probstat-english.dict

%
% Dictionary for the Probability and Statistics Cookbook.
%
% Language: English (default)
% Author: Matthias Vallentin
%

\ProvidesDictionary{probstat}{english}

%
% Preamble
%

\newtranslation{maintitle}{Probability and Statistics}
\newtranslation{subtitle}{Cookbook}
\newtranslation{disclaimer}{%
  This cookbook integrates a variety of topics in probability theory and
  statistics. It is based on literature \cite{Hoel72,Wasserman03,Shumway06} and
  in-class material from courses of the statistics department at the University
  of California in Berkeley but also influenced by other sources
  \cite{Steger01,Steger02}. If you find errors or have suggestions for further
  topics, I would appreciate if you send me an \href{mailto:\email}{email}. The
  most recent version of this document is available at \web. To reproduce,
  please contact me.}

%
% Global (sorted alphabetically)
%

\newtranslation{be the}{be the}
\newtranslation{and}{and}
\newtranslation{Continuous}{Continuous}
\newtranslation{continuous}{continuous}
\newtranslation{Definition}{Definition}
\newtranslation{Definitions}{Definitions}
\newtranslation{Discrete}{Discrete}
\newtranslation{discrete}{discrete}
\newtranslation{Estimate}{Estimate}
\newtranslation{Example}{Example}
\newtranslation{for some}{for some}
\newtranslation{If}{If}
\newtranslation{In particular}{In particular}
\newtranslation{is the}{is the}
\newtranslation{large}{large}
\newtranslation{Let}{Let}
\newtranslation{let}{let}
\newtranslation{of}{of}
\newtranslation{Properties}{Properties}
\newtranslation{properties}{properties}
\newtranslation{Sample mean}{Sample mean}
\newtranslation{Sample variance}{Sample variance}
\newtranslation{small}{small}
\newtranslation{Suppose}{Suppose}
\newtranslation{Then}{Then}
\newtranslation{when}{when}
\newtranslation{where}{where}
\newtranslation{with}{with}
\newtranslation{under}{under}

%
% Sections (sorted by occurrence; each paragraph covers one section)
%

\newtranslation{Distribution Overview}{Distribution Overview}
\newtranslation{Discrete Distributions}{Discrete Distributions}
\newtranslation{Notation}{Notation}
\newtranslation{Uniform}{Uniform}
\newtranslation{Bernoulli}{Bernoulli}
\newtranslation{Binomial}{Binomial}
\newtranslation{Multinomial}{Multinomial}
\newtranslation{Hypergeometric}{Hypergeometric}
\newtranslation{Negative Binomial}{Negative Binomial}
\newtranslation{Geometric}{Geometric}
\newtranslation{Poisson}{Poisson}
\newtranslation{Footnote about notation}{%
  We use the notation $\gamma(s,x)$ and $\Gamma(x)$ to refer to the Gamma
  functions (see \S\ref{sec:math:gamma}), and use $\text{B}(x,y)$ and $I_x$ to
  refer to the Beta functions (see \S\ref{sec:math:beta}).}

\newtranslation{Continuous Distributions}{Continuous Distributions}
\newtranslation{Normal}{Normal}
\newtranslation{Log-Normal}{Log-Normal}
\newtranslation{Multivariate Normal}{Multivariate Normal}
\newtranslation{Student's t}{Student's $t$}
\newtranslation{Chi-square}{Chi-square}
\newtranslation{Exponential}{Exponential}
\newtranslation{Inverse Gamma}{Inverse Gamma}

\newtranslation{Probability Theory}{Probability Theory}
\newtranslation{Sample space}{Sample space}
\newtranslation{Outcome}{Outcome (point or element)}
\newtranslation{Event}{Event}
\newtranslation{Sigma-algebra}{$\sigma$-algebra}
\newtranslation{Probability distribution}{Probability distribution}
\newtranslation{Probability space}{Probability space}
\newtranslation{Continuity of Probabilities}{Continuity of Probabilities}
\newtranslation{Independence}{Independence}
\newtranslation{Conditional Probability}{Conditional Probability}
\newtranslation{Law of Total Probability}{Law of Total Probability}
\newtranslation{Bayes' Theorem}{Bayes' Theorem}
\newtranslation{Inclusion-Exclusion Principle}{Inclusion-Exclusion Principle}

\newtranslation{Random Variables}{Random Variables}
\newtranslation{Random Variable}{Random Variable (RV)}
\newtranslation{Probability Mass Function}{Probability Mass Function (PMF)}
\newtranslation{Probability Density Function}{
  Probability Density Function (PDF)}
\newtranslation{Cumulative Distribution Function}{
  Cumulative Distribution Function (CDF)}
\newtranslation{Nondecreasing}{Nondecreasing}
\newtranslation{Normalized}{Normalized}
\newtranslation{Right-Continuous}{Right-Continuous}
\newtranslation{Transformations}{Transformations}
\newtranslation{Transformation function}{Transformation function}
\newtranslation{Special case if transform strictly monotone}{%
  Special case if $\transform$ strictly monotone}
\newtranslation{The Rule of the Lazy Statistician}{%
  The Rule of the Lazy Statistician}
\newtranslation{Convolution}{Convolution}

\newtranslation{Expectation}{Expectation}
\newtranslation{inequality}{inequality}
\newtranslation{Conditional expectation}{Conditional expectation}

\newtranslation{Variance}{Variance}
\newtranslation{Standard deviation}{Standard deviation}
\newtranslation{Covariance}{Covariance}
\newtranslation{Correlation}{Correlation}
\newtranslation{Conditional variance}{Conditional variance}

\newtranslation{Inequalities}{Inequalities}
\newtranslation{convex}{convex}

\newtranslation{Distribution Relationships}{Distribution Relationships}
\newtranslation{far from 0 and 1}{far from 0 and 1}
\newtranslation{Memoryless property}{Memoryless property}
\newtranslation{Upper quantile of}{Upper quantile of}

\newtranslation{Probability and Moment Generating Functions}{%
  Probability and Moment Generating Functions}

\newtranslation{Multivariate Distributions}{Multivariate Distributions} 
\newtranslation{Standard Bivariate Normal}{Standard Bivariate Normal}
\newtranslation{Joint density}{Joint density}
\newtranslation{Conditionals}{Conditionals}
\newtranslation{Bivariate Normal}{Bivariate Normal}
\newtranslation{Conditional mean and variance}{Conditional mean and variance}
\newtranslation{Multivariate Normal}{Multivariate Normal}
\newtranslation{Covariance Matrix}{Covariance matrix}
\newtranslation{Precision Matrix}{Precision matrix}

\newtranslation{Convergence}{Convergence}
\newtranslation{Convergence Introduction}{%
  Let $\{X_1,X_2,\ldots\}$ be a sequence of \rv's and let $X$ be another \rv.
  Let $F_n$ denote the \cdf of $X_n$ and let $F$ denote the \cdf of $X$.}
\newtranslation{Types of Convergence}{Types of convergence}
\newtranslation{In distribution}{In distribution (weakly, in law)}
\newtranslation{In probability}{In probability}
\newtranslation{Almost surely}{Almost surely (strongly)}
\newtranslation{In quadratic mean}{In quadratic mean ($L_2$)}
\newtranslation{Relationships}{Relationships}
\newtranslation{Slutzky's Theorem}{\textsc{Slutzky's Theorem}}
\newtranslation{In general}{In general}
\newtranslation{Law of Large Numbers}{Law of Large Numbers (LLN)}
\newtranslation{WLLN}{Weak (WLLN)}
\newtranslation{SLLN}{Strong (SLLN)}
\newtranslation{LLN Introduction}{%
  Let $\{X_1,\ldots,X_n\}$ be a sequence of \iid \rv's, $\E{X_1}=\mu$.}
\newtranslation{Central Limit Theorem}{Central Limit Theorem (CLT)}
\newtranslation{CLT Introduction}{%
  Let $\{X_1,\ldots,X_n\}$ be a sequence of \iid \rv's, $\E{X_1}=\mu$, and 
  $\V{X_1} = \sigma^2$.}
\newtranslation{CLT notations}{CLT notations}
\newtranslation{Continuity correction}{Continuity correction}
\newtranslation{Delta method}{Delta method}

\newtranslation{Statistical Inference}{Statistical Inference}
\newtranslation{Statistical Inference Introduction}{%
  Let $X_1,\cdots,X_n \distiid F$ if not otherwise noted.}
\newtranslation{Point Estimation}{Point Estimation}
\newtranslation{Point estimator}{Point estimator $\that_n$ of $\theta$ is a \rv}
\newtranslation{Consistency}{Consistency}
\newtranslation{Sampling distribution}{Sampling distribution}
\newtranslation{Standard error}{Standard error}
\newtranslation{Mean squared error}{Mean squared error}
\newtranslation{is consistent}{is consistent}
\newtranslation{Asymptotic normality}{Asymptotic normality}
\newtranslation{Slutzky Note}{%
  \translate{Slutzky's Theorem} often lets us replace $\se(\that_n)$ by some
  (weakly) consistent estimator $\shat_n$.}
\newtranslation{Normal-based Confidence Interval}{%
  Normal-based Confidence Interval}
\newtranslation{Empirical distribution}{Empirical distribution}
\newtranslation{Empirical Distribution Function}{%
  Empirical Distribution Function (ECDF)}
\newtranslation{for any fixed x}{for any fixed $x$}
\newtranslation{Dvoretzky-Kiefer-Wolfowitz inequality}{%
  \textsc{Dvoretzky-Kiefer-Wolfowitz} (DKW) inequality}
\newtranslation{Nonparametric confidence band}{%
  Nonparametric $1-\alpha$ confidence band for $F$}
\newtranslation{Statistical Functionals}{Statistical Functionals}
\newtranslation{Statistical functional}{Statistical functional}
\newtranslation{Plug-in estimator of theta}{%
  Plug-in estimator of $\theta = (F)$}
\newtranslation{Linear functional}{Linear functional}
\newtranslation{Plug-in estimator for linear functional}{%
  Plug-in estimator for linear functional}
\newtranslation{Often}{Often}
\newtranslation{quantile}{quantile}

\newtranslation{Parametric Inference}{Parametric Inference}
\newtranslation{Parametric Inference Introduction}{%
  Let $\mathfrak{F} = \bigl\{ f(x;\theta) : \theta\in\Theta \bigr\}$ be a
  parametric model with parameter space $\Theta \subset \R^k$ and parameter 
  $\theta = (\theta_1,\dots,\theta_k)$.}
\newtranslation{Method of moments}{Method of Moments}
\newtranslation{jth moment}{$j^{\mathrm{th}}$ moment}
\newtranslation{jth sample moment}{$j^{\mathrm{th}}$ sample moment}
\newtranslation{Method of Moments estimator}{Method of moments estimator (MoM)}
\newtranslation{Properties of the MoM estimator}{%
  Properties of the MoM estimator}
\newtranslation{exists with probability tending to 1}{%
  exists with probability tending to 1}
\newtranslation{Maximum likelihood}{Maximum likelihood}
\newtranslation{Likelihood}{Likelihood}
\newtranslation{Log-likelihood}{Log-likelihood}
\newtranslation{Maximum likelihood estimator}{%
  Maximum likelihood estimator (\mle)}
\newtranslation{Score function}{Score function}
\newtranslation{Fisher information}{Fisher information}
\newtranslation{exponential family}{exponential family}
\newtranslation{Observed Fisher information}{Observed Fisher information}
\newtranslation{Properties of the MLE}{Properties of the \mle}
\newtranslation{Equivariance}{Equivariance}
\newtranslation{Asymptotic optimality}{%
  Asymptotic optimality (or efficiency), i.e., smallest variance for large
  samples. If $\ttil_n$ is any other estimator, the asymptotic relative
  efficiency is}
\newtranslation{Approximately the Bayes estimator}{%
  Approximately the Bayes estimator}
\newtranslation{is differentiable}{is differentiable}
\newtranslation{Multiparameter Models}{Multiparameter Models}
\newtranslation{Fisher information matrix}{Fisher information matrix}
\newtranslation{Under appropriate regularity conditions}{%
  Under appropriate regularity conditions}
\newtranslation{Note on component convergence}{%
  Further, if $\that_j$ is the $j^{\mathrm{th}}$ component of $\theta$,
  then \[\frac{(\that_j-\theta_j)}{\sehat_j} \dconv \norm[0,1]\]
  where $\sehat_j^2 = J_n(j,j)$ and $\cov{\that_j,\that_k} = J_n(j,k)$}
\newtranslation{Multiparameter delta method}{Multiparameter delta method}
\newtranslation{let the gradient of transform be}{%
  let the gradient of \transform be}
\newtranslation{Parametric Bootstrap}{Parametric Bootstrap}
\newtranslation{Parametric Bootstrap Note}{%
  Sample from $f(x;\that_n)$ instead of from $\Fnhat$, 
  where $\that_n$ could be the \mle or method of moments estimator.}

\newtranslation{Hypothesis Testing}{Hypothesis Testing}
\newtranslation{versus}{versus}
\newtranslation{Null hypothesis}{Null hypothesis}
\newtranslation{Alternative hypothesis}{Alternative hypothesis}
\newtranslation{Simple hypothesis}{Simple hypothesis}
\newtranslation{Composite hypothesis}{Composite hypothesis}
\newtranslation{Two-sided test}{Two-sided test}
\newtranslation{One-sided test}{One-sided test}
\newtranslation{Critical value}{Critical value}
\newtranslation{Test statistic}{Test statistic}
\newtranslation{Rejection region}{Rejection region}
\newtranslation{Power function}{Power function}
\newtranslation{Power of a test}{Power of a test}
\newtranslation{Test size}{Test size}
\newtranslation{Retain}{Retain}
\newtranslation{Reject}{Reject}
\newtranslation{true}{true}
\newtranslation{Type I error}{Type I error}
\newtranslation{Type II error}{Type II error}
\newtranslation{power}{power}
\newtranslation{p-value}{p-value}
\newtranslation{evidence}{evidence}
\newtranslation{very strong evidence against}{very strong evidence against}
\newtranslation{strong evidence against}{strong evidence against}
\newtranslation{weak evidence against}{weak evidence against}
\newtranslation{little or no evidence against}{little or no evidence against}
\newtranslation{Wald test}{Wald test}
\newtranslation{Likelihood ratio test}{Likelihood ratio test (LRT)}
\newtranslation{Multinomial LRT}{Multinomial LRT}
\newtranslation{The approximate size alpha LRT rejects H0}{%
  The approximate size $\alpha$ LRT rejects $H_0$ when
  $\lambda(X) \ge \chi_{k-1,\alpha}^2$}
\newtranslation{Pearson}{Pearson}
\newtranslation{Pearson Chi-square}{Pearson $\chi^2$}
\newtranslation{Pearson Chi-square test}{\translate{Pearson Chi-square} test}
\newtranslation{Faster convergence than LRT, hence preferable for small n}{%
  Faster $\dconv X_{k-1}^2$ than LRT, hence preferable for small $n$}
\newtranslation{Independence testing}{Independence testing}
\newtranslation{rows}{rows}
\newtranslation{columns}{columns}
\newtranslation{multinomial sample of size}{multinomial sample of size}
\newtranslation{unconstrained}{unconstrained}
\newtranslation{LRT}{LRT}

\newtranslation{Bayesian Inference}{Bayesian Inference}
\newtranslation{Prior density}{Prior density}
\newtranslation{joint density of the data}{joint density of the data}
\newtranslation{Posterior density}{Posterior density}
\newtranslation{Normalizing constant}{Normalizing constant}
\newtranslation{Kernel: part of a density that depends on theta}{%
  Kernel: part of a density that depends on $\theta$}
\newtranslation{Posterior mean}{Posterior mean}
\newtranslation{Credible Intervals}{Credible Intervals}
\newtranslation{Posterior interval}{Posterior interval}
\newtranslation{Equal-tail credible interval}{Equal-tail credible interval}
\newtranslation{Highest posterior density (HPD) region}{%
  Highest posterior density (HPD) region}
\newtranslation{is unimodal}{is unimodal}
\newtranslation{is an interval}{is an interval}
\newtranslation{Function of parameters}{Function of parameters}
\newtranslation{Posterior CDF for tau}{Posterior CDF for $\tau$}
\newtranslation{Bayesian delta method}{Bayesian delta method}
\newtranslation{Priors}{Priors}
\newtranslation{Choice}{Choice}
\newtranslation{Subjective Bayesianism}{%
  Subjective Bayesianism: prior should incorporate as much detail as
  possible the research's a priori knowledge --- via \emph{prior elicitation}}
\newtranslation{Objective Bayesianism}{%
  Objective Bayesianism: prior should incorporate as little detail as possible
  (\emph{non-informative} prior)}
\newtranslation{Robust Bayesianism}{%
  Robust Bayesianism: consider various priors and determine \emph{sensitivity}
  of our inferences to changes in the prior}
\newtranslation{Types}{Types}
\newtranslation{Flat}{Flat}
\newtranslation{Proper}{Proper}
\newtranslation{Improper}{Improper}
\newtranslation{transformation-invariant}{transformation-invariant}
\newtranslation{Jeffrey's Prior}{\textsc{Jeffrey}'s prior}
\newtranslation{Conjugate}{Conjugate}
\newtranslation{belong to the same parametric family}{%
  belong to the same parametric family}
\newtranslation{Conjugate Priors}{Conjugate Priors}
\newtranslation{Discrete likelihood}{Discrete likelihood}
\newtranslation{Posterior hyperparameters}{Posterior hyperparameters}
\newtranslation{Conjugate prior}{Conjugate prior}
\newtranslation{Continuous likelihood}{%
  Continuous likelihood (subscript $c$ denotes constant)}
\newtranslation{Scaled Inverse Chi-square}{Scaled Inverse Chi-square}
\newtranslation{Normal-scaled Inverse Gamma}{Normal-scaled Inverse Gamma}
\newtranslation{Inverse-Wishart}{Inverse-Wishart}
\newtranslation{Bayesian Testing}{Bayesian Testing}
\newtranslation{Prior probability}{Prior probability}
\newtranslation{Posterior probability}{Posterior probability}
\newtranslation{Let H_0...H_K-1 be k hypotheses}{%
Let $H_0,\ldots,H_{K-1}$ be $K$ hypotheses}
\newtranslation{Marginal likelihood}{Marginal likelihood}
\newtranslation{Posterior odds}{Posterior odds (of $H_i$ relative to $H_j$)}
\newtranslation{Bayes factor}{Bayes factor}
\newtranslation{Weak}{Weak}
\newtranslation{Moderate}{Moderate}
\newtranslation{Strong}{Strong}
\newtranslation{Decisive}{Decisive}

\newtranslation{Exponential Family}{Exponential Family}
\newtranslation{Scalar parameter}{Scalar parameter}
\newtranslation{Vector parameter}{Vector parameter}
\newtranslation{Natural form}{Natural form}

\newtranslation{Sampling Methods}{Sampling Methods}
\newtranslation{The Bootstrap}{The Bootstrap}
\newtranslation{be a statistic}{be a statistic}
\newtranslation{Approximate}{Approximate}
\newtranslation{using simulation}{using simulation}
\newtranslation{Repeat the following B times}{%
  Repeat the following $B$ times to get $T_{n,1}^*,\dots,T_{n,B}^*$,
  an \iid sample from the sampling distribution implied by $\Fnhat$}
\newtranslation{Sample uniformly}{Sample uniformly}
\newtranslation{Bootstrap Confidence Intervals}{Bootstrap Confidence Intervals}
\newtranslation{Normal-based interval}{Normal-based interval}
\newtranslation{Pivotal interval}{Pivotal interval}
\newtranslation{Location parameter}{Location parameter}
\newtranslation{Pivot}{Pivot}
\newtranslation{using bootstrap}{using bootstrap}
\newtranslation{beta sample quantile of}{$\beta$ sample quantile of}
\newtranslation{Approximate 1-alpha CI}{%
  Approximate $1 - \alpha$ confidence interval}
\newtranslation{Percentile interval}{Percentile interval}
\newtranslation{Rejection Sampling}{Rejection Sampling}
\newtranslation{Setup}{Setup}
\newtranslation{We can easily sample from g}{%
  We can easily sample from $g(\theta)$}
\newtranslation{We want to sample from h, but it is difficult}{%
  We want to sample from $h(\theta)$, but it is difficult}
\newtranslation{We know h up to a proportional constant}{%
  We know $h(\theta)$ up to a proportional constant}
\newtranslation{Envelope condition: we can find M > 0 such that}{%
  Envelope condition: we can find $M > 0$ such that}
\newtranslation{Algorithm}{Algorithm}
\newtranslation{Draw}{Draw}
\newtranslation{Generate}{Generate}
\newtranslation{Accept}{Accept}
\newtranslation{We can easily sample from the prior}{%
  We can easily sample from the prior}
\newtranslation{Repeat until B values of theta-cand have been accepted}{%
  Repeat until $B$ values of $\theta^{cand}$ have been accepted}
\newtranslation{Target is the posterior}{Target is the posterior}
\newtranslation{Envelope condition}{Envelope condition}
\newtranslation{Importance Sampling}{Importance Sampling}
\newtranslation{Importance Sampling Introduction}{%
  Sample from an importance function $g$ rather than target density $h$.\\
  Algorithm to obtain an approximation to $\E{q(\theta) \giv x^n}$:}
\newtranslation{Sample from the prior}{Sample from the prior}

\newtranslation{Decision Theory}{Decision Theory}
\newtranslation{Unknown quantity affecting our decision}{%
  Unknown quantity affecting our decision}
\newtranslation{Decision rule: synonymous for an estimator theta hat}{%
  Decision rule: synonymous for an estimator $\that$}
\newtranslation{Action a in A}{%
  Action $a \in \mathcal{A}$: possible value of the decision rule. In the
  estimation context, the action is just an estimate of $\theta$, $\that(x)$.}
\newtranslation{Loss function L}{%
  Loss function $L$: consequences of taking action $a$ when true state is
  $\theta$ or discrepancy between $\theta$ and $\that$}
\newtranslation{Loss functions}{Loss functions}
\newtranslation{Squared error loss}{Squared error loss}
\newtranslation{Linear loss}{Linear loss}
\newtranslation{Absolute error loss}{Absolute error loss}
\newtranslation{linear loss with}{linear loss with}
\newtranslation{loss}{loss}
\newtranslation{Zero-one loss}{Zero-one loss}
\newtranslation{Risk}{Risk}
\newtranslation{Posterior risk}{Posterior risk}
\newtranslation{(Frequentist) risk}{(Frequentist) risk}
\newtranslation{Bayes risk}{Bayes risk}
\newtranslation{Admissibility}{Admissibility}
\newtranslation{dominates}{dominates}
\newtranslation{is inadmissible if there is at least one other estimator}{%
  is inadmissible if there is at least one other estimator $\that'$ that
  dominates it. Otherwise it is called admissible.}
\newtranslation{Bayes Rule}{Bayes Rule}
\newtranslation{Bayes rule}{Bayes rule}
\newtranslation{or Bayes estimator}{or Bayes estimator}
\newtranslation{Theorems}{Theorems}
\newtranslation{posterior mean}{posterior mean}
\newtranslation{posterior median}{posterior median}
\newtranslation{posterior mode}{posterior mode}
\newtranslation{Minimax Rules}{Minimax Rules}
\newtranslation{Maximum risk}{Maximum risk}
\newtranslation{Minimax rule}{Minimax rule}
\newtranslation{Least favorable prior}{Least favorable prior}

\newtranslation{Linear Regression}{Linear Regression}
\newtranslation{Response variable}{Response variable}
\newtranslation{Covariate}{Covariate}
\newtranslation{aka predictor variable or feature}{%
  aka predictor variable or feature}
\newtranslation{Simple Linear Regression}{Simple Linear Regression}
\newtranslation{Model}{Model}
\newtranslation{Fitted line}{Fitted line}
\newtranslation{Predicted (fitted) values}{Predicted (fitted) values}
\newtranslation{Residuals}{Residuals}
\newtranslation{Residual sums of squares}{Residual sums of squares}
\newtranslation{Least square estimates}{Least square estimates}
\newtranslation{unbiased estimate}{unbiased estimate}
\newtranslation{Further properties}{Further properties}
\newtranslation{Approximate 1-alpha CI for beta0 and beta1}{%
  Approximate $1-\alpha$ confidence intervals for $\beta_0$ and $\beta_1$}
\newtranslation{reject}{reject}
\newtranslation{Likelihood}{Likelihood}
\newtranslation{Under the assumption of Normality, the least squares estimator
  is also the MLE}{%
  Under the assumption of Normality, the least squares estimator is also the
  \mle}
\newtranslation{Prediction}{Prediction}
\newtranslation{Observe X = x* of the covariate and want to predict their 
  outcome Y*}{%
  Observe $X = x_*$ of the covariate and want to predict their outcome $Y_*$}
\newtranslation{Prediction interval}{Prediction interval}
\newtranslation{Multiple Regression}{Multiple Regression}
\newtranslation{If the k by k matrix X'X is invertible}{%
  If the $(k \times k)$ matrix $X^TX$ is invertible}
\newtranslation{Estimate regression function}{Estimate regression function}
\newtranslation{Unbiased estimate for sigma-squared}{%
  Unbiased estimate for $\sigma^2$}
\newtranslation{Confidence interval}{Confidence interval}
\newtranslation{Model Selection}{Model Selection}
\newtranslation{Model Selection Introduction}{
  Consider predicting a new observation $Y^*$ for covariates $X^*$ and let $S
  \subset J$ denote a subset of the covariates in the model, where $|S| = k$
  and $|J| = n$}
\newtranslation{Issues}{Issues}
\newtranslation{Underfitting: too few covariates yields high bias}{%
  Underfitting: too few covariates yields high bias}
\newtranslation{Overfitting: too many covariates yields high variance}{%
  Overfitting: too many covariates yields high variance}
\newtranslation{Procedure}{Procedure}
\newtranslation{Assign a score to each model}{Assign a score to each model}
\newtranslation{Search through all models to find the one with the highest 
  score}{Search through all models to find the one with the highest score}
\newtranslation{Hypothesis testing}{Hypothesis testing}
\newtranslation{Mean squared prediction error (MSPE)}{%
  Mean squared prediction error (\mspe)}
\newtranslation{Prediction risk}{Prediction risk}
\newtranslation{Training error}{Training error}
\newtranslation{The training error is a downward-biased estimate of the 
  prediction risk}{%
  The training error is a downward-biased estimate of the prediction risk}
\newtranslation{Adjusted R2}{Adjusted $R^2$}
\newtranslation{Mallow's Cp statistic}{\textsc{Mallow's} $C_p$ statistic}
\newtranslation{lack of fit}{lack of fit}
\newtranslation{complexity penalty}{complexity penalty}
\newtranslation{Akaike Information Criterion (AIC)}{%
  \textsc{Akaike} Information Criterion (AIC)}
\newtranslation{Bayesian Information Criterion (BIC)}{%
  Bayesian Information Criterion (BIC)}
\newtranslation{Validation and training}{Validation and training}
\newtranslation{validation data}{validation data}
\newtranslation{often}{often}
\newtranslation{Leave-one-out cross-validation}{Leave-one-out cross-validation}

\newtranslation{Non-parametric Function Estimation}{%
  Non-parametric Function Estimation}
\newtranslation{Density Estimation}{Density Estimation}
\newtranslation{Integrated square error (ISE)}{Integrated square error (\ise)}
\newtranslation{Frequentist risk}{Frequentist risk}
\newtranslation{Histograms}{Histograms}
\newtranslation{Number of bins}{Number of bins}
\newtranslation{Binwidth}{Binwidth}
\newtranslation{Bin}{Bin}
\newtranslation{has}{has}
\newtranslation{observations}{observations}
\newtranslation{Define}{Define}
\newtranslation{Histogram estimator}{Histogram estimator}
\newtranslation{Cross-validation estimate of}{Cross-validation estimate of}
\newtranslation{Kernel Density Estimator (KDE)}{Kernel Density Estimator (KDE)}
\newtranslation{Kernel}{Kernel}
\newtranslation{KDE}{KDE}
\newtranslation{Epanechnikov Kernel}{\textsc{Epanechnikov} Kernel}
\newtranslation{otherwise}{otherwise}
\newtranslation{Non-parametric Regression}{Non-parametric Regression}
\newtranslation{Consider pairs of points (x1,Y1),...,(xn,Yn) related by}{%
  Consider pairs of points $(x_1,Y_1),\dots,(x_n,Y_n)$ related by}
\newtranslation{k-nearest Neighbor Estimator}{$k$-nearest Neighbor Estimator}
\newtranslation{values of}{values of}
\newtranslation{closest to}{closest to}
\newtranslation{Nadaraya-Watson Kernel Estimator}{%
  \textsc{Nadaraya-Watson} Kernel Estimator}
\newtranslation{Smoothing Using Orthogonal Functions}{%
  Smoothing Using Orthogonal Functions}
\newtranslation{Approximation}{Approximation}
\newtranslation{Multivariate regression}{Multivariate regression}
\newtranslation{Least squares estimator}{Least squares estimator}
\newtranslation{for equally spaced observations only}{%
  for equally spaced observations only}

\newtranslation{Stochastic Processes}{Stochastic Processes}
\newtranslation{Stochastic Process}{Stochastic Process}
\newtranslation{Notations}{Notations}
\newtranslation{State space}{State space}
\newtranslation{Index set}{Index set}
\newtranslation{Markov Chains}{Markov Chains}
\newtranslation{Markov chain}{Markov chain}
\newtranslation{Transition probabilities}{Transition probabilities}
\newtranslation{n-step}{n-step}
\newtranslation{Transition matrix}{Transition matrix}
\newtranslation{element is}{element is}
\newtranslation{Marginal probability}{Marginal probability}
\newtranslation{initial distribution}{initial distribution}
\newtranslation{Poisson Processes}{Poisson Processes}
\newtranslation{Poisson process}{Poisson process}
\newtranslation{number of events up to and including time}{%
  number of events up to and including time}
\newtranslation{Independent increments}{Independent increments}
\newtranslation{Intensity function}{Intensity function}
\newtranslation{Homogeneous Poisson process}{Homogeneous Poisson process}
\newtranslation{Waiting times}{Waiting times}
\newtranslation{time at which Xt occurs}{time at which $X_t$ occurs}
\newtranslation{Interarrival times}{Interarrival times}

\newtranslation{Time Series}{Time Series}
\newtranslation{Mean function}{Mean function}
\newtranslation{Autocovariance function}{Autocovariance function}
\newtranslation{Autocorrelation function (ACF)}{Autocorrelation function (ACF)}
\newtranslation{Cross-covariance function (CCV)}{%
  Cross-covariance function (CCV)}
\newtranslation{Cross-correlation function (CCF)}{%
  Cross-correlation function (CCF)}
\newtranslation{Backshift operator}{Backshift operator}
\newtranslation{Difference operator}{Difference operator}
\newtranslation{White noise}{White noise}
\newtranslation{Gaussian}{Gaussian}
\newtranslation{Random walk}{Random walk}
\newtranslation{Drift}{Drift}
\newtranslation{Symmetric moving average}{Symmetric moving average}
\newtranslation{Stationary Time Series}{Stationary Time Series}
\newtranslation{Strictly stationary}{Strictly stationary}
\newtranslation{Weakly stationary}{Weakly stationary}
\newtranslation{Autocovariance function}{Autocovariance function}
\newtranslation{Autocorrelation function (ACF)}{Autocorrelation function (ACF)}
\newtranslation{Jointly stationary time series}{Jointly stationary time series}
\newtranslation{Linear process}{Linear process}
\newtranslation{Estimation of Correlation}{Estimation of Correlation}
\newtranslation{Sample autocovariance function}{Sample autocovariance function}
\newtranslation{Estimation of Correlation}{Estimation of Correlation}
\newtranslation{Sample autocovariance function}{Sample autocovariance function}
\newtranslation{Sample autocorrelation function}{%
  Sample autocorrelation function}
\newtranslation{Sample cross-variance function}{Sample cross-variance function}
\newtranslation{Sample cross-correlation function}{%
  Sample cross-correlation function}
\newtranslation{is white noise}{is white noise}
\newtranslation{Non-Stationary Time Series}{Non-Stationary Time Series}
\newtranslation{Classical decomposition model}{Classical decomposition model}
\newtranslation{trend}{trend}
\newtranslation{seasonal component}{seasonal component}
\newtranslation{random noise term}{random noise term}
\newtranslation{Detrending}{Detrending}
\newtranslation{Least squares}{Least squares}
\newtranslation{Choose trend model}{Choose trend model}
\newtranslation{Minimize RSS to obtain trend estimate}{%
  Minimize \rss to obtain trend estimate}
\newtranslation{Residuals}{Residuals}
\newtranslation{noise}{noise}
\newtranslation{Moving average}{Moving average}
\newtranslation{The low-pass filter vt is a symmetric moving average}{%
  The \emph{low-pass} filter $v_t$ is a symmetric moving average $m_t$ with
  $a_j = \frac{1}{2k+1}$}
\newtranslation{a linear trend function mut passes without distortion}{%
  a linear trend function $\mu_t = \beta_0 + \beta_1t$ passes without
  distortion}
\newtranslation{Differencing}{Differencing}
\newtranslation{ARIMA models}{ARIMA models}
\newtranslation{Autoregressive polynomial}{Autoregressive polynomial}
\newtranslation{Autoregressive operator}{Autoregressive operator}
\newtranslation{Autoregressive model order p}{Autoregressive model order $p$}
\newtranslation{Moving average polynomial}{Moving average polynomial}
\newtranslation{Moving average operator}{Moving average operator}
\newtranslation{moving average model order}{moving average model order}
\newtranslation{Partial autocorrelation function (PACF)}{%
  Partial autocorrelation function (PACF)}
\newtranslation{regression of xi on}{regression of $x_i$ on}
\newtranslation{Exponentially Weighted Moving Average (EWMA)}{%
  Exponentially Weighted Moving Average (EWMA)}
\newtranslation{Seasonal ARIMA}{Seasonal ARIMA}
\newtranslation{Denoted by}{Denoted by}
\newtranslation{Causality and Invertibility}{Causality and Invertibility}
\newtranslation{is causal (future-independent)}{is causal (future-independent)}
\newtranslation{such that}{such that}
\newtranslation{is invertible}{is invertible}
\newtranslation{causal}{causal}
\newtranslation{roots of phi(z) lie outside the unit circle}{
  roots of $\phi(z)$ lie outside the unit circle}
\newtranslation{invertible}{invertible}
\newtranslation{roots of theta(z) lie outside the unit circle}{%
  roots of $\theta(z)$ lie outside the unit circle}
\newtranslation{Behavior of the ACF and PACF for causal and invertible ARMA
  models}{%
  Behavior of the ACF and PACF for causal and invertible ARMA models}
\newtranslation{tails off}{tails off}
\newtranslation{cuts off after lag}{cuts off after lag}
\newtranslation{Spectral Analysis}{Spectral Analysis}
\newtranslation{Periodic process}{Periodic process}
\newtranslation{Frequency index omega (cycles per unit time)}{%
  Frequency index $\omega$ (cycles per unit time)}
\newtranslation{period}{period}
\newtranslation{Amplitude}{Amplitude}
\newtranslation{Phase}{Phase}
\newtranslation{often normally distributed RV's}{%
  often normally distributed \rv's}
\newtranslation{Periodic mixture}{Periodic mixture}
\newtranslation{are independent zero-mean RV's with variances}{%
  are independent zero-mean \rv's with variances}
\newtranslation{Spectral representation of a periodic process}{%
  Spectral representation of a periodic process}
\newtranslation{Spectral distribution function}{Spectral distribution function}
\newtranslation{Spectral density}{Spectral density}
\newtranslation{Needs}{Needs}
\newtranslation{Discrete Fourier Transform (DFT)}{%
  Discrete Fourier Transform (DFT)}
\newtranslation{Fourier/Fundamental frequencies}{%
  Fourier/Fundamental frequencies}
\newtranslation{Inverse DFT}{Inverse DFT}
\newtranslation{Periodogram}{Periodogram}
\newtranslation{Scaled Periodogram}{Scaled Periodogram}

\newtranslation{Gamma Function}{Gamma Function}
\newtranslation{Ordinary}{Ordinary}
\newtranslation{Upper incomplete}{Upper incomplete}
\newtranslation{Lower incomplete}{Lower incomplete}
\newtranslation{Beta Function}{Beta Function}
\newtranslation{Incomplete}{Incomplete}
\newtranslation{Regularized incomplete}{Regularized incomplete}
\newtranslation{Series}{Series}
\newtranslation{Finite}{Finite}
\newtranslation{Binomial}{Binomial}
\newtranslation{Vandermonde's Identity}{\textsc{Vandermonde}'s Identity}
\newtranslation{Binomial Theorem}{Binomial Theorem}
\newtranslation{Infinite}{Infinite}
\newtranslation{Combinatorics}{Combinatorics}
\newtranslation{Sampling}{Sampling}
\newtranslation{k out of n}{$k$ out of $n$}
\newtranslation{w/o replacement}{w/o replacement}
\newtranslation{w/ replacement}{w/ replacement}
\newtranslation{ordered}{ordered}
\newtranslation{unordered}{unordered}
\newtranslation{Stirling numbers, 2nd kind}{Stirling numbers, $2^{nd}$ kind}
\newtranslation{Partitions}{Partitions}
\newtranslation{Balls and Urns}{Balls and Urns}
\newtranslation{distinguishable}{distinguishable}
\newtranslation{indistinguishable}{indistinguishable}
\newtranslation{arbitrary}{arbitrary}
\newtranslation{injective}{injective}
\newtranslation{surjective}{surjective}
\newtranslation{bijective}{bijective}
\newtranslation{Univariate distribution relationships, 
  courtesy Leemis and McQueston}{%
  Univariate distribution relationships, courtesy Leemis and McQueston}

% vim: ft=tex