%\setlength\topmargin{-0.25in} \setlength\oddsidemargin{-0.25in}
%\setlength\textheight{9.0in} \setlength\textwidth{7.0in}
%\setlength\columnsep{0.375in} \newlength\titlebox
%\setlength\titlebox{2.25in} \setlength\headheight{0pt}
%\setlength\headsep{0pt}

\setlength{\evensidemargin}{-0.25 in} \setlength{\oddsidemargin}{-0.25 in}
\setlength{\textwidth}{6.8 in} \setlength{\topmargin}{-0.75 in}
\setlength{\textheight}{9.3 in}

\renewcommand{\baselinestretch}{0.9}

%\renewcommand{\labelenumi}{\arabic{enumi}}
%\renewcommand{\labelenumii}{(\alph{enumii})}
%\renewcommand{\labelenumiii}{(\roman{enumiii})}

\makeatletter
\let\orig@Itemize =\itemize
\let\orig@Enumerate =\enumerate
\let\orig@Description =\description % Zero the vertical spacing parameters
\def\MySpacing{\itemsep=0.0ex\topsep=0.0ex\partopsep=0.0ex\parskip=0.0ex\parsep=0.0ex}
\def\WideSpacing{\itemsep=2.0ex\topsep=0.5ex\partopsep=0.0ex\parskip=0.0ex\parsep=0.0ex}
% Redefine the environments in terms of the original values
\renewenvironment{itemize}{\orig@Itemize\MySpacing}{\endlist}
\newenvironment{Myenumerate}{\orig@Enumerate\MySpacing}{\endlist}
\newenvironment{enumerateWide}{\orig@Enumerate\WideSpacing}{\endlist}
\renewenvironment{description}{\orig@Description\MySpacing}{\endlist}

\newenvironment{qenumerate}{\parindent 0in \orig@Enumerate}{\endlist}

\def\qitemize{%
  \ifnum \@itemdepth >\thr@@\@toodeep\else
    \advance\@itemdepth\@ne
    \edef\@itemitem{labelitem\romannumeral\the\@itemdepth}%
    \expandafter
    \list
      \csname\@itemitem\endcsname
%      {\def\makelabel##1{\hss\llap{##1}}}%
       {\def\makelabel##1{\llap{##1}}}%
  \fi}

\def\hsp{\vspace{0.1 in}}
\def\fsp{\vspace{0.2 in}}

\makeatother


\renewcommand\rmdefault{ppl}

\newcommand\textsclarge[1]{\fontshape{sc}\fontsize{14}{15} \selectfont{#1}}
\newcommand\textscnormal[1]{\fontshape{sc}\fontsize{12}{15} \selectfont{#1}}

%\newcommand{\mathsfsmall}{\fontshape{sf}\fontsize{12}{15} \selectfont{#1}}
%\def\transpose{{\mathsfsmall{T}}}
\DeclareOldFontCommand{\sf}{\small\it\sffamily}{\mathsf}
\def\transpose{\top}


\DeclareSymbolFont{AMSb}{U}{msb}{m}{n}
\DeclareMathSymbol{\N}{\mathbin}{AMSb}{"4E}
\DeclareMathSymbol{\Z}{\mathbin}{AMSb}{"5A}
\DeclareMathSymbol{\R}{\mathbin}{AMSb}{"52}
\DeclareMathSymbol{\Q}{\mathbin}{AMSb}{"51}
\DeclareMathSymbol{\I}{\mathbin}{AMSb}{"49}
\DeclareMathSymbol{\C}{\mathbin}{AMSb}{"43}

\def\argmax{\arg \max}
\def\argmin{\arg \min}
\def\One{\mathbbm{1}}
\def\One{\mathbb{1}}
\def\Zero{\mathbf{0}}

\def\Student{\text{Student}}
\def\FisherF{\text{Fisher}}


\def\betahat{\widehat{\beta}}
\def\X{\mathbb{X}}
\def\CalC{\mathscr{C}}
\def\CalH{\mathcal{H}}
\def\CalY{\mathcal{Y}}
\def\calY{\mathcal{Y}}
\def\calX{\mathcal{X}}

\def\calA{\mathscr{A}}
\def\calAN{\mathcal{A N}}

\def\calB{\mathscr{B}}
\def\calE{\mathscr{E}}

\def\calN{\mathcal{N}}


\def\Pr{\text{Pr}}
\def\P{\text{P}}

\def\Expect{\mathbbmss{{E}}}
\def\E{\mathbb{E}}
\def\Var{\text{Var}}

\def\KL{\mathbbmss{{K}}}
\def\Hell{\mathbbmss{{H}}}

\def\Cov{\text{Cov}}
\def\Corr{\text{Corr}}

\def\Ind{\mathbb{{I}}}
\def\Ident{\mathbf{{I}}}

\def\Fbar{\overline{F}}

%\def\dsum#1#2{\displaystyle{\sum_{#1}^{#2}}~}
%\def\dint#1#2{\displaystyle{\int_{#1}^{#2}}}
\def\Prob#1{\mbox{P}\left (~#1~\right )}
\def\Exp#1#2{\mbox{E}_{#1} [~#2~]}
\def\bigchoose#1#2{\displaystyle{#1 \choose #2}}

\newcommand{\undertilde}[1]{\underset{\widetilde{}}{#1}}

\newcommand\Normal{\mathcal{N}}

\def\by{\bm{y}}
\def\bx{\mathbf{x}}
\def\bY{\mathbf{Y}}
\def\bX{\mathbf{X}}
\def\bH{\mathbf{H}}
\def\bV{\mathbf{V}}
\def\bM{\mathbf{M}}
\def\bt{\mathbf{t}}
\def\ba{\mathbf{a}}
\def\bg{\mathbf{g}}

\def\bz{\mathbf{z}}
\def\bZ{\mathbf{Z}}
\def\muvec{\mathbf{\mu}}

\def\Xvecn{{\mathbf{X_n}}}
\def\xvec{{\mathbf{x}}}
\def\xunder{{\utilde{x}}}
\newcommand\Xvec{\mathbf{X}}
\newcommand\Yvec{\mathbf{Y}}
\def\yvec{{\bm{y}}}
\def\tvec{{\undertilde{t}}}
\def\gvec{{\utilde{g}}}
\def\thvec{{\mathbf{\theta}}}
\def\phivec{{\mathbf{\phi}}}
\def\etavec{{\utilde{\eta}}}
\def\wvec{{\undertilde{w}}}
\def\Svec{{\undertilde{S}}}
\def\zerovec{{\mathbf{0}}}
\def\Psimat{{\bm{\Psi}}}
\def\avec{{\undertilde{a}}}

\def\Xveca{{\mathbf{X}}_1}
\def\Xvecb{{\mathbf{X}}_2}
\def\xveca{{\mathbf{x}}_1}
\def\xvecb{{\mathbf{x}}_2}
\def\mvec{{\mathbf{m}}}
\def\cvec{{\mathbf{c}}}


\def\dotb{\dot{b}}
\def\ddotb{\ddot{b}}
\def\dotg{\dot{g}}
\def\ddotg{\ddot{g}}
\def\dotf{\dot{f}}
\def\ddotf{\ddot{f}}

\newcommand\Xbarn{{\overline{X}_n}}


\newcommand\pivec{{\mathbf{\pi}}}

\def\lra{\longrightarrow}
\def\Lra{\Longrightarrow}
\def\Llra{\Longleftrightarrow}


\def\CiP{\stackrel{p}{\lra}}
\def\CiR{\stackrel{r}{\lra}}
\newcommand\CRM[1]{\stackrel{r = #1}{\lra}}
\newcommand\CiM[1]{\stackrel{#1}{\lra}}
\newcommand\Cqm{\stackrel{q.m.}{\lra}}
\def\Cas{\stackrel{a.s.}{\lra}}
\def\CiL{\stackrel{\mathfrak{L}}{\lra}}
\def\eqdef{\stackrel{def}{=}}
\def\CiD{\stackrel{d}{\lra}}
\def\Cae{\stackrel{a.e.}{\lra}}
\def\Cwp{\stackrel{w.p. 1}{\lra}}

\def\nlra{\nrightarrow}

\def\nCas{\stackrel{a.s.}{\nlra}}
\def\nCiP{\stackrel{p}{\nlra}}


\def\approxsim{\; \; \raisebox{-0.3ex}{.} \\ \hspace{-0.11in} \sim \\ \hspace{-0.11in} \raisebox{0.95ex}{.} \; \; }

\newcommand*{\approxdist}{\mathrel{\vcenter{\offinterlineskip
\vskip-.25ex\hbox{\hskip.55ex$\cdot$}\vskip-.25ex\hbox{$\sim$}
\vskip-.5ex\hbox{\hskip.55ex$\cdot$}}}}



\def\nlranew{{\; \; \raisebox{-0.3ex}{.} \\ \hspace{-0.11in} \sim \\ \hspace{-0.105in} \raisebox{0.95ex}{.} \; \;
}}

\def\eqdist{\stackrel{d}{=}}


\newcommand\Xvecsub[1]{{\mathbf{X}}_{\! #1}}
\newcommand\Zvecsub[1]{{\mathbf{Z}}_{\! #1}}
\newcommand\Zvec{\mathbf{Z}}
\newcommand\Uvec{\mathbf{U}}

\newcommand\Xbarvecn{{\overline{\mathbf{X}}_{\!n}}}

\def\Xmean{\overline{X}}
\def\xmean{\overline{x}}


\def\O{\text{O}}
\def\o{\text{o}}

\def\expit{\text{expit}}
\def\logit{\text{logit}}

\def\EF{Exponential Family }

\def\tildethn{\skew2\widetilde{\thvec}_n}
\def\hatthn{\skew2\hat{\thvec}_n}
\def\tildephin{\skew2\widetilde{\bmphi}_n}
\def\Fisher{\mathcal{I}}
\def\FisherI{\mathcal{I}}
\def\FisherItilde{\widetilde{\mathcal{I}}}
\def\FisherIhat{\widehat{I}_n}
\def\FisherIhatzzo{\widehat{I}_{n00.1}}
\def\FisherIhatzz{\widehat{I}_{n00}}
\def\FisherIhatzo{\widehat{I}_{n01}}
\def\FisherIhatoz{\widehat{I}_{n10}}
\def\FisherIhatoo{\widehat{I}_{n11}}

\def\tildethetan{{\widetilde{\theta}_n}}
\def\Xseq{X_1,\ldots,X_n}
\def\xseq{x_1,\ldots,x_n}
\def\Yseq{Y_1,\ldots,Y_n}
\def\yseq{y_1,\ldots,y_n}
\def\Zseq{Z_1,\ldots,Z_n}
\def\zseq{z_1,\ldots,z_n}
\def\lvec{{\mathbf{l}}}
\def\lvecn{{\mathbf{l}_n}}
\def\dotlvec{{\mathbf{\dot{l}}}}
\def\dotlvecn{{{\dot{l}}_n}}
\def\ddotlvecn{{{\ddot{l}}_n}}
\def\dddotlvecn{{{\dddot{l}}_n}}
\def\thvec{{\mathbf{\theta}}}
\def\thveczero{{\mathbf{\theta}_0}}
\def\thvecone{{\mathbf{\theta}_1}}
\def\thvectwo{{\mathbf{\theta}_2}}
\def\phivec{{\mathbf{\phi}}}
\def\etavec{{\undertilde{\eta}}}
\def\betavec{{\mathbf{\beta}}}
\def\omegavec{{\mathbf{\omega}}}
\def\wvec{{\undertilde{w}}}
\def\Svec{{\undertilde{S}}}

\def\Psimat{{\bm{\Psi}}}
\def\avec{{\undertilde{a}}}
\def\tildethn{\skew2\widetilde{\thvec}_n}
\def\hatthn{\skew2\hat{\thvec}_n}
\def\tildephin{\skew2\widetilde{\bmphi}_n}

\def\thnstar{\bm{\theta}_n^\star}

\def\thtilde{\widetilde{\theta}_n}
\def\thhat{\widehat{\theta}_n}
\def\thhatzero{\widehat{\theta}_{n0}}
\def\thhatone{\widehat{\theta}_{n1}}
%\def\dsum#1#2{\displaystyle{\sum_{#1}^{#2}}~}
%\def\dint#1#2{\displaystyle{\int_{#1}^{#2}}}
\def\thetahat{\widehat{\theta}}
\def\sigmahat{\widehat{\sigma}}
\def\tauhat{\widehat{\tau}}
\def\lambdahat{\widehat{\lambda}}


\def\xbar{\overline{x}}

\def\d{\;\text{d}}

\def\Zhat{\widehat Z}

\def\calS{\mathcal{S}}
\def\calR{\mathcal{R}}
\def\pihat{\widehat{\pi}}
\def\Rpack{\texttt{R}}
\def\thetavec{\boldsymbol{\theta}}
\def\bu{\mathbf{u}}
\def\Ihat{\widehat{I}}
\def\Zhat{\widehat{Z}}

\def\pitilde{\widetilde{\pi}}
\def\ptilde{\widetilde{p}}
\def\ybar{\overline{y}}
\def\phat{\widehat{p}}
\def\Lhat{\widehat{\mathcal{L}}}

\def\Post{\pi}
\def\Prop{q}

\def\btheta{\boldsymbol{\theta }}
\def\balpha{\boldsymbol{\alpha }}
\def\bbeta{\boldsymbol{\beta }}
\def\bvarphi{\boldsymbol{\varphi }}
\def\bx{\mathbf{x}}
\def\by{\mathbf{y}}
\def\bu{\mathbf{u}}
\def\bv{\mathbf{v}}
\def\bg{\mathbf{g}}
\def\bh{\mathbf{h}}
\def\bt{\mathbf{t}}
\def\bz{\mathbf{z}}
\def\bc{\mathbf{c}}

\def\bA{\mathbf{A}}
\def\bW{\mathbf{W}}
\def\bB{\mathbf{B}}
\def\bX{\mathbf{X}}

\def\R{\mathbb{R}}

\def\dint{\displaystyle{\int}}
\def\dsum{\displaystyle{\sum}}

\def\pitilde{\widetilde{\pi}}
\def\ptilde{\widetilde{p}}
\def\qtilde{\widetilde{q}}

\def\bomega{\mathbf{\omega}}
\def\bkappa{\mathbf{\kappa}}

\def\calX{\mathcal{X}}

\newcommand\IndState[1][1]{\State\hspace{#1\algorithmicindent}}

\def\ftilde{\widetilde{f}}

\def\Dset{\mathcal{D}}
\def\cald{\delta}

\newcommand{\independent}{\perp\mkern-9.5mu\perp}
\newcommand{\notindependent}{\slashed{\independent}}

\def\rz{\textcolor{red}{\textsf{z}}}
\def\ra{\textcolor{red}{\textsf{a}}}
\def\rZ{\textcolor{red}{\textsf{Z}}}
\def\rx{\textcolor{red}{\textsf{x}}}
\newcommand\rxa[1]{\textcolor{red}{\textsf{x}_{#1}}}
\newcommand\rza[1]{\textcolor{red}{\textsf{z}_{#1}}}

\def\rtmt{\textcolor{red}{1}}
\def\rnotmt{\textcolor{red}{0}}

\def\binst{\textcolor{blue}{1}}
\def\bnoinst{\textcolor{blue}{0}}

\def\cinst{\textcolor{cyan}{1}}
\def\cnoinst{\textcolor{cyan}{0}}

\def\bw{\textcolor{blue}{\textsf{w}}}
\def\cm{\textcolor{cyan}{\textsf{m}}}
\def\rzast{\textcolor{red}{\textsf{z}^\ast}}


%\DeclareMathOperator*{\plim}{\textrm{plim}}

\def\true{{\scaleto{\text{TRUE}}{3pt}}}
\def\eff{{\scaleto{\text{eff}}{5pt}}}
\def\OR{{\scaleto{\text{OR}}{3pt}}}
\def\MATCH{{\scaleto{\text{MATCH}}{3pt}}}
\def\IPW{{\scaleto{\text{IPW}}{3pt}}}
\def\AIPW{{\scaleto{\text{AIPW}}{3pt}}}
\def\AOR{{\scaleto{\text{AOR}}{3pt}}}
\def\Gest{{\scaleto{\text{G}}{3pt}}}
\def\eff{{\scaleto{\text{eff}}{6pt}}}
\def\res{{\scaleto{\text{res}}{3pt}}}

\def\bD{\mathbf{D}}
\def\bC{\mathbf{C}}


\def\given{ \ | \ }

\def\Pa{\text{P}\scaleto{\text{A}}{5pt}}

\def\ps{\text{e}}

\def\calI{\mathcal{I}}
\def\calJ{\mathcal{J}}
\def\calH{\mathcal{H}}
\def\calT{\mathcal{T}}
\def\calU{\mathcal{U}}

\def\avar{\text{avar}}
\def\var{\text{var}}