Add supplemental documentation for turnbull

docs/turnbull.tex

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+\documentclass[a4paper,12pt]{article}
+
+\usepackage[math-style=ISO, bold-style=ISO]{unicode-math}
+\setmainfont{TeX Gyre Termes}
+\setmathfont{TeX Gyre Termes Math}
+
+\usepackage{parskip}
+\usepackage{microtype}
+\usepackage[left=2cm,right=2cm,top=2cm,bottom=2cm]{geometry}
+\frenchspacing
+\setlength{\emergencystretch}{3em}
+
+\usepackage[hidelinks]{hyperref}
+\usepackage{mathtools}
+
+\newcommand{\bbeta}{\kern -0.1em\symbf{β}}
+\newcommand{\blambda}{\kern -0.1em\symbf{Λ}}
+\newcommand{\nablasub}[1]{\nabla_{\kern -0.15em #1}}
+
+\begin{document}
+	{\centering\bfseries Supplemental documentation for hpstat \textit{turnbull} command\par}
+	
+	The hpstat \textit{turnbull} command implements Turnbull's nonparametric survival curve estimation for interval-censored observations [1]. This documentation discusses technical details of the implementation.
+	
+	Let $\hat{F}(t)$ be a maximum likelihood estimator for the cumulative distribution function for failure times. Turnbull [1] demonstrated that $\hat{F}(t)$ decreases only on the set of what are now called ‘Turnbull intervals’, or ‘innermost intervals’, $(q_j, p_j]$ for $j = 1, 2, …, m$.
+	
+	Let $s_j$ be the probability of failure within the interval $(q_j, p_j]$. We seek a maximum likelihood estimator for the vector $\symbf{s} = (s_1, s_2, …, s_m)^\mathrm{T}$.
+	
+	Take the $i$-th observation, $1 ≤ i ≤ n$ , whose failure time falls in $(L_i, R_i]$. Let $α_{i,j} = \mathrm{I}\left((q_j, p_j] \subseteq (L_i, R_i]\right)$.
+	
+	As discussed by Turnbull [1], noting that we consider only the case of no truncation, we commence with an arbitrary initial guess for $\hat{\symbf{s}}$, and iteratively apply:
+	%
+	\begin{align*}
+		μ_{ij}(\hat{\symbf{s}}) &= \frac{α_{i,j} \hat{s}_j}{\sum_{k=1}^m α_{i,k} \hat{s}_k} \\
+		π_j(\hat{\symbf{s}}) &= \frac{\sum_{i=1}^n μ_{ij}(\hat{\symbf{s}})}{n} \\
+		\hat{s}_j &\leftarrow π_j(\hat{\symbf{s}}), \qquad \text{for all $j = 1, 2, …, m$}
+	\end{align*}
+	%
+	This yields the maximum likelihood estimator $\hat{\symbf{s}}$.
+	
+	Now let $\hat{F}_0 = 0 ≤ \hat{F}_1 ≤ \hat{F}_2 ≤ … ≤ \hat{F}_m = 1$ be the values of $\hat{F}(t)$ outside the Turnbull intervals, such that $\hat{s}_j = \hat{F}_j - \hat{F}_{j-1}$. We seek the standard errors of these $\hat{\symbf{F}} = (\hat{F}_1, \hat{F}_2, …, \hat{F}_{m-1})^\mathrm{T}$.
+	%
+	Note that the log-likelihood $\mathcal{L}_i$ for the $i$-th observation is:
+	%
+	\begin{align*}
+		\mathcal{L}_i &= \log\left(\sum_{j=1}^m α_{i,j} \hat{s}_j\right) \\
+		&= \log\left(\sum_{j=1}^m α_{i,j} (\hat{F}_j - \hat{F}_{j-1})\right)
+	\end{align*}
+	%
+	Note the gradient $\nablasub{\hat{\symbf{F}}} \mathcal{L}_i$ is the vector whose $h$-th element is:
+	%
+	\begin{align*}
+		\frac{\partial \mathcal{L}_i}{\partial \hat{F}_h} &= \frac{α_{i,h} - α_{i,h+1}}{\sum_{j=1}^m α_{i,j} (\hat{F}_j - \hat{F}_{j-1})}
+	\end{align*}
+	%
+	And so the Hessian $\nablasub{\hat{\symbf{F}}} \mathcal{L}_i$ has $(h, k)$-th elements:
+	%
+	\begin{align*}
+		\frac{\partial \mathcal{L}_i}{\partial \hat{F}_h \partial \hat{F}_k} &= - \frac{( α_{i,h} - α_{i,h+1} ) ( α_{i,k} - α_{i,k+1} )}{\left( \sum_{j=1}^m α_{i,j} (\hat{F}_j - \hat{F}_{j-1}) \right)^2}
+	\end{align*}
+	%
+	The sum of all $\nablasub{\hat{\symbf{F}}} \mathcal{L}_i$ yields the Hessian of the log-likelihood $\nablasub{\hat{\symbf{F}}} \mathcal{L}$.
+	
+	The covariance matrix of $\hat{\symbf{F}}$ is given by the inverse of $-\nablasub{\hat{\symbf{F}}} \mathcal{L}$. The standard errors for each of $\hat{\symbf{F}}$ are the square roots of the diagonal elements of the covariance matrix, as required.
+	
+	{\vspace{0.5cm}\scshape\centering References\par}
+	
+	\begin{enumerate}
+		\item Turnbull BW. The empirical distribution function with arbitrarily grouped, censored and truncated data. \textit{Journal of the Royal Statistical Society, Series B (Methodological)}. 1976;38(3):290–5. \href{https://doi.org/10.1111/j.2517-6161.1976.tb01597.x}{doi: 10.1111\slash j.2517-6161.1976.tb01597.x}
+	\end{enumerate}
+	
+\end{document}