# Models as Instruments
Jan de Leeuw
Version 0.01, December 17, 2015
$$
\newcommand{\defi}{\mathop{=}\limits^{\Delta}} % mathop for define
\def\Rav#1{\underline #1} % random variables are underlined
\def\Exp#1{{\bf E}\,(#1)} % expected values are bold
\def\Var#1{{\bf V}\,(#1)} % variances are bold
\def\Cov#1#2{{\bf C}\,(#1,#2)} % covariances are bold
\def\Cor#1#2{{\bf R}\,(#1,#2)} % correlations are bold
\def\Rx{\Rav{x}} % random variables are underlined
\def\Ry{\Rav{y}}
\def\Rz{\Rav{z}}
\def\Rp{\Rav{p}} % and so are random proportions
\def\Rq{\Rav{q}}
\newcommand{\mR}{\mathbb{R}} % reals
\newcommand{\mN}{\mathbb{N}} % natural numbers
\newcommand{\mS}{\mathbb{S}} % unit simplex
\newcommand{\mC}{\mathbb{C}} % complex numbers
\newcommand{\mQ}{\mathbb{Q}} % rational numbers
\newcommand{\mZ}{\mathbb{Z}} % integers
\newcommand{\cA}{\mathcal{A}}
\newcommand{\cB}{\mathcal{B}}
\newcommand{\cC}{\mathcal{C}}
\newcommand{\cD}{\mathcal{D}}
\newcommand{\cE}{\mathcal{E}}
\newcommand{\cF}{\mathcal{F}}
\newcommand{\cG}{\mathcal{G}}
\newcommand{\cH}{\mathcal{H}}
\newcommand{\cI}{\mathcal{I}}
\newcommand{\cJ}{\mathcal{J}}
\newcommand{\cK}{\mathcal{K}}
\newcommand{\cL}{\mathcal{L}}
\newcommand{\cM}{\mathcal{M}}
\newcommand{\cN}{\mathcal{N}}
\newcommand{\cO}{\mathcal{O}}
\newcommand{\cP}{\mathcal{P}}
\newcommand{\cQ}{\mathcal{Q}}
\newcommand{\cR}{\mathcal{R}}
\newcommand{\cS}{\mathcal{S}}
\newcommand{\cT}{\mathcal{T}}
\newcommand{\cU}{\mathcal{U}}
\newcommand{\cV}{\mathcal{V}}
\newcommand{\cW}{\mathcal{W}}
\newcommand{\cX}{\mathcal{X}}
\newcommand{\cY}{\mathcal{Y}}
\newcommand{\cZ}{\mathcal{Z}}
\newcommand{\del}{\delta}
\newcommand{\gam}{\gamma}
\newcommand{\lbd}{\lambda}
\newcommand{\ome}{\omega}
\newcommand{\eps}{\epsilon}
\newcommand{\sig}{\sigma}
\newcommand{\alp}{\alpha}
\newcommand{\bet}{\beta}
\newcommand{\Del}{\Delta}
\newcommand{\Gam}{\Gamma}
\newcommand{\Lbd}{\Lambda}
\newcommand{\Ome}{\Omega}
\newcommand{\Eps}{\Epsilon}
\newcommand{\Sig}{\Sigma}
\newcommand{\Alp}{\Alpha}
\newcommand{\vat}{\vartheta}
\newcommand{\bul}{\bullet}
\def\claw{\mathop{\Longrightarrow}\limits^{\cL}}
\def\cprob{\mathop{\Longrightarrow}\limits^{\cP}}
\def\isad{\mathop{=}\limits^{a.d.}}
$$
![cannot handle the truth.](images.jpeg)
Note: This is a working paper which will be expanded/updated frequently. The directory [gifi.stat.ucla.edu/truth](http://gifi.stat.ucla.edu/truth) has a pdf copy of this article, the complete Rmd file with all code chunks, and R and C files with the code.
#Introduction
In this paper, I distinguish various types of errors an estimate can have, and
I estimate these errors using Delta method and Jackknife techniques. The paper is based on, but extends and generalizes, @deleeuw_C_88a and @deleeuw_C_97a.
The notion of a distance-type loss function plays a central role. I evaluate
the distance between the target and the sequence of estimates. This is called the
*overall error*. I also compute the distance
between the estimates and a sequence of variables with the
same distribution as the target sequence, but independent
of the target sequence. This is called the *prediction
error*. Then I compute estimates of the distance between
the target and the limit of the estimation sequence, which is
the *specification error*. Finally, in the computations
I need the distance between the target sequence and the
estimation sequence, which I call the *deviance*.
Just words, of course, but chosen to suggest the various
types of errors that are involved.
The paper is closely related to the work of @linhart_zucchini_86,
which has been used by @cudeck_browne_83
and @browne_cudeck_89, @browne_cudeck_92.
A related Bayesian approach, also based on predictive
distribution, has been discussed by @gelfand_dey_chang_92.
Related research in covariance structure modeling has been
published by @mcdonald_89, @mcdonald_marsh_90, and @bentler_90.
#Estimation Problem
We start, as in De Leeuw \cite{model}, with simple
experiments in which we estimate a
vector of proportions (this could also be a multidimensional
cross table, or a discreticized multivariate distribution). This
will be generalized slightly in a later section.\par
%
Thus we start by considering
an \emph{estimation problem} with the following ingredients.
%
**Definition:**
The \emph{data} is sequence of random vectors of \emph{proportions} $\underline{p}_n,$
taking values in the unit simplex $\mathbb{S}^m.$ The \emph{target} is a
fixed element $\pi$ of $\mathbb{S}^m.$
Of course this is highly suggestive, and to some extent misleading,
terminology. Actual data are not a random variable, and certainly
not a sequence of random variables. To arrive at this representation
we must first embed the actual data in a \emph{framework of replication},
which is modeled by a single random variable, and we then have to
embed that random variable in a sequence to get our asymptotics
under way. Thus the distance from what we call \emph{data} to what an ordinary
person would call \emph{data} is quite large.
In the same way calling a particular point in parameter space \emph{the
target} must be seen as have some heuristic value, at the most.
We still have to connect the data and the target in some
convenient way. The three assumptions below are enough to
justify our expansions and expectations, and they also
serve to simplify the notation.
**Assumption:**
For all $n\in\mathcal{N}$ we have $\mathbf{E}(\underline{p}_n)=\pi.$
**Assumption:**
There is a positive semi-definite $V(\pi)$ such that for all $n\in\mathbb{N}$
we have $n\Var{\Rp_n}=V(\pi).$ $V(\pi)$ is of rank $m-1,$ and only has the
constant vectors in its null-space.
**Assumption:**
For all $j=1,\cdots,m$ we have
$\textbf{E}\mid\Rp_{jn}-\pi_j\mid^6=\cO(n^{-3})$
By Liaponoff's Inequality on absolute moments, Assumption \ref{A:speed}
implies that $\textbf{E}\mid\Rp_{jn}-\pi_j\mid^s=\cO(n^{-\frac{s}{2})}$
for all $1\leq s\leq 6.$ Our assumptions do \emph{not} imply
asymptotic normality, and they do \emph{not} assert that the
$\Rp_n$ are averages of a sequence of independent indicators. In
the multinomial case, of course, the assumptions are true, and we
have $V(\pi)=\Pi-\pi\pi',$ where $\Pi$ is a diagonal matrix with the
elements of $\pi$ along the diagonal.
**Definition:**
An \emph{estimator} $\Phi$ is a function from $\mathbb{S}^m$ to
$\mathbb{S}^m.$
Definition \ref{D:est}
is somewhat non-standard, because often estimators map
data into lower-dimensional (parameter) spaces. Thus
$\Phi:\mS^m\Rightarrow\mR^r,$ with $r\leq m.$
But in fact this amounts to much the same thing, it merely
means that the range of our estimator $\Phi$ is a manifold of dimension
$r.$ We assume
**Assumption:**
$\Phi$ is three times totally differentiable in a neighborhood
of $\pi,$ and the derivatives are all bounded in that
neighborhood.
To measure quality of estimators, we basically look at the
distance to the target. Thus we need another definition.
%
**Definition:**
A \emph{loss function} $\Delta$ is a real-valued function on
$\mS^m\times\mS^m$ satisfying
$$
\Delta(x,y)=
\begin{cases}
\geq 0 & \text{for all $x,y\in\mS^m,$}\\
=0& \text{for all $x,y\in\mS^m$ with $x=y$.}
\end{cases}
$$
The function $\Delta$ is distance-like,
but it need not be symmetric, nor need it satisfy the
triangular inequality. For technical purposes we assume
**Assumption:**
$\Delta$ is three times totally differentiable in a
neighborhood of $(\pi,\Phi(\pi)),$ and the derivatives
are bounded in that neighborhood.
In De Leeuw \cite{model} it is assumed that the
estimate $\Phi$ is a \emph{minimum distance
estimate}, i.e. it is computed by minimizing
$\Delta(\Rp_n,p)$ over $p$ in some sort of
\emph{model} $\Omega,$ where a model is just
a subset of $\mS^m.$ In the work of Brown and
Cudeck, it is assumed, in addition, that the
discrepancy function is \emph{appropriate} for the
multinomial experiment, in the sense that it
gives efficient estimates if the model is true.
We do not make any of these assumptions in this
paper, in fact we do not even assume that
we actually deal with a multinomial or even
an asymptotically normal experiment. In our setup, there
can be dependence or overdispersion, and some of the higher
order moments need not even exist. The only requirement
is that our three assumptions \ref{A:unbiased},
\ref{A:variance}, \ref{A:speed} are true.
#Quality of Estimators
We measure the quality of the
estimate by using the loss function \ref{E:loss} to
measure the distance between the target
and the estimate.
%
**Definition:**
The *overall error* is
$$\Rav{\beta}_n\defi\Delta(\pi,\Phi(\Rp_n)).$$
We also define $\beta_n\defi\Exp{\Rav{\beta}_n},$
the *expected overall error* or EOE.
**Remark:**
Obviously the errors depend on $\pi,$ but we surpress
that dependence in our notation. Linhart and
Zucchini calls the EOE the *overall discrepancy*, De Leeuw simply calls it the *bias*.
Now suppose $\Rq_n$ is a second sequence of
random proportions with the same asymptotic
distribution as $\Rp_n,$ but independent of $\Rp_n.$
We call $\Rq_n$ the \emph{replication}, i.e. it is
the outcome of an independent replication of
our experiment.
Actually, we merely need to assume that Assumptions
\ref{A:unbiased}, \ref{A:variance}, and
\ref{A:speed} are true for $\Rq_n$ as well, i.e. the
$\Rq_n$ are an independent sequence of random variables,
converging to the same target as $\Rp_n$, with the same
speed.
%
**Definition:**
The \emph{prediction error} is defined as
$$\Rav{\mu}_n\defi\Delta(\Rq_n,\Phi(\Rp_n)).$$
The \emph{expected prediction error} or EPE is, obviously,
$\mu_n\defi\Exp{\Rav{\mu}_n},$
where expectation is both over $\Rq_n$ and $\Rp_n.$
**Remark:**
The EPE is called the \emph{distortion} by De Leeuw.
Linhart and Zucchini do not
use prediction errors in their work. Cudeck and Browne
\cite{cudbro} call the prediction error the
\emph{cross validation index}, and give ways to
approximate it in \cite{brocud89}.
**Definition:**
The \emph{specification error} is defined as
$$\delta\defi\Delta(\pi,\Phi(\pi)).$$
**Remark:**
Linhart and Zucchini \cite{linzuc} calls this the
\emph{discrepancy due to approximation}.
The specification error is a non-random quantity, so there is no need
for expectations. It also does not vary with $n$.
**Definition:**
The \emph{deviance} is
$$\Rav{\lambda}_n\defi\Delta(\Rp_n,\Phi(\Rp_n)).$$
And of course we also have the \emph{expected deviance} or EDE
$\lambda_n\defi\Exp{\Rav{\lambda}_n}.$
**Remark:**
The deviance does not seem to be a very interesting
quantity in itself, but in our interpretation
of the estimation situation we only observe $\Rp_n,$
or at least a realization of $\Rp_n.$ We do not
observe $\Rq_n$ or $\pi,$ and thus the prediction
error and specification error cannot be observed
directly. There is, of course, a familiar trick
to emulate an observable $\Rq_n.$ If we split our
data in two halves, we can use the first half as
a realization of $\Rp_{n/2}$ and the second half
as a realization of $\Rq_{n/2}.$ We shall use
other, Jackknife-based, techniques below, because
they seem to be somewhat more systematic.
**Definition:**
The \emph{estimation error} is defined as
$$\Rav{\eps}_n\defi\Delta(\Phi(\pi),\Phi(\Rp_n)).$$
It also has
an \emph{expected estimation error} or EEE, which is
$\eps_n\defi\Exp{\Rav{\eps}_n}.$
**Remark:**
Linhart and Zucchini call this the *discrepancy due to estimation*.
Of course generally the various error measures we have defined above
do not only
have expectations, but also variances, and if suitably
normed perhaps even asymptotic distributions. In this paper,
as in \cite{model}, we concentrate on the
expectations.
#The Distance Geometry of Estimation
A study of the definitions in Section \ref{S:quality} shows that
we have six basic quantities between which we measure discrepancies.
%
There is the target $\pi$ and its image $\Phi(\pi),$ the
data $\Rp_n$ and its image $\Phi(\Rp_n),$ and the replication
$\Rq_n$ and its image $\Phi(\Rq_n).$ These six basic quantities
have $30$ nontrivial discrepancies between them. If the loss
function is symmetric in its two arguments, then there are
still $15$ discrepancies. The situation is graphically shown
in Figure 4.1.\par
%
Some of these $30$ discrepancies, and their expected values,
were actually named in Section \ref{S:quality}, but many remained
anonymous. Nevertheless, even the anonymous
ones sometimes are of interest.
For instance, $\Delta(\Rp_n,\Rq_n)$ could be used as a
measure of dispersion, and it could be compared with
$\Delta(\Phi(\Rp_n),\Phi(\Rq_n))$ to show that estimation
enhances precision. Also, as we shall see further on,
we can also put Jackknifed or Bootstrapped versions
of $\Rp_n$ in this same space.\par
%
The \emph{idea} behind out setup is that
is that $\hat\pi_n\defi\Phi(\Rp_n)$
estimates $\pi,$ but this notion is not really defined anywhere.
In fact, we could say that the \emph{estimation sequence}
$\Phi(\Rp_n)$ really estimates $\Phi(\pi),$ because
obviously
$$
\Phi(\Rp_n)\cprob\Phi(\pi).
$$
This is why $\eps_n(\pi)$ is called the estimation
error.
But to a certain extent, this is cheating. Although
trivially we are estimating $\Phi(\pi),$ we want to
know how good our estimate of $\Phi(\pi)$ is
as an estimator of $\pi.$ To assert that we are
estimating what we are estimating, and then by
definition consistently, has been referred to as
\emph{debilitating relativism}.
We can make things a little bit
more intuitive by defining the \emph{model} $\Omega$
associated with an estimator $\Phi$ as the set
of fixed points of $\Phi.$ Thus
$$
\Omega=\{p\in\mS^m\mid\Phi(p)=p\}.
$$
Since $\Phi(\Rp_n)\cprob\Phi(\pi),$ we know that
if $\pi\in\Omega,$ then $\Phi(\Rp_n)\cprob\pi,$
i.e. we have consistency if $\pi$ is on the model.
This is sometimes expressed as saying that
\emph{the model is true}.\par
%
%
On the other hand we can also start with the model
$\Omega,$ i.e. add it as an ingredient to our
estimation problem, and then define $\Phi$ to be
\emph{F-consistent\/} for $\Omega$ if
$\Phi(p)=p$ for all $p\in\Omega.$ There is a standard
way of guaranteeing F-consistency, which we already
indicated in Section \ref{S:estim}. We define $\Phi$
by the minimum distance or minimum discrepancy
principle, which means that we choose $\Phi(\Rp_n)$
as the argmin of $\Delta(\Rp_n,p)$ over $p\in\Omega.$
Or, to put it more geometrically, $\Phi(\Rp_n)$ is
the \emph{projection} of $\Rp_n$ on the model, in
the metric defined by the loss function. This is
illustrated in Figure 4.2.\par
%
For most of this paper, the notion of a model is
not needed, however. It suffices to consider the
situation where an estimate of $\pi$ has magically appeared
out of the statisticians hat.
#Some Simple Expansions
It is clear from the previous sections that we are interested
in statistics of the form $\Delta(F_1(\Rx_n),F_2(\Ry_n)),$
where $F_1$ and $F_2$ can be either $\Phi$ or the identity,
and $\Rx_n$ and $\Ry_n$ can be either $\Rp_n, \Rq_n$ or $\pi.$
Choosing all combinations give a total of $2\times 2\times 3\times 3=36$
possibilities, of which $6$ give a zero discrepancy because
both $F_1=F_2$ and $\Rx_n=\Ry_n.$\par
To derive our basic result, we need some simple abbreviations.
The first partials of the loss function are
$$
\begin{split}
t(x,y)&\defi\cD_1\Delta(x,y),\\
u(x,y)&\defi\cD_2\Delta(x,y),
\end{split}
$$
and second partials are
$$
\begin{split}
A(x,y)&\defi\cD_{11}\Delta(x,y),\\
B(x,y)&\defi\cD_{12}\Delta(x,y),\\
C(x,y)&\defi\cD_{22}\Delta(x,y).
\end{split}
$$
We also need
$$
\begin{split}
G_1(p)&\defi\cD F_1(p),\\
G_2(p)&\defi\cD F_2(p),
\end{split}
$$
and
$$
\begin{split}
H_1(p)&\defi\cD G_1(p),\\
H_2(p)&\defi\cD G_2(p).
\end{split}
$$
Observe that for $k=1,2$ the function $H_k(\bullet)$ maps $\mR^m\otimes\mR^m$
into $\mR^m,$ which means that $H_k(\bullet)(z,z)$ can be
written as
$$
H_k(\bullet)(z,z)=\left[
\begin{array}{c}
z'H_{k1}(\bullet)z\\
\vdots\\
z'H_{km}(\bullet)z
\end{array}\right]
$$
with each of the $H_{kj}(\bullet)$ square symmetric matrices. The fact
that the $H_k$ are not matrices
is actually only a minor nuisance, because the quantities we really
need are
$$
\begin{split}
\Gamma_1(x,y)&\defi\sum_{j=1}^m t_j(x,y)H_{1j}(\pi),\\
\Gamma_2(x,y)&\defi\sum_{j=1}^m u_j(x,y)H_{2j}(\pi),
\end{split}
$$
and these are matrices.
We continue the orgy of definitions with
$$
\begin{split}
V_{11}(\pi)&\defi n\Cov{\Rx_n}{\Rx_n},\\
V_{12}(\pi)&\defi n\Cov{\Rx_n}{\Ry_n},\\
V_{22}(\pi)&\defi n\Cov{\Ry_n}{\Ry_n},
\end{split}
$$
and
$$
\begin{split}
W_{11}(x,y)&\defi \Gamma_1(x,y)+G_1(\pi)A(x,y)G_1'(\pi),\\
W_{12}(x,y)&\defi G_1(\pi)B(x,y)G_2'(\pi),\\
W_{22}(x,y)&\defi \Gamma_2(x,y)+G_2(\pi)C(x,y)G_2'(\pi).
\end{split}
$$
Finally, $V_{21}(\pi)$ and $W_{21}(x,y)$ are defined
by symmetry, and the four submatrices from Equations
\ref{vdef:a}-\ref{vdef:c} and \ref{wdef:a}-\ref{wdef:c} are collected
in matrices $\overline{V}(\pi)$ and $\overline{W}(x,y).$
We are now finally read to state the main result
of this section.
**Theorem:**
$$
2n\Exp{\Delta(F_1(\Rx_n),F_2(\Ry_n))-\Delta(F_1(\pi),F_2(\pi))}\Rightarrow
\text{ tr }\overline{W}(F_1(\pi),F_2(\pi))\overline{V}(\pi).
$$
**Proof:**
Apply the first half of
Hurt's Theorem, from the Appendix, with $q=2.$
#Estimating the Various Errors
We now apply Theorem \ref{T:expans} to some of the expected
error measures we have defined earlier. Unfortunately, we
need more definitions. Let
$$
G(r)\defi \left.\frac{\partial\Phi}{\partial p}\right|_{p=r},
$$
**Corollary:**
For the expected overall error
$$
2n(\beta_n-\delta)\Rightarrow
\text{ tr }
[\Gamma(\pi,\Phi(\pi))+G(\pi)C(\pi,\Phi(\pi))G'(\pi)]V(\pi).
$$
**Proof:**
Here $F_1$ is the identity and $F_2=\Phi.$ Also $\Rx_n=\pi,$
and $\Ry_n=\Rp_n.$
**Corollary:**
For the expected prediction error
$$
2n(\mu_n-\delta)\Rightarrow
\text{ tr }
[A(\pi,\Phi(\pi))+\Gamma(\pi,\Phi(\pi))+
G(\pi)C(\pi,\Phi(\pi))G'(\pi)]V(\pi)
$$
**Proof:**
Here $F_1$ is the identity, $F_2=\Phi,$ $\Rx_n=\Rq_n,$ and
$\Ry_n=\Rp_n.$
**Corollary:**
For the expected deviance
$$
\begin{multline*}
2n(\lambda_n-\delta)\Rightarrow
\text{ tr }
[A(\pi,\Phi(\pi))+
B(\pi,\Phi(\pi))G'(\pi)+G(\pi)B'(\pi,\Phi(\pi))+\\
\Gamma(\pi,\Phi(\pi))+G(\pi)C(\pi,\Phi(\pi))G'(\pi)]V(\pi)
\end{multline*}
$$
**Proof:**
Here $F_1$ is the identity, $F_2=\Phi,$ and $\Rx_n=\Ry_n=\Rp_n.$
**Corollary:**
For the expected estimation error
$$
2n\eps_n\Rightarrow\text{ tr }
[G(\pi)C(\Phi(\pi),\Phi(\pi))G'(\pi)]V(\pi).
$$
**Proof:**
Here $F_1=F_2=\Phi,$ $\Rx_n=\pi,$ and $\Ry_n=\Rp_n.$ Moreover we use
$\Gamma_2(\Phi(\pi),\Phi(\pi))=0,$ because $u_j(\Phi(\pi),\Phi(\pi))=0$
for all $j=1,\cdots,m.$
The formula's in the corrollaries are a bit hard to use, partly
because they involve the complicated matrix $\Gamma$ of second
derivatives of the estimator. As an example, we give the
following result.
**Theorem:**
Define
$$
\begin{multline}\notag
2n\Rav{\cD}_n\defi
2n\Rav{\lambda}_n-
\text{ tr }
[A(\Rp,\Phi(\Rp))+
B(\Rp,\Phi(\Rp))G'(\Rp)+G(\Rp)B'(\Rp,\Phi(\Rp))+\\
\Gamma(\Rp,\Phi(\Rp))+G(\Rp)C(\Rp,\Phi(\Rp))G'(\Rp)]V(\Rp).
\end{multline}
$$
Then
$$
n\Exp{\Rav{\cD}_n}\Rightarrow\delta.
$$
**Proof:**
Direct from Corollary \ref{C:EDE}.
Another problem is that the formulas require
values for some of the unobserved quantities.
But, as explained in @deleeuw_C_97a,
there is a simple and fairly straightforward way around this
problem.
**THeorem:**
Define
$$
2n\Rav{\cP}_n\defi 2n\Rav{\lambda_n}-
\text{ tr }[B(\Rp_n,\Phi(\Rp_n))G'(\Rp_n)+
G(\Rp_n)B'(\Rp_n,\Phi(\Rp_n))]V(\Rp_n),
$$
$$
2n\Rav{\cB}_n\defi 2n\Rav{\lambda_n}-
\text{ tr }[A(\Rp_n,\Phi(\Rp_n))+B(\Rp_n,\Phi(\Rp_n))G'(\Rp_n)+
G(\Rp_n)B'(\Rp_n,\Phi(\Rp_n))]V(\Rp_n).
$$
Then
$$
\begin{split}
\Exp{\Rav{\cP}_n}&\Rightarrow \mu_n,\notag\\
\Exp{\Rav{\cB}_n}&\Rightarrow \beta_n.\notag
\end{split}
$$
**Proof:**
From Corollaries \ref{C:EPE} and \ref{C:EOE}.
#Use of the Jackknife
Although Theorems \ref{T:SPE} and \ref{T:EEE} can be applied in considerable
generality, they do require quite heavy derivative-type calculations.
This may not seem to be a large disadvantage, since differentiation
is usually straightforward, although tedious. The major problem, however,
is that automating differentiation is not simple. Unless sophisticated
symbolic mathematics packages are used, differentiation
has to be done by hand,
and then programmed into subroutines which are problem-specific.
Alternatively, we can do numerical differentiation. This is automatic,
and comparatively easy to program in full generality, but we still
have to make some strategic choices on how to implement it precisely. In
this paper we use the Jackknife for our numerical differentiation,
because it is specifically designed for statistical problems, and
in particular for statistical problems dealing with proportions.\par
Jackknife-based techniques are somewhat less general than the similar
delta-method expansions, because they actually suppose a strictly
multinomial model. I am sure they can be adapted quite easily,
but here we simply assume that $V(\pi)=\Pi-\pi\pi'.$
If we want to use the Jackknife,
we use perturbations in our differentiation of the form
$$
p_{n:j}\defi p_n+\frac{1}{n-1}(p_n-e_j),
$$
where the $e_j$ are the unit-vectors in $\mR^m$. Thus
$$
p_n=\sum_{j=1}^m p_{nj}e_j.
$$
We also set
$$
V(p_n)\defi\sum_{j=1}^m p_{nj}(p_n-e_j)(p_n-e_j)'=P_n-p_np_n',
$$
with $P_n$ the diagonal matrix with the $p_n.$
Define the following functions. They are defined to mimic the expansions
of the error measures we have derived in the previous sections.
**Definition:**
$$
\begin{split}
\cJ_n(p_n)&\defi 2(n-1)^2\{\sum_{j=1}^m p_{nj}\Delta(p_{n:j},\Phi(p_n))-
\Delta(p_n,\Phi(p_n))\},\\
\cK_n(p_n)&\defi 2(n-1)^2\{\sum_{j=1}^m p_{nj}\Delta(p_n,\Phi(p_{n:j}))-
\Delta(p_n,\Phi(p_n))\},\\
\cL_n(p_n)&\defi 2(n-1)^2\{\sum_{j=1}^m p_{nj}\Delta(p_{n:j},\Phi(p_{n:j}))-
\Delta(p_n,\Phi(p_n))\}.
\end{split}
$$
**Lemma:**
$$
\begin{split}
\cJ_n(p_n)&\Rightarrow\text{ tr }A(p_n,\Phi(p_n))V(p_n),\\
\cK_n(p_n)&\Rightarrow\text{ tr }[\Gamma(p_n,\Phi(p_n))+
G(p_n)C(p_n,\Phi(p_n))G'(p_n)]V(p_n),
\end{split}
$$
and
$$
\begin{multline*}
\cL_n(p_n)\Rightarrow\text{ tr }[A(p_n,\Phi(p_n))+
\Gamma(p_n,\Phi(p_n))+\\G(p_n)B(p_n,\Phi(p_n))+
B(p_n,\Phi(p_n))G'(p_n)+\\
G(p_n)C(p_n,\Phi(p_n))G'(p_n)]V(p_n).
\end{multline*}
$$
**Proof:**
These are just simple expansions.
**Theorem:**
$$
\Exp{2n\Rav{\lambda}_n-\cL_n(\Rp_n)}\Rightarrow\delta
$$
**Proof:**
From Lemma \ref{L:jack} and Theorem \ref{T:SPE}.
**Theorem:**
$$
\begin{split}
\Exp{2n\Rav{\lambda}_n-\{\cL_n(\Rp_n)-\cK_n(\Rp_n)\}}
&\Rightarrow\beta_n,\\
\Exp{2n\Rav{\lambda}_n-\{\cJ_n(\Rp_n)+\cK_n(\Rp_n)-\cL_n(\Rp_n)\}}
&\Rightarrow\mu_n.
\end{split}
$$
**Proof:**
From Lemma \ref{L:jack} and Theorem \ref{T:EEE}.
It is clear from the form of Theorems \ref{T:JEC} and \ref{T:JEE},
and the type of reasoning that is used,
that they generalize without modification to the more general
situation explained in Section \ref{S:moment}.
#Tables
$$
\begin{split}
\hline
Case&\mathcal{S}_n&\mathcal{T}_n&\Delta_n&\mathcal{D}_1\Psi_n&\mathcal{D}_2\Psi_n\\
\hline
1&\mu&\mu&0&0&0\\
2&\mu&x&\Delta_n(\mu,x)&h_n(\mu,x)&0\\
3&\mu&y&\Delta_n(\mu,y)&0&h_n(\mu,y)\\
4&x&\mu&\Delta_n(x,\mu)&g_n(x,\mu)&0\\
5&x&x&0&0&0\\
6&x&y&\Delta_n(x,y)&g_n(x,y)&h_n(x,y)\\
7&y&\mu&\Delta_n(y,\mu)&0&g_n(y,\mu)\\
8&y&x&\Delta_n(y,x)&h_n(y,x)&g_n(y,x)\\
9&y&y&0&0&0\\
10&\mu&\Phi_n(\mu)&\Delta_n(\mu,\Phi_n(\mu))&0&0\\
11&\mu&\Phi_n(x)&\Delta_n(\mu,\Phi_n(x))&h_n(\mu,\Phi_n(x))G_n(x)&0\\
12&\mu&\Phi_n(y)&\Delta_n(\mu,\Phi_n(y))&0&h_n(\mu,\Phi_n(y))G_n(y)\\
13&x&\Phi_n(\mu)&\Delta_n(x,\Phi_n(\mu))&g_n(x,\Phi_n(\mu))&0\\
14&x&\Phi_n(x)&\Delta_n(x,\Phi_n(x))&g_n(x,\Phi_n(x))+h(x,\Phi_n(x))G_n(x)&0\\
15&x&\Phi_n(y)&\Delta_n(x,\Phi_n(y))&g_n(x,\Phi_n(y))&h_n(x,\Phi_n(y))G_n(y)\\
16&y&\Phi_n(\mu)&\Delta_n(y,\Phi_n(\mu))&0&g_n(y,\Phi_n(\mu))\\
17&y&\Phi_n(x)&\Delta_n(y,\Phi_n(x))&h_n(y,\Phi_n(x))G_n(x)&g_n(y,\Phi_n(x))\\
18&y&\Phi_n(y)&\Delta_n(y,\Phi_n(y))&0&g_n(y,\Phi_n(x))+h_n(y,\Phi_n(y))G_n(y)\\
19&\Phi_n(\mu)&\mu&\Delta_n(\Phi_n(\mu),\mu)&0&0\\
20&\Phi_n(\mu)&x&\Delta_n(\Phi_n(\mu),x)&h_n(\Phi_n(\mu),x)&0\\
21&\Phi_n(\mu)&y&\Delta_n(\Phi_n(\mu),y)&0&h_n(\Phi_n(\mu),y)\\
22&\Phi_n(x)&\mu&\Delta_n(\Phi_n(x),\mu)&g_n(\Phi_n(x),\mu)G_n(x)&0\\
23&\Phi_n(x)&x&\Delta_n(\Phi_n(x),x)&g_n(\Phi_n(x),x)G_n(x)+h_n(\Phi_n(x),x)&0\\
24&\Phi_n(x)&y&\Delta_n(\Phi_n(x),y)&g_n(\Phi_n(x),y)G_n(x)&h_n(\Phi_n(x),y)\\
25&\Phi_n(y)&\mu&\Delta_n(\Phi_n(y),\mu)&0&g_n(\Phi_n(y),\mu)G_n(y)\\
26&\Phi_n(y)&x&\Delta_n(\Phi_n(y),x)&h_n(\Phi_n(y),x)&g_n(\Phi_n(y),y)G_n(y)\\
27&\Phi_n(y)&y&\Delta_n(\Phi_n(y),y)&0&g_n(\Phi_n(y),y)G_n(y)+h_n(\Phi_n(y),y)\\
28&\Phi_n(\mu)&\Phi_n(\mu)&0&0&0\\
29&\Phi_n(\mu)&\Phi_n(x)&\Delta_n(\Phi_n(\mu),\Phi_n(x))&h_n(\Phi_n(\mu),\Phi_n(x))G_n(x)&0\\
30&\Phi_n(\mu)&\Phi_n(y)&\Delta_n(\Phi_n(\mu),\Phi_n(y))&0&h_n(\Phi_n(\mu),\Phi_n(y))G_n(y)\\
31&\Phi_n(x)&\Phi_n(\mu)&\Delta_n(\Phi_n(x),\Phi_n(\mu))&g_n(\Phi_n(x),\Phi_n(\mu))G_n(x)&0\\
32&\Phi_n(x)&\Phi_n(x)&0&0&0\\
33&\Phi_n(x)&\Phi_n(y)&\Delta_n(\Phi_n(x),\Phi_n(y))&g_n(\Phi_n(x),\Phi_n(y))G_n(x)&
h_n(\Phi_n(x),\Phi_n(y))G_n(y)\\
34&\Phi_n(y)&\Phi_n(\mu)&\Delta_n(\Phi_n(y),\Phi_n(\mu))&0&g_n(\Phi_n(y),\Phi_n(\mu))G_n(y)\\
35&\Phi_n(y)&\Phi_n(x)&\Delta_n(\Phi_n(y),\Phi_n(x))&h_n(\Phi_n(y),\Phi_n(x))G_n(x)&g_n(\Phi_n(y),\Phi_n(x))G_n(y)\\
36&\Phi_n(y)&\Phi_n(y)&0&0&0\\
\hline
\end{split}
$$
#Figures
#Appendix: Hurt's Theorem
In this Appendix we give a relevant theorem that can be used to justify
the approximations to the expected values and standard errors used in the paper. It
is, to some extent, classical, but a particularly clear and
useful statement appears in two publications by Jan Hurt
[@hurt_76, @hurt_78].
$$
\DeclareMathOperator{\sumsum}{\sum\cdots\sum}
\DeclareMathOperator{\summer}{\sum\ \sum}
$$
**Theorem** : Suppose
* \{$\phi_n(x)$\} is a sequence of real-valued functions
on $\mathbb{R}^m,$
* \{$\underline{x}_n$\}=\{$(\underline{x}_{1n},\cdots\underline{x}_{mn})$\}
is a sequence of m-dimensional statistics.
Assume that
* For all $n, \phi_n$ is $(q+1)$-times totally differentiable
with respect to the $x_j$ in the interval $\mathbb{K}=\prod_{j=1}^m
[\mu_j-\delta_j,\mu_j+\delta_j]$, with $\delta_j>0,$ and $\delta_j$
independent of $n.$
* For all $n, \phi_n$ is bounded on $\mathbb{R}^m.$
* For all $n,$ the derivatives $\phi_n^{(1)},\cdots,\phi_n^{(q+1)}$ are bounded on $\mathbb{K}.$
* For all $j$ and all $n,$ $\underline{x}_{jn}$ has finite absolute moments up to the order $2(q+1).$
* For all $j=1,\cdots,m$ we have $\textbf{E}\mid
\underline{x}_{jn}-\mu_j\mid^{2(q+1)}=\mathcal{O}(n^{-(q+1)}).$
Then
$$
\mathbf{E}(\phi_n(\underline{x}_n)-\phi_n(\mu))=
\sum_{j=1}^q \frac{1}{j!}
\sumsum_{\substack{i_1+\cdots+i_m=j\\i_1,\cdots,i_m\geq 0}}
\left(\frac{\partial^j\phi_n}{\partial x^{i_1}_1\cdots\partial
x^{i_m}_m}\right)_{x=\mu}\times
\mathbf{E}(\underline{x}_{1n}-\mu_1)^{i_1}\cdots(\underline{x}_{mn}-\mu_m)^{i_m})+
\mathcal{O}(n^{-\frac{(q+1)}{2}}).
$$
and
$$
\begin{split}\notag
\mathbf{V}(\phi_n(\underline{x}_n)-\phi_n(\mu))=
\summer_{\substack{j,k=1,\cdots,q\\j+k\leq q+1}}\frac{1}{j!}\frac{1}{k!}
\sumsum_{\substack{i_1+\cdots+i_m=j\\i_1,\cdots,i_m\geq 0}}
\sumsum_{\substack{\ell_1+\cdots+\ell_m=j\\\ell_1,\cdots,\ell_m\geq 0}}
\times
\left(\frac{\partial^j\phi_n}{\partial x^{i_1}_1\cdots\partial
x^{i_m}_m}\right)_{x=\mu}
\left(\frac{\partial^k\phi_n}{\partial x^{\ell_1}_1\cdots\partial
x^{\ell_m}_m}\right)_{x=\mu}\times\\
\times
\mathbf{C}((\underline{x}_{1n}-\mu_1)^{i_1}\cdots(\underline{x}_{mn}-\mu_m)^{i_m})
((\underline{x}_{1n}-\mu_1)^{\ell_1}\cdots(\underline{x}_{mn}-\mu_m)^{\ell_m})
+\mathcal{O}(n^{-\frac{(q+2)}{2}}).
\end{split}
$$
**Remark:** In @hurt_78 the same result is proved, but the residuals
are not expressed in terms of powers of $n$, but in terms
of the absolute moments. This extends the result to a more
general class of statistics.
##References