<?xml version="1.0" encoding="utf-8"?>
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head>
<!-- 2023-06-03 Sat 16:55 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<title>Equations for the rstpm2 package</title>
<meta name="generator" content="Org mode" />
<meta name="author" content="Mark Clements" />
<style type="text/css">
 <!--/*--><![CDATA[/*><!--*/
  .title  { text-align: center;
             margin-bottom: .2em; }
  .subtitle { text-align: center;
              font-size: medium;
              font-weight: bold;
              margin-top:0; }
  .todo   { font-family: monospace; color: red; }
  .done   { font-family: monospace; color: green; }
  .priority { font-family: monospace; color: orange; }
  .tag    { background-color: #eee; font-family: monospace;
            padding: 2px; font-size: 80%; font-weight: normal; }
  .timestamp { color: #bebebe; }
  .timestamp-kwd { color: #5f9ea0; }
  .org-right  { margin-left: auto; margin-right: 0px;  text-align: right; }
  .org-left   { margin-left: 0px;  margin-right: auto; text-align: left; }
  .org-center { margin-left: auto; margin-right: auto; text-align: center; }
  .underline { text-decoration: underline; }
  #postamble p, #preamble p { font-size: 90%; margin: .2em; }
  p.verse { margin-left: 3%; }
  pre {
    border: 1px solid #ccc;
    box-shadow: 3px 3px 3px #eee;
    padding: 8pt;
    font-family: monospace;
    overflow: auto;
    margin: 1.2em;
  }
  pre.src {
    position: relative;
    overflow: visible;
    padding-top: 1.2em;
  }
  pre.src:before {
    display: none;
    position: absolute;
    background-color: white;
    top: -10px;
    right: 10px;
    padding: 3px;
    border: 1px solid black;
  }
  pre.src:hover:before { display: inline;}
  /* Languages per Org manual */
  pre.src-asymptote:before { content: 'Asymptote'; }
  pre.src-awk:before { content: 'Awk'; }
  pre.src-C:before { content: 'C'; }
  /* pre.src-C++ doesn't work in CSS */
  pre.src-clojure:before { content: 'Clojure'; }
  pre.src-css:before { content: 'CSS'; }
  pre.src-D:before { content: 'D'; }
  pre.src-ditaa:before { content: 'ditaa'; }
  pre.src-dot:before { content: 'Graphviz'; }
  pre.src-calc:before { content: 'Emacs Calc'; }
  pre.src-emacs-lisp:before { content: 'Emacs Lisp'; }
  pre.src-fortran:before { content: 'Fortran'; }
  pre.src-gnuplot:before { content: 'gnuplot'; }
  pre.src-haskell:before { content: 'Haskell'; }
  pre.src-hledger:before { content: 'hledger'; }
  pre.src-java:before { content: 'Java'; }
  pre.src-js:before { content: 'Javascript'; }
  pre.src-latex:before { content: 'LaTeX'; }
  pre.src-ledger:before { content: 'Ledger'; }
  pre.src-lisp:before { content: 'Lisp'; }
  pre.src-lilypond:before { content: 'Lilypond'; }
  pre.src-lua:before { content: 'Lua'; }
  pre.src-matlab:before { content: 'MATLAB'; }
  pre.src-mscgen:before { content: 'Mscgen'; }
  pre.src-ocaml:before { content: 'Objective Caml'; }
  pre.src-octave:before { content: 'Octave'; }
  pre.src-org:before { content: 'Org mode'; }
  pre.src-oz:before { content: 'OZ'; }
  pre.src-plantuml:before { content: 'Plantuml'; }
  pre.src-processing:before { content: 'Processing.js'; }
  pre.src-python:before { content: 'Python'; }
  pre.src-R:before { content: 'R'; }
  pre.src-ruby:before { content: 'Ruby'; }
  pre.src-sass:before { content: 'Sass'; }
  pre.src-scheme:before { content: 'Scheme'; }
  pre.src-screen:before { content: 'Gnu Screen'; }
  pre.src-sed:before { content: 'Sed'; }
  pre.src-sh:before { content: 'shell'; }
  pre.src-sql:before { content: 'SQL'; }
  pre.src-sqlite:before { content: 'SQLite'; }
  /* additional languages in org.el's org-babel-load-languages alist */
  pre.src-forth:before { content: 'Forth'; }
  pre.src-io:before { content: 'IO'; }
  pre.src-J:before { content: 'J'; }
  pre.src-makefile:before { content: 'Makefile'; }
  pre.src-maxima:before { content: 'Maxima'; }
  pre.src-perl:before { content: 'Perl'; }
  pre.src-picolisp:before { content: 'Pico Lisp'; }
  pre.src-scala:before { content: 'Scala'; }
  pre.src-shell:before { content: 'Shell Script'; }
  pre.src-ebnf2ps:before { content: 'ebfn2ps'; }
  /* additional language identifiers per "defun org-babel-execute"
       in ob-*.el */
  pre.src-cpp:before  { content: 'C++'; }
  pre.src-abc:before  { content: 'ABC'; }
  pre.src-coq:before  { content: 'Coq'; }
  pre.src-groovy:before  { content: 'Groovy'; }
  /* additional language identifiers from org-babel-shell-names in
     ob-shell.el: ob-shell is the only babel language using a lambda to put
     the execution function name together. */
  pre.src-bash:before  { content: 'bash'; }
  pre.src-csh:before  { content: 'csh'; }
  pre.src-ash:before  { content: 'ash'; }
  pre.src-dash:before  { content: 'dash'; }
  pre.src-ksh:before  { content: 'ksh'; }
  pre.src-mksh:before  { content: 'mksh'; }
  pre.src-posh:before  { content: 'posh'; }
  /* Additional Emacs modes also supported by the LaTeX listings package */
  pre.src-ada:before { content: 'Ada'; }
  pre.src-asm:before { content: 'Assembler'; }
  pre.src-caml:before { content: 'Caml'; }
  pre.src-delphi:before { content: 'Delphi'; }
  pre.src-html:before { content: 'HTML'; }
  pre.src-idl:before { content: 'IDL'; }
  pre.src-mercury:before { content: 'Mercury'; }
  pre.src-metapost:before { content: 'MetaPost'; }
  pre.src-modula-2:before { content: 'Modula-2'; }
  pre.src-pascal:before { content: 'Pascal'; }
  pre.src-ps:before { content: 'PostScript'; }
  pre.src-prolog:before { content: 'Prolog'; }
  pre.src-simula:before { content: 'Simula'; }
  pre.src-tcl:before { content: 'tcl'; }
  pre.src-tex:before { content: 'TeX'; }
  pre.src-plain-tex:before { content: 'Plain TeX'; }
  pre.src-verilog:before { content: 'Verilog'; }
  pre.src-vhdl:before { content: 'VHDL'; }
  pre.src-xml:before { content: 'XML'; }
  pre.src-nxml:before { content: 'XML'; }
  /* add a generic configuration mode; LaTeX export needs an additional
     (add-to-list 'org-latex-listings-langs '(conf " ")) in .emacs */
  pre.src-conf:before { content: 'Configuration File'; }

  table { border-collapse:collapse; }
  caption.t-above { caption-side: top; }
  caption.t-bottom { caption-side: bottom; }
  td, th { vertical-align:top;  }
  th.org-right  { text-align: center;  }
  th.org-left   { text-align: center;   }
  th.org-center { text-align: center; }
  td.org-right  { text-align: right;  }
  td.org-left   { text-align: left;   }
  td.org-center { text-align: center; }
  dt { font-weight: bold; }
  .footpara { display: inline; }
  .footdef  { margin-bottom: 1em; }
  .figure { padding: 1em; }
  .figure p { text-align: center; }
  .equation-container {
    display: table;
    text-align: center;
    width: 100%;
  }
  .equation {
    vertical-align: middle;
  }
  .equation-label {
    display: table-cell;
    text-align: right;
    vertical-align: middle;
  }
  .inlinetask {
    padding: 10px;
    border: 2px solid gray;
    margin: 10px;
    background: #ffffcc;
  }
  #org-div-home-and-up
   { text-align: right; font-size: 70%; white-space: nowrap; }
  textarea { overflow-x: auto; }
  .linenr { font-size: smaller }
  .code-highlighted { background-color: #ffff00; }
  .org-info-js_info-navigation { border-style: none; }
  #org-info-js_console-label
    { font-size: 10px; font-weight: bold; white-space: nowrap; }
  .org-info-js_search-highlight
    { background-color: #ffff00; color: #000000; font-weight: bold; }
  .org-svg { width: 90%; }
  /*]]>*/-->
</style>
<script type="text/javascript">
/*
@licstart  The following is the entire license notice for the
JavaScript code in this tag.

Copyright (C) 2012-2020 Free Software Foundation, Inc.

The JavaScript code in this tag is free software: you can
redistribute it and/or modify it under the terms of the GNU
General Public License (GNU GPL) as published by the Free Software
Foundation, either version 3 of the License, or (at your option)
any later version.  The code is distributed WITHOUT ANY WARRANTY;
without even the implied warranty of MERCHANTABILITY or FITNESS
FOR A PARTICULAR PURPOSE.  See the GNU GPL for more details.

As additional permission under GNU GPL version 3 section 7, you
may distribute non-source (e.g., minimized or compacted) forms of
that code without the copy of the GNU GPL normally required by
section 4, provided you include this license notice and a URL
through which recipients can access the Corresponding Source.


@licend  The above is the entire license notice
for the JavaScript code in this tag.
*/
<!--/*--><![CDATA[/*><!--*/
 function CodeHighlightOn(elem, id)
 {
   var target = document.getElementById(id);
   if(null != target) {
     elem.cacheClassElem = elem.className;
     elem.cacheClassTarget = target.className;
     target.className = "code-highlighted";
     elem.className   = "code-highlighted";
   }
 }
 function CodeHighlightOff(elem, id)
 {
   var target = document.getElementById(id);
   if(elem.cacheClassElem)
     elem.className = elem.cacheClassElem;
   if(elem.cacheClassTarget)
     target.className = elem.cacheClassTarget;
 }
/*]]>*///-->
</script>
<script type="text/x-mathjax-config">
    MathJax.Hub.Config({
        displayAlign: "center",
        displayIndent: "0em",

        "HTML-CSS": { scale: 100,
                        linebreaks: { automatic: "false" },
                        webFont: "TeX"
                       },
        SVG: {scale: 100,
              linebreaks: { automatic: "false" },
              font: "TeX"},
        NativeMML: {scale: 100},
        TeX: { equationNumbers: {autoNumber: "AMS"},
               MultLineWidth: "85%",
               TagSide: "right",
               TagIndent: ".8em"
             }
});
</script>
<script type="text/javascript"
        src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.0/MathJax.js?config=TeX-AMS_HTML"></script>
</head>
<body>
<div id="content">
<h1 class="title">Equations for the rstpm2 package</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#orgfa17b91">1. Generalised survival model</a></li>
<li><a href="#org92fd60c">2. Mixture cure models</a></li>
<li><a href="#orgab1b650">3. Integral equation for AFT models</a></li>
<li><a href="#org9f78459">4. Integral equation for AFT models with functional constraints on gamma</a></li>
<li><a href="#orgf26dba5">5. Non-integral equation for AFT models</a></li>
<li><a href="#org3f1b7de">6. Non-integral equation for AFT models without time-varying effect</a></li>
<li><a href="#org806037c">7. Integral equations for AFT models with \(H = B(\ldots)\)</a></li>
</ul>
</div>
</div>

<div id="outline-container-orgfa17b91" class="outline-2">
<h2 id="orgfa17b91"><span class="section-number-2">1</span> Generalised survival model</h2>
<div class="outline-text-2" id="text-1">
<div class="org-src-container">
<pre class="src src-maxima"><span style="color: #a020f0;">logexpand:all</span>;
<span style="color: #a020f0;">derivsubst:</span><span style="color: #008b8b;">true</span>;
<span style="color: #483d8b;">load</span>(pdiff)$
<span style="color: #483d8b;">assume</span>(t<span style="color: #a020f0;">&gt;</span><span style="color: #008b8b;">0</span>)$
S <span style="color: #a020f0;">:</span> G(eta(t,z,theta))<span style="color: #a020f0;">*</span>S0(t);
H <span style="color: #a020f0;">:</span> <span style="color: #a020f0;">-</span><span style="color: #483d8b;">log</span>(S);
h <span style="color: #a020f0;">:</span> <span style="color: #483d8b;">diff</span>(H,t);
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"H:"</span>);
<span style="color: #483d8b;">tex</span>(H);
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"h:"</span>);
<span style="color: #483d8b;">tex</span>(h);

<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(H,theta):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(H,theta));

<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(log(h),theta):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(<span style="color: #483d8b;">log</span>(h),theta));

<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"ll:"</span>);
ll <span style="color: #a020f0;">:</span> <span style="color: #483d8b;">delta</span><span style="color: #a020f0;">*</span><span style="color: #483d8b;">log</span>(h) <span style="color: #a020f0;">-</span> H <span style="color: #a020f0;">-</span>  <span style="color: #483d8b;">diff</span>(eta(t,z,<span style="color: #483d8b;">beta</span>),t)<span style="color: #a020f0;">^</span><span style="color: #008b8b;">2</span>;
<span style="color: #483d8b;">tex</span>(ll);
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(ll, beta)"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(ll, <span style="color: #483d8b;">beta</span>));
</pre>
</div>

<p>
H: 
\[-\log G\left(\eta\left(t , z , \vartheta\right)\right)-\log S_{0}
 \left(t\right)\]
h: 
\[-{{\eta_{\left(1 , 0 , 0\right)}(t,z,\vartheta)\,G_{\left(1\right)}
 (\eta\left(t , z , \vartheta\right))}\over{G\left(\eta\left(t , z , 
 \vartheta\right)\right)}}-{{S_{0}_{\left(1\right)}(t)}\over{S_{0}
 \left(t\right)}}\]
diff(H,theta): 
\[-{{\eta_{\left(0 , 0 , 1\right)}(t,z,\vartheta)\,G_{\left(1\right)}
 (\eta\left(t , z , \vartheta\right))}\over{G\left(\eta\left(t , z , 
 \vartheta\right)\right)}}\]
diff(log(h),theta): 
\[{{-{{\eta_{\left(0 , 0 , 1\right)}(t,z,\vartheta)\,\eta_{\left(1 , 
 0 , 0\right)}(t,z,\vartheta)\,G_{\left(2\right)}(\eta\left(t , z , 
 \vartheta\right))}\over{G\left(\eta\left(t , z , \vartheta\right)
 \right)}}+{{\eta_{\left(0 , 0 , 1\right)}(t,z,\vartheta)\,\eta_{
 \left(1 , 0 , 0\right)}(t,z,\vartheta)\,G_{\left(1\right)}(\eta
 \left(t , z , \vartheta\right))^2}\over{G\left(\eta\left(t , z , 
 \vartheta\right)\right)^2}}-{{\eta_{\left(1 , 0 , 1\right)}(t,z,
 \vartheta)\,G_{\left(1\right)}(\eta\left(t , z , \vartheta\right))
 }\over{G\left(\eta\left(t , z , \vartheta\right)\right)}}}\over{-{{
 \eta_{\left(1 , 0 , 0\right)}(t,z,\vartheta)\,G_{\left(1\right)}(
 \eta\left(t , z , \vartheta\right))}\over{G\left(\eta\left(t , z , 
 \vartheta\right)\right)}}-{{S_{0}_{\left(1\right)}(t)}\over{S_{0}
 \left(t\right)}}}}\]
ll: 
\[\delta\,\log \left(-{{\eta_{\left(1 , 0 , 0\right)}(t,z,\vartheta)
 \,G_{\left(1\right)}(\eta\left(t , z , \vartheta\right))}\over{G
 \left(\eta\left(t , z , \vartheta\right)\right)}}-{{S_{0}_{\left(1
 \right)}(t)}\over{S_{0}\left(t\right)}}\right)+\log G\left(\eta
 \left(t , z , \vartheta\right)\right)+\log S_{0}\left(t\right)-\eta
 _{\left(1 , 0 , 0\right)}(t,z,\beta)^2\]
diff(ll, beta) 
\[-2\,\eta_{\left(1 , 0 , 0\right)}(t,z,\beta)\,\eta_{\left(1 , 0 , 1
 \right)}(t,z,\beta)\]
</p>
</div>
</div>


<div id="outline-container-org92fd60c" class="outline-2">
<h2 id="org92fd60c"><span class="section-number-2">2</span> Mixture cure models</h2>
<div class="outline-text-2" id="text-2">
<div class="org-src-container">
<pre class="src src-maxima"><span style="color: #a020f0;">logexpand:all</span>;
<span style="color: #a020f0;">derivsubst:</span><span style="color: #008b8b;">true</span>;
<span style="color: #483d8b;">load</span>(pdiff)$
S <span style="color: #a020f0;">:</span> S0(t)<span style="color: #a020f0;">*</span>(pi(theta)<span style="color: #a020f0;">+</span>(<span style="color: #008b8b;">1</span><span style="color: #a020f0;">-</span>pi(theta))<span style="color: #a020f0;">*</span><span style="color: #483d8b;">exp</span>(<span style="color: #a020f0;">-</span>Hu(t,theta)));
H <span style="color: #a020f0;">:</span> <span style="color: #a020f0;">-</span><span style="color: #483d8b;">log</span>(S);
h <span style="color: #a020f0;">:</span> <span style="color: #483d8b;">diff</span>(H,t);
h <span style="color: #a020f0;">:</span> <span style="color: #483d8b;">subst</span>(hu(t,theta), <span style="color: #483d8b;">diff</span>(Hu(t,theta),t), h);

<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"H:"</span>);
<span style="color: #483d8b;">tex</span>(H);
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"h:"</span>);
<span style="color: #483d8b;">tex</span>(h);

<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(H,theta):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(H,theta));

<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(log(h),theta):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(<span style="color: #483d8b;">log</span>(h),theta));
<span style="color: #483d8b;">diff</span>(<span style="color: #483d8b;">log</span>(h),theta) <span style="color: #a020f0;">-</span> <span style="color: #483d8b;">diff</span>(H,theta);
</pre>
</div>

<p>
H: 
\[-\log \left(\pi\left(\vartheta\right)+e^ {- {\it Hu}\left(t , 
 \vartheta\right) }\,\left(1-\pi\left(\vartheta\right)\right)\right)-
 \log S_{0}\left(t\right)\]
h: 
\[{{e^ {- {\it Hu}\left(t , \vartheta\right) }\,{\it hu}\left(t , 
 \vartheta\right)\,\left(1-\pi\left(\vartheta\right)\right)}\over{\pi
 \left(\vartheta\right)+e^ {- {\it Hu}\left(t , \vartheta\right) }\,
 \left(1-\pi\left(\vartheta\right)\right)}}-{{S_{0}_{\left(1\right)}(
 t)}\over{S_{0}\left(t\right)}}\]
diff(H,theta): 
\[-{{-e^ {- {\it Hu}\left(t , \vartheta\right) }\,\pi_{\left(1\right)
 }(\vartheta)+\pi_{\left(1\right)}(\vartheta)-e^ {- {\it Hu}\left(t
  , \vartheta\right) }\,{\it Hu}_{\left(0 , 1\right)}(t,\vartheta)\,
 \left(1-\pi\left(\vartheta\right)\right)}\over{\pi\left(\vartheta
 \right)+e^ {- {\it Hu}\left(t , \vartheta\right) }\,\left(1-\pi
 \left(\vartheta\right)\right)}}\]
diff(log(h),theta): 
\[{{-{{e^ {- {\it Hu}\left(t , \vartheta\right) }\,{\it hu}\left(t , 
 \vartheta\right)\,\left(1-\pi\left(\vartheta\right)\right)\,\left(-e
 ^ {- {\it Hu}\left(t , \vartheta\right) }\,\pi_{\left(1\right)}(
 \vartheta)+\pi_{\left(1\right)}(\vartheta)-e^ {- {\it Hu}\left(t , 
 \vartheta\right) }\,{\it Hu}_{\left(0 , 1\right)}(t,\vartheta)\,
 \left(1-\pi\left(\vartheta\right)\right)\right)}\over{\left(\pi
 \left(\vartheta\right)+e^ {- {\it Hu}\left(t , \vartheta\right) }\,
 \left(1-\pi\left(\vartheta\right)\right)\right)^2}}-{{e^ {- {\it Hu}
 \left(t , \vartheta\right) }\,{\it hu}\left(t , \vartheta\right)\,
 \pi_{\left(1\right)}(\vartheta)}\over{\pi\left(\vartheta\right)+e
 ^ {- {\it Hu}\left(t , \vartheta\right) }\,\left(1-\pi\left(
 \vartheta\right)\right)}}+{{e^ {- {\it Hu}\left(t , \vartheta\right)
  }\,{\it hu}_{\left(0 , 1\right)}(t,\vartheta)\,\left(1-\pi\left(
 \vartheta\right)\right)}\over{\pi\left(\vartheta\right)+e^ {- 
 {\it Hu}\left(t , \vartheta\right) }\,\left(1-\pi\left(\vartheta
 \right)\right)}}-{{e^ {- {\it Hu}\left(t , \vartheta\right) }\,
 {\it Hu}_{\left(0 , 1\right)}(t,\vartheta)\,{\it hu}\left(t , 
 \vartheta\right)\,\left(1-\pi\left(\vartheta\right)\right)}\over{\pi
 \left(\vartheta\right)+e^ {- {\it Hu}\left(t , \vartheta\right) }\,
 \left(1-\pi\left(\vartheta\right)\right)}}}\over{{{e^ {- {\it Hu}
 \left(t , \vartheta\right) }\,{\it hu}\left(t , \vartheta\right)\,
 \left(1-\pi\left(\vartheta\right)\right)}\over{\pi\left(\vartheta
 \right)+e^ {- {\it Hu}\left(t , \vartheta\right) }\,\left(1-\pi
 \left(\vartheta\right)\right)}}-{{S_{0}_{\left(1\right)}(t)}\over{
 S_{0}\left(t\right)}}}}\]
</p>
</div>
</div>

<div id="outline-container-orgab1b650" class="outline-2">
<h2 id="orgab1b650"><span class="section-number-2">3</span> Integral equation for AFT models</h2>
<div class="outline-text-2" id="text-3">
<div class="org-src-container">
<pre class="src src-maxima"><span style="color: #a020f0;">logexpand:all</span>;
<span style="color: #a020f0;">derivsubst:</span><span style="color: #008b8b;">true</span>;
<span style="color: #483d8b;">load</span>(pdiff)$
<span style="color: #483d8b;">assume</span>(t<span style="color: #a020f0;">&gt;</span><span style="color: #008b8b;">0</span>)$
S <span style="color: #a020f0;">:</span> <span style="color: #483d8b;">exp</span>(<span style="color: #a020f0;">-</span><span style="color: #483d8b;">exp</span>(B(<span style="color: #483d8b;">log</span>(<span style="color: #483d8b;">integrate</span>(<span style="color: #483d8b;">exp</span>(<span style="color: #a020f0;">-</span>x(v)<span style="color: #a020f0;">*</span><span style="color: #483d8b;">beta</span>),v,<span style="color: #008b8b;">0</span>,t)),<span style="color: #483d8b;">gamma</span>)));
H <span style="color: #a020f0;">:</span> <span style="color: #a020f0;">-</span><span style="color: #483d8b;">log</span>(S);
h <span style="color: #a020f0;">:</span> <span style="color: #483d8b;">diff</span>(H,t);
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"H:"</span>);
<span style="color: #483d8b;">tex</span>(H);
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"h:"</span>);
<span style="color: #483d8b;">tex</span>(h);
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"log(h):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">log</span>(h));

<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(H,beta):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(H,<span style="color: #483d8b;">beta</span>));
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(H,gamma):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(H,<span style="color: #483d8b;">gamma</span>));

<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(log(h),beta):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(<span style="color: #483d8b;">log</span>(h),<span style="color: #483d8b;">beta</span>));
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(log(h),gamma):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(<span style="color: #483d8b;">log</span>(h),<span style="color: #483d8b;">gamma</span>));

<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"ll:"</span>);
ll <span style="color: #a020f0;">:</span> <span style="color: #483d8b;">delta</span><span style="color: #a020f0;">*</span><span style="color: #483d8b;">log</span>(h) <span style="color: #a020f0;">-</span> H <span style="color: #a020f0;">-</span> (<span style="color: #483d8b;">subst</span>(u<span style="color: #a020f0;">=</span><span style="color: #483d8b;">log</span>(<span style="color: #483d8b;">integrate</span>(<span style="color: #483d8b;">exp</span>(<span style="color: #a020f0;">-</span>x(v)<span style="color: #a020f0;">*</span><span style="color: #483d8b;">beta</span>),v,<span style="color: #008b8b;">0</span>,t)), <span style="color: #483d8b;">diff</span>(B(u,<span style="color: #483d8b;">gamma</span>),u)))<span style="color: #a020f0;">^</span><span style="color: #008b8b;">2</span>;
<span style="color: #483d8b;">tex</span>(ll);
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(ll, beta)"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(ll, <span style="color: #483d8b;">beta</span>));
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(ll, gamma)"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(ll, <span style="color: #483d8b;">gamma</span>));
</pre>
</div>

<p>
H: 
\[e^{B\left(\log \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv} , 
 \gamma\right)}\]
h: 
\[{{e^{B\left(\log \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv}
  , \gamma\right)-\beta\,x\left(t\right)}\,B_{\left(1 , 0\right)}(
 \log \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv},\gamma)}\over{
 \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv}}}\]
log(h): 
\[\log B_{\left(1 , 0\right)}(\log \int_{0}^{t}{e^ {- \beta\,x\left(v
 \right) }\;dv},\gamma)+B\left(\log \int_{0}^{t}{e^ {- \beta\,x\left(
 v\right) }\;dv} , \gamma\right)-\log \int_{0}^{t}{e^ {- \beta\,x
 \left(v\right) }\;dv}-\beta\,x\left(t\right)\]
diff(H,beta): 
\[-{{\left(\int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\,x\left(v
 \right)\;dv}\right)\,e^{B\left(\log \int_{0}^{t}{e^ {- \beta\,x
 \left(v\right) }\;dv} , \gamma\right)}\,B_{\left(1 , 0\right)}(\log 
 \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv},\gamma)}\over{
 \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv}}}\]
diff(H,gamma): 
\[e^{B\left(\log \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv} , 
 \gamma\right)}\,B_{\left(0 , 1\right)}(\log \int_{0}^{t}{e^ {- \beta
 \,x\left(v\right) }\;dv},\gamma)\]
diff(log(h),beta): 
\[-{{\left(\int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\,x\left(v
 \right)\;dv}\right)\,B_{\left(2 , 0\right)}(\log \int_{0}^{t}{e^ {- 
 \beta\,x\left(v\right) }\;dv},\gamma)}\over{\left(\int_{0}^{t}{e
 ^ {- \beta\,x\left(v\right) }\;dv}\right)\,B_{\left(1 , 0\right)}(
 \log \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv},\gamma)}}-{{
 \left(\int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\,x\left(v\right)
 \;dv}\right)\,B_{\left(1 , 0\right)}(\log \int_{0}^{t}{e^ {- \beta\,
 x\left(v\right) }\;dv},\gamma)}\over{\int_{0}^{t}{e^ {- \beta\,x
 \left(v\right) }\;dv}}}+{{\int_{0}^{t}{e^ {- \beta\,x\left(v\right)
  }\,x\left(v\right)\;dv}}\over{\int_{0}^{t}{e^ {- \beta\,x\left(v
 \right) }\;dv}}}-x\left(t\right)\]
diff(log(h),gamma): 
\[{{B_{\left(1 , 1\right)}(\log \int_{0}^{t}{e^ {- \beta\,x\left(v
 \right) }\;dv},\gamma)}\over{B_{\left(1 , 0\right)}(\log \int_{0}^{t
 }{e^ {- \beta\,x\left(v\right) }\;dv},\gamma)}}+B_{\left(0 , 1
 \right)}(\log \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv},
 \gamma)\]
ll: 
\[\delta\,\left(\log B_{\left(1 , 0\right)}(\log \int_{0}^{t}{e^ {- 
 \beta\,x\left(v\right) }\;dv},\gamma)+B\left(\log \int_{0}^{t}{e
 ^ {- \beta\,x\left(v\right) }\;dv} , \gamma\right)-\log \int_{0}^{t
 }{e^ {- \beta\,x\left(v\right) }\;dv}-\beta\,x\left(t\right)\right)-
 B_{\left(1 , 0\right)}(\log \int_{0}^{t}{e^ {- \beta\,x\left(v
 \right) }\;dv},\gamma)^2-e^{B\left(\log \int_{0}^{t}{e^ {- \beta\,x
 \left(v\right) }\;dv} , \gamma\right)}\]
diff(ll, beta) 
\[\delta\,\left(-{{\left(\int_{0}^{t}{e^ {- \beta\,x\left(v\right) }
 \,x\left(v\right)\;dv}\right)\,B_{\left(2 , 0\right)}(\log \int_{0
 }^{t}{e^ {- \beta\,x\left(v\right) }\;dv},\gamma)}\over{\left(\int_{
 0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv}\right)\,B_{\left(1 , 0
 \right)}(\log \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv},
 \gamma)}}-{{\left(\int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\,x
 \left(v\right)\;dv}\right)\,B_{\left(1 , 0\right)}(\log \int_{0}^{t
 }{e^ {- \beta\,x\left(v\right) }\;dv},\gamma)}\over{\int_{0}^{t}{e
 ^ {- \beta\,x\left(v\right) }\;dv}}}+{{\int_{0}^{t}{e^ {- \beta\,x
 \left(v\right) }\,x\left(v\right)\;dv}}\over{\int_{0}^{t}{e^ {- 
 \beta\,x\left(v\right) }\;dv}}}-x\left(t\right)\right)+{{2\,\left(
 \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\,x\left(v\right)\;dv}
 \right)\,B_{\left(1 , 0\right)}(\log \int_{0}^{t}{e^ {- \beta\,x
 \left(v\right) }\;dv},\gamma)\,B_{\left(2 , 0\right)}(\log \int_{0
 }^{t}{e^ {- \beta\,x\left(v\right) }\;dv},\gamma)}\over{\int_{0}^{t
 }{e^ {- \beta\,x\left(v\right) }\;dv}}}+{{\left(\int_{0}^{t}{e^ {- 
 \beta\,x\left(v\right) }\,x\left(v\right)\;dv}\right)\,e^{B\left(
 \log \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv} , \gamma
 \right)}\,B_{\left(1 , 0\right)}(\log \int_{0}^{t}{e^ {- \beta\,x
 \left(v\right) }\;dv},\gamma)}\over{\int_{0}^{t}{e^ {- \beta\,x
 \left(v\right) }\;dv}}}\]
diff(ll, gamma) 
\[\delta\,\left({{B_{\left(1 , 1\right)}(\log \int_{0}^{t}{e^ {- 
 \beta\,x\left(v\right) }\;dv},\gamma)}\over{B_{\left(1 , 0\right)}(
 \log \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv},\gamma)}}+B_{
 \left(0 , 1\right)}(\log \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }
 \;dv},\gamma)\right)-2\,B_{\left(1 , 0\right)}(\log \int_{0}^{t}{e
 ^ {- \beta\,x\left(v\right) }\;dv},\gamma)\,B_{\left(1 , 1\right)}(
 \log \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv},\gamma)-e^{B
 \left(\log \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv} , \gamma
 \right)}\,B_{\left(0 , 1\right)}(\log \int_{0}^{t}{e^ {- \beta\,x
 \left(v\right) }\;dv},\gamma)\]
</p>
</div>
</div>

<div id="outline-container-org9f78459" class="outline-2">
<h2 id="org9f78459"><span class="section-number-2">4</span> Integral equation for AFT models with functional constraints on gamma</h2>
<div class="outline-text-2" id="text-4">
<div class="org-src-container">
<pre class="src src-maxima"><span style="color: #a020f0;">logexpand:all</span>;
<span style="color: #a020f0;">derivsubst:</span><span style="color: #008b8b;">true</span>;
<span style="color: #483d8b;">load</span>(pdiff)$
<span style="color: #483d8b;">assume</span>(t<span style="color: #a020f0;">&gt;</span><span style="color: #008b8b;">0</span>)$
<span style="color: #483d8b;">gamma</span> <span style="color: #a020f0;">:</span> gamma0 <span style="color: #a020f0;">+</span> <span style="color: #483d8b;">exp</span>(alpha);
S <span style="color: #a020f0;">:</span> <span style="color: #483d8b;">exp</span>(<span style="color: #a020f0;">-</span><span style="color: #483d8b;">exp</span>(B(<span style="color: #483d8b;">log</span>(<span style="color: #483d8b;">integrate</span>(<span style="color: #483d8b;">exp</span>(<span style="color: #a020f0;">-</span>x(v)<span style="color: #a020f0;">*</span><span style="color: #483d8b;">beta</span>),v,<span style="color: #008b8b;">0</span>,t)),<span style="color: #483d8b;">gamma</span>)));
H <span style="color: #a020f0;">:</span> <span style="color: #a020f0;">-</span><span style="color: #483d8b;">log</span>(S);
h <span style="color: #a020f0;">:</span> <span style="color: #483d8b;">diff</span>(H,t);
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"H:"</span>);
<span style="color: #483d8b;">tex</span>(H);
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"h:"</span>);
<span style="color: #483d8b;">tex</span>(h);

<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(H,beta):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(H,<span style="color: #483d8b;">beta</span>));
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(H,alpha):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(H,alpha));
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(H,gamma0):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(H,gamma0));

<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(log(h),beta):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(<span style="color: #483d8b;">log</span>(h),<span style="color: #483d8b;">beta</span>));
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(log(h),alpha):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(<span style="color: #483d8b;">log</span>(h),alpha));
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(log(h),gamma0):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(<span style="color: #483d8b;">log</span>(h),gamma0));
</pre>
</div>

<p>
H: 
\[e^{B\left(\log \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv} , 
 \gamma_{0}+e^{\alpha}\right)}\]
h: 
\[{{e^{B\left(\log \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv}
  , \gamma_{0}+e^{\alpha}\right)-\beta\,x\left(t\right)}\,B_{\left(1
  , 0\right)}(\log \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv},
 \gamma_{0}+e^{\alpha})}\over{\int_{0}^{t}{e^ {- \beta\,x\left(v
 \right) }\;dv}}}\]
diff(H,beta): 
\[-{{\left(\int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\,x\left(v
 \right)\;dv}\right)\,e^{B\left(\log \int_{0}^{t}{e^ {- \beta\,x
 \left(v\right) }\;dv} , \gamma_{0}+e^{\alpha}\right)}\,B_{\left(1 , 
 0\right)}(\log \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv},
 \gamma_{0}+e^{\alpha})}\over{\int_{0}^{t}{e^ {- \beta\,x\left(v
 \right) }\;dv}}}\]
diff(H,alpha): 
\[e^{B\left(\log \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv} , 
 \gamma_{0}+e^{\alpha}\right)+\alpha}\,B_{\left(0 , 1\right)}(\log 
 \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv},\gamma_{0}+e^{
 \alpha})\]
diff(H,gamma0): 
\[e^{B\left(\log \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv} , 
 \gamma_{0}+e^{\alpha}\right)}\,B_{\left(0 , 1\right)}(\log \int_{0
 }^{t}{e^ {- \beta\,x\left(v\right) }\;dv},\gamma_{0}+e^{\alpha})\]
diff(log(h),beta): 
\[-{{\left(\int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\,x\left(v
 \right)\;dv}\right)\,B_{\left(2 , 0\right)}(\log \int_{0}^{t}{e^ {- 
 \beta\,x\left(v\right) }\;dv},\gamma_{0}+e^{\alpha})}\over{\left(
 \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv}\right)\,B_{\left(1
  , 0\right)}(\log \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv},
 \gamma_{0}+e^{\alpha})}}-{{\left(\int_{0}^{t}{e^ {- \beta\,x\left(v
 \right) }\,x\left(v\right)\;dv}\right)\,B_{\left(1 , 0\right)}(\log 
 \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv},\gamma_{0}+e^{
 \alpha})}\over{\int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv}}}+{{
 \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\,x\left(v\right)\;dv}
 }\over{\int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv}}}-x\left(t
 \right)\]
diff(log(h),alpha): 
\[{{e^{\alpha}\,B_{\left(1 , 1\right)}(\log \int_{0}^{t}{e^ {- \beta
 \,x\left(v\right) }\;dv},\gamma_{0}+e^{\alpha})}\over{B_{\left(1 , 0
 \right)}(\log \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv},
 \gamma_{0}+e^{\alpha})}}+e^{\alpha}\,B_{\left(0 , 1\right)}(\log 
 \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv},\gamma_{0}+e^{
 \alpha})\]
diff(log(h),gamma0): 
\[{{B_{\left(1 , 1\right)}(\log \int_{0}^{t}{e^ {- \beta\,x\left(v
 \right) }\;dv},\gamma_{0}+e^{\alpha})}\over{B_{\left(1 , 0\right)}(
 \log \int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv},\gamma_{0}+e^{
 \alpha})}}+B_{\left(0 , 1\right)}(\log \int_{0}^{t}{e^ {- \beta\,x
 \left(v\right) }\;dv},\gamma_{0}+e^{\alpha})\]
</p>
</div>
</div>


<div id="outline-container-orgf26dba5" class="outline-2">
<h2 id="orgf26dba5"><span class="section-number-2">5</span> Non-integral equation for AFT models</h2>
<div class="outline-text-2" id="text-5">
<div class="org-src-container">
<pre class="src src-maxima"><span style="color: #a020f0;">logexpand:all</span>;
<span style="color: #a020f0;">derivsubst:</span><span style="color: #008b8b;">true</span>;
<span style="color: #483d8b;">load</span>(pdiff)$
<span style="color: #483d8b;">assume</span>(t<span style="color: #a020f0;">&gt;</span><span style="color: #008b8b;">0</span>)$
S <span style="color: #a020f0;">:</span> <span style="color: #483d8b;">exp</span>(<span style="color: #a020f0;">-</span><span style="color: #483d8b;">exp</span>(B(<span style="color: #483d8b;">log</span>(t<span style="color: #a020f0;">*</span><span style="color: #483d8b;">exp</span>(<span style="color: #a020f0;">-</span>eta(X,<span style="color: #483d8b;">log</span>(t),<span style="color: #483d8b;">beta</span>))),<span style="color: #483d8b;">gamma</span>)));
H <span style="color: #a020f0;">:</span> <span style="color: #a020f0;">-</span><span style="color: #483d8b;">log</span>(S);
h <span style="color: #a020f0;">:</span> <span style="color: #483d8b;">diff</span>(H,t);
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"H:"</span>);
<span style="color: #483d8b;">tex</span>(H);
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"h:"</span>);
<span style="color: #483d8b;">tex</span>(h);
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"log(h):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">log</span>(h));

<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(H,beta):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(H,<span style="color: #483d8b;">beta</span>));
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(H,gamma):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(H,<span style="color: #483d8b;">gamma</span>));

<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(log(h),beta):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(<span style="color: #483d8b;">log</span>(h),<span style="color: #483d8b;">beta</span>));
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(log(h),gamma):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(<span style="color: #483d8b;">log</span>(h),<span style="color: #483d8b;">gamma</span>));

<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"ll:"</span>);
ll <span style="color: #a020f0;">:</span> <span style="color: #483d8b;">delta</span><span style="color: #a020f0;">*</span><span style="color: #483d8b;">log</span>(h) <span style="color: #a020f0;">-</span> H <span style="color: #a020f0;">-</span> (<span style="color: #008b8b;">1</span><span style="color: #a020f0;">-</span><span style="color: #483d8b;">subst</span>(u<span style="color: #a020f0;">=</span><span style="color: #483d8b;">log</span>(t), <span style="color: #483d8b;">diff</span>(eta(X,u,<span style="color: #483d8b;">beta</span>),u)))<span style="color: #a020f0;">^</span><span style="color: #008b8b;">2</span> <span style="color: #a020f0;">-</span>
<span style="color: #483d8b;">subst</span>(u<span style="color: #a020f0;">=</span><span style="color: #483d8b;">log</span>(t)<span style="color: #a020f0;">-</span>eta(X,<span style="color: #483d8b;">log</span>(t),<span style="color: #483d8b;">beta</span>), <span style="color: #483d8b;">diff</span>(B(u,<span style="color: #483d8b;">gamma</span>),u))<span style="color: #a020f0;">^</span><span style="color: #008b8b;">2</span>;
<span style="color: #483d8b;">tex</span>(ll);
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(ll, beta)"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(ll, <span style="color: #483d8b;">beta</span>));
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(ll, gamma)"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(ll, <span style="color: #483d8b;">gamma</span>));

</pre>
</div>

<p>
H: 
\[e^{B\left(\log t-\eta\left(X , \log t , \beta\right) , \gamma
 \right)}\]
h: 
\[\left({{1}\over{t}}-{{\eta_{\left(0 , 1 , 0\right)}(X,\log t,\beta)
 }\over{t}}\right)\,e^{B\left(\log t-\eta\left(X , \log t , \beta
 \right) , \gamma\right)}\,B_{\left(1 , 0\right)}(\log t-\eta\left(X
  , \log t , \beta\right),\gamma)\]
log(h): 
\[\log B_{\left(1 , 0\right)}(\log t-\eta\left(X , \log t , \beta
 \right),\gamma)+B\left(\log t-\eta\left(X , \log t , \beta\right) , 
 \gamma\right)+\log \left({{1}\over{t}}-{{\eta_{\left(0 , 1 , 0
 \right)}(X,\log t,\beta)}\over{t}}\right)\]
diff(H,beta): 
\[-\eta_{\left(0 , 0 , 1\right)}(X,\log t,\beta)\,e^{B\left(\log t-
 \eta\left(X , \log t , \beta\right) , \gamma\right)}\,B_{\left(1 , 0
 \right)}(\log t-\eta\left(X , \log t , \beta\right),\gamma)\]
diff(H,gamma): 
\[e^{B\left(\log t-\eta\left(X , \log t , \beta\right) , \gamma
 \right)}\,B_{\left(0 , 1\right)}(\log t-\eta\left(X , \log t , \beta
 \right),\gamma)\]
diff(log(h),beta): 
\[-{{\eta_{\left(0 , 0 , 1\right)}(X,\log t,\beta)\,B_{\left(2 , 0
 \right)}(\log t-\eta\left(X , \log t , \beta\right),\gamma)}\over{B
 _{\left(1 , 0\right)}(\log t-\eta\left(X , \log t , \beta\right),
 \gamma)}}-\eta_{\left(0 , 0 , 1\right)}(X,\log t,\beta)\,B_{\left(1
  , 0\right)}(\log t-\eta\left(X , \log t , \beta\right),\gamma)-{{
 \eta_{\left(0 , 1 , 1\right)}(X,\log t,\beta)}\over{\left({{1}\over{
 t}}-{{\eta_{\left(0 , 1 , 0\right)}(X,\log t,\beta)}\over{t}}\right)
 \,t}}\]
diff(log(h),gamma): 
\[{{B_{\left(1 , 1\right)}(\log t-\eta\left(X , \log t , \beta\right)
 ,\gamma)}\over{B_{\left(1 , 0\right)}(\log t-\eta\left(X , \log t , 
 \beta\right),\gamma)}}+B_{\left(0 , 1\right)}(\log t-\eta\left(X , 
 \log t , \beta\right),\gamma)\]
ll: 
\[\delta\,\left(\log B_{\left(1 , 0\right)}(\log t-\eta\left(X , 
 \log t , \beta\right),\gamma)+B\left(\log t-\eta\left(X , \log t , 
 \beta\right) , \gamma\right)+\log \left({{1}\over{t}}-{{\eta_{\left(
 0 , 1 , 0\right)}(X,\log t,\beta)}\over{t}}\right)\right)-B_{\left(1
  , 0\right)}(\log t-\eta\left(X , \log t , \beta\right),\gamma)^2-e
 ^{B\left(\log t-\eta\left(X , \log t , \beta\right) , \gamma\right)}
 -\left(1-\eta_{\left(0 , 1 , 0\right)}(X,\log t,\beta)\right)^2\]
diff(ll, beta) 
\[\delta\,\left(-{{\eta_{\left(0 , 0 , 1\right)}(X,\log t,\beta)\,B_{
 \left(2 , 0\right)}(\log t-\eta\left(X , \log t , \beta\right),
 \gamma)}\over{B_{\left(1 , 0\right)}(\log t-\eta\left(X , \log t , 
 \beta\right),\gamma)}}-\eta_{\left(0 , 0 , 1\right)}(X,\log t,\beta)
 \,B_{\left(1 , 0\right)}(\log t-\eta\left(X , \log t , \beta\right),
 \gamma)-{{\eta_{\left(0 , 1 , 1\right)}(X,\log t,\beta)}\over{\left(
 {{1}\over{t}}-{{\eta_{\left(0 , 1 , 0\right)}(X,\log t,\beta)}\over{
 t}}\right)\,t}}\right)+2\,\eta_{\left(0 , 0 , 1\right)}(X,\log t,
 \beta)\,B_{\left(1 , 0\right)}(\log t-\eta\left(X , \log t , \beta
 \right),\gamma)\,B_{\left(2 , 0\right)}(\log t-\eta\left(X , \log t
  , \beta\right),\gamma)+\eta_{\left(0 , 0 , 1\right)}(X,\log t,\beta
 )\,e^{B\left(\log t-\eta\left(X , \log t , \beta\right) , \gamma
 \right)}\,B_{\left(1 , 0\right)}(\log t-\eta\left(X , \log t , \beta
 \right),\gamma)+2\,\left(1-\eta_{\left(0 , 1 , 0\right)}(X,\log t,
 \beta)\right)\,\eta_{\left(0 , 1 , 1\right)}(X,\log t,\beta)\]
diff(ll, gamma) 
\[\delta\,\left({{B_{\left(1 , 1\right)}(\log t-\eta\left(X , \log t
  , \beta\right),\gamma)}\over{B_{\left(1 , 0\right)}(\log t-\eta
 \left(X , \log t , \beta\right),\gamma)}}+B_{\left(0 , 1\right)}(
 \log t-\eta\left(X , \log t , \beta\right),\gamma)\right)-2\,B_{
 \left(1 , 0\right)}(\log t-\eta\left(X , \log t , \beta\right),
 \gamma)\,B_{\left(1 , 1\right)}(\log t-\eta\left(X , \log t , \beta
 \right),\gamma)-e^{B\left(\log t-\eta\left(X , \log t , \beta\right)
  , \gamma\right)}\,B_{\left(0 , 1\right)}(\log t-\eta\left(X , \log 
 t , \beta\right),\gamma)\]
</p>
</div>
</div>


<div id="outline-container-org3f1b7de" class="outline-2">
<h2 id="org3f1b7de"><span class="section-number-2">6</span> Non-integral equation for AFT models without time-varying effect</h2>
<div class="outline-text-2" id="text-6">
<div class="org-src-container">
<pre class="src src-maxima"><span style="color: #a020f0;">logexpand:all</span>;
<span style="color: #a020f0;">derivsubst:</span><span style="color: #008b8b;">true</span>;
<span style="color: #483d8b;">load</span>(pdiff)$
<span style="color: #483d8b;">assume</span>(t<span style="color: #a020f0;">&gt;</span><span style="color: #008b8b;">0</span>)$
S <span style="color: #a020f0;">:</span> <span style="color: #483d8b;">exp</span>(<span style="color: #a020f0;">-</span><span style="color: #483d8b;">exp</span>(B(<span style="color: #483d8b;">log</span>(t<span style="color: #a020f0;">*</span><span style="color: #483d8b;">exp</span>(<span style="color: #a020f0;">-</span>eta(X,<span style="color: #483d8b;">beta</span>))),<span style="color: #483d8b;">gamma</span>)));
H <span style="color: #a020f0;">:</span> <span style="color: #a020f0;">-</span><span style="color: #483d8b;">log</span>(S);
h <span style="color: #a020f0;">:</span> <span style="color: #483d8b;">diff</span>(H,t);
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"H:"</span>);
<span style="color: #483d8b;">tex</span>(H);
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"h:"</span>);
<span style="color: #483d8b;">tex</span>(h);
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"log(h):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">log</span>(h));

<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(H,beta):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(H,<span style="color: #483d8b;">beta</span>));
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(H,gamma):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(H,<span style="color: #483d8b;">gamma</span>));

<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(log(h),beta):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(<span style="color: #483d8b;">log</span>(h),<span style="color: #483d8b;">beta</span>));
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(log(h),gamma):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(<span style="color: #483d8b;">log</span>(h),<span style="color: #483d8b;">gamma</span>));

<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"ll:"</span>);
ll <span style="color: #a020f0;">:</span> <span style="color: #483d8b;">delta</span><span style="color: #a020f0;">*</span><span style="color: #483d8b;">log</span>(h) <span style="color: #a020f0;">-</span> H <span style="color: #a020f0;">-</span> (<span style="color: #008b8b;">1</span><span style="color: #a020f0;">-</span><span style="color: #483d8b;">subst</span>(u<span style="color: #a020f0;">=</span><span style="color: #483d8b;">log</span>(t), <span style="color: #483d8b;">diff</span>(eta(X,u,<span style="color: #483d8b;">beta</span>),u)))<span style="color: #a020f0;">^</span><span style="color: #008b8b;">2</span> <span style="color: #a020f0;">-</span>
<span style="color: #483d8b;">subst</span>(u<span style="color: #a020f0;">=</span><span style="color: #483d8b;">log</span>(t)<span style="color: #a020f0;">-</span>eta(X,<span style="color: #483d8b;">log</span>(t),<span style="color: #483d8b;">beta</span>), <span style="color: #483d8b;">diff</span>(B(u,<span style="color: #483d8b;">gamma</span>),u))<span style="color: #a020f0;">^</span><span style="color: #008b8b;">2</span>;
<span style="color: #483d8b;">tex</span>(ll);
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(ll, beta)"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(ll, <span style="color: #483d8b;">beta</span>));
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(ll, gamma)"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(ll, <span style="color: #483d8b;">gamma</span>));

</pre>
</div>

<p>
H: 
\[e^{B\left(\log t-\eta\left(X , \beta\right) , \gamma\right)}\]
h: 
\[{{e^{B\left(\log t-\eta\left(X , \beta\right) , \gamma\right)}\,B_{
 \left(1 , 0\right)}(\log t-\eta\left(X , \beta\right),\gamma)}\over{
 t}}\]
log(h): 
\[\log B_{\left(1 , 0\right)}(\log t-\eta\left(X , \beta\right),
 \gamma)+B\left(\log t-\eta\left(X , \beta\right) , \gamma\right)-
 \log t\]
diff(H,beta): 
\[-\eta_{\left(0 , 1\right)}(X,\beta)\,e^{B\left(\log t-\eta\left(X
  , \beta\right) , \gamma\right)}\,B_{\left(1 , 0\right)}(\log t-\eta
 \left(X , \beta\right),\gamma)\]
diff(H,gamma): 
\[e^{B\left(\log t-\eta\left(X , \beta\right) , \gamma\right)}\,B_{
 \left(0 , 1\right)}(\log t-\eta\left(X , \beta\right),\gamma)\]
diff(log(h),beta): 
\[-{{\eta_{\left(0 , 1\right)}(X,\beta)\,B_{\left(2 , 0\right)}(\log 
 t-\eta\left(X , \beta\right),\gamma)}\over{B_{\left(1 , 0\right)}(
 \log t-\eta\left(X , \beta\right),\gamma)}}-\eta_{\left(0 , 1\right)
 }(X,\beta)\,B_{\left(1 , 0\right)}(\log t-\eta\left(X , \beta\right)
 ,\gamma)\]
diff(log(h),gamma): 
\[{{B_{\left(1 , 1\right)}(\log t-\eta\left(X , \beta\right),\gamma)
 }\over{B_{\left(1 , 0\right)}(\log t-\eta\left(X , \beta\right),
 \gamma)}}+B_{\left(0 , 1\right)}(\log t-\eta\left(X , \beta\right),
 \gamma)\]
ll: 
\[\delta\,\left(\log B_{\left(1 , 0\right)}(\log t-\eta\left(X , 
 \beta\right),\gamma)+B\left(\log t-\eta\left(X , \beta\right) , 
 \gamma\right)-\log t\right)-B_{\left(1 , 0\right)}(\log t-\eta\left(
 X , \log t , \beta\right),\gamma)^2-e^{B\left(\log t-\eta\left(X , 
 \beta\right) , \gamma\right)}-\left(1-\eta_{\left(0 , 1 , 0\right)}(
 X,\log t,\beta)\right)^2\]
diff(ll, beta) 
\[2\,\eta_{\left(0 , 0 , 1\right)}(X,\log t,\beta)\,B_{\left(1 , 0
 \right)}(\log t-\eta\left(X , \log t , \beta\right),\gamma)\,B_{
 \left(2 , 0\right)}(\log t-\eta\left(X , \log t , \beta\right),
 \gamma)+\delta\,\left(-{{\eta_{\left(0 , 1\right)}(X,\beta)\,B_{
 \left(2 , 0\right)}(\log t-\eta\left(X , \beta\right),\gamma)}\over{
 B_{\left(1 , 0\right)}(\log t-\eta\left(X , \beta\right),\gamma)}}-
 \eta_{\left(0 , 1\right)}(X,\beta)\,B_{\left(1 , 0\right)}(\log t-
 \eta\left(X , \beta\right),\gamma)\right)+\eta_{\left(0 , 1\right)}(
 X,\beta)\,e^{B\left(\log t-\eta\left(X , \beta\right) , \gamma
 \right)}\,B_{\left(1 , 0\right)}(\log t-\eta\left(X , \beta\right),
 \gamma)+2\,\left(1-\eta_{\left(0 , 1 , 0\right)}(X,\log t,\beta)
 \right)\,\eta_{\left(0 , 1 , 1\right)}(X,\log t,\beta)\]
diff(ll, gamma) 
\[-2\,B_{\left(1 , 0\right)}(\log t-\eta\left(X , \log t , \beta
 \right),\gamma)\,B_{\left(1 , 1\right)}(\log t-\eta\left(X , \log t
  , \beta\right),\gamma)+\delta\,\left({{B_{\left(1 , 1\right)}(\log 
 t-\eta\left(X , \beta\right),\gamma)}\over{B_{\left(1 , 0\right)}(
 \log t-\eta\left(X , \beta\right),\gamma)}}+B_{\left(0 , 1\right)}(
 \log t-\eta\left(X , \beta\right),\gamma)\right)-e^{B\left(\log t-
 \eta\left(X , \beta\right) , \gamma\right)}\,B_{\left(0 , 1\right)}(
 \log t-\eta\left(X , \beta\right),\gamma)\]
</p>
</div>
</div>


<div id="outline-container-org806037c" class="outline-2">
<h2 id="org806037c"><span class="section-number-2">7</span> Integral equations for AFT models with \(H = B(\ldots)\)</h2>
<div class="outline-text-2" id="text-7">
<div class="org-src-container">
<pre class="src src-maxima"><span style="color: #a020f0;">logexpand:all</span>;
<span style="color: #a020f0;">derivsubst:</span><span style="color: #008b8b;">true</span>;
<span style="color: #483d8b;">load</span>(pdiff)$
<span style="color: #483d8b;">assume</span>(t<span style="color: #a020f0;">&gt;</span><span style="color: #008b8b;">0</span>)$
S <span style="color: #a020f0;">:</span> <span style="color: #483d8b;">exp</span>(<span style="color: #a020f0;">-</span>B(<span style="color: #483d8b;">integrate</span>(<span style="color: #483d8b;">exp</span>(<span style="color: #a020f0;">-</span>x(v)<span style="color: #a020f0;">*</span><span style="color: #483d8b;">beta</span>),v,<span style="color: #008b8b;">0</span>,t),<span style="color: #483d8b;">gamma</span>));
H <span style="color: #a020f0;">:</span> <span style="color: #a020f0;">-</span><span style="color: #483d8b;">log</span>(S);
h <span style="color: #a020f0;">:</span> <span style="color: #483d8b;">diff</span>(H,t);
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"H:"</span>);
<span style="color: #483d8b;">tex</span>(H);
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"h:"</span>);
<span style="color: #483d8b;">tex</span>(h);

<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(H,beta):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(H,<span style="color: #483d8b;">beta</span>));
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(H,gamma):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(H,<span style="color: #483d8b;">gamma</span>));

<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(log(h),beta):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(<span style="color: #483d8b;">log</span>(h),<span style="color: #483d8b;">beta</span>));
<span style="color: #483d8b;">print</span>(<span style="color: #8b2252;">"diff(log(h),gamma):"</span>);
<span style="color: #483d8b;">tex</span>(<span style="color: #483d8b;">diff</span>(<span style="color: #483d8b;">log</span>(h),<span style="color: #483d8b;">gamma</span>));
</pre>
</div>

<p>
H: 
\[B\left(\int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv} , \gamma
 \right)\]
h: 
\[e^ {- \beta\,x\left(t\right) }\,B_{\left(1 , 0\right)}(\int_{0}^{t
 }{e^ {- \beta\,x\left(v\right) }\;dv},\gamma)\]
diff(H,beta): 
\[-\left(\int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\,x\left(v\right)
 \;dv}\right)\,B_{\left(1 , 0\right)}(\int_{0}^{t}{e^ {- \beta\,x
 \left(v\right) }\;dv},\gamma)\]
diff(H,gamma): 
\[B_{\left(0 , 1\right)}(\int_{0}^{t}{e^ {- \beta\,x\left(v\right) }
 \;dv},\gamma)\]
diff(log(h),beta): 
\[-{{\left(\int_{0}^{t}{e^ {- \beta\,x\left(v\right) }\,x\left(v
 \right)\;dv}\right)\,B_{\left(2 , 0\right)}(\int_{0}^{t}{e^ {- \beta
 \,x\left(v\right) }\;dv},\gamma)}\over{B_{\left(1 , 0\right)}(\int_{
 0}^{t}{e^ {- \beta\,x\left(v\right) }\;dv},\gamma)}}-x\left(t\right)\]
diff(log(h),gamma): 
\[{{B_{\left(1 , 1\right)}(\int_{0}^{t}{e^ {- \beta\,x\left(v\right)
  }\;dv},\gamma)}\over{B_{\left(1 , 0\right)}(\int_{0}^{t}{e^ {- 
 \beta\,x\left(v\right) }\;dv},\gamma)}}\]
</p>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Mark Clements</p>
<p class="date">Created: 2023-06-03 Sat 16:55</p>
</div>
</body>
</html>
