{{2.8}{310}{\label {sec:tsnormal} Test Statistics (TS) Method\relax }{equation.2.8}{}} \newlabel{eq:hnorm}{{2.9}{310}{\label {sec:tsnormal} Test Statistics (TS) Method\relax }{equation.2.9}{}} \newlabel{eq:exbaranorm}{{2.10}{310}{\label {sec:tsnormal} Test Statistics (TS) Method\relax }{equation.2.10}{}} \newlabel{eq:exbarrnorm}{{2.11}{310}{\label {sec:tsnormal} Test Statistics (TS) Method\relax }{equation.2.11}{}} \newlabel{eq:tsequation}{{2.12}{310}{\label {sec:tsnormal} Test Statistics (TS) Method\relax }{equation.2.12}{}} \newlabel{eq:xnormts}{{2.13}{310}{\label {sec:tsnormal} Test Statistics (TS) Method\relax }{equation.2.13}{}} \newlabel{eq:ynormts}{{2.14}{310}{\label {sec:tsnormal} Test Statistics (TS) Method\relax }{equation.2.14}{}} \newlabel{sec:emnormal}{{2.3}{311}{\label {sec:emnormal} EM Method\relax }{subsection.2.3}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {2.3} EM Method}{311}{subsection.2.3}} \newlabel{sec:betamodel}{{3}{311}{\label {sec:betamodel} Beta Model\relax }{section.3}{}} \@writefile{toc}{\contentsline {section}{\numberline {3} Beta Model}{311}{section.3}} \newlabel{eq:mixbeta}{{3.1}{311}{\label {sec:betamodel} Beta Model\relax }{equation.3.1}{}} \newlabel{sec:hckbeta}{{3.1}{312}{\label {sec:hckbeta} Hsueh, Chen, and Kodell (HCK) Method\relax }{subsection.3.1}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {3.1} Hsueh, Chen, and Kodell (HCK) Method}{312}{subsection.3.1}} \newlabel{eq:betabeta}{{3.2}{312}{\label {sec:hckbeta} Hsueh, Chen, and Kodell (HCK) Method\relax }{equation.3.2}{}} \newlabel{eq:xybetahck}{{3.3}{312}{\label {sec:hckbeta} Hsueh, Chen, and Kodell (HCK) Method\relax }{equation.3.3}{}} \newlabel{eq:startvaluesbeta}{{3.5}{312}{\label {sec:hckbeta} Hsueh, Chen, and Kodell (HCK) Method\relax }{equation.3.4}{}} \newlabel{sec:tsbeta}{{3.2}{312}{\label {sec:tsbeta} Test Statistics (TS) Method\relax }{subsection.3.2}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {3.2} Test Statistics (TS) Method}{312}{subsection.3.2}} \newlabel{lem:expbeta}{{3}{312}{\label {sec:tsbeta} Test Statistics (TS) Method\relax }{lemma.3}{}} \newlabel{lem:epbarbeta}{{4}{313}{\label {sec:tsbeta} Test Statistics (TS) Method\relax }{lemma.4}{}} \newlabel{eq:epbarabeta}{{3.5}{313}{\label {sec:tsbeta} Test Statistics (TS) Method\relax }{equation.3.5}{}} \newlabel{eq:epbarrbeta}{{3.6}{313}{\label {sec:tsbeta} Test Statistics (TS) Method\relax }{equation.3.6}{}} \newlabel{sec:embeta}{{3.3}{313}{\label {sec:embeta} EM Method\relax }{subsection.3.3}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {3.3} EM Method}{313}{subsection.3.3}} \newlabel{sec:fdrcontrol}{{4}{314}{\label {sec:fdrcontrol} Adaptive Control of FDR\relax }{section.4}{}} \@writefile{toc}{\contentsline {section}{\numberline {4} Adaptive Control of FDR}{314}{section.4}} \newlabel{eq:storeyfdrest}{{4.1}{314}{\label {sec:fdrcontrol} Adaptive Control of FDR\relax }{equation.4.1}{}} \newlabel{eq:fdrest}{{4.2}{314}{\label {sec:fdrcontrol} Adaptive Control of FDR\relax }{equation.4.2}{}} \newlabel{lem:fdrnormsolve}{{5}{315}{\label {sec:fdrcontrol} Adaptive Control of FDR\relax }{lemma.5}{}} \newlabel{lem:fdrbetasolve}{{6}{315}{\label {sec:fdrcontrol} Adaptive Control of FDR\relax }{lemma.6}{}} \newlabel{eq:gamma}{{4.3}{315}{\label {sec:fdrcontrol} Adaptive Control of FDR\relax }{equation.4.3}{}} \newlabel{sec:simresults}{{5}{315}{\label {sec:simresults} Simulation Results\relax }{section.5}{}} \@writefile{toc}{\contentsline {section}{\numberline {5} Simulation Results}{315}{section.5}} \newlabel{sec:simnorm}{{5.1}{316}{\label {sec:simnorm} Simulation Results for Normal Model\relax }{subsection.5.1}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {5.1} Simulation Results for Normal Model}{316}{subsection.5.1}} \newlabel{eq:storeybias}{{5.1}{316}{\label {sec:simnorm} Simulation Results for Normal Model\relax }{equation.5.1}{}} \newlabel{eq:storeyvar}{{5.2}{316}{\label {sec:simnorm} Simulation Results for Normal Model\relax }{equation.5.2}{}} \@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Bias and $\sqrt {\unhbox \voidb@x \hbox {MSE}}$ of $\unhbox \voidb@x \hbox {$\setbox \z@ \hbox {\frozen@everymath \@emptytoks \mathsurround \z@ $\textstyle \pi $}\mathaccent "0362{\pi }$}_0$ for ST, HCK, TS and EM Estimators for Normal Model (Data Generated by Normal Model with Fixed $\unhbox \voidb@x \hbox {$\delta $}$)}}{317}{figure.1}} \newlabel{fig:pi0biasmsefixdelta}{{1}{317}{\label {fig:pi0biasmsefixdelta} Bias and $\sqrt {\mse }$ of $\hpi _0$ for ST, HCK, TS and EM Estimators for Normal Model (Data Generated by Normal Model with Fixed $\del $)\relax }{figure.1}{}} \@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Bias and $\sqrt {\unhbox \voidb@x \hbox {MSE}}$ of $\unhbox \voidb@x \hbox {$\setbox \z@ \hbox {\frozen@everymath \@emptytoks \mathsurround \z@ $\textstyle \unhbox \voidb@x \hbox {FDR}$}\mathaccent "0362{\unhbox \voidb@x \hbox {FDR}}$}$ for ST, HCK, TS and EM Estimators for Normal Model (Data Generated by Normal Model with Fixed $\unhbox \voidb@x \hbox {$\delta $}$)}}{317}{figure.2}} \newlabel{fig:fdrbiasmsefixdelta}{{2}{317}{\label {fig:fdrbiasmsefixdelta} Bias and $\sqrt {\mse }$ of $\hfdr $ for ST, HCK, TS and EM Estimators for Normal Model (Data Generated by Normal Model with Fixed $\del $)\relax }{figure.2}{}} \@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces Bias and $\sqrt {\unhbox \voidb@x \hbox {MSE}}$ of $\unhbox \voidb@x \hbox {$\setbox \z@ \hbox {\frozen@everymath \@emptytoks \mathsurround \z@ $\textstyle \pi $}\mathaccent "0362{\pi }$}_0$ for ST, HCK, TS and EM Estimators for Normal Model (Data Generated by Normal Model with Random $\unhbox \voidb@x \hbox {$\delta $}$)}}{318}{figure.3}} \newlabel{fig:pi0biasmserandelta}{{3}{318}{\label {fig:pi0biasmserandelta} Bias and $\sqrt {\mse }$ of $\hpi _0$ for ST, HCK, TS and EM Estimators for Normal Model (Data Generated by Normal Model with Random $\del $)\relax }{figure.3}{}} \@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Bias and $\sqrt {\unhbox \voidb@x \hbox {MSE}}$ of $\unhbox \voidb@x \hbox {$\setbox \z@ \hbox {\frozen@everymath \@emptytoks \mathsurround \z@ $\textstyle \unhbox \voidb@x \hbox {FDR}$}\mathaccent "0362{\unhbox \voidb@x \hbox {FDR}}$}$ for ST, HCK, TS and EM Estimators for Normal Model (Data Generated by Normal Model with Random $\unhbox \voidb@x \hbox {$\delta $}$)}}{318}{figure.4}} \newlabel{fig:fdrbiasmserandelta}{{4}{318}{\label {fig:fdrbiasmserandelta} Bias and $\sqrt {\mse }$ of $\hfdr $ for ST, HCK, TS and EM Estimators for Normal Model (Data Generated by Normal Model with Random $\del $)\relax }{figure.4}{}} \@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces Bias and $\sqrt {\unhbox \voidb@x \hbox {MSE}}$ of $\unhbox \voidb@x \hbox {$\setbox \z@ \hbox {\frozen@everymath \@emptytoks \mathsurround \z@ $\textstyle \pi $}\mathaccent "0362{\pi }$}_0$ for ST, HCK, TS and EM Estimators for Normal Model (Data Generated by Beta Model)}}{319}{figure.5}} \newlabel{fig:pi0biasmsenormalbeta}{{5}{319}{\label {fig:pi0biasmsenormalbeta} Bias and $\sqrt {\mse }$ of $\hpi _0$ for ST, HCK, TS and EM Estimators for Normal Model (Data Generated by Beta Model)\relax }{figure.5}{}} \@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces Bias and $\sqrt {\unhbox \voidb@x \hbox {MSE}}$ of $\unhbox \voidb@x \hbox {$\setbox \z@ \hbox {\frozen@everymath \@emptytoks \mathsurround \z@ $\textstyle \unhbox \voidb@x \hbox {FDR}$}\mathaccent "0362{\unhbox \voidb@x \hbox {FDR}}$}$ for ST, HCK, TS and EM Estimators for Normal Model (Data Generated by Beta Model)}}{319}{figure.6}} \newlabel{fig:fdrbiasmsenormalbeta}{{6}{319}{\label {fig:fdrbiasmsenormalbeta} Bias and $\sqrt {\mse }$ of $\hfdr $ for ST, HCK, TS and EM Estimators for Normal Model (Data Generated by Beta Model)\relax }{figure.6}{}} \newlabel{sec:simbeta}{{5.2}{320}{\label {sec:simbeta} Simulation Results for Beta Model\relax }{subsection.5.2}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {5.2} Simulation Results for Beta Model}{320}{subsection.5.2}} \@writefile{lof}{\contentsline {figure}{\numberline {7}{\ignorespaces Bias and $\sqrt {\unhbox \voidb@x \hbox {MSE}}$ of $\unhbox \voidb@x \hbox {$\setbox \z@ \hbox {\frozen@everymath \@emptytoks \mathsurround \z@ $\textstyle \pi $}\mathaccent "0362{\pi }$}_0$ for ST, HCK, TS and EM Estimators for Beta Model (Data Generated by Beta Model)}}{320}{figure.7}} \newlabel{fig:pi0biasmsebeta}{{7}{320}{\label {fig:pi0biasmsebeta} Bias and $\sqrt {\mse }$ of $\hpi _0$ for ST, HCK, TS and EM Estimators for Beta Model (Data Generated by Beta Model)\relax }{figure.7}{}} \@writefile{lof}{\contentsline {figure}{\numberline {8}{\ignorespaces Bias and $\sqrt {\unhbox \voidb@x \hbox {MSE}}$ of $\unhbox \voidb@x \hbox {$\setbox \z@ \hbox {\frozen@everymath \@emptytoks \mathsurround \z@ $\textstyle \unhbox \voidb@x \hbox {FDR}$}\mathaccent "0362{\unhbox \voidb@x \hbox {FDR}}$}$ for ST, HCK, TS and EM Estimators for Beta Model (Data Generated by Beta Model)}}{321}{figure.8}} \newlabel{fig:fdrbiasmsebeta}{{8}{321}{\label {fig:fdrbiasmsebeta} Bias and $\sqrt {\mse }$ of $\hfdr $ for ST, HCK, TS and EM Estimators for Beta Model (Data Generated by Beta Model)\relax }{figure.8}{}} \@writefile{lof}{\contentsline {figure}{\numberline {9}{\ignorespaces Bias and $\sqrt {\unhbox \voidb@x \hbox {MSE}}$ of $\unhbox \voidb@x \hbox {$\setbox \z@ \hbox {\frozen@everymath \@emptytoks \mathsurround \z@ $\textstyle \pi $}\mathaccent "0362{\pi }$}_0$ for ST, HCK, TS and EM Estimators for Beta Model (Data Generated by Normal Model)}}{321}{figure.9}} \newlabel{fig:pi0biasmsebetanormal}{{9}{321}{\label {fig:pi0biasmsebetanormal} Bias and $\sqrt {\mse }$ of $\hpi _0$ for ST, HCK, TS and EM Estimators for Beta Model (Data Generated by Normal Model)\relax }{figure.9}{}} \newlabel{sec:example}{{6}{321}{\label {sec:example} Example\relax }{section.6}{}} \@writefile{toc}{\contentsline {section}{\numberline {6} Example}{321}{section.6}} \@writefile{lof}{\contentsline {figure}{\numberline {10}{\ignorespaces Bias and $\sqrt {\unhbox \voidb@x \hbox {MSE}}$ of $\unhbox \voidb@x \hbox {$\setbox \z@ \hbox {\frozen@everymath \@emptytoks \mathsurround \z@ $\textstyle \unhbox \voidb@x \hbox {FDR}$}\mathaccent "0362{\unhbox \voidb@x \hbox {FDR}}$}$ for ST, HCK, TS and EM Estimators for Beta Model (Data Generated by Normal Model)}}{322}{figure.10}} \newlabel{fig:fdrbiasmsebetanormal}{{10}{322}{\label {fig:fdrbiasmsebetanormal} Bias and $\sqrt {\mse }$ of $\hfdr $ for ST, HCK, TS and EM Estimators for Beta Model (Data Generated by Normal Model)\relax }{figure.10}{}} \@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces Estimates of NAEP Trial State Assessment }}{322}{table.1}} \newlabel{tab:exampleest}{{1}{322}{Estimates of NAEP Trial State Assessment \relax }{table.1}{}} \@writefile{lof}{\contentsline {figure}{\numberline {11}{\ignorespaces Histogram of the $p$-Values for the NAEP Data}}{323}{figure.11}} \newlabel{fig:pvaluehist}{{11}{323}{\label {fig:pvaluehist} Histogram of the $p$-Values for the NAEP Data\relax }{figure.11}{}} \newlabel{sec:summary}{{7}{323}{\label {sec:summary} Concluding Remarks\relax }{section.7}{}} \@writefile{toc}{\contentsline {section}{\numberline {7} Concluding Remarks}{323}{section.7}} \@writefile{lot}{\contentsline {table}{\numberline {2}{\ignorespaces Test Results for NAEP Trial State Assessment (Normal Model) }}{324}{table.2}} \newlabel{tab:exampletest}{{2}{324}{Test Results for NAEP Trial State Assessment (Normal Model) \relax }{table.2}{}} \@writefile{toc}{\contentsline {section}{Acknowledgment}{324}{section*.3}} \@writefile{toc}{\contentsline {section}{Appendix A}{324}{section*.5}} \bibcite{r1}{{1}{}{{}}{{}}} \newlabel{eq:fdrbetasolve1}{{A.1}{325}{Appendix A\relax }{equation.7.1}{}} \bibcite{r2}{{2}{}{{}}{{}}} \bibcite{r3}{{3}{}{{}}{{}}} \bibcite{r4}{{4}{}{{}}{{}}} \bibcite{r5}{{5}{}{{}}{{}}} \bibcite{r6}{{6}{}{{}}{{}}} \bibcite{r7}{{7}{}{{}}{{}}} \bibcite{r8}{{8}{}{{}}{{}}} \bibcite{r9}{{9}{}{{}}{{}}} \bibcite{r10}{{10}{}{{}}{{}}} \bibcite{r11}{{11}{}{{}}{{}}} \bibcite{r12}{{12}{}{{}}{{}}} \bibcite{r13}{{13}{}{{}}{{}}} \bibcite{r14}{{14}{}{{}}{{}}} \bibcite{r15}{{15}{}{{}}{{}}} \bibcite{r16}{{16}{}{{}}{{}}} \bibcite{r16b}{{17}{}{{}}{{}}} \bibcite{r17}{{18}{}{{}}{{}}} \bibcite{r18}{{19}{}{{}}{{}}} \bibcite{r19}{{20}{}{{}}{{}}} \bibcite{r20}{{21}{}{{}}{{}}} \bibcite{r20b}{{22}{}{{}}{{}}} \bibcite{r21}{{23}{}{{}}{{}}} \bibcite{r22}{{24}{}{{}}{{}}} \global\NAT@numberstrue \@writefile{toc}{\contentsline {section}{References}{326}{section*.7}}