%\VignetteEngine{knitr::knitr}
%\VignetteIndexEntry{The sadists package}
%\VignetteKeyword{distributions}
%\VignettePackage{sadists}
\documentclass[10pt,a4paper,english]{article}

% front matter%FOLDUP
\usepackage[hyphens]{url}
\usepackage{amsmath}
\usepackage{amsfonts}
% for therefore
\usepackage{amssymb}
% for theorems?
\usepackage{amsthm}

\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{example}{Example}[section]

\theoremstyle{remark}
\newtheorem*{remark}{Remark}
\newtheorem*{caution}{Caution}
\newtheorem*{note}{Note}

% see http://tex.stackexchange.com/a/3034/2530
\PassOptionsToPackage{hyphens}{url}\usepackage{hyperref}
\usepackage{hyperref}
\usepackage[square,numbers]{natbib}
%\usepackage[authoryear]{natbib}
%\usepackage[iso]{datetime}
%\usepackage{datetime}

%compactitem and such:
\usepackage[newitem,newenum,increaseonly]{paralist}

\makeatletter
\makeatother

%\input{sr_defs.tex}
\usepackage{sadists}

%\providecommand{\sideWarning}[1][0.5]{\marginpar{\hfill\includegraphics[width=#1\marginparwidth]{warning}}}
% see: https://stat.ethz.ch/pipermail/r-help/2007-November/144810.html
\newcommand{\pkg}[1]{{\normalfont\fontseries{b}\selectfont #1}}
\let\proglang=\textsf
\let\code=\texttt
\newcommand{\CRANpkg}[1]{\href{http://CRAN.R-project.org/package=#1}{\pkg{#1}}}
\newcommand{\sadists}{\CRANpkg{sadists}\xspace}
\newcommand{\PDQutils}{\CRANpkg{PDQutils}\xspace}

% knitr setup%FOLDUP

<<'preamble', include=FALSE, warning=FALSE, message=FALSE, cache=FALSE>>=
library(knitr)

# set the knitr options ... for everyone!
# if you unset this, then vignette build bonks. oh, joy.
#opts_knit$set(progress=TRUE)
opts_knit$set(eval.after='fig.cap')
# for a package vignette, you do want to echo.
# opts_chunk$set(echo=FALSE,warning=FALSE,message=FALSE)
opts_chunk$set(warning=FALSE,message=FALSE)
#opts_chunk$set(results="asis")
opts_chunk$set(cache=TRUE,cache.path="cache/sadists")

#opts_chunk$set(fig.path="figure/",dev=c("pdf","cairo_ps"))
opts_chunk$set(fig.path="figure/sadists",dev=c("pdf"))
#opts_chunk$set(fig.width=4.0,fig.height=6,dpi=200)
# see http://stackoverflow.com/q/23419782/164611
opts_chunk$set(fig.width=5.0,fig.height=3.0,out.width="5in",out.height="3in",dpi=200)

# doing this means that png files are made of figures;
# the savings is small, and it looks like shit:
#opts_chunk$set(fig.path="figure/",dev=c("png","pdf","cairo_ps"))
#opts_chunk$set(fig.width=4,fig.height=4)
# for figures? this is sweave-specific?
#opts_knit$set(eps=TRUE)

# this would be for figures:
#opts_chunk$set(out.width='.8\\textwidth')
# for text wrapping:
options(width=64,digits=2)
#opts_chunk$set(size="tiny")
opts_chunk$set(size="small")
opts_chunk$set(tidy=TRUE,tidy.opts=list(width.cutoff=36,blank=TRUE))

compile.time <- Sys.time()

# from the environment

# only recompute if FORCE_RECOMPUTE=True w/out case match.
FORCE_RECOMPUTE <- 
	(toupper(Sys.getenv('FORCE_RECOMPUTE',unset='False')) == "TRUE")

# compiler flags!

# not used yet
LONG.FORM <- FALSE

mc.resolution <- ifelse(LONG.FORM,1000,200)
mc.resolution <- max(mc.resolution,100)

library(sadists)

gen_norm <- rnorm
lseq <- function(from,to,length.out) { 
	exp(seq(log(from),log(to),length.out = length.out))
}
@
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    %UNFOLD
%UNFOLD

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% document incantations%FOLDUP
\begin{document}

\title{The \pkg{sadists} package}
\author{Steven E. Pav}% \thanks{\email{shabbychef@gmail.com}}}
%\date{\today, \currenttime}

\maketitle
%UNFOLD

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{abstract}%FOLDUP
The \sadists package includes `dpqr' functions for some obscure
distributions, mostly involving sums and ratios of (non-central)
chi-squares, chis, and normals.
\end{abstract}%UNFOLD
\section{Introduction}%FOLDUP

The \sadists package provides density, distribution, quantile and random
generation functions (the `dpqr' functions) for some obscure distributions.
For all of these, the `dpq' functions are 
approximated via the Edgeworth and Cornish-Fisher expansions. As such, this
package is a showcase for the capabilities of the \PDQutils package,
which does the heavy lifting once the cumulants have been computed. \cite{PDQutils-Manual}

It should be noted that the functions provided by \sadists do \emph{not}
recycle their distribution parameters against the 
\Rcode{x, p, q} or \Rcode{n} parameters. This is in contrast to the 
common \Rlang idiom, and may cause some confusion. This is mostly for reasons
of performance, but also because some of the distributions have vector-valued
parameters; recycling over these would require the user to provide \emph{lists}
of parameters, which would be unpleasant.

First, a function which will evaluate the `dpq' functions versus random
draws of the variable:

<<'setup',echo=TRUE,eval=TRUE>>=
require(ggplot2)
require(grid)
testf <- function(dpqr,nobs,...) {
	rv <- sort(dpqr$r(nobs,...))
	data <- data.frame(draws=rv,pvals=dpqr$p(rv,...))
	text.size <- 6   # sigh

	# http://stackoverflow.com/a/5688125/164611
	p1 <- qplot(rv, geom = 'blank') +   
		geom_line(aes(y = ..density.., colour = 'Empirical'), stat = 'density') +  
		stat_function(fun = function(x) { dpqr$d(x,...) }, aes(colour = 'Theoretical')) +                       
		geom_histogram(aes(y = ..density..), alpha = 0.3) +                        
		scale_colour_manual(name = 'Density', values = c('red', 'blue')) +
		theme(text=element_text(size=text.size)) + 
		labs(title="Density (tests dfunc)")

	# Q-Q plot
	p2 <- ggplot(data, aes(sample = draws)) + stat_qq(distribution=function(p) { dpqr$q(p,...) }) +
		geom_abline(slope=1,intercept=0,colour='red') + 
		theme(text=element_text(size=text.size)) + 
		labs(title="Q-Q plot (tests qfunc)")

	# empirical CDF of the p-values; should be uniform
	p3 <- ggplot(data, aes(sample = pvals)) + stat_qq(distribution=qunif) + 
		geom_abline(slope=1,intercept=0,colour='red') + 
		theme(text=element_text(size=text.size)) + 
		labs(title="P-P plot (tests pfunc)")

	# Define grid layout to locate plots and print each graph
	pushViewport(viewport(layout = grid.layout(2, 2)))
	print(p1, vp = viewport(layout.pos.row = 1, layout.pos.col = 1:2))
	print(p2, vp = viewport(layout.pos.row = 2, layout.pos.col = 1))
	print(p3, vp = viewport(layout.pos.row = 2, layout.pos.col = 2))
}
@
%UNFOLD

\section{Sum of (non-central) chi-squares to a power}%FOLDUP

Let $X_i \sim \chi^2_{\nu_i}\left(\delta_i\right)$ be independent non-central
chi-square variates, where $\delta_i$ are the non-centrality parameters and
$\nu_i$ are the degrees of freedom. Let $w_i, p_i$ be given constants. Then
$$
Y = \sum_i w_i X_i^{p_i}
$$
follows a weighted sum of non-central chi-squares to a power distribution. 
This is not a common distribution. 
However, its cumulants can be easily computed, so the `pdq' functions can be 
approximated by classical expansions. 
Moreover, its CDF and quantile functions can be used to compute
those of the doubly non-central F, and it is related to the upsilon
distribution.

<<'sumchisqpow',echo=TRUE,eval=TRUE,fig.cap="Confirming the dpqr functions of the sum of chi-squares to a power distribution.">>=
require(sadists)
wts <- c(-1,1,3,-3)
df <- c(100,200,100,50)
ncp <- c(0,1,0.5,2)
pow <- c(1,0.5,2,1.5)
testf(list(d=dsumchisqpow,p=psumchisqpow,q=qsumchisqpow,r=rsumchisqpow),nobs=2^14,wts,df,ncp,pow)
@
%UNFOLD
\section{K-prime distribution}%FOLDUP

\label{sec:kprime_distribution}

Let $X_i \sim \chi^2_{\nu_i}$ be chi-square random variables with $\nu_i$
degrees of freedom for $i=1,2$, independent of $Z\sim\mathcal{N}\left(0,1\right)$,
a standard normal. Suppose $a, b$ are given constants. Then
$$
Y = \frac{b Z + a \sqrt{X_1 / \nu_1}}{\sqrt{X_2 / \nu_2}}
$$
follows a K-prime distribution with degrees of freedom 
$\asrowvec{\nu_1, \nu_2}$ and parameters $a, b$. \cite{PoitLecoutre2010}
Depending on these four parameters, the K-prime generalizes the following:
\begin{itemize}
\item The normal distribution, when $b=1, a=0, \nu_2 = \infty$.
\item The Lambda-prime distribution (see \secref{lamprime_distribution}),
when $b=1, a\ne 0, \nu_2 = \infty$.
\item The (central) t-distribution, when $b=1, a=0, \nu_2 < \infty$.
\item The square-root of the F-distribution, when 
$b = 0,a = 1$.
\item The (central) chi-distribution, 
when $b=0, a = 1, \nu_2 = \infty$.
\end{itemize}

<<'kprime',echo=TRUE,eval=TRUE,fig.cap="Confirming the dpqr functions of the K-prime distribution.">>=
require(sadists)
v1 <- 50
v2 <- 80
a <- 0.5
b <- 1.5
testf(list(d=dkprime,p=pkprime,q=qkprime,r=rkprime),nobs=2^14,v1,v2,a,b)
@

%UNFOLD
\section{Lambda prime distribution}%FOLDUP

\label{sec:lamprime_distribution}

Let $X \sim \chi^2_{\nu}$ be a chi-square random variable with 
$\nu$ degrees of freedom, independent of $Z\sim\mathcal{N}\left(0,1\right)$,
a standard normal. Then
$$
Y = Z + t \sqrt{X/\nu}
$$
follows a Lambda-prime distribution with parameter $t$ and degrees of freedom 
$\nu$.  \cite{Lecoutre1999} It is a special case of the K-prime distribution
(\secref{kprime_distribution}) and of the upsilon distribution
(\secref{upsilon_distribution}). 

<<'lambdap',echo=TRUE,eval=TRUE,fig.cap="Confirming the dpqr functions of the Lambda-prime distribution.">>=
require(sadists)
df <- 50
ts <- 1.5
testf(list(d=dlambdap,p=plambdap,q=qlambdap,r=rlambdap),nobs=2^14,df,ts)
@

%UNFOLD
\section{Upsilon distribution}%FOLDUP
\label{sec:upsilon_distribution}

Let $X_i \sim \chi^2_{\nu_i}$ be independent central
chi-square variates, where $\nu_i$ are the degrees of freedom. 
Let $Z\sim\mathcal{N}\left(0,1\right)$ be a standard normal,
independent of the $X_i$. 
Let $t_i$ be given constants. Then
$$
Y = Z + \sum_i t_i \sqrt{X_i}
$$
follows an upsilon distribution with parameter
$\asrowvec{t_1,t_2,\ldots,t_k}$ and 
degrees of freedom $\asrowvec{\nu_1,\nu_2,\ldots,\nu_k}$.

<<'upsilon',echo=TRUE,eval=TRUE,fig.cap="Confirming the dpqr functions of the upsilon distribution.">>=
require(sadists)
df <- c(30,50,100,20,10)
ts <- c(-3,2,5,-4,1)
testf(list(d=dupsilon,p=pupsilon,q=qupsilon,r=rupsilon),nobs=2^14,df,ts)
@
%UNFOLD
\section{Doubly non-central F distribution}%FOLDUP

\label{sec:dnf}

The doubly non-central F distribution generalizes the F distribution to the case where the denominator
chi-square is non-central. 
For $i=1,2$, 
let $X_i \sim \chi^2_{\nu_i}\left(\delta_i\right)$ be independent non-central
chi-square variates, where $\delta_i$ are the non-centrality parameters and
$\nu_i$ are the degrees of freedom. 
Then
$$
Y = \frac{X_1/\nu_1}{X_2/\nu_2}
$$
follows a doubly non-central F distribution.

<<'dnf',echo=TRUE,eval=TRUE,fig.cap="Confirming the dpqr functions of the doubly non-central F distribution.">>=
require(sadists)
df1 <- 40
df2 <- 80
ncp1 <- 1.5
ncp2 <- 2.5
testf(list(d=ddnf,p=pdnf,q=qdnf,r=rdnf),nobs=2^14,df1,df2,ncp1,ncp2)
@
%UNFOLD
\section{Doubly non-central t distribution}%FOLDUP
\label{sec:dnt}

The doubly non-central t distribution
generalizes the t distribution to the case where the denominator chi-square is non-central.
Let $X_2 \sim \chi^2_{\nu_2}\left(\delta_2\right)$ be a non-central
chi-square variate, with non-centrality parameter $\delta_2$ and $\nu_2$
degrees of freedom. Let $X_2$ be independent of $Z$, a standard normal. Then
$$
Y = \frac{Z + \delta_1}{\sqrt{X_2/\nu_2}}
$$
follows a doubly non-central t distribution with degrees of freedom $\nu_2$ and
non-centrality parameters $\delta_1, \delta_2$. The square of a doubly
non-central t is, up to scaling, a doubly non-central F, see \secref{dnf}.

<<'dnt',echo=TRUE,eval=TRUE,fig.cap="Confirming the dpqr functions of the doubly non-central t distribution.">>=
require(sadists)
df <- 75
ncp1 <- 2
ncp2 <- 3
testf(list(d=ddnt,p=pdnt,q=qdnt,r=rdnt),nobs=2^14,df,ncp1,ncp2)
@
%UNFOLD
\section{Doubly non-central Beta distribution}%FOLDUP

\label{sec:dnbeta}

The doubly non-central Beta distribution generalizes the Beta distribution to the case where the denominator
chi-square is non-central. 
For $i=1,2$, 
let $X_i \sim \chi^2_{\nu_i}\left(\delta_i\right)$ be independent non-central
chi-square variates, where $\delta_i$ are the non-centrality parameters and
$\nu_i$ are the degrees of freedom. 
Then
$$
Y = \frac{X_1}{X_1 + X_2}
$$
follows a doubly non-central Beta distribution. Note that
$$
F = \frac{\nu_2}{\nu_1}\frac{Y}{1-Y}
$$
follows a doubly non-central F distribution. The `PDQ' functions use this
relationship.

<<'dnbeta',echo=TRUE,eval=TRUE,fig.cap="Confirming the dpqr functions of the doubly non-central Beta distribution.">>=
require(sadists)
df1 <- 40
df2 <- 80
ncp1 <- 1.5
ncp2 <- 2.5
testf(list(d=ddnbeta,p=pdnbeta,q=qdnbeta,r=rdnbeta),nobs=2^14,df1,df2,ncp1,ncp2)
@
%UNFOLD
\section{Doubly non-central Eta distribution}%FOLDUP

\label{sec:dneta}

The doubly non-central Eta distribution is to the doubly non-central Beta what
the doubly non-central t is to the doubly non-central F. 
Let $X_2 \sim \chi^2_{\nu_2}\left(\delta_2\right)$ be a non-central
chi-square variate, with non-centrality parameter $\delta_2$ and $\nu_2$
degrees of freedom. Let $X_2$ be independent of $Z$, a standard normal. Then
$$
Y = \frac{Z}{\sqrt{Z^2 + X_2}}
$$
follows a doubly non-central Eta distribution with degrees of freedom $\nu_2$ and
non-centrality parameters $\delta_1, \delta_2$. The square of a doubly
non-central Eta is a doubly non-central Beta, see \secref{dnbeta}.
Note that
$$
t = {\sqrt{\nu_2}}\frac{Y}{\sqrt{1-Y^2}}
$$
follows a doubly non-central t distribution. The `PDQ' functions use this
relationship.

<<'dneta',echo=TRUE,eval=TRUE,fig.cap="Confirming the dpqr functions of the doubly non-central Eta distribution.">>=
require(sadists)
df <- 100
ncp1 <- 0.5
ncp2 <- 2.5
testf(list(d=ddneta,p=pdneta,q=qdneta,r=rdneta),nobs=2^14,df,ncp1,ncp2)
@
%UNFOLD
\section{Sum of logs of (non-central) chi-squares}%FOLDUP
\label{sec:sumlogchisq}

Let $X_i \sim \chi^2_{\nu_i}\left(\delta_i\right)$ be independent non-central
chi-square variates, where $\delta_i$ are the non-centrality parameters and
$\nu_i$ are the degrees of freedom. Let $w_i$ be given constants. Then
$$
Y = \sum_i w_i \log{X_i}
$$
follows a weighted sum of log of non-central chi-squares distribution. 
This is not a common distribution. However, its cumulants can easily
be computed.  \cite{pav2015lnc}

<<'sumlogchisq',echo=TRUE,eval=TRUE,fig.cap="Confirming the dpqr functions of the sum of log of chi-squares distribution.">>=
require(sadists)
wts <- c(5,-4,10,-15)
df <- c(100,200,100,50)
ncp <- c(0,1,0.5,2)
testf(list(d=dsumlogchisq,p=psumlogchisq,q=qsumlogchisq,r=rsumlogchisq),nobs=2^14,wts,df,ncp)
@
%UNFOLD
\section{Product of (non-central) chi-squares to a power}%FOLDUP

Let $X_i \sim \chi^2_{\nu_i}\left(\delta_i\right)$ be independent non-central
chi-square variates, where $\delta_i$ are the non-centrality parameters and
$\nu_i$ are the degrees of freedom. Let $p_i$ be given constants. Then
$$
Y = \prod_i X_i^{p_i}
$$
follows a product of non-central chi-squares to a power distribution. 
This is not a common distribution. 
The `PDQ' functions are computed using a transform of the 
logs of chi-squares distribution, see \secref{sumlogchisq}.

<<'prodchisqpow',echo=TRUE,eval=TRUE,fig.cap="Confirming the dpqr functions of the product of chi-squares to a power distribution.">>=
require(sadists)
df <- c(100,200,100,50)
ncp <- c(0,1,0.5,2)
pow <- c(1,0.5,2,1.5)
testf(list(d=dprodchisqpow,p=pprodchisqpow,q=qprodchisqpow,r=rprodchisqpow),nobs=2^14,df,ncp,pow)
@
%UNFOLD
\section{Product of doubly non-central F variates}%FOLDUP

Let $X_j \sim F_{\nu_{1,j},\nu_{2,j}}\left(\delta_{1,j},\delta_{2,j}\right)$ 
be independent doubly non-central F variates, where $\delta_{i,j}$ are the 
non-centrality parameters and $\nu_{i,j}$ are the degrees of freedom. 
Then
$$
Y = \prod_j X_j
$$
follows a product of doubly non-central Fs distribution. 
This is not a common distribution. 
The `PDQ' functions are computed using a transform of the 
logs of chi-squares distribution, see \secref{sumlogchisq}.

<<'proddnf',echo=TRUE,eval=TRUE,fig.cap="Confirming the dpqr functions of the product of doubly non-central Fs distribution.">>=
require(sadists)
df1 <- c(10,20,5)
df2 <- c(1000,500,150)
ncp1 <- c(1,0,2.5)
ncp2 <- c(0,1.5,5)
testf(list(d=dproddnf,p=pproddnf,q=qproddnf,r=rproddnf),nobs=2^14,df1,df2,ncp1,ncp2)
@
%UNFOLD
%\section{Sum of generalized gamma variates}%FOLDUP

%Let $X_i \sim G\left(a_i, d_i, p_i\right)$ 
%be independently distributed generalized gamma variates with parameters
%$a_i, d_i$ and inverse power $p_i$. \cite{stacey1962}
%Let $w_i$ be given constants. Then
%$$
%Y = \sum_i w_i X_i
%$$
%follows a sum of generalized gammas distribution.
%This is not a common distribution. 

%<<'sumgengamma',echo=TRUE,eval=TRUE,fig.cap="Confirming the dpqr functions of the sum of generalized gammas distribution.">>=
%require(sadists)
%wts <- c(8,-10)
%a <- c(10,2)
%d <- c(2,1)
%pow <- c(0.5,1.5)
%testf(list(d=dsumgengamma,p=psumgengamma,q=qsumgengamma,r=rsumgengamma),nobs=2^14,wts,a,d,pow)
%@
%%UNFOLD

\nocite{paolella2007intermediate,Lecoutre1999,PoitLecoutre2010}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% bibliography%FOLDUP
\nocite{markowitz1952portfolio,markowitz1999early,markowitz2012foundations}
%\bibliographystyle{jss}
%\bibliographystyle{siam}
%\bibliographystyle{ieeetr}
\bibliographystyle{plainnat}
%\bibliographystyle{acm}
\bibliography{sadists,rauto}
%UNFOLD

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\appendix%FOLDUP

%\section{Confirming the scalar Gaussian case}

%\begin{example}%FOLDUP
%To sanity check 
%\theoremref{theta_asym_var_gaussian},
%consider the $\nlatf = 1$ Gaussian case. In this case, 
%$$
%\fvech{\pvsm} = \asvec{1,\pmu,\psig^2 + \pmu^2},\quad\mbox{and}\quad
%\fvech{\minv{\pvsm}} =
 %\asvec{1 + \frac{\pmu^2}{\psig^2},- \frac{\pmu}{\psig^2},\oneby{\psig^2}}.
%$$
%Let $\smu, \ssig^2$ be the unbiased sample estimates. By well 
%known results \cite{spiegel2007schaum}, $\smu$ and $\ssig^2$ are independent,
%and have asymptotic variances of $\psig^2/\ssiz$ and $2\psig^4/\ssiz$ 
%respectively. By the delta method, the asymptotic variance of
%$\Unun \fvech{\svsm}$ and $\fvech{\minv{\svsm}}$ can be computed as
%\begin{equation}
%\begin{split}
%\VAR{\Unun \fvech{\svsm}} &\rightsquigarrow 
%\oneby{\ssiz}\qform{\twobytwossym{\psig^2}{0}{2\psig^4}}{\twobytwo{1}{2\pmu}{0}{1}},\\
%&= \oneby{\ssiz}\twobytwossym{\psig^2}{2\pmu\psig^2}{4\pmu^2\psig^2 +
%2\psig^4}.
%\label{eqn:gauss_confirm_theta}
%\end{split}
%\end{equation}
%\begin{equation}
%\begin{split}
%\VAR{\fvech{\minv{\svsm}}} &\rightsquigarrow 
%\oneby{\ssiz}\qform{\twobytwossym{\psig^2}{0}{2\psig^4}}{\twobythree{\frac{2\pmu}{\psig^2}}{-\frac{1}{\psig^2}}{0}{-\frac{\pmu^2}{\psig^4}}{\frac{\pmu}{\psig^4}}{-\frac{1}{\psig^4}}},\\
%&=
%\oneby{\ssiz}\gram{\twobythree{2\psnr}{-\frac{1}{\psig}}{0}{-\sqrt{2}\psnr^2}{\sqrt{2}\frac{\psnr}{\psig}}{-\frac{\sqrt{2}}{\psig^2}}},\\
%&= \oneby{\ssiz}\threebythreessym{2\psnr^2\wrapParens{2 + \psnr^2}}{-\frac{2\psnr}{\psig}\wrapParens{1+\psnr^2}}{2\frac{\psnr^2}{\psig^2}}{\frac{1 +
%2\psnr^2}{\psig^2}}{-\frac{2\psnr}{\psig^3}}{\frac{2}{\psig^4}}.
%\label{eqn:gauss_confirm_itheta}
%\end{split}
%\end{equation}

%Now it remains to compute $\VAR{\Unun \fvech{\svsm}}$ via 
%\theoremref{theta_asym_var_gaussian}, and then 
%\VAR{\fvech{\minv{\svsm}}} via
%\theoremref{inv_distribution}, and confirm they match the values above.
%This is a rather tedious computation best left to a computer. Below is 
%an excerpt of an iPython notebook using Sympy \cite{PER-GRA:2007,sympy}
%which performs this computation. 
%This notebook is available online. \cite{SEP2013example}

%% SYMPY from here out%FOLDUP
    %\begin{Verbatim}[commandchars=\\\{\}]
%{\color{incolor}In [{\color{incolor}1}]:} \PY{c}{\PYZsh{} confirm the asymptotic distribution of Theta}
        %\PY{c}{\PYZsh{} for scalar Gaussian case.}
        %\PY{k+kn}{from} \PY{n+nn}{\PYZus{}\PYZus{}future\PYZus{}\PYZus{}} \PY{k+kn}{import} \PY{n}{division}
        %\PY{k+kn}{from} \PY{n+nn}{sympy} \PY{k+kn}{import} \PY{o}{*}
        %\PY{k+kn}{from} \PY{n+nn}{sympy.physics.quantum} \PY{k+kn}{import} \PY{n}{TensorProduct}
        %\PY{n}{init\PYZus{}printing}\PY{p}{(}\PY{n}{use\PYZus{}unicode}\PY{o}{=}\PY{n+nb+bp}{False}\PY{p}{,} \PY{n}{wrap\PYZus{}line}\PY{o}{=}\PY{n+nb+bp}{False}\PY{p}{,} \PYZbs{}
                      %\PY{n}{no\PYZus{}global}\PY{o}{=}\PY{n+nb+bp}{True}\PY{p}{)}
        %\PY{n}{mu} \PY{o}{=} \PY{n}{symbols}\PY{p}{(}\PY{l+s}{\PYZsq{}}\PY{l+s}{\PYZbs{}}\PY{l+s}{mu}\PY{l+s}{\PYZsq{}}\PY{p}{)}
        %\PY{n}{sg} \PY{o}{=} \PY{n}{symbols}\PY{p}{(}\PY{l+s}{\PYZsq{}}\PY{l+s}{\PYZbs{}}\PY{l+s}{sigma}\PY{l+s}{\PYZsq{}}\PY{p}{)} 
        %\PY{c}{\PYZsh{} the elimination, duplication and U\PYZus{}\PYZob{}\PYZhy{}1\PYZcb{} matrices:}
        %\PY{n}{Elim} \PY{o}{=} \PY{n}{Matrix}\PY{p}{(}\PY{l+m+mi}{3}\PY{p}{,}\PY{l+m+mi}{4}\PY{p}{,}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}  \PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}   \PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{)}
        %\PY{n}{Dupp} \PY{o}{=} \PY{n}{Matrix}\PY{p}{(}\PY{l+m+mi}{4}\PY{p}{,}\PY{l+m+mi}{3}\PY{p}{,}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{)}
        %\PY{n}{Unun} \PY{o}{=} \PY{n}{Matrix}\PY{p}{(}\PY{l+m+mi}{2}\PY{p}{,}\PY{l+m+mi}{3}\PY{p}{,}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}  \PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{)}
        %\PY{k}{def} \PY{n+nf}{Qform}\PY{p}{(}\PY{n}{A}\PY{p}{,}\PY{n}{x}\PY{p}{)}\PY{p}{:}
            %\PY{l+s+sd}{\PYZdq{}\PYZdq{}\PYZdq{}compute the quadratic form x\PYZsq{}Ax\PYZdq{}\PYZdq{}\PYZdq{}}
            %\PY{k}{return} \PY{n}{x}\PY{o}{.}\PY{n}{transpose}\PY{p}{(}\PY{p}{)} \PY{o}{*} \PY{n}{A} \PY{o}{*} \PY{n}{x}
%\end{Verbatim}

    %\begin{Verbatim}[commandchars=\\\{\}]
%{\color{incolor}In [{\color{incolor}2}]:} \PY{n}{Theta} \PY{o}{=} \PY{n}{Matrix}\PY{p}{(}\PY{l+m+mi}{2}\PY{p}{,}\PY{l+m+mi}{2}\PY{p}{,}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{,}\PY{n}{mu}\PY{p}{,}\PY{n}{mu}\PY{p}{,}\PY{n}{mu}\PY{o}{*}\PY{o}{*}\PY{l+m+mi}{2} \PY{o}{+} \PY{n}{sg}\PY{o}{*}\PY{o}{*}\PY{l+m+mi}{2}\PY{p}{]}\PY{p}{)}
        %\PY{n}{Theta}
%\end{Verbatim}
%\texttt{\color{outcolor}Out[{\color{outcolor}2}]:}
    
    
        %\begin{equation*}
        %\left[\begin{matrix}1 & \mu\\\mu & \mu^{2} + \sigma^{2}\end{matrix}\right]
        %\end{equation*}

    

    %\begin{Verbatim}[commandchars=\\\{\}]
%{\color{incolor}In [{\color{incolor}3}]:} \PY{c}{\PYZsh{} compute tensor products and }
        %\PY{c}{\PYZsh{} the derivative d vech(Theta\PYZca{}\PYZhy{}1) / d vech(Theta)}
        %\PY{c}{\PYZsh{} see also \theoremref{inv_distribution}}
        %\PY{n}{Theta\PYZus{}Theta} \PY{o}{=} \PY{n}{TensorProduct}\PY{p}{(}\PY{n}{Theta}\PY{p}{,}\PY{n}{Theta}\PY{p}{)}
        %\PY{n}{iTheta\PYZus{}iTheta} \PY{o}{=} \PY{n}{TensorProduct}\PY{p}{(}\PY{n}{Theta}\PY{o}{.}\PY{n}{inv}\PY{p}{(}\PY{p}{)}\PY{p}{,}\PY{n}{Theta}\PY{o}{.}\PY{n}{inv}\PY{p}{(}\PY{p}{)}\PY{p}{)}
        %\PY{n}{theta\PYZus{}i\PYZus{}deriv} \PY{o}{=} \PY{n}{Elim} \PY{o}{*} \PY{p}{(}\PY{n}{iTheta\PYZus{}iTheta}\PY{p}{)} \PY{o}{*} \PY{n}{Dupp}
%\end{Verbatim}

    %\begin{Verbatim}[commandchars=\\\{\}]
%{\color{incolor}In [{\color{incolor}4}]:} \PY{c}{\PYZsh{} towards \theoremref{theta_asym_var_gaussian}}
        %\PY{n}{DTTD} \PY{o}{=} \PY{n}{Qform}\PY{p}{(}\PY{n}{Theta\PYZus{}Theta}\PY{p}{,}\PY{n}{Dupp}\PY{p}{)}
        %\PY{n}{D\PYZus{}DTTD\PYZus{}D} \PY{o}{=} \PY{n}{Qform}\PY{p}{(}\PY{n}{DTTD}\PY{p}{,}\PY{n}{theta\PYZus{}i\PYZus{}deriv}\PY{p}{)}
        %\PY{n}{iOmega} \PY{o}{=} \PY{n}{Qform}\PY{p}{(}\PY{n}{D\PYZus{}DTTD\PYZus{}D}\PY{p}{,}\PY{n}{Unun}\PY{o}{.}\PY{n}{transpose}\PY{p}{(}\PY{p}{)}\PY{p}{)}
        %\PY{n}{Omega} \PY{o}{=} \PY{l+m+mi}{2} \PY{o}{*} \PY{n}{iOmega}\PY{o}{.}\PY{n}{inv}\PY{p}{(}\PY{p}{)}
        %\PY{n}{simplify}\PY{p}{(}\PY{n}{Omega}\PY{p}{)}
%\end{Verbatim}
%\texttt{\color{outcolor}Out[{\color{outcolor}4}]:}
    
    
        %\begin{equation*}
        %\left[\begin{matrix}\sigma^{2} & 2 \mu \sigma^{2}\\2 \mu \sigma^{2} & 2 \sigma^{2} \left(2 \mu^{2} + \sigma^{2}\right)\end{matrix}\right]
        %\end{equation*}

    

    %\begin{Verbatim}[commandchars=\\\{\}]
%{\color{incolor}In [{\color{incolor}5}]:} \PY{c}{\PYZsh{} this matches the computation in \eqnref{gauss_confirm_theta}}
        %\PY{c}{\PYZsh{} on to the inverse:}
        %\PY{c}{\PYZsh{} actually use \theoremref{inv_distribution}}
        %\PY{n}{theta\PYZus{}i\PYZus{}deriv\PYZus{}t} \PY{o}{=} \PY{n}{theta\PYZus{}i\PYZus{}deriv}\PY{o}{.}\PY{n}{transpose}\PY{p}{(}\PY{p}{)}
        %\PY{n}{theta\PYZus{}inv\PYZus{}var} \PY{o}{=} \PY{n}{Qform}\PY{p}{(}\PY{n}{Qform}\PY{p}{(}\PY{n}{Omega}\PY{p}{,}\PY{n}{Unun}\PY{p}{)}\PY{p}{,}\PY{n}{theta\PYZus{}i\PYZus{}deriv\PYZus{}t}\PY{p}{)}
        %\PY{n}{simplify}\PY{p}{(}\PY{n}{theta\PYZus{}inv\PYZus{}var}\PY{p}{)}
%\end{Verbatim}
%\texttt{\color{outcolor}Out[{\color{outcolor}5}]:}
    
    
        %\begin{equation*}
        %\left[\begin{matrix}\frac{2 \mu^{2}}{\sigma^{4}} \left(\mu^{2} + 2 \sigma^{2}\right) & - \frac{2 \mu}{\sigma^{4}} \left(\mu^{2} + \sigma^{2}\right) & \frac{2 \mu^{2}}{\sigma^{4}}\\- \frac{2 \mu}{\sigma^{4}} \left(\mu^{2} + \sigma^{2}\right) & \frac{1}{\sigma^{4}} \left(2 \mu^{2} + \sigma^{2}\right) & - \frac{2 \mu}{\sigma^{4}}\\\frac{2 \mu^{2}}{\sigma^{4}} & - \frac{2 \mu}{\sigma^{4}} & \frac{2}{\sigma^{4}}\end{matrix}\right]
        %\end{equation*}

    

    %\begin{Verbatim}[commandchars=\\\{\}]
%{\color{incolor}In [{\color{incolor}6}]:} \PY{c}{\PYZsh{} this matches the computation in \eqnref{gauss_confirm_itheta}}
        %\PY{c}{\PYZsh{} now check \conjectureref{theta_asym_var_gaussian}}
        %\PY{n}{conjec} \PY{o}{=} \PY{n}{Qform}\PY{p}{(}\PY{n}{Theta\PYZus{}Theta}\PY{p}{,}\PY{n}{Dupp}\PY{p}{)}
        %\PY{n}{e1} \PY{o}{=} \PY{n}{Matrix}\PY{p}{(}\PY{l+m+mi}{3}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{,}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{)}
        %\PY{n}{convar} \PY{o}{=} \PY{l+m+mi}{2} \PY{o}{*} \PY{p}{(}\PY{n}{conjec}\PY{o}{.}\PY{n}{inv}\PY{p}{(}\PY{p}{)} \PY{o}{\PYZhy{}} \PY{n}{e1} \PY{o}{*} \PY{n}{e1}\PY{o}{.}\PY{n}{transpose}\PY{p}{(}\PY{p}{)}\PY{p}{)}
        %\PY{n}{simplify}\PY{p}{(}\PY{n}{convar}\PY{p}{)}
%\end{Verbatim}
%\texttt{\color{outcolor}Out[{\color{outcolor}6}]:}
    
    
        %\begin{equation*}
        %\left[\begin{matrix}\frac{2 \mu^{2}}{\sigma^{4}} \left(\mu^{2} + 2 \sigma^{2}\right) & - \frac{2 \mu}{\sigma^{4}} \left(\mu^{2} + \sigma^{2}\right) & \frac{2 \mu^{2}}{\sigma^{4}}\\- \frac{2 \mu}{\sigma^{4}} \left(\mu^{2} + \sigma^{2}\right) & \frac{1}{\sigma^{4}} \left(2 \mu^{2} + \sigma^{2}\right) & - \frac{2 \mu}{\sigma^{4}}\\\frac{2 \mu^{2}}{\sigma^{4}} & - \frac{2 \mu}{\sigma^{4}} & \frac{2}{\sigma^{4}}\end{matrix}\right]
        %\end{equation*}

    

    %\begin{Verbatim}[commandchars=\\\{\}]
%{\color{incolor}In [{\color{incolor}7}]:} \PY{c}{\PYZsh{} are they the same?}
        %\PY{n}{simplify}\PY{p}{(}\PY{n}{theta\PYZus{}inv\PYZus{}var} \PY{o}{\PYZhy{}} \PY{n}{convar}\PY{p}{)}
%\end{Verbatim}
%\texttt{\color{outcolor}Out[{\color{outcolor}7}]:}
    
    
        %\begin{equation*}
        %\left[\begin{matrix}0 & 0 & 0\\0 & 0 & 0\\0 & 0 & 0\end{matrix}\right]
        %\end{equation*}

    


    %%UNFOLD
    
%%%  OLD%FOLDUP
    %%\begin{Verbatim}[commandchars=\\\{\}]
%%{\color{incolor}In [{\color{incolor}1}]:} \PY{k+kn}{from} \PY{n+nn}{\PYZus{}\PYZus{}future\PYZus{}\PYZus{}} \PY{k+kn}{import} \PY{n}{division}
        %%\PY{k+kn}{from} \PY{n+nn}{sympy} \PY{k+kn}{import} \PY{o}{*}
        %%\PY{k+kn}{from} \PY{n+nn}{sympy.physics.quantum} \PY{k+kn}{import} \PY{n}{TensorProduct}
        %%\PY{n}{init\PYZus{}printing}\PY{p}{(}\PY{n}{use\PYZus{}unicode}\PY{o}{=}\PY{n+nb+bp}{False}\PY{p}{,} \PY{n}{wrap\PYZus{}line}\PY{o}{=}\PY{n+nb+bp}{False}\PY{p}{,} \PY{n}{no\PYZus{}global}\PY{o}{=}\PY{n+nb+bp}{True}\PY{p}{)}
        %%\PY{n}{mu} \PY{o}{=} \PY{n}{symbols}\PY{p}{(}\PY{l+s}{\PYZsq{}}\PY{l+s}{\PYZbs{}}\PY{l+s}{mu}\PY{l+s}{\PYZsq{}}\PY{p}{)}
        %%\PY{n}{sg} \PY{o}{=} \PY{n}{symbols}\PY{p}{(}\PY{l+s}{\PYZsq{}}\PY{l+s}{\PYZbs{}}\PY{l+s}{sigma}\PY{l+s}{\PYZsq{}}\PY{p}{)} 
        %%\PY{c}{\PYZsh{} the elimination, duplication and U\PYZus{}\PYZob{}\PYZhy{}1\PYZcb{} matrices:}
        %%\PY{n}{Elim} \PY{o}{=} \PY{n}{Matrix}\PY{p}{(}\PY{l+m+mi}{3}\PY{p}{,}\PY{l+m+mi}{4}\PY{p}{,}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}  \PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}   \PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{)}
        %%\PY{n}{Dupp} \PY{o}{=} \PY{n}{Matrix}\PY{p}{(}\PY{l+m+mi}{4}\PY{p}{,}\PY{l+m+mi}{3}\PY{p}{,}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{)}
        %%\PY{n}{Unun} \PY{o}{=} \PY{n}{Matrix}\PY{p}{(}\PY{l+m+mi}{2}\PY{p}{,}\PY{l+m+mi}{3}\PY{p}{,}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}  \PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{)}
        %%\PY{k}{def} \PY{n+nf}{quad\PYZus{}form}\PY{p}{(}\PY{n}{A}\PY{p}{,}\PY{n}{x}\PY{p}{)}\PY{p}{:}
            %%\PY{l+s+sd}{\PYZdq{}\PYZdq{}\PYZdq{}compute the quadratic form x\PYZsq{}Ax\PYZdq{}\PYZdq{}\PYZdq{}}
            %%\PY{k}{return} \PY{n}{x}\PY{o}{.}\PY{n}{transpose}\PY{p}{(}\PY{p}{)} \PY{o}{*} \PY{n}{A} \PY{o}{*} \PY{n}{x}
%%\end{Verbatim}

    %%\begin{Verbatim}[commandchars=\\\{\}]
%%{\color{incolor}In [{\color{incolor}2}]:} \PY{n}{Theta} \PY{o}{=} \PY{n}{Matrix}\PY{p}{(}\PY{l+m+mi}{2}\PY{p}{,}\PY{l+m+mi}{2}\PY{p}{,}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{,}\PY{n}{mu}\PY{p}{,}\PY{n}{mu}\PY{p}{,}\PY{n}{mu}\PY{o}{*}\PY{o}{*}\PY{l+m+mi}{2} \PY{o}{+} \PY{n}{sg}\PY{o}{*}\PY{o}{*}\PY{l+m+mi}{2}\PY{p}{]}\PY{p}{)}
        %%\PY{n}{Theta}
%%\end{Verbatim}
%%\texttt{\color{outcolor}Out[{\color{outcolor}2}]:}
    
    
        %%\begin{equation*}
        %%\left[\begin{matrix}1 & \mu\\\mu & \mu^{2} + \sigma^{2}\end{matrix}\right]
        %%\end{equation*}

    

    %%\begin{Verbatim}[commandchars=\\\{\}]
%%{\color{incolor}In [{\color{incolor}3}]:} \PY{c}{\PYZsh{} compute tensor products and }
        %%\PY{c}{\PYZsh{} the derivative d vech(Theta\PYZca{}\PYZhy{}1) / d vech(Theta)}
        %%\PY{n}{Theta\PYZus{}Theta} \PY{o}{=} \PY{n}{TensorProduct}\PY{p}{(}\PY{n}{Theta}\PY{p}{,}\PY{n}{Theta}\PY{p}{)}
        %%\PY{n}{iTheta\PYZus{}iTheta} \PY{o}{=} \PY{n}{TensorProduct}\PY{p}{(}\PY{n}{Theta}\PY{o}{.}\PY{n}{inv}\PY{p}{(}\PY{p}{)}\PY{p}{,}\PY{n}{Theta}\PY{o}{.}\PY{n}{inv}\PY{p}{(}\PY{p}{)}\PY{p}{)}
        %%\PY{n}{theta\PYZus{}i\PYZus{}deriv} \PY{o}{=} \PY{n}{Elim} \PY{o}{*} \PY{p}{(}\PY{n}{iTheta\PYZus{}iTheta}\PY{p}{)} \PY{o}{*} \PY{n}{Dupp}
%%\end{Verbatim}

    %%\begin{Verbatim}[commandchars=\\\{\}]
%%{\color{incolor}In [{\color{incolor}4}]:} \PY{c}{\PYZsh{} towards thm \ref{theorem:theta_asym_var_gaussian}}
        %%\PY{n}{DTTD} \PY{o}{=} \PY{n}{quad\PYZus{}form}\PY{p}{(}\PY{n}{Theta\PYZus{}Theta}\PY{p}{,}\PY{n}{Dupp}\PY{p}{)}
        %%\PY{n}{D\PYZus{}DTTD\PYZus{}D} \PY{o}{=} \PY{n}{quad\PYZus{}form}\PY{p}{(}\PY{n}{DTTD}\PY{p}{,}\PY{n}{theta\PYZus{}i\PYZus{}deriv}\PY{p}{)}
        %%\PY{n}{iOmega} \PY{o}{=} \PY{n}{quad\PYZus{}form}\PY{p}{(}\PY{n}{D\PYZus{}DTTD\PYZus{}D}\PY{p}{,}\PY{n}{Unun}\PY{o}{.}\PY{n}{transpose}\PY{p}{(}\PY{p}{)}\PY{p}{)}
        %%\PY{n}{Omega} \PY{o}{=} \PY{l+m+mi}{2} \PY{o}{*} \PY{n}{iOmega}\PY{o}{.}\PY{n}{inv}\PY{p}{(}\PY{p}{)}
        %%\PY{n}{simplify}\PY{p}{(}\PY{n}{Omega}\PY{p}{)}
%%\end{Verbatim}
%%\texttt{\color{outcolor}Out[{\color{outcolor}4}]:}
    
    
        %%\begin{equation*}
        %%\left[\begin{matrix}\sigma^{2} & 2 \mu \sigma^{2}\\2 \mu \sigma^{2} & 2 \sigma^{2} \left(2 \mu^{2} + \sigma^{2}\right)\end{matrix}\right]
        %%\end{equation*}

%%This matches the computation in \eqnref{gauss_confirm_theta}. On to the
%%inverse:
    

    %%\begin{Verbatim}[commandchars=\\\{\}]
%%{\color{incolor}In [{\color{incolor}5}]:} \PY{c}{\PYZsh{} now use theorem \ref{theorem:inv_distribution}}
        %%\PY{n}{theta\PYZus{}inv\PYZus{}var} \PY{o}{=} \PY{n}{quad\PYZus{}form}\PY{p}{(}\PY{n}{quad\PYZus{}form}\PY{p}{(}\PY{n}{Omega}\PY{p}{,}\PY{n}{Unun}\PY{p}{)}\PY{p}{,}\PY{n}{theta\PYZus{}i\PYZus{}deriv}\PY{o}{.}\PY{n}{transpose}\PY{p}{(}\PY{p}{)}\PY{p}{)}
        %%\PY{n}{simplify}\PY{p}{(}\PY{n}{theta\PYZus{}inv\PYZus{}var}\PY{p}{)}
%%\end{Verbatim}
%%\texttt{\color{outcolor}Out[{\color{outcolor}5}]:}
    
    
        %%\begin{equation*}
        %%\left[\begin{matrix}\frac{2 \mu^{2}}{\sigma^{4}} \left(\mu^{2} + 2 \sigma^{2}\right) & - \frac{2 \mu}{\sigma^{4}} \left(\mu^{2} + \sigma^{2}\right) & \frac{2 \mu^{2}}{\sigma^{4}}\\- \frac{2 \mu}{\sigma^{4}} \left(\mu^{2} + \sigma^{2}\right) & \frac{1}{\sigma^{4}} \left(2 \mu^{2} + \sigma^{2}\right) & - \frac{2 \mu}{\sigma^{4}}\\\frac{2 \mu^{2}}{\sigma^{4}} & - \frac{2 \mu}{\sigma^{4}} & \frac{2}{\sigma^{4}}\end{matrix}\right]
        %%\end{equation*}

%%This matches the computation in \eqnref{gauss_confirm_itheta}.  Check
%%the conjecture:
    

    %%\begin{Verbatim}[commandchars=\\\{\}]
%%{\color{incolor}In [{\color{incolor}6}]:} \PY{c}{\PYZsh{} now conjecture \ref{conjecture:theta_asym_var_gaussian}}
        %%\PY{n}{conjec} \PY{o}{=} \PY{n}{quad\PYZus{}form}\PY{p}{(}\PY{n}{Theta\PYZus{}Theta}\PY{p}{,}\PY{n}{Dupp}\PY{p}{)}
        %%\PY{n}{convar} \PY{o}{=} \PY{l+m+mi}{2} \PY{o}{*} \PY{p}{(}\PY{n}{conjec}\PY{o}{.}\PY{n}{inv}\PY{p}{(}\PY{p}{)} \PY{o}{\PYZhy{}} \PY{n}{Matrix}\PY{p}{(}\PY{l+m+mi}{3}\PY{p}{,}\PY{l+m+mi}{3}\PY{p}{,}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{)}\PY{p}{)}
        %%\PY{n}{simplify}\PY{p}{(}\PY{n}{convar}\PY{p}{)}
%%\end{Verbatim}
%%\texttt{\color{outcolor}Out[{\color{outcolor}6}]:}
    
    
        %%\begin{equation*}
        %%\left[\begin{matrix}\frac{2 \mu^{2}}{\sigma^{4}} \left(\mu^{2} + 2 \sigma^{2}\right) & - \frac{2 \mu}{\sigma^{4}} \left(\mu^{2} + \sigma^{2}\right) & \frac{2 \mu^{2}}{\sigma^{4}}\\- \frac{2 \mu}{\sigma^{4}} \left(\mu^{2} + \sigma^{2}\right) & \frac{1}{\sigma^{4}} \left(2 \mu^{2} + \sigma^{2}\right) & - \frac{2 \mu}{\sigma^{4}}\\\frac{2 \mu^{2}}{\sigma^{4}} & - \frac{2 \mu}{\sigma^{4}} & \frac{2}{\sigma^{4}}\end{matrix}\right]
        %%\end{equation*}

    

    %%\begin{Verbatim}[commandchars=\\\{\}]
%%{\color{incolor}In [{\color{incolor}7}]:} \PY{c}{\PYZsh{} are they the same?}
        %%\PY{n}{simplify}\PY{p}{(}\PY{n}{theta\PYZus{}inv\PYZus{}var} \PY{o}{\PYZhy{}} \PY{n}{convar}\PY{p}{)}
%%\end{Verbatim}
%%\texttt{\color{outcolor}Out[{\color{outcolor}7}]:}
    
    
        %%\begin{equation*}
        %%\left[\begin{matrix}0 & 0 & 0\\0 & 0 & 0\\0 & 0 & 0\end{matrix}\right]
        %%\end{equation*}

    %%%UNFOLD

%\end{example}%UNFOLD
%%UNFOLD

\end{document}
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% vim:fdm=marker:fmr=FOLDUP,UNFOLD:cms=%%s:syn=rnoweb:ft=rnoweb:nu
