probstat.tex

\usepackage{amsmath,amssymb}
\usepackage{dsfont}
\usepackage{cancel}
\usepackage{graphicx}
\usepackage{xargs}
\usepackage{xspace}

% =============================================================================
%                                  Formatting
% =============================================================================

% Make a note on the margin.
\newcommand{\marnote}[1]{
  \reversemarginpar
  \marginpar[\raggedleft\footnotesize\textit{\\[3ex]#1}]%
      {\raggedright\footnotesize\textit{\\[3ex]#1}}
  \normalmarginpar
}

\newcommand{\pwiseii}[1]{\ensuremath{\left\{\begin{array}{ll}#1\end{array}}}
\newcommand{\pwiseiii}[1]{\ensuremath{\left\{\begin{array}{ll}#1\end{array}}}
\newcommand{\prn}[1]{\ensuremath{\left(#1\right)}}
\newcommand{\brk}[1]{\ensuremath{\left[#1\right]}}
\newcommand{\brc}[1]{\ensuremath{\left\{#1\right\}}}
\newcommand{\x}[1]{\ensuremath{\cancel{#1}}}

% =============================================================================
%                                 General Math
% =============================================================================

% Special functions and operators
\DeclareMathOperator{\erf}{erf}
\DeclareMathOperator{\logit}{logit}
\DeclareMathOperator{\sign}{sign}
\DeclareMathOperator*{\argmin}{\arg\!\min}

% Definitions
\def\define{:=}
\def\defined{=:}
\def\eqdef{\triangleq}

% Proofs
\def\qed{\ifhmode\unskip\nobreak\fi\hfill \ensuremath{\square}}

% Standard transformation function
\def\transform{\ensuremath{\varphi}\xspace}

% Logic
\newcommand{\comp}[1]{\neg{#1}}
\newcommand{\imp}{\ensuremath{\;\Longrightarrow\;}}
\newcommand{\pmi}{\ensuremath{\;\Longleftarrow\;}}
\newcommand{\nimp}{\ensuremath{\;\not\!\!\Longrightarrow\;}}
\newcommand{\npmi}{\ensuremath{\;\not\!\!\Longleftarrow\;}}
\newcommand{\eqv}{\ensuremath{\;\Longleftrightarrow\;}}

% Numbers.
\def\C{\mathbb{C}}
\def\N{\mathbb{N}}
\def\R{\mathbb{R}}
\def\Z{\mathbb{Z}}

% Matrices
\newcommand{\eyeii}{\ensuremath{\left(\begin{matrix}1 & 0 \\ 0 & 1\end{matrix}\right)}}
\newcommand{\eyeiii}{\ensuremath{\left(\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{matrix}\right)}}

% Limits
\newcommand{\Lim}[2]{\ensuremath{\lim_{#1\to #2}}}
\newcommand{\limx}[1][\infty]{\ensuremath{\lim_{x\to #1}}}
\newcommand{\limn}[1][\infty]{\ensuremath{\lim_{n\to #1}}}

% Sums and products
\newcommand{\Sum}[2][i=1]{\ensuremath{\sum_{#1}^{#2}}}
\newcommand{\sumin}{\ensuremath{\sum_{i=1}^n}}
\newcommand{\sumiN}{\ensuremath{\sum_{i=1}^N}}
\newcommand{\sumim}{\ensuremath{\sum_{i=1}^m}}
\newcommand{\sumjk}{\ensuremath{\sum_{j=1}^k}}
\newcommand{\sumjn}{\ensuremath{\sum_{j=1}^n}}
\newcommand{\sumjm}{\ensuremath{\sum_{j=1}^m}}
\newcommand{\isum}[1][n]{\ensuremath{\sum_{#1}^\infty}}
\newcommand{\dsum}[4][i=1]{\ensuremath{\sum_{#1}^{#2}\sum_{#3}^{#4}}}
\newcommand{\Prod}[2][i=1]{\ensuremath{\prod_{#1}^{#2}}}
\newcommand{\prodin}{\ensuremath{\prod_{i=1}^n}}
\newcommand{\prodjn}{\ensuremath{\prod_{j=1}^n}}

% Derivatives
\newcommand{\der}[2][]{\ensuremath{\frac{d #1}{d #2}}}
\newcommand{\dder}[2][]{\ensuremath{\frac{d^2 #1}{d #2^2}}}
\newcommand{\pder}[2][]{\ensuremath{\frac{\partial #1}{\partial #2}}}
\newcommand{\pdder}[2][]{\ensuremath{\frac{\partial^2 #1}{\partial #2^2}}}
\newcommand{\mpder}[3][]{%
  \ensuremath{\frac{\partial^2 #1}{\partial #2 \partial #3}}}

% Differentials
%\renewcommand{\d}[1]{\,\mathrm{d}#1}
\renewcommand{\d}[1]{\,d#1}
\def\ds{\d{s}}
\def\dt{\d{t}}
\def\dtheta{\d{\theta}}
\def\du{\d{u}}
\def\dx{\d{x}}
\def\dy{\d{y}}
\def\dfx{\d{F_X(x)}}
\def\dfy{\d{F_Y(y)}}
\def\dfhatx{\d{\widehat{F}_n(x)}}

% Transcendentals w/ extended arguments.
\newcommand{\Exp}[1]{\ensuremath{\exp\left\{#1\right\}}}
\newcommand{\Log}[1]{\ensuremath{\log\left\{#1\right\}}}

% =============================================================================
%                          Probability and Statistics
% =============================================================================

% Formatted terminology.
\def\bias{\textsf{bias}\xspace}
\def\se{\textsf{se}\xspace}
\def\pdf{\textsc{pdf}\xspace}
\def\cdf{\textsc{cdf}\xspace}
\def\ise{\textsc{ise}\xspace}
\def\pgf{\textsc{pgf}\xspace}
\def\mgf{\textsc{mgf}\xspace}
\def\mse{\textsc{mse}\xspace}
\def\mspe{\textsc{mspe}\xspace}
\def\mle{\textsc{mle}\xspace}
\def\mom{\textsc{mom}\xspace}
\def\are{\textsc{are}\xspace}
\def\rss{\textsc{rss}\xspace}
\def\ess{\textsc{ess}\xspace}
\def\tss{\textsc{tss}\xspace}

% Naming shortcuts.
\def\ahat{\ensuremath{\widehat{\alpha}}}
\def\atil{\ensuremath{\tilde{\alpha}}}
\def\bhat{\ensuremath{\widehat{\beta}}}
\def\btil{\ensuremath{\tilde{\beta}}}
\def\dhat{\ensuremath{\widehat{\delta}}}
\def\ehat{\ensuremath{\hat{\epsilon}}}
\def\ghat{\ensuremath{\widehat{\gamma}}}
\def\khat{\ensuremath{\widehat{\kappa}}}
\def\lhat{\ensuremath{\widehat{\lambda}}}
\def\ltil{\ensuremath{\tilde{\lambda}}}
\def\mhat{\ensuremath{\widehat{\mu}}}
\def\nhat{\ensuremath{\widehat{\nu}}}
\def\mtil{\ensuremath{\tilde{\mu}}}
\def\psihat{\ensuremath{\widehat{\psi}}}
\def\shat{\ensuremath{\widehat{\sigma}}}
\def\stil{\ensuremath{\tilde{\sigma}}}
\def\that{\ensuremath{\widehat{\theta}}}
\def\ttil{\ensuremath{\widetilde{\theta}}}
\def\rhohat{\widehat{\rho}}
\def\xihat{\widehat{\xi}}

\def\sehat{\ensuremath{\widehat{\se}}}
\def\fhat{\ensuremath{\widehat{f}}}
\def\Fhat{\ensuremath{\widehat{F}}}
\def\fnhat{\ensuremath{\widehat{f}_n}}
\def\Fnhat{\ensuremath{\widehat{F}_n}}
\def\Jhat{\ensuremath{\widehat{J}}}
\def\phat{\ensuremath{\widehat{p}}}
\def\ptil{\ensuremath{\tilde{p}}}
\def\rhat{\widehat{r}}
\def\Rbar{\bar{R}}
\def\Rhat{\widehat{R}}
\def\Qbar{\bar{Q}}
\def\Qhat{\widehat{Q}}
\def\Xhat{\widehat{X}}
\def\xbar{\bar{x}}
\def\Xbar{\bar{X}}
\def\Xsqbar{\overline{X^2}}
\def\xnbar{\overline{x}_n}
\def\Xnbar{\overline{X}_n}
\def\Yhat{\widehat{Y}}
\def\ybar{\overline{y}}
\def\Ybar{\overline{Y}}
\def\Ynbar{\overline{Y}_n}

% Random variables.
\def\rv{\textsc{rv}\xspace}
\def\iid{\ensuremath{\textsc{iid}}\xspace}
\def\dist{\ensuremath{\sim}\xspace}
\def\disteq{\ensuremath{\stackrel{D}{=}}\xspace}
\def\distiid{\ensuremath{\stackrel{iid}{\sim}}\xspace}
\def\ind{\ensuremath{\perp\!\!\!\perp}\xspace}
\def\nind{\ensuremath{\perp\!\!\!\!\big\vert\!\!\!\!\perp}\xspace}
\def\Xon{\ensuremath{X_1,\dots,X_n}\xspace}
\def\xon{\ensuremath{x_1,\dots,x_n}\xspace}
\def\giv{\ensuremath{\,|\,}}
\def\Giv{\ensuremath{\,\big|\,}}
\def\GIV{\ensuremath{\,\Big|\,}}
\newcommand{\indicator}[1]{\mathds{1}_{\left\{#1\right\}}}

% Probability, expectation, and variance.
\def\prob{\mathbb{P}}
\renewcommand{\Pr}[2][]{\ensuremath{\prob_{#1}\left[#2\right]}\xspace}
\newcommand{\E}[2][]{\ensuremath{\mathbb{E}_{#1}\left[#2\right]}}
\newcommand{\V}[2][]{\ensuremath{\mathbb{V}_{#1}\left[#2\right]}}
\newcommand{\cov}[2][]{\ensuremath{\mathrm{Cov}_{#1}\left[#2\right]}}
\newcommand{\corr}[2][]{\ensuremath{\rho_{#1}\left[#2\right]}}
\def\sd{\ensuremath{\textsf{sd}}\xspace}
\def\samplemean{\ensuremath{\bar{X}_n}\xspace}
\def\samplevar{\ensuremath{S^2}\xspace}
\def\za{\ensuremath{z_{\alpha}}}
\def\zat{\ensuremath{z_{\alpha/2}}}

% Inference
\def\Ll{\ensuremath{\mathcal{L}}\xspace}
\def\Lln{\ensuremath{\Ll_n}\xspace}
\def\ll{\ensuremath{\ell}}
\def\lln{\ensuremath{\ll_n}}

% Hypothesis testing
\newcommand{\hyp}[2]{
\ensuremath{H_0:#1 \ifhmode\quad\text{versus}\quad\fi\text{ vs. } H_1:#2}}

% Convergence.
\def\conv{\rightarrow}
\def\convinf{\rightarrow_{n\to\infty}}
\def\pconv{\stackrel{\text{\tiny{P}}}{\rightarrow}}
\def\npconv{\stackrel{\text{\tiny{P}}}{\nrightarrow}}
\def\dconv{\stackrel{\text{\tiny{D}}}{\rightarrow}}
\def\ndconv{\stackrel{\text{\tiny{D}}}{\nrightarrow}}
\def\qmconv{\stackrel{\text{\tiny{qm}}}{\rightarrow}}
\def\nqmconv{\stackrel{\text{\tiny{qm}}}{\nrightarrow}}
\def\asconv{\stackrel{\text{\tiny{as}}}{\rightarrow}}
\def\nasconv{\stackrel{\text{\tiny{as}}}{\nrightarrow}}

%
% Distributions
%
\newcommandx{\unif}[1][1={a,b}]{\textrm{Unif}\left({#1}\right)}
\newcommandx{\unifd}[1][1={a,\ldots,b}]{\textrm{Unif}\left\{{#1}\right\}}
\newcommandx{\dunif}[3][1=x,2=a,3=b]{\frac{I(#2<#1<#3)}{#3-#2}}
\newcommandx{\dunifd}[3][1=x,2=a,3=b]{\frac{I(#2<#1<#3)}{#3-#2+1}}
\newcommandx{\punif}[3][1=x,2=a,3=b]{
\begin{cases} 0 & #1 < #2 \\ \frac{#1-#2}{#3-#2} & #2 < #1 < #3 \\ 1 & #1 > #3\\\end{cases}}
\newcommandx{\punifd}[3][1=x,2=a,3=b]{
\begin{cases} 0 & #1 < #2\\ \frac{\lfloor#1\rfloor-#2+1}{#3-#2} & #2 \le #1 \le #3 \\ 1 & #1 > #3\\ \end{cases}}

% Bernoulli
\newcommandx\bern[1][1=p]{\textrm{Bern}\left({#1}\right)}
\newcommandx\dbern[2][1=x,2=p]{#2^{#1} \left(1-#2\right)^{1-#1}}
\newcommandx\pbern[2][1=x,2=p]{\left(1-#2\right)^{1-#1}}

% Binomial
\newcommandx\bin[1][1={n,p}]{\textrm{Bin}\left(#1\right)}
\newcommandx\dbin[3][1=x,2=n,3=p]{\binom{#2}{#1}#3^#1\left(1-#3\right)^{#2-#1}}

% Multinomial
\newcommandx\mult[1][1={n,p}]{\textrm{Mult}\left(#1\right)}
\newcommandx\dmult[3][1=x,2=n,3=p]{\frac{#2!}{#1_1!\ldots#1_k!}#3_1^{#1_1}\cdots#3_k^{#1_k}}

% Hypergeometric
\newcommandx\hyper[1][1={N,m,n}]{\textrm{Hyp}\left({#1}\right)}
\newcommandx\dhyper[4][1=x,2=N,3=m,4=n]{\frac{\binom{#3}{#1}\binom{#3-#1}{#4-#1}}{\binom{#2}{#1}}}

% Negative Binomial
\newcommandx\nbin[1][1={r,p}]{\textrm{NBin}\left({#1}\right)}
\newcommandx\dnbin[3][1=x,2=r,3=p]{\binom{#1+#2-1}{#2-1}#3^#2(1-#3)^#1}
\newcommandx\pnbin[3][1=x,2=r,3=p]{I_#3(#2,#1+1)}

% Geometric
\newcommandx\geo[1][1=p]{\textrm{Geo}\left(#1\right)}
\newcommandx\dgeo[2][1=x,2=p]{#2(1-#2)^{#1-1}}
\newcommandx\pgeo[2][1=x,2=p]{1-(1-#2)^#1}

% Poisson
\newcommandx\pois[1][1=\lambda]{\textrm{Po}\left({#1}\right)}
\newcommandx\dpois[2][1=x,2=\lambda]{\frac{#2^#1 e^{-#2}}{#1!}}
\newcommandx\ppois[2][1=x,2=\lambda]{e^{-#2}\sum_{i=0}^#1\frac{#2^i}{i!}}

% Normal
\newcommandx\norm[1][1={\mu,\sigma^2}]{\mathcal{N}\left({#1}\right)}
\newcommandx\dnorm[3][1=x,2=\mu,3=\sigma]%
{\frac{1}{#3\sqrt{2\pi}}\Exp{-\frac{\left(#1-#2\right)^2}{2 #3^2}}}
\newcommandx\pnorm[1][1=x]{\Phi\left({#1}\right)}
\newcommandx\qnorm[1]{\Phi^{-1}\left({#1}\right)}

% Multivariate Normal
\newcommandx\mvn[1][1={\mu,\Sigma}]{\mathrm{MVN}\left({#1}\right)}

% Exponential
\newcommandx\ex[1][1=\beta]{\textrm{Exp}\left(#1\right)}
\newcommandx\dex[2][1=x,2=\beta]{\frac{1}{#2}e^{-#1/#2}}
\newcommandx\pex[2][1=x,2=\beta]{1-e^{-#1/#2}}

% Gamma
\newcommandx\gam[1][1={\alpha,\beta}]{\textrm{Gamma}\left({#1}\right)}
\newcommandx\dgamma[3][1=x,2=\alpha,3=\beta]%
{\frac{1}{\Gamma\left( #2 \right) #3^{#2}} #1^{#2 -1}e^{- #1 / #3}}

% InverseGamma
\newcommandx\invgamma[1][1={\alpha,\beta}]{\textrm{InvGamma}\left({#1}\right)}
\newcommandx\dinvgamma[3][1=x,2=\alpha,3=\beta]%
{\frac{#3^{#2}}{\Gamma\left(#2\right)}#1^{-#2-1}e^{-#3/#1}}
\newcommandx\pinvgamma[3][1=x,2=\alpha,3=\beta]%
{\frac{\Gamma\left(#2,\frac{#3}{#1}\right)}{\Gamma\left(#2\right)}}

% Beta
\newcommandx\bet[1][1={\alpha,\beta}]{\textrm{Beta}\left(#1\right)}
\newcommandx\dbeta[3][1=x,2=\alpha,3=\beta]
{\frac{\Gamma\left(#2+#3\right)}{\Gamma\left(#2\right)\Gamma\left(#3\right)}#1^{#2-1}\left(1-#1\right)^{#3-1}}

% Dirichlet
\newcommandx\dir[1][1={\alpha}]{\textrm{Dir}\left(#1\right)}
\newcommandx\ddir[3][1=x,2=\alpha]{\frac{\Gamma\left(\sum_{i=1}^k #2_i\right)}{\prod_{i=1}^k\Gamma\left(#2_i\right)}\prod_{i=1}^k #1_i^{#2_i-1}}

% Weibull
\newcommandx\weibull[1][1={\alpha}]{\textrm{Dir}\left(#1\right)}
\newcommandx\dweibull[3][1=x,2=\lambda,3=k]{\frac{#3}{#2}
\left(\frac{#1}{#2}\right)^{#3-1} e^{-(#1/#2)^k}}

% Chi-squard
\newcommandx\chisq[1][1=k]{\chi_{#1}^2}

% Zeta
\newcommandx\zet[1][1=s]{\textrm{Zeta}\left(#1\right)}
\newcommandx\dzeta[2][1=x,2=s]{\frac{#1^{-#2}}{\zeta\left(#2\right)}}

% Time Series
\newcommandx\AR[1][1=p]{\mathsf{AR}\left({#1}\right)}
\newcommandx\MA[1][1=q]{\mathsf{MA}\left({#1}\right)}
\newcommandx\ARMA[1][1={p,q}]{\mathsf{ARMA}\left({#1}\right)}
\newcommandx\ARIMA[1][1={p,d,q}]{\mathsf{ARIMA}\left({#1}\right)}
\newcommandx\SARIMA[3][1={p,d,q},2={P,D,Q},3=s]{\mathsf{ARIMA}\left(#1\right) \times \left(#2\right)_{#3}}


% =============================================================================
%                                 Algorithms
% =============================================================================

\newcommandx\step[1][1=t]{^{(#1)}}