Notation

Abbreviations

$\textstyle \parbox{0.24\textwidth}{cdf}$$\textstyle \parbox{0.74\textwidth}{cumulative distribution function}$

$\textstyle \parbox{0.24\textwidth}{df}$$\textstyle \parbox{0.74\textwidth}{degrees of freedom}$

$\textstyle \parbox{0.24\textwidth}{iff}$$\textstyle \parbox{0.74\textwidth}{if and only if}$

$\textstyle \parbox{0.24\textwidth}{i.i.d.}$$\textstyle \parbox{0.74\textwidth}{independent and identically distributed}$

$\textstyle \parbox{0.24\textwidth}{w.r.t.}$$\textstyle \parbox{0.74\textwidth}{with respect to}$

$\textstyle \parbox{0.24\textwidth}{pdf}$$\textstyle \parbox{0.74\textwidth}{probability density function}$

$\textstyle \parbox{0.24\textwidth}{ADE}$$\textstyle \parbox{0.74\textwidth}{average derivative estimator}$

$\textstyle \parbox{0.24\textwidth}{AM}$$\textstyle \parbox{0.74\textwidth}{additive model}$

$\textstyle \parbox{0.24\textwidth}{AMISE}$$\textstyle \parbox{0.74\textwidth}{asymptotic MISE}$

$\textstyle \parbox{0.24\textwidth}{AMSE}$$\textstyle \parbox{0.74\textwidth}{asymptotic MSE}$

$\textstyle \parbox{0.24\textwidth}{APLM}$$\textstyle \parbox{0.74\textwidth}{additive partial linear model}$

$\textstyle \parbox{0.24\textwidth}{ASE}$$\textstyle \parbox{0.74\textwidth}{averaged squared error}$

$\textstyle \parbox{0.24\textwidth}{ASH}$$\textstyle \parbox{0.74\textwidth}{average shifted histogram}$

$\textstyle \parbox{0.24\textwidth}{CHARN}$$\textstyle \parbox{0.74\textwidth}{conditional heteroscedastic autoregressive
nonlinear}$

$\textstyle \parbox{0.24\textwidth}{CV}$$\textstyle \parbox{0.74\textwidth}{cross-validation}$

$\textstyle \parbox{0.24\textwidth}{DM}$$\textstyle \parbox{0.74\textwidth}{Deutsche Mark}$

$\textstyle \parbox{0.24\textwidth}{GAM}$$\textstyle \parbox{0.74\textwidth}{generalized additive model}$

$\textstyle \parbox{0.24\textwidth}{GAPLM}$$\textstyle \parbox{0.74\textwidth}{generalized additive partial linear model}$

$\textstyle \parbox{0.24\textwidth}{GLM}$$\textstyle \parbox{0.74\textwidth}{generalized linear model}$

$\textstyle \parbox{0.24\textwidth}{GPLM}$$\textstyle \parbox{0.74\textwidth}{generalized partial linear model}$

$\textstyle \parbox{0.24\textwidth}{ISE}$$\textstyle \parbox{0.74\textwidth}{integrated squared error}$

$\textstyle \parbox{0.24\textwidth}{IRLS}$$\textstyle \parbox{0.74\textwidth}{iteratively reweighted least squares}$

$\textstyle \parbox{0.24\textwidth}{LR}$$\textstyle \parbox{0.74\textwidth}{likelihood ratio}$

$\textstyle \parbox{0.24\textwidth}{LS}$$\textstyle \parbox{0.74\textwidth}{least squares}$

$\textstyle \parbox{0.24\textwidth}{MASE}$$\textstyle \parbox{0.74\textwidth}{mean averaged squared error}$

$\textstyle \parbox{0.24\textwidth}{MISE}$$\textstyle \parbox{0.74\textwidth}{mean integrated squared error}$

$\textstyle \parbox{0.24\textwidth}{ML}$$\textstyle \parbox{0.74\textwidth}{maximum likelihood}$

$\textstyle \parbox{0.24\textwidth}{MLE}$$\textstyle \parbox{0.74\textwidth}{maximum likelihood estimator}$

$\textstyle \parbox{0.24\textwidth}{MSE}$$\textstyle \parbox{0.74\textwidth}{mean squared error}$

$\textstyle \parbox{0.24\textwidth}{PLM}$$\textstyle \parbox{0.74\textwidth}{partial linear model}$

$\textstyle \parbox{0.24\textwidth}{PMLE}$$\textstyle \parbox{0.74\textwidth}{pseudo maximum likelihood estimator}$

$\textstyle \parbox{0.24\textwidth}{RSS}$$\textstyle \parbox{0.74\textwidth}{residual sum of squares}$

$\textstyle \parbox{0.24\textwidth}{S.D.}$$\textstyle \parbox{0.74\textwidth}{standard deviation}$

$\textstyle \parbox{0.24\textwidth}{S.E.}$$\textstyle \parbox{0.74\textwidth}{standard error}$

$\textstyle \parbox{0.24\textwidth}{SIM}$$\textstyle \parbox{0.74\textwidth}{single index model}$

$\textstyle \parbox{0.24\textwidth}{SLS}$$\textstyle \parbox{0.74\textwidth}{semiparametric least squares}$

$\textstyle \parbox{0.24\textwidth}{USD}$$\textstyle \parbox{0.74\textwidth}{US Dollar}$

$\textstyle \parbox{0.24\textwidth}{WADE}$$\textstyle \parbox{0.74\textwidth}{weighted average derivative estimator}$

$\textstyle \parbox{0.24\textwidth}{WSLS}$$\textstyle \parbox{0.74\textwidth}{weighted semiparametric least squares}$

Scalars, Vectors and Matrices

$\textstyle \parbox{0.24\textwidth}{$X$, $Y$}$$\textstyle \parbox{0.74\textwidth}{random variables}$

$\textstyle \parbox{0.24\textwidth}{$x$, $y$}$$\textstyle \parbox{0.74\textwidth}{scalars (realizations of $X$, $Y$)}$

$\textstyle \parbox{0.24\textwidth}{$X_{1},\ldots,X_{n}$}$$\textstyle \parbox{0.74\textwidth}{random sample of size $n$}$

$\textstyle \parbox{0.24\textwidth}{$X_{(1)},\ldots,X_{(n)}$}$$\textstyle \parbox{0.74\textwidth}{ordered random sample of size $n$}$

$\textstyle \parbox{0.24\textwidth}{$x_{1},\ldots,x_{n}$}$$\textstyle \parbox{0.74\textwidth}{realizations of $X_{1},\ldots,X_{n}$}$

$\textstyle \parbox{0.24\textwidth}{${\boldsymbol{X}}$}$$\textstyle \parbox{0.74\textwidth}{vector of variables}$

$\textstyle \parbox{0.24\textwidth}{${\boldsymbol{x}}$}$$\textstyle \parbox{0.74\textwidth}{vector (realizations of ${\boldsymbol{X}}$)}$

$\textstyle \parbox{0.24\textwidth}{$x_0$}$$\textstyle \parbox{0.74\textwidth}{origin (of histogram)}$

$\textstyle \parbox{0.24\textwidth}{$h$}$$\textstyle \parbox{0.74\textwidth}{binwidth or bandwidth}$

$\textstyle \parbox{0.24\textwidth}{$\widetilde h$}$$\textstyle \parbox{0.74\textwidth}{auxiliary bandwidth in marginal integration}$

$\textstyle \parbox{0.24\textwidth}{${\mathbf{H}}$}$$\textstyle \parbox{0.74\textwidth}{bandwidth matrix}$

$\textstyle \parbox{0.24\textwidth}{${\mathbf{I}}$}$$\textstyle \parbox{0.74\textwidth}{identity matrix}$

$\textstyle \parbox{0.24\textwidth}{${\mathbf{X}}$}$$\textstyle \parbox{0.74\textwidth}{data or design matrix}$

$\textstyle \parbox{0.24\textwidth}{${\boldsymbol{Y}}$}$$\textstyle \parbox{0.74\textwidth}{vector of observations $Y_1,\ldots,Y_n$}$

$\textstyle \parbox{0.24\textwidth}{$\beta$}$$\textstyle \parbox{0.74\textwidth}{parameter}$

$\textstyle \parbox{0.24\textwidth}{${\boldsymbol{\beta}}$}$$\textstyle \parbox{0.74\textwidth}{parameter vector}$

$\textstyle \parbox{0.24\textwidth}{${\boldsymbol{e}}_0$}$$\textstyle \parbox{0.74\textwidth}{first unit vector, i.e. ${\boldsymbol{e}}_0=(1,0,\ldots,0)^\top$}$

$\textstyle \parbox{0.24\textwidth}{${\boldsymbol{e}}_j$}$$\textstyle \parbox{0.74\textwidth}{$(j+1)$th unit vector, i.e. ${\boldsymbol{e}}_j=(0,\ldots,0,
\begin{array}[t]{c}1\\ {\small j}\end{array},0,\ldots,0)^\top$}$

$\textstyle \parbox{0.24\textwidth}{${{ {1}\hskip-4pt{1}}}_n$}$$\textstyle \parbox{0.74\textwidth}{vector of ones of length $n$}$

$\textstyle \parbox{0.24\textwidth}{${\boldsymbol{\mu}}$}$$\textstyle \parbox{0.74\textwidth}{vector of expectations of $Y_1,\ldots,Y_n$ in generalized\\ mo\-dels}$

$\textstyle \parbox{0.24\textwidth}{${\boldsymbol{\eta}}$}$$\textstyle \parbox{0.74\textwidth}{vector of index values
${\boldsymbol{X}}_1^\top\beta,\ldots,{\boldsymbol{X}}_n^\top\beta$ in generalized\\ models}$

$\textstyle \parbox{0.24\textwidth}{$LR$}$$\textstyle \parbox{0.74\textwidth}{likelihood ratio test statistic}$

$\textstyle \parbox{0.24\textwidth}{${\boldsymbol{U}}$}$$\textstyle \parbox{0.74\textwidth}{vector of variables (linear part of the model)}$

$\textstyle \parbox{0.24\textwidth}{${\boldsymbol{T}}$}$$\textstyle \parbox{0.74\textwidth}{vector of continuous variables (nonparametric part of the model)}$

$\textstyle \parbox{0.24\textwidth}{${\boldsymbol{X}}_{\underline{\alpha}}$}$$\textstyle \parbox{0.74\textwidth}{random vector of all but $\alpha$th component}$

$\textstyle \parbox{0.24\textwidth}{${\boldsymbol{X}}_{\underline{\alpha j}}$}$$\textstyle \parbox{0.74\textwidth}{random vector of all but $\alpha$th and
$j$th component}$

$\textstyle \parbox{0.24\textwidth}{${\mathbf{S}}$, ${\mathbf{S}}^P$, ${\mathbf{S}}_\alpha$}$$\textstyle \parbox{0.74\textwidth}{smoother matrices}$

$\textstyle \parbox{0.24\textwidth}{${\boldsymbol{m}}$}$$\textstyle \parbox{0.74\textwidth}{vector of regression values
$m({\boldsymbol{X}}_1),\ldots,m({\boldsymbol{X}}_n)$}$

$\textstyle \parbox{0.24\textwidth}{${\boldsymbol{g}}_\alpha$}$$\textstyle \parbox{0.74\textwidth}{vector of additive component function values\\
$g_\alpha({\boldsymbol{X}}_1)$,\ldots,$g_\alpha({\boldsymbol{X}}_n)$}$

Matrix algebra

$\textstyle \parbox{0.24\textwidth}{$\mathop{\hbox{tr}}({\mathbf{A}})$}$$\textstyle \parbox{0.74\textwidth}{trace of matrix ${\mathbf{A}}$}$

$\textstyle \parbox{0.24\textwidth}{$\mathop{\hbox{diag}}({\mathbf{A}})$}$$\textstyle \parbox{0.74\textwidth}{diagonal of matrix ${\mathbf{A}}$}$

$\textstyle \parbox{0.24\textwidth}{$\mathop{\rm{det}}({\mathbf{A}})$}$$\textstyle \parbox{0.74\textwidth}{determinant matrix ${\mathbf{A}}$}$

$\textstyle \parbox{0.24\textwidth}{$\mathop{\rm{rank}}({\mathbf{A}})$}$$\textstyle \parbox{0.74\textwidth}{rank of matrix ${\mathbf{A}}$}$

$\textstyle \parbox{0.24\textwidth}{${\mathbf{A}}^{-1}$}$$\textstyle \parbox{0.74\textwidth}{inverse of matrix ${\mathbf{A}}$}$

$\textstyle \parbox{0.24\textwidth}{$\Vert{\boldsymbol{u}}\Vert$}$$\textstyle \parbox{0.74\textwidth}{norm of vector ${\boldsymbol{u}}$, i.e. $\sqrt{{\boldsymbol{u}}^\top {\boldsymbol{u}}}$}$

Functions

$\textstyle \parbox{0.24\textwidth}{$\log$}$$\textstyle \parbox{0.74\textwidth}{logarithm (base $e$)}$

$\textstyle \parbox{0.24\textwidth}{$\varphi$}$$\textstyle \parbox{0.74\textwidth}{pdf of standard normal distribution}$

$\textstyle \parbox{0.24\textwidth}{$\Phi$}$$\textstyle \parbox{0.74\textwidth}{cdf of standard normal distribution}$

$\textstyle \parbox{0.24\textwidth}{$\Ind$}$$\textstyle \parbox{0.74\textwidth}{indicator function, i.e. $\Ind(A)=1$\ if $A$\ holds, $0$\ otherwise}$

$\textstyle \parbox{0.24\textwidth}{$K$}$$\textstyle \parbox{0.74\textwidth}{kernel function (univariate)}$

$\textstyle \parbox{0.24\textwidth}{$K_h$}$$\textstyle \parbox{0.74\textwidth}{scaled kernel function, i.e. $K_h(u)=K(u/h)/h$}$

$\textstyle \parbox{0.24\textwidth}{${\mathcal{K}}$}$$\textstyle \parbox{0.74\textwidth}{kernel function (multivariate)}$

$\textstyle \parbox{0.24\textwidth}{${\mathcal{K}}_{\mathbf{H}}$}$$\textstyle \parbox{0.74\textwidth}{scaled kernel function, i.e. ${\mathcal{K}}_...
...athcal{K}}({\mathbf{H}}^{-1}{\boldsymbol{u}})/\mathop{\rm{det}}({\mathbf{H}})$}$

$\textstyle \parbox{0.24\textwidth}{$\mu_2(K)$}$$\textstyle \parbox{0.74\textwidth}{second moment of $K$, i.e. $\int u^2 K(u)\,du$}$

$\textstyle \parbox{0.24\textwidth}{$\mu_p(K)$}$$\textstyle \parbox{0.74\textwidth}{$p$th moment of $K$, i.e. $\int u^p K(u)\,du$}$

$\textstyle \parbox{0.24\textwidth}{$\Vert K\Vert^2_2$}$$\textstyle \parbox{0.74\textwidth}{squared $L_2$\ norm of $K$, i.e. $\int \{K(u)\}^2\,du$}$

$\textstyle \parbox{0.24\textwidth}{$f$}$$\textstyle \parbox{0.74\textwidth}{probability density function (pdf)}$

$\textstyle \parbox{0.24\textwidth}{$f_X$}$$\textstyle \parbox{0.74\textwidth}{pdf of $X$}$

$\textstyle \parbox{0.24\textwidth}{$f(x,y)$}$$\textstyle \parbox{0.74\textwidth}{joint density of $X$\ and $Y$}$

$\textstyle \parbox{0.24\textwidth}{$\gradi_f$}$$\textstyle \parbox{0.74\textwidth}{gradient vector (partial first derivatives)}$

$\textstyle \parbox{0.24\textwidth}{${\mathcal{H}}_f$}$$\textstyle \parbox{0.74\textwidth}{Hessian matrix (partial second derivatives)}$

$\textstyle \parbox{0.24\textwidth}{$K\star K$}$$\textstyle \parbox{0.74\textwidth}{convolution of $K$, i.e. $K\star K (u) =\int
K(u-v)K(v)\,dv$}$

$\textstyle \parbox{0.24\textwidth}{$w$, $\widetilde{w}$}$$\textstyle \parbox{0.74\textwidth}{weight functions}$

$\textstyle \parbox{0.24\textwidth}{$m$}$$\textstyle \parbox{0.74\textwidth}{unknown function (to be estimated)}$

$\textstyle \parbox{0.24\textwidth}{$m^{(\nu)}$}$$\textstyle \parbox{0.74\textwidth}{$\nu$th derivative (to be estimated)}$

$\textstyle \parbox{0.24\textwidth}{$\ell$, $\ell_i$}$$\textstyle \parbox{0.74\textwidth}{log-likelihood, individual log-likelihood}$

$\textstyle \parbox{0.24\textwidth}{$G$}$$\textstyle \parbox{0.74\textwidth}{known link function}$

$\textstyle \parbox{0.24\textwidth}{$g$}$$\textstyle \parbox{0.74\textwidth}{unknown link function (to be estimated)}$

$\textstyle \parbox{0.24\textwidth}{$a$, $b$, $c$}$$\textstyle \parbox{0.74\textwidth}{exponential family characteristics in generalized models}$

$\textstyle \parbox{0.24\textwidth}{$V$}$$\textstyle \parbox{0.74\textwidth}{variance function of $Y$\ in generalized models}$

$\textstyle \parbox{0.24\textwidth}{$g_\alpha$}$$\textstyle \parbox{0.74\textwidth}{additive component (to be estimated)}$

$\textstyle \parbox{0.24\textwidth}{$g_\alpha^{(\nu)}$}$$\textstyle \parbox{0.74\textwidth}{$\nu$th derivative (to be estimated)}$

$\textstyle \parbox{0.24\textwidth}{$f_\alpha$}$$\textstyle \parbox{0.74\textwidth}{pdf of $X_\alpha$}$

Moments

$\textstyle \parbox{0.24\textwidth}{$E X$}$$\textstyle \parbox{0.74\textwidth}{mean value of $X$}$

$\textstyle \parbox{0.24\textwidth}{$\sigma^2=\mathop{\mathit{Var}}(X)$}$$\textstyle \parbox{0.74\textwidth}{variance of $X$, i.e. $\mathop{\mathit{Var}}(X)=E(X-E X)^2$}$

$\textstyle \parbox{0.24\textwidth}{$E(Y\vert X)$}$$\textstyle \parbox{0.74\textwidth}{conditional mean $Y$\ given $X$\ (random variable)}$

$\textstyle \parbox{0.24\textwidth}{$E(Y\vert X=x)$}$$\textstyle \parbox{0.74\textwidth}{conditional mean $Y$\ given $X=x$\ (realization of $E(Y\vert X)$) }$

$\textstyle \parbox{0.24\textwidth}{$E(Y\vert x)$}$$\textstyle \parbox{0.74\textwidth}{same as $E(Y\vert X=x)$}$

$\textstyle \parbox{0.24\textwidth}{$\sigma^2(x)$}$$\textstyle \parbox{0.74\textwidth}{conditional variance of $Y$\ given $X=x$\ (realization of
$\mathop{\mathit{Var}}(Y\vert X)$) }$

$\textstyle \parbox{0.24\textwidth}{$E_{X_1} g(X_1,X_2)$}$$\textstyle \parbox{0.74\textwidth}{mean of $g(X_1,X_2)$\ w.r.t. $X_1$\ only}$

$\textstyle \parbox{0.24\textwidth}{$\med(Y\vert X)$}$$\textstyle \parbox{0.74\textwidth}{conditional median $Y$\ given $X$\ (random variable)}$

$\textstyle \parbox{0.24\textwidth}{$\mu$}$$\textstyle \parbox{0.74\textwidth}{same as $E(Y\vert X)$\ in generalized models}$

$\textstyle \parbox{0.24\textwidth}{$V(\mu)$}$$\textstyle \parbox{0.74\textwidth}{variance function of $Y$\ in generalized models}$

$\textstyle \parbox{0.24\textwidth}{$\psi$}$$\textstyle \parbox{0.74\textwidth}{nuisance (dispersion) parameter in generalized models}$

$\textstyle \parbox{0.24\textwidth}{$\mse_x$}$$\textstyle \parbox{0.74\textwidth}{$\mse$\ at the point $x$}$

$\textstyle \parbox{0.24\textwidth}{${\mathcal{P}}_\alpha$}$$\textstyle \parbox{0.74\textwidth}{conditional expectation function $E(\bullet\vert X_\alpha)$}$

Distributions

$\textstyle \parbox{0.24\textwidth}{$U[0,1]$}$$\textstyle \parbox{0.74\textwidth}{uniform distribution on $[0,1]$}$

$\textstyle \parbox{0.24\textwidth}{$U[a,b]$}$$\textstyle \parbox{0.74\textwidth}{uniform distribution on $[a,b]$}$

$\textstyle \parbox{0.24\textwidth}{$N(0,1)$}$$\textstyle \parbox{0.74\textwidth}{standard normal or Gaussian distribution }$

$\textstyle \parbox{0.24\textwidth}{$N(\mu,\sigma^2)$}$$\textstyle \parbox{0.74\textwidth}{normal distribution with mean $\mu$\ and variance $\sigma^2$}$

$\textstyle \parbox{0.24\textwidth}{$N({\boldsymbol{\mu}},{\boldsymbol{\Sigma}})$}$$\textstyle \parbox{0.74\textwidth}{multi-dimensional normal distribution
with mean ${\boldsymbol{\mu}}$\ and covariance matrix ${\boldsymbol{\Sigma}}$}$

$\textstyle \parbox{0.24\textwidth}{$\chi^2_m$}$$\textstyle \parbox{0.74\textwidth}{$\chi^2$\ distribution with $m$\ degrees of freedom}$

$\textstyle \parbox{0.24\textwidth}{$t_{m}$}$$\textstyle \parbox{0.74\textwidth}{$t$-distribution with $m$\ degrees of freedom}$

Estimates

$\textstyle \parbox{0.24\textwidth}{$\widehat{\beta}$}$$\textstyle \parbox{0.74\textwidth}{estimated coefficient}$

$\textstyle \parbox{0.24\textwidth}{$\widehat{{\boldsymbol{\beta}}}$}$$\textstyle \parbox{0.74\textwidth}{estimated coefficient vector}$

$\textstyle \parbox{0.24\textwidth}{$\widehat{f}_h$}$$\textstyle \parbox{0.74\textwidth}{estimated density function}$

$\textstyle \parbox{0.24\textwidth}{$\widehat{f}_{h,-i}$}$$\textstyle \parbox{0.74\textwidth}{estimated density function when leaving out
observation~$i$}$

$\textstyle \parbox{0.24\textwidth}{$\widehat{m}_h$}$$\textstyle \parbox{0.74\textwidth}{estimated regression function}$

$\textstyle \parbox{0.24\textwidth}{$\widehat{m}_{p,h}$}$$\textstyle \parbox{0.74\textwidth}{estimated regression function using local
polynomials of degree $p$\ and bandwidth $h$}$

$\textstyle \parbox{0.24\textwidth}{$\widehat{m}_{p,{\mathbf{H}}}$}$$\textstyle \parbox{0.74\textwidth}{estimated multivariate
regression function using local polynomials of degree $p$ and bandwidth matrix ${\mathbf{H}}$}$

Convergence

$\textstyle \parbox{0.24\textwidth}{$o(\bullet)$}$$\textstyle \parbox{0.74\textwidth}{$a=o(b)$\ iff $a/b\to 0$\ as $n\to\infty$\ or $h\to 0$}$

$\textstyle \parbox{0.24\textwidth}{$O(\bullet)$}$$\textstyle \parbox{0.74\textwidth}{$a=O(b)$\ iff $a/b\to$\ constant
as $n\to\infty$\ or $h\to 0$}$

$\textstyle \parbox{0.24\textwidth}{$o_p(\bullet)$}$$\textstyle \parbox{0.74\textwidth}{$U=o_p(V)$\ iff for all $\epsilon>0$\ holds
$P(\vert U/V\vert>\epsilon)\to 0$}$

$\textstyle \parbox{0.24\textwidth}{$O_p(\bullet)$}$$\textstyle \parbox{0.74\textwidth}{$U=O_p(V)$\ iff for all $\epsilon>0$\ exists...
...ert>c)<\epsilon$\ as $n$\ is sufficiently large
or $h$\ is sufficiently small}$

$\textstyle \parbox{0.24\textwidth}{$\mathrel{\mathop{\longrightarrow}\limits_{}^{a.s.}}$}$$\textstyle \parbox{0.74\textwidth}{almost sure convergence}$

$\textstyle \parbox{0.24\textwidth}{$\mathrel{\mathop{\longrightarrow}\limits_{}^{P}}$}$$\textstyle \parbox{0.74\textwidth}{convergence in probability}$

$\textstyle \parbox{0.24\textwidth}{$\mathrel{\mathop{\longrightarrow}\limits_{}^{L}}$}$$\textstyle \parbox{0.74\textwidth}{convergence in distribution}$

$\textstyle \parbox{0.24\textwidth}{$\approx$}$$\textstyle \parbox{0.74\textwidth}{asymptotically equal}$

$\textstyle \parbox{0.24\textwidth}{$\sim$}$$\textstyle \parbox{0.74\textwidth}{asymptotically proportional}$

Other

$\textstyle \parbox{0.24\textwidth}{$\mathbb{N}$}$$\textstyle \parbox{0.74\textwidth}{natural numbers}$

$\textstyle \parbox{0.24\textwidth}{$\mathbb{Z}$}$$\textstyle \parbox{0.74\textwidth}{integers}$

$\textstyle \parbox{0.24\textwidth}{$\mathbb{R}$}$$\textstyle \parbox{0.74\textwidth}{real numbers}$

$\textstyle \parbox{0.24\textwidth}{$\mathbb{R}^d$}$$\textstyle \parbox{0.74\textwidth}{$d$-dimensional real space}$

$\textstyle \parbox{0.24\textwidth}{$\propto$}$$\textstyle \parbox{0.74\textwidth}{proportional}$

$\textstyle \parbox{0.24\textwidth}{$\equiv$}$$\textstyle \parbox{0.74\textwidth}{constantly equal}$

$\textstyle \parbox{0.24\textwidth}{$\char93 $}$$\textstyle \parbox{0.74\textwidth}{number of elements of a set}$

$\textstyle \parbox{0.24\textwidth}{$B_j$}$$\textstyle \parbox{0.74\textwidth}{$j$th bin, i.e. $[x_0+(j-1)h,x_0+jh)$}$

$\textstyle \parbox{0.24\textwidth}{$m_j$}$$\textstyle \parbox{0.74\textwidth}{bin center of $B_j$, i.e. $m_j=x_0+(j-\frac12)h$}$