
\usepackage[utf8x]{inputenc}
\usepackage{mathrsfs}
\usepackage{amssymb}
\usepackage{amsfonts}
\usepackage{amsmath}
\usepackage{amsthm,verbatim}
%\usepackage[sort]{natbib}
\usepackage{wrapfig}
\usepackage{multirow}
\usepackage{longtable}
\usepackage{enumerate}
\usepackage{color, graphicx}
\usepackage{tikz}

\def\th{^{th}}

%b stands for bold, B stands for cap bold, m stands matchcal, t stands for tilde

\newcommand{\Del}[1]{{\color{red} {\sout{{\color{green} #1}}}}}
\newcommand{\Add}[1]{{\color{blue}  \bf {\color{blue} #1}}}
\newcommand{\red}{\textcolor{black}}
\newcommand{\blue}{\textcolor{blue}}
\newcommand{\green}{\textcolor{green}}

\newcommand{\mpi}{\cal \pi}

\newcommand{\bA}{{\boldsymbol A}}
\newcommand{\bC}{{\boldsymbol C}}
\newcommand{\bE}{{\boldsymbol E}}
\newcommand{\bG}{{\boldsymbol G}}
\newcommand{\bH}{{\boldsymbol H}}
\newcommand{\bM}{{\boldsymbol M}}
\newcommand{\bI}{{\boldsymbol I}}
\newcommand{\bP}{{\boldsymbol P}}
\newcommand{\bQ}{{\boldsymbol Q}}
\newcommand{\bR}{{\boldsymbol R}}
\newcommand{\bT}{{\boldsymbol T}}
\newcommand{\bU}{{\boldsymbol U}}
\newcommand{\bV}{{\boldsymbol V}}
\newcommand{\bW}{{\boldsymbol W}}
\newcommand{\bX}{{\boldsymbol X}}
\newcommand{\bY}{{\boldsymbol Y}}
\newcommand{\bZ}{{\boldsymbol Z}}

\newcommand{\bSigma}{{\boldsymbol \Sigma}}
\newcommand{\bLambda}{{\boldsymbol \Lambda}}
\newcommand{\bOmega}{{\boldsymbol \Omega}}
\newcommand{\boldeta}{{\boldsymbol \eta}}

\newcommand{\bOnes}{{\boldsymbol 1}}
\newcommand{\bZeros}{{\boldsymbol 0}}

\newcommand{\ba}{{\boldsymbol a}}
\newcommand{\bb}{{\boldsymbol b}}
\newcommand{\bc}{{\boldsymbol c}}
\newcommand{\bd}{{\boldsymbol d}}
\newcommand{\be}{{\boldsymbol e}}
\newcommand{\bbf}{{\boldsymbol f}}
\newcommand{\bg}{{\boldsymbol g}}
\newcommand{\bp}{{\boldsymbol p}}
\newcommand{\br}{{\boldsymbol r}}
\newcommand{\bs}{{\boldsymbol s}}
\newcommand{\bt}{{\boldsymbol t}}
\newcommand{\bu}{{\boldsymbol u}}
\newcommand{\bv}{{\boldsymbol v}}
\newcommand{\bw}{{\boldsymbol w}}
\newcommand{\bx}{{\boldsymbol x}}
\newcommand{\by}{{\boldsymbol y}}
\newcommand{\bz}{{\boldsymbol z}}

\newcommand{\BN}{\mathbb{N}}
\newcommand{\BR}{\mathbb{R}}
\newcommand{\BE}{\mathbb{E}}
\newcommand{\BI}{\mathbb{I}}
\newcommand{\BP}{\mathbb{P}}
\newcommand{\BT}{\mathbb{T}}
\newcommand{\BX}{\mathbb{X}}

\newcommand{\mA}{{\cal A}}
\newcommand{\mB}{{\cal B}}
\newcommand{\mC}{{\cal C}}
\newcommand{\mE}{{\cal E}}
\newcommand{\mF}{{\cal F}}
\newcommand{\mH}{{\cal H}}
\newcommand{\mI}{{\cal I}}
\newcommand{\mK}{{\cal K}}
\newcommand{\mL}{{\cal L}}
\newcommand{\mM}{{\cal M}}
\newcommand{\mP}{{\cal P}}
\newcommand{\mS}{{\cal S}}
\newcommand{\mU}{{\cal U}}
\newcommand{\mV}{{\cal V}}
\newcommand{\mW}{{\cal W}}
\newcommand{\mX}{{\cal X}}
\newcommand{\mY}{{\cal Y}}



%\newcommand{\var}{{\mbox{Var}}}
\newcommand{\cov}{{\mbox{Cov}}}
\newcommand{\sZ}{{Z_{\xi}}}

\newcommand{\balpha}{{\boldsymbol \alpha}}
\newcommand{\bbeta}{{\boldsymbol \beta}}
\newcommand{\bgamma}{{\boldsymbol \gamma}}
\newcommand{\btheta}{{\boldsymbol \theta}}
\newcommand{\bepsilon}{{\boldsymbol \epsilon}}
\newcommand{\bvarepsilon}{{\boldsymbol \varepsilon}}
\newcommand{\blambda}{{\boldsymbol \lambda}}
\newcommand{\bmu}{{\boldsymbol \mu}}
\newcommand{\bnu}{{\boldsymbol \nu}}
\newcommand{\bxi}{{\boldsymbol \xi}}
\newcommand{\bpi}{{\boldsymbol \pi}}
\newcommand{\brho}{{\boldsymbol \rho}}
\newcommand{\bpsi}{{\boldsymbol \psi}}
\newcommand{\bomega}{{\boldsymbol \omega}}

\newcommand{\bGamma}{{\boldsymbol \Gamma}}

\newcommand{\tvarsigma}{\widetilde{\varsigma}}
\newcommand{\seta}{\sigma_{\eta}}
\newcommand{\mFdr}{\mbox{Fdr}}
\newcommand{\utwi}[1]{\mbox{\boldmath $#1$}}
\def\boxit#1{\vbox{\hrule\hbox{\vrule\kern6pt
          \vbox{\kern6pt#1\kern6pt}\kern6pt\vrule}\hrule}}
%\setlength{\parskip}{3mm}

\newtheorem{assumption}{Assumption}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{definition}{Definition}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{formula}{Formula}[section]
\newtheorem{proposition}{Proposition}[section]
\newtheorem{example}{Example}[section]

\newtheorem{remark}{Remark}

%%%%%%%%%%%% Journal definitions %%%%%%%%%%%%%%%%%%%%%%%%%
\def\AJE{{\it American Journal of Epidemiology}}
\def\ANNALS{{\it Annals of Statistics}}
\def\BIOK{{\it Biometrika}}
\def\BMCS{{\it Biometrics}}
\def\STATMED{{\it Statistics in Medicine}}
\def\STATSCI{{\it Statistical Science}}
\def\JSPI{{\it Journal of Statistical Planning \& Inference}}
\def\JRSSB{{\it Journal of the Royal Statistical Society, Series B}}
\def\COMMS{{\it Communications in Statistics, Theory \& Methods}}
\def\JQT{{\it Journal of Quality Technology}}
\def\TECH{{\it Technometrics}}
\def\JASA{{\it Journal of the American Statistical Association}}
\def\CSDA{{\it Computational Statistics \& Data Analysis}}
\def\JCGS{{\it Journal of Computational and Graphical Statistics}}
\def\JCB{{\it Journal of Computational Biology}}
\def\BIOINF{{\it Bioinformatics}}
\def\JAMA{{\it Journal of the American Medical Association}}
\def\JNUTR{{\it Journal of Nutrition}}
\def\LETTERS{{\it Letters in Probability and Statistics}}
\def\JABES{{\it Journal of Agricultural, Biological and
                      Environmental Statistics}}
\def\TECH{{\it Technometrics}}
\def\COMMS{{\it Communications in Statistics, Series A}}
\def\SCAN{{\it Scandinavian Journal of Statistics}}
\def\PNAS{{\it Proceedings of the National Academy of Sciences USA}}

\def\c{\textcolor{red}}
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