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\begin{document}

\title{Sequential Construction of a Confidence of Given Width}
\author{R.C.H. Cheng}
\maketitle

\section{Confidence Intervals of Given Width; $\protect\sigma ^{2}$ Known}

We consider the sequential construction of a confidence interval of \textit{%
given} width $w,$ with \textit{given} level of confidence.

Consider a sequence of observations%
\[
X_{1},X_{2},...,X_{n}
\]%
where%
\[
X_{i}=\mu +\varepsilon _{i}
\]%
with all the $\varepsilon _{i}$ having the same distribution with mean $0$
and variance $\sigma ^{2}$. (Typically but not necessarily normal.) Let the
sample mean after $n$ observations be%
\begin{equation}
D_{n}=n^{-1}\sum_{i=1}^{n}X_{i}=\mu +n^{-1}\sum_{i=1}^{n}\varepsilon _{i}.
\label{Dn}
\end{equation}

Consider the construction of a confidence interval for the unknown $\mu $ of
form $(D_{n}-h_{1},$ $D_{n}+h_{2})$ where $h_{1},h_{2}>0$ so that the width
of the interval is%
\[
w=h_{1}+h_{2}. 
\]%
We can obtain such a confidence interval from the probability statement%
\begin{equation}
\Pr (z_{\beta }\leq \sqrt{n}(D_{n}-\mu )/\sigma \leq z_{1-\alpha })=1-\alpha
-\beta  \label{Prob1}
\end{equation}%
where $\sqrt{n}(D_{n}-\mu )/\sigma $ is the standardised form of $D_{n}$ so
that it is distributed as a $N(0,1)$ variable, and where $z_{\beta }$ and $%
z_{1-\alpha }$ are the $\beta $ and $1-\alpha $ quantiles of the $N(0,1)$
distribution.

We shall assume that 
\begin{equation}
\alpha ,\ \beta <1/2  \label{AlphaBetaCondition}
\end{equation}%
so that $z_{\beta }<0$ and $z_{1-\alpha }>0.$ The probability statement (\ref%
{Prob1}) can be written as%
\[
\Pr (D_{n}-h_{1}\leq \mu \leq D_{n}+h_{2})=1-\alpha -\beta 
\]%
provided we set%
\begin{equation}
h_{1}=-\frac{z_{1-\alpha }}{z_{\beta }}h_{2}  \label{h1}
\end{equation}%
and%
\begin{equation}
h_{2}=\sigma z_{\beta }/\sqrt{n}.  \label{h2}
\end{equation}%
Both are positive under assumption (\ref{AlphaBetaCondition}) and the width
of the interval is then%
\[
w=h_{1}+h_{2}. 
\]%
which is dependent only on $\alpha ,$ $\beta \,,$\ $\sigma ^{2}$ and $n$.
The values of $\alpha $ and $\beta $ are user selected. Thus if $\sigma ^{2}$
is known, a confidence interval of given width $w$ can be obtained simply by
selecting $n$ to satisfy (\ref{h2}). (Rounding up may be needed to ensure
that $n$ is integer.): We call this the \emph{Known Sigma Stopping Rule}:%
\begin{equation}
\text{Stop at }n=\sigma ^{2}z_{\beta }^{2}/h_{2}^{2}  \label{StopRule0}
\end{equation}

The relationship between $\sigma $ and $n$ can be made explicit by combining
(\ref{h1}) and (\ref{h2}) to eliminate $h_{2}.$This yields%
\begin{equation}
n=\frac{(z_{1-\alpha }-z_{\beta })^{2}\sigma ^{2}}{w^{2}}.
\label{sigsquared}
\end{equation}

\section{Case $\protect\sigma ^{2}$ Unknown}

Consider now the situation where $\sigma ^{2}$ is not known.

The sample variance, 
\[
s_{n}^{2}=\frac{1}{n-1}\sum_{i=1}^{n}(d_{i}-D_{n})^{2} 
\]%
can now be used as a first order approximation for $\sigma ^{2}$. This
changes with $n.$ A simple way to determine $n$ is by the stopping rule:%
\begin{equation}
\text{Take }N\text{ as the first }n\text{ for which }s_{n}^{2}\leq \frac{%
nw^{2}}{(z_{1-\alpha }-z_{\beta })^{2}}.  \label{StopRule1}
\end{equation}%
Here $N$ is a random variable. Chow and Robbins (1965) give the theoretical
justification for selecting $N$ in this way, at least for the case $\alpha
=\beta $. Further properties of the distribution of $N$ are discussed by
Starr (1966a and 1966b) and Woodroofe (1977) (again for the case $\alpha
=\beta )$. In particular Starr (1966a) showed that use of $N$ gives
reasonably consistent coverage and that it is more efficient than the
well-known two-stage procedure proposed by Stein (1945). However this choice
of $N$ does seem liable to produce somewhat narrow confidence intervals so
that the coverage is less than the nominal value of $(1-2\alpha ).$Use of $N$
based on (\ref{StopRule1}) does not seem to be much used in practice.

Anscombe (1953) provides a second order accurate version. The method is not
often mentioned but performs significantly better and moreover is very
easily implemented. It is this version that we recommend and which we now
describe.

We transform the $X_{1},$ $X_{2},....X_{n}$ of (\ref{Dn}) to independent
(actually $\sigma \chi _{1}^{2})$ variates%
\begin{equation}
U_{i}=\frac{1}{i(i+1)}\{iX_{i+1}-\sum_{j=1}^{i}X_{j}\}^{2}\ \ \
i=1,2,...,n-1.  \label{Ui}
\end{equation}%
so that if additional observations $X_{n+1},$ $X_{n+2},...$ are observed,
additional $U_{n},U_{n+1},...$can be added without changing the previous $%
U_{i}.$ Anscombe (1953) discusses different second order versions of the
rule (\ref{StopRule1}). We use Anscombe's (2.6) rule, only for general $%
\alpha ,$ $\beta :\,$%
\begin{eqnarray}
\text{Take }N\text{ as the first }n\text{ }( &\geq &3)\text{ for which}
\label{StopRule2} \\
\sum_{i=1}^{n-1}U_{i} &\leq &\frac{w^{2}}{(z_{1-\alpha }-z_{\beta })^{2}}%
(n-3)(n+0.324-\frac{1}{8}(z_{1-\alpha }-z_{\beta })^{2})  \nonumber
\end{eqnarray}

\section{References}

Anscombe, F.J. 1953. Sequential Estimation. J. Roy. Statist. Soc. Series B,
15,1, 1-29.

Chow, Y.S. and Robbins, H. 1965. On the asymptotic theory of fixed-width
sequential confidence intervals for the mean. Ann. Math. Stat. 36, 2,
457-462.

Starr, N. 1966a. The performance of a sequential procedure for the
fixed-width interval estimation of the mean. Ann. Math. Stat. 37, 1, 36-50.

Starr, N. 1996b. On the asymptotic efficiency of a sequential procedure for
estimating the mean. Ann.Math.Stat. 37, 5, 1173-1185.

Stein, C. 1945. A two-sample test for a linear hypothesis whose power is
independent of the variance. Ann. Math. Stat. 16, 243-258.

Woodroofe, M. 1977. Second order approximations for sequential point and
interval estimation. Ann. Math. Stat. 5,5 984-995.

\end{document}
