Appendix D — Notation
This appendix collects every symbol this book uses, so you have one place to look when a formula in a later chapter reuses a letter from an earlier one. It describes this book’s choices and not a universal standard, because there is no universal standard: statistics notation varies by field, by textbook, and sometimes by chapter within one textbook. Expect other sources to write the sample mean as \(\bar{x}\) where this book writes \(\hat{M}\), and to use \(P(\cdot)\) where this book uses \(Prob(\cdot)\). The organizing idea here is a single distinction, carried consistently: a hat marks a quantity computed from data, and a bare letter marks the theoretical quantity it corresponds to. Once you have that, most formulas can be read before you know the details, because the decoration tells you whether you are looking at something you can compute or something you can only estimate. The tables below are grouped by topic in roughly the order the book introduces them.
D.1 Hats and No Hats
The book distinguishes the sample (the data you observed) from the population (the theoretical quantity you want to know) using hats.
For example: \(M\) has a theoretical value of \(\mu\) but varies across samples and \(\hat{M}\) is the one number from your sample data.
The table below pairs sample quantities with their population or theoretical counterparts.
| Concept | Sample (data) | Population (theory) |
|---|---|---|
| Observation | \(\hat{X}_i\), \(\hat{Y}_i\) | \(X_i\), \(Y_i\) |
| Cumulative Distribution Function | \(\hat{F}(x)\), ECDF | \(F(x)\) or \(Prob(X_i \leq x)\) |
| Probability Density Function (continuous) | \(\hat{f}(x)\), histogram or kernel density | \(f(x)\) |
| Proportion | \(\hat{P}(x)\) | \(p(x)\) or \(Prob(X_i = x)\) |
| Mean | \(\hat{M}\) | \(\mu\) or \(\mathbb{E}[X_i]\) |
| Variance, Std deviation | \(\hat{V}\), \(\hat{S}\) | \(\sigma^2\) or \(\mathbb{V}[X_i]\), \(\sigma\) |
| Median | \(\tilde{M}=\hat{F}^{-1}(0.5)\) | \(F^{-1}(0.5)\) |
| Interquartile range | \(\hat{IQR}\) | \(F^{-1}(0.75)-F^{-1}(0.25)\) |
| Joint probability | \(\hat{P}(x, y)\) | \(p(x, y)\) or \(Prob(X_i = x ~\text{and}~ Y_i = y)\) |
| Conditional probability | \(\hat{P}(x \mid y)\) | \(p(x \mid y)\) or \(Prob(X_i = x \mid Y_i = y)\) |
| Conditional mean function | \(\hat{M}_Y(x,h)\), mean of \(Y\) within window \([x-h, x+h]\) or generalization | \(M(x)\) or \(\mathbb{E}[Y_i \mid X_i=x]\) |
| Covariance | \(\hat{C}_{XY}\) | \(\rho\) or \(\mathbb{C}[X_i, Y_i]\) |
| Regression coefficient | \(\hat{B}_k\) | \(\beta_k\) |
| Residual / error | \(\hat{E}_i\) | \(\epsilon_i\) |
Note that \(\mathbb{E}\), \(\mathbb{V}\), and \(\mathbb{C}\) are functions computed only on random variables.
D.2 Indices and Sub/Superscripts
| Pattern | Meaning | Example |
|---|---|---|
| \(_i\) | Observation in sample or population ( \(i = 1, \ldots, n, n + 1, \ldots N\)) | \(\hat{X}_i\) is value of the \(i\)th observation |
| \(_j\) | Secondary observation index (pairwise sums only) | \(\sum_{j \neq i} \hat{X}_j\) |
| \(_k\) | Variable index in \(K\) variables (\(k=1,\ldots,K\)) | \(\hat{B}_k\), \(\hat{X}_{ik}\) |
| \(_g\) | Group index in \(G\) groups (\(g=1,\ldots,G\)) | \(\hat{M}_g\) |
| \(^*\) | Optimal or equilibrium value | \(P^*\), equilibrium price |
| \(^{(b)}\) | Bootstrap replicate | \(\hat{X}_i^{(b)}\), \(\hat{M}^{(b)}\) |
| \(^{(-i)}\) | Leave-one-out replicate, omitting observation \(i\) | \(\hat{M}^{(-i)}\), \(\hat{y}_j^{(-i)}\) |
| \(^{\text{boot}~,~ \text{jack}}\) | Bootstrap, Jackknife variant | \(\hat{SE}^{\text{boot}}\), \(\hat{SE}^{\text{jack}}\) |
| \(_0\) | Null hypothesis | \(\hat{F}_0\) |
In \(\hat{y}_j^{(-i)}\), the subscript \(j\) identifies the observation being predicted, and the superscript \((-i)\) identifies the observation omitted from the fit.
D.3 Regression
Fitted simple regression:
\[\hat{Y}_i = \hat{B}_0 + \hat{B}_1 \hat{X}_i + \hat{E}_i\]
Fitted local linear regression at \(x\):
\[\hat{y}(x) = \hat{B}_0(x,h) + \hat{B}_1(x,h)x\]
Here \(\hat{y}(x)\) is the fitted value at \(x\), and the local coefficients depend on \(x\) and the bandwidth \(h\).
Fitted multiple regression:
\[\hat{Y}_i = \sum_{k=0}^{K} \hat{B}_k \hat{X}_{ik} + \hat{E}_i\]
Here \(\hat{X}_{i0}=1\) supplies the intercept, \(\hat{y}_i=\sum_{k=0}^{K}\hat{B}_k\hat{X}_{ik}\) is the fitted value, and \(\hat{E}_i=\hat{Y}_i-\hat{y}_i\) is the fitted sample residual. In least-squares minimization, lowercase \(b_k\) denotes a trial coefficient, and \(\hat{E}_i=\hat{Y}_i-\sum_{k=0}^{K}b_k\hat{X}_{ik}\) depends on those trial values until the fit is chosen. The coefficient estimator \(B_k\) varies across samples, while \(\hat{B}_k\) is its value in the observed sample and \(\beta_k\) is the population coefficient it estimates. The population error \(\epsilon_i\) is an unobserved random disturbance, whereas the sample residual \(\hat{E}_i\) is computed from the fitted model.
| Symbol | Meaning |
|---|---|
| \(b_0\), \(b_k\) | Trial intercept and coefficients in minimization |
| \(B_k\) | Coefficient estimator before seeing data |
| \(\hat{B}_k\), \(\hat{B}_k(x, h)\) | Fitted intercept, coefficient |
| \(\beta_k\), \(\hat{B}_k(\mathbf{x})\) | Population coefficient, marginal effect at \(\mathbf{x}\) |
| \(h\) or \(h_k\) | Bandwidth (half-width of a window around \(x\)) |
| \(\mathbf{1}(\cdot)\) | Indicator function |
| \(k(\hat{X}_i, x, h)\) | Kernel weight function |
| \(W_x\), \(w_x\) | Unnormalized weight, normalized weight (\(\sum_{x} w_x = 1\)) |
| \(\epsilon_i\) | Population error term |
| \(\hat{E}_i\) | Sample residual |
| \(\hat{R}^2\) | Coefficient of determination |
| \(\hat{R}^2_{adj}\) | Adjusted R-squared |
Bold \(\hat{\mathbf{B}}\) collects the fitted coefficients, and bold \(\hat{\mathbf{E}}\) collects the sample residuals. In contingency tables, \(\hat{E}_{kj}\) instead denotes an expected cell count; the context and indices distinguish it from a regression residual.
The letter \(h\) is a half-width everywhere it appears, measured from a point \(x\) out to the edge of the window. So a histogram bin is \(2h\) wide, a kernel gives zero weight once \(|\hat{X}_{i}-x| > h\), and a local regression at \(x\) fits on the observations within \(h\) of it. These are all the same idea, which is how much data around a point gets used. LOESS is the one exception: its span fixes the share of observations in each window instead of the width, so the window widens where data are sparse rather than staying at a fixed \(h\).
The letter \(k\) does double duty here, distinguished by whether it takes arguments. A bare subscript \(k\) is the variable index, so \(h_{k}\) is the bandwidth for variable \(X_{k}\) and \(\hat{B}_{k}(\mathbf{x})\) is the marginal effect of that variable at the point \(\mathbf{x}\). Written with arguments, \(k(\cdot)\) is the kernel weight function. Capital \(K\) is the number of explanatory variables, and bold \(\mathbf{K}(\mathbf{x})\) is the matrix of kernel weights.
D.4 Hypothesis Testing and Sampling
| Symbol | Meaning |
|---|---|
| \(\hat{M}\), \(\hat{D}\), \(\hat{B}_k\) | Estimate of the population mean (realized, after seeing data), difference-in-means between two groups, or regression coefficient |
| \(SE(D)\), \(SE(D)\), \(SE(B_k)\) | Theoretical standard error of the estimators |
| \(\widehat{SE}(M)\), \(\widehat{SE}(D)\), \(\widehat{SE}(B_k)\) | Estimated standard error of the estimator |
| \(\hat{M}^{(b)}\), \(\hat{M}^{(-i)}\) | Bootstrap replicate, Jackknife (Leave-one-out) mean |
| \(\hat{SE}^{\text{boot}}\), \(\hat{SE}^{\text{jack}}\) | Bootstrap, Jackknife method of standard error estimate |
| \(\hat{t}\), \(\hat{F}\) | t-statistic, F-statistic |
| \(H_0\), \(H_A\) | Null, Alternative hypothesis |
| \(q(\alpha/2)\) | Critical quantile |
For a regression coefficient estimator, \(SE(B_k)=\sqrt{\mathbb{V}[B_k]}\), and \(\widehat{SE}(B_k)\) is its estimate computed from data. The method can be specified as \(\widehat{SE}^{\text{boot}}(B_k)\) or \(\widehat{SE}^{\text{jack}}(B_k)\).
Further Reading
- Code Style – how these symbols map onto object names in R.
- Sampling – where the \(M\) against \(\hat{M}\) distinction is introduced and used.