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\title{Some puzzles appearing in statistical inference}

\alttitle{Quelques énigmes apparaissant dans l'inférence statistique}

\author{\firstname{Seyf} \lastname{Alemam}}

\address{Department of Statistics, University of Tabriz, P. O. Box 51666-17766, Tabriz, Iran}

\email{saif.alemams@gmail.com}

\author{\firstname{Hazhir} \lastname{Homei}}

\address[1]{Department of Statistics, University of Tabriz, P. O. Box 51666-17766, Tabriz, Iran}

\email{homei@tabrizu.ac.ir}

\author{\firstname{Saralees} \lastname{Nadarajah}\IsCorresp}

\address{Department of Mathematics, University of Manchester, Manchester M13 9PL, UK} 

\email{mbbsssn2@manchester.ac.uk}

%%%
\subjclass{62F10, 62F99}

\keywords{\kwd{Complete minimal sufficient statistic} \kwd{Incomplete minimal sufficient statistic}
\kwd{Minimal sufficient statistic} \kwd{Rao--Blackwell theorem} \kwd{Sufficient statistic}}

\altkeywords{\kwd{Statistique suffisante minimale complète} \kwd{statistique suffisante minimale incomplète} \kwd{statistique suffisante minimale} \kwd{théorème de Rao--Blackwell} \kwd{statistique suffisante}}

%%%


\begin{abstract}
Rao-Blackwell theorem is widely known to be a mathematically
powerful technique that can be used to improve the precision of an estimator.
The procedure entails exploiting a sufficient statistic to obtain an
improved estimator or a uniformly minimum variance unbiased estimator.
A modification of sufficient statistics is introduced here which can be applied
for Rao-Blackwell theorem along with some fruitful applications that illustrate its properties.
Also some theorems have been rewritten in statistical inference.
\end{abstract}

\begin{altabstract}
Le théorème de Rao-Blackwell est largement connu pour être une technique mathématique puissante qui peut être utilisée pour améliorer la précision d'un estimateur. La procédure consiste à exploiter une statistique suffisante pour obtenir un estimateur amélioré ou un estimateur sans biais à variance uniformément minimale.
Nous présentons ici une modification des statistiques suffisantes qui peut être appliquée au théorème de Rao-Blackwell ainsi que quelques applications fructueuses qui illustrent ses propriétés. De plus, certains théorèmes ont été réécrits dans le cadre de l'inférence statistique.
\end{altabstract}



\begin{document}



\maketitle

\section{Introduction}


Fisher~\cite{F} introduced sufficient statistics in 1920; since then sufficient statistics have been used more and
more in statistical inference.
We should understand them better to derive uniformly
minimum variance unbiased estimators.
Statisticians have cleverly embedded sufficient statistics into estimators,
which is the main idea of Rao-Blackwell theorem; see~\cite{Bl} and~\cite{Ra2}.
Uniformly minimum variance unbiased estimators
can be calculated by complete sufficient statistics,
leading to Lehmann-Scheff\'{e} theorem; see~\cite{KV, LS1} and~\cite{LS2}.
The application of these theorems is still seen in the literature; see~\cite{KV}.
However, when a complete sufficient statistic is lacking, there
may be nonconstant functions that can be uniformly minimum variance unbiased estimators.


Some researchers focus on the
examples given by \cite[Problem~5.11]{Ra2} or \cite[p.~76-77]{LS1, LS2}
and try to solve these kinds of problems by the following theorem.



\begin{theorem}[Lehmann-Scheff\'{e} theorem]
\label{thm1}
Let $X = \left[X_{1}, \ldots, X_{n}\right]$ and suppose $X_1, \ldots, X_n$
are random variables having distribution $P_{\theta}$, $\theta \in \Theta$.
A necessary and sufficient condition for a statistic $T(X)$ to be
uniformly minimum variance unbiased estimator of its mean is that
$E\left[T(X)U(X)\right] = 0$ for all $\theta \in \Theta$ and all $U\in\mathcal{U}_{0}$,
where $\mathcal{U}_{0}$ denotes the set of all the unbiased estimators of $0$.
\end{theorem}


This theorem can be used widely when there are no complete sufficient statistics.
It is a strong competitor to the theorem of Rao-Blackwell.



This fact is very seldom pointed out and
exemplified in undergraduate or graduate textbooks; see, for example,~\cite{Bo}.
The motivation to introduce a new concept of
sufficient statistic called $\mathscr{H}$-sufficient statistic comes from the above discussion.
We investigate the properties of
$\mathscr{H}$-sufficient statistic and compare them with those of
sufficient statistic.
Then Rao-Blackwell theorem (RBT) and
Lehmann-Scheff\'{e} theorem (LST) will be generalized in a way
which can solve some of the problems where UMVUE exists
but there are no complete
sufficient statistics; cf.
\cite[Problem~5.11]{Ra1},~\cite[p.~76-77]{Le2},~\cite[Example~3.7, p.~167]{Sh},
\cite[Example~10, p.~366]{RE},
\cite[Section~7.6.1, p.~377]{Mu},~\cite[p.~243]{PR},~\cite[Section~12.4, p.~293 ]{Ru} and
\cite[p~330-331, Remark]{MGB}.
Some of the theorems are restated
and proved by using the newly introduced $\mathscr{H}$-sufficient statistic.



Some researchers~\cite{bh1998} state that: ``If a minimal sufficient statistic is
not complete, then by the suggestion of Fisherian tradition we
should consider condition on ancillary statistics for the purposes
of inference.
This approach runs into problems because there are
many situations where several ancillary statistics exists but there
are no maximal `ancillaries'.
Of course, when a complete sufficient
statistic exists, Basu's theorem assures us that we need not worry
about conditioning on ancillary statistics since they are all
independent of the complete sufficient statistic''.
We suggest complete $\mathscr{H} $-sufficient statistics for the purposes of
inference when there are no complete sufficient statistics.
Theorem~\ref{thm1} assures that we need not worry
about ancillary statistics since they are all uncorrelated of
complete $\mathscr{H} $-sufficient statistics.



\subsection{The main contribution}

If the minimal sufficient statistic is not complete,
then the RBT and LST will not be of much use,
as has been explicitly stated in various books and articles.
The main contribution of this article is a generalization of
RBT and LST, resulting in the use of the
newly introduced $\mathscr{H}$-sufficient statistics.
This enables us to obtain uniformly minimum variance unbiased estimators
even when the minimal sufficient statistic is not complete,
in which case RBT and LST are not directly applicable.

\subsection{Definitions}

Consider a statistical model $\left(\mathscr{X}, \mathscr{A}, \mathcal{P}
=\left\{ P_{\theta}:\theta \in \Theta \right\}\right)$.
We assume here that the
family of probability measures on the sample space $\mathscr{X}$ has
the form $\left\{ P_{\theta}:\theta \in \Theta \right\}$.
A random element in
$\mathscr{X}$ is denoted by $X = \left[X_{1}, \ldots, X_{n} \right]$.
The probability measure $P_{\theta} \in \mathcal{P}$ is called the
population.
The random element $X = \left[X_{1}, \ldots, X_{n}\right]$ that
produces the data is called a sample from $\left\{ P_{\theta}:\theta \in \Theta \right\}$.
Let $ X :$ $\mathscr{X}$ $\to $ $\mathscr{X}$ denote the identity
mapping, $\left(\mathscr{Y}; \mathscr{B}\right)$ a measurable space, and
$T : \mathscr{X} \to \mathscr{Y}$ a
$\mathscr{A}-\mathscr{B}-$ measurable mapping (that is, $T^{-1}B \in \mathscr{A}$
for all $B \in \mathscr{B}$).
Here, $T(X)$ is called a statistic to $\left(\mathscr{Y}; \mathscr{B}\right)$, and we write
$T: \left(\mathscr{X}, \mathscr{A}\right) \to \left(\mathscr{Y}; \mathscr{B}\right)$.

To understand the role that a $\mathscr{H}$-sufficient statistic plays in inference, we first need to define
some basic concepts.


\begin{definition}
\label{def:1}
{\rm Let $X = \left[X_{1}, \ldots, X_{n}\right]$
and suppose $X_1, \ldots, X_n$
are random variables from an unknown
population $P_{\theta} \in \mathcal{P}$ and $\mathfrak{a}$ a real
valued parameter, $\mathfrak{a}:\Theta \to \mathbb{R}$,
related to $P_{\theta}$.
An estimator $\delta(X)$,
$\delta:\mathscr{X}\to \mathbb{R}$, of $\mathfrak{a}$ is
unbiased if and only if $E \left[\delta(X)\right] = \mathfrak{a}$ for every
$P_{\theta} \in \mathcal{P}.$}
\end{definition}

If there exists an unbiased estimator of $\mathfrak{a}$, then
$\mathfrak{a}$ is called a U-estimable parameter.

\begin{definition}
\label{def:11}
{\rm Let $X = \left[X_{1}, \ldots, X_{n}\right]$
and suppose $X_1, \ldots, X_n$
are random variables from an unknown
population $P \in \mathcal{P}$.
A statistic $T(X)$, $T : \mathscr{X} \to \mathbb{R}$,
is said to be complete for $P \in \mathcal{P}$
if and only if for any Borel-measurable function $f$
from $\mathbb{R}$ to $\mathbb{R}$, $E\left[f(T)\right] = 0$ for all $P \in \mathcal{P}$
implies $f(T) = 0$ almost surely $\mathcal{P}$.}
\end{definition}

\begin{definition}
\label{def:222}
{\rm Let $X = \left[X_{1}, \ldots, X_{n}\right]$
and suppose $X_1, \ldots, X_n$
are random variables from an unknown
population $P_{\theta} \in \mathcal{P}$.
An unbiased estimator
$T(X)$ of $\mathfrak{a}$ is called a uniformly minimum variance unbiased
estimator (UMVUE) if and only if Var$\left[T(X)\right] \leq$ Var$\left[\delta(X)\right]$ for
every $P_{\theta} \in \mathcal{P}$ (or for every $\theta \in \Theta$)
and any other unbiased estimator $\delta(X)$ of $\mathfrak{a}$.}
\end{definition}



Throughout this note we assume that $X = \left[X_{1}, \ldots, X_{n}\right]$
and suppose $X_1, \ldots, X_n$
are random variables
from an unknown population $P_{\theta} \in \mathcal{P}$
and there exists an unbiased estimator for $\mathfrak{a}$.
Let $\mathcal{U}_{\mathfrak{a} }$ denote the class of unbiased
estimators $\delta\colon\mathscr{X}\to \mathbb{R}$ for $\mathfrak{a}$;
and $\mathcal{U}_{0} \left(\mathscr{H}_{\mathfrak{a}}\right)$ is the set of all the
unbiased estimators of $0$, which is a function of
$\mathscr{H}$-sufficient statistics for $\mathfrak{a}$, see Definition~\ref{def:2}.
all the considered estimators are assumed to have finite variances.
The space used in this note is $\mathbb{R}^{n}$ and the
elements of $\mathcal{B}$ are Borel sets.
For related notation and discussions, we refer the reader to~\cite{Sh}.


\section{Sufficient statistics}


The concept of sufficient statistic plays a fundamental role in all
areas of statistical inference; see~\cite{CB}.

\begin{definition}
\label{def:0}
{\rm Let $X = \left[X_{1}, \ldots, X_{n}\right]$
and suppose $X_1, \ldots, X_n$
are random variables from an unknown
population $P_{\theta} \in \mathcal{P}$, where $\mathcal{P}$ is a
family of populations.
A statistic $T(X)$ is called a sufficient
statistic for $\mathcal{P}$ (or for $\theta$) if there exists a
Markov kernel $k : \mathfrak{T} \times \mathcal{C} \to [0, 1]$
such that for every $\theta \in \Theta$, $k$ is a version of
a regular conditional distribution of $X$ given $T(X)$ under
$P_{\theta}$.}
\end{definition}

Two weaker concepts of sufficiency, which are
tailored to a given unbiased estimable aspect $\mathfrak{a}:\Theta \to \mathbb{R}$
are introduced and discussed in the
following.
Some properties of these statistics are studied in the sequel.


\subsection{\texorpdfstring{$\mathscr{H}$}{H}-sufficient statistic in distribution}

\begin{definition}
\label{def:12}
{\rm Let $X = \left[X_{1}, \ldots, X_{n}\right]$
and suppose $X_1, \ldots, X_n$
are random variables from an
unknown population $P_{\theta} \in \mathcal{P}$.
A statistic $T(X)$
is called a $\mathscr{H}$-sufficient in distribution for
$\mathfrak{a}$ if for all $\delta(X) \in \mathcal{U}_{\mathfrak{a}}$
there is a Markov kernel $k_{\mathfrak{a}, \delta(X)} : \mathfrak{T} \times \mathcal{B} \left(\mathbb{R}\right) \to [0, 1]$
such that for
every $\theta \in \Theta$, $k_{\mathfrak{a}, \delta}$ is a version of
a regular conditional distribution of $\delta(X)$ given $T(X)$ under
$P_{\theta}$.}
\end{definition}



\begin{example}[Example of~\cite{Me}]
\label{ex:1}
Let $X$ be a Poisson random variable with $E(X)=\lambda$.
We note that $k(-1)^{X}$ is a $\mathscr{H}$-sufficient
statistic in distribution for $e^{-2\lambda}$, where $k$ is a constant.
We can check that $(-1)^{X}$ is a UMVUE for $e^{-2\lambda}$.
\end{example}



\subsection{\texorpdfstring{$\mathscr{H}$}{H}-sufficient statistic}


To derive UMVUEs when there are no complete sufficient statistics,
we need to introduce a new concept named $\mathscr{H}$-sufficient
statistic for $\mathfrak{a}$.
It is defined as follows.




\begin{definition}
\label{def:2}
{\rm Let $X = \left[X_{1}, \ldots, X_{n}\right]$
and suppose $X_1, \ldots, X_n$
are random variables from an
unknown population $P_{\theta} \in \mathcal{P}$.
A statistic $T(X)$ is called $\mathscr{H}$-sufficient for $\mathfrak{a}$
if for all $\delta(X) \in \mathcal{U}_{\mathfrak{a} }$ there is a
measurable mapping $h_{\mathfrak{a}, \delta} : \mathfrak{T} \to \mathbb{R}$
such that for every $\theta \in \Theta$ we have
$E_{\theta} \left[\delta(X) \mid T \right] = h_{\mathfrak{a}, \delta} \circ T$ almost surely $P_{\theta}$.}
\end{definition}


\begin{example}
\label{ex:2}
{\rm Let $X$ be a discrete
random variable from $P_{\theta}$ with the probability mass function
\begin{equation*}
P_{\theta}(X=-1)=\theta,
\
P_{\theta} (X=k)=(1-\theta)^2\theta^k,
\
k=0,1,2,\ldots,
\end{equation*}
where $\theta \in (0, 1)$ is unknown.
We note that $I_{\{0\}}(X)$ is a $\mathscr{H}$-sufficient
statistic for $(1-\theta)^2$ since for every $\theta \in (0, 1)$ and
every $\alpha\in\mathbb{R}$, we have
\begin{equation*}
E_{\theta}\left[I_{\{0\}}(X)+\alpha X\mid I_{\{0\}}(X)\right] =I_{\{0\}}(X)
\end{equation*}
almost surely $P_{\theta}$.
Here, $X$ is
not complete, although it is still a minimal sufficient statistic for $(1-\theta)^2$.
We also note that
$I_{\{0\}}(X)$ is not a $\mathscr{H}$-sufficient statistic in
distribution for $(1-\theta)^2$.}
\end{example}



Some properties of $\mathscr{H}$-sufficient statistics are
discussed in the following.



\begin{theorem}
\label{thm:0}
Let $\mathcal{P}=\left\{ P_{\theta}:\theta \in \Theta \right\}$
be a family of distributions.
Consider
\begin{enumerate}[label=(\roman*)]

\item\label{theo10_i}
a sufficient statistic for $\mathcal{P}$ (or $\theta$),

\item\label{theo10_ii}
a $\mathscr{H}$-sufficient statistic in distribution for
$\mathfrak{a}$,

\item\label{theo10_iii}
a $\mathscr{H}$-sufficient statistic for $\mathfrak{a}$.

\end{enumerate}
Then, we have
\begin{enumerate}[label=(\alph*)]

\item\label{theo10_a}
any sufficient statistic for $\mathcal{P}$ is a
$\mathscr{H}$-sufficient statistic in distribution for
$\mathfrak{a}$;

\item\label{theo10_b}
any $\mathscr{H}$-sufficient statistic in distribution for
$\mathfrak{a}$ is a $\mathscr{H}$-sufficient statistic for
$\mathfrak{a}$;

\item\label{theo10_c}
any sufficient statistic for $\mathcal{P}$ is a
$\mathscr{H}$-sufficient statistic for $\mathfrak{a}$.

\end{enumerate}
\end{theorem}

\begin{proof}
Since the conditional distribution of samples given
a sufficient statistic does not depend on $\theta$,
the conditional expectation of any statistic given a sufficient
statistic does not depend on $\theta$.
So,~\ref{theo10_c}~follows.
The proofs of~\ref{theo10_a} and~\ref{theo10_b} are
similar.
\end{proof}


\begin{remark}
\label{rem:1}
In general, the converse of none of the three parts of Theorem~\ref{thm:0}
hold (see Examples~\ref{ex:1} and~\ref{ex:2}).
\end{remark}




It is clear from Theorem~\ref{thm:0} and Remark~\ref{rem:1} that
the class of $\mathscr{H}$-sufficient statistics for $\mathfrak{a}$
contains sufficient statistics for $\theta$.
Also we can conclude from Theorem~\ref{thm:0}
that the jointly sufficient statistics are $\mathscr{H}$-sufficient statistics.


\begin{proposition}
Let $X = \left[X_{1}, \ldots, X_{n}\right]$
and suppose $X_1, \ldots, X_n$
are random variables from an
unknown population $P_{\theta} \in \mathcal{P}$.
If an unbiased estimator $T(X)$ is unique for $\mathfrak{a}$,
then $T(X)$ is a $\mathscr{H}$-sufficient statistic for
$\mathfrak{a}$.
\end{proposition}

\begin{proof}
It is obvious that $E_{\theta}\left[T(X)\mid T(X)\right]=T(X)$ almost surely $\mathcal{P}$.
\end{proof}


\begin{proposition}
Let $X = \left[X_{1}, \ldots, X_{n}\right]$
and suppose $X_1, \ldots, X_n$
are random variables from an
unknown population $P_{\theta} \in \mathcal{P}$.
Let $T(X)$ be a $\mathscr{H}$-sufficient statistic for
$\mathfrak{a}$ such that $S(X)= g \left(T(X)\right)$, where $S(X)$ is another
statistic and $g$ is a one-to-one measurable function.
Then $S(X)$ is a $\mathscr{H}$-sufficient statistic for $\mathfrak{a}$.
\end{proposition}

\begin{proof}
Let $U(X)$ be an arbitrary unbiased estimator for $\mathfrak{a}$.
Then, we have $E_{\theta} \left[U(X)\mid S(X)\right] = E_{\theta} \left[U(X)\mid T(X)\right]$
almost surely $\mathcal{P}$,
which shows that $E_{\theta} \left[U(X)\mid S(X)\right]$ does not depend on the parameter.
\end{proof}


\begin{remark}
Let $X = \left[X_{1}, \ldots, X_{n}\right]$
and suppose $X_1, \ldots, X_n$
are random variables from an
unknown population $P_{\theta} \in \mathcal{P}$.
Let $S(X)$ be a $\mathscr{H}$-sufficient statistic for
$\mathfrak{a}$ and $U(X)$ another statistic such that
$S(X)=g\left(U(X)\right)$ for a measurable function $g$.
We expect $U(X)$ to be a $\mathscr{H}$-sufficient statistic for $\mathfrak{a}$,
but actually it is not.
Consider Example~\ref{ex:2} again:
Let $ S(X)= I_{0}(X)$ and $U(X)= 1$, $0$ and 2 (or any value other than 0 and 1) for $x=0, -1$ and $ x>1$, respectively.
Then, verify that (i) $S(X)$ is $\mathscr{H}$-sufficient,
(ii) $S(X)$ is a function of $U(X)$, but (iii) $U(X)$ is not $\mathscr{H}$-sufficient.
\end{remark}



\section{A generalization of RBT and LST}

We now apply the RBT for arbitrary $\mathscr{H}$-sufficient
statistics for $\mathfrak{a}$ to obtain a better estimator.


\begin{theorem}
\label{thm:1}
Let $X = \left[X_{1}, \ldots, X_{n}\right]$
and suppose $X_1, \ldots, X_n$
are random variables from an
unknown population
$P_{\theta} \in \mathcal{P} =\left\{ P_{\theta'}:\theta' \in \Theta \right\}$.
Let $H(X)$ be a $\mathscr{H}$-sufficient statistic for $\mathfrak{a}$.
Let $\delta(X)$ be an unbiased estimator of a U-estimable $\mathfrak{a}$,
and the loss function $L(\theta, \delta(X))$ be a strictly convex function of $\delta(X)$.
Then, if $\delta(X)$ has finite expectation and risk, we have
$R \left(\theta, \delta(X)\right) = EL \left[\theta, \delta(X)\right] < \infty$,
and if $\psi \left(\mathfrak{h}\right) = E \left[\delta(X)\mid H(X)=\mathfrak{h}\right]$
then the risk of the estimator $\psi\left(H(X)\right)$
satisfies $R \left(\theta, \psi \left(H(X)\right)\right) < R \left(\theta, \delta(X)\right)$
unless $\delta(X) = \psi \left(H(X)\right)$ almost surely $\mathcal{P}$.
\end{theorem}


\begin{proof}
The proof is an easy application of Jensen's inequality and
Definition~\ref{def:2}; see~\cite{LC}.
\end{proof}

We now reexpress Lemma 1.10 in~\cite{LC} within the new framework.



\begin{lemma}
\label{lem:0}
Let $X = \left[X_{1}, \ldots, X_{n}\right]$
and suppose $X_1, \ldots, X_n$
are random variables from an
unknown population $\mathcal{P} =\left\{ P_{\theta}:\theta \in \Theta \right\}$.
Let $H(X)$ be a complete $\mathscr{H}$-sufficient statistic for $\mathfrak{a}$.
Then, every U-estimable $\mathfrak{a}$ has one and
only one unbiased estimator that is a function of $H(X)$.
Of course, uniqueness here means that any two such functions agree almost surely $\mathcal{P}$.
\end{lemma}

\begin{proof}
The uniqueness of the unbiased estimator follows from completeness of $H(X)$;\break see~\cite{LC}.
\end{proof}

The generalization of LST \cite[Theorem~5.1]{LS1}  by using a
complete $\mathscr{H}$-sufficient statistic for $\mathfrak{a}$ is as follows.



\begin{theorem}
\label{thm:2}
Let $X = \left[X_{1}, \ldots, X_{n}\right]$
and suppose $X_1, \ldots, X_n$
are random variables from an
unknown population $\mathcal{P} =\left\{ P_{\theta}:\theta \in \Theta \right\}$.
Suppose
that $H(X)$ is a complete $\mathscr{H}$-sufficient statistic for $\mathfrak{a}$.
Then we have the following:

\begin{enumerate}[label=(\roman*)]
\item\label{theo17_i}
For every U-estimable $\mathfrak{a}$, there exists an unbiased
estimator that uniformly minimizes the risk for any loss function
$L(\theta, \delta)$ which is convex in $\delta$; therefore, this
estimator in particular is UMVUE of $\mathfrak{a}$.


\item\label{theo17_ii}
The UMVU estimator of~\ref{theo17_i} is a unique unbiased estimator
and is a function of $H(X)$;
it is a unique unbiased estimator
with minimum risk, provided its risk is finite and $ L(\theta, \delta)$ is strictly convex in $\delta$.

\end{enumerate}
\end{theorem}

\begin{proof}
\ref{theo17_i}~is obvious by Theorem~\ref{thm:1}.
For~\ref{theo17_ii}, see~\cite{LC} and Lemma~\ref{lem:0}.
\end{proof}



\begin{theorem}
Let $X = \left[X_{1}, \ldots, X_{n}\right]$
and suppose $X_1, \ldots, X_n$
are random variables from an
unknown population $P_{\theta}$, $\theta \in \Theta$.
Let $T(X)$ be an unbiased estimator for $\mathfrak{a}$ and $H(X)$
a $\mathscr{H}$-sufficient statistic for $\mathfrak{a}$ such
that $T(X) = g\left(H(X)\right)$ for a measurable function $g$.
Then a necessary and sufficient condition for $T(X)$
to be a UMVUE of $\mathfrak{a}$ is that
$E_{\theta} \left[ T(X)U^\ast(X)\right] = 0$ for all
$U^\ast(X)\in\mathcal{U}_{0} \left(\mathscr{H}_{\mathfrak{a}}\right)$ and $\theta \in \Theta$.
\end{theorem}

\begin{proof}
Let $\mathcal{U}_{0} \left(\mathscr{H}_{\mathfrak{a}}\right)$ and $U(X)$ be in $\mathcal{U}_{0}$.
Suppose that $U(X)\in\mathcal{U}_{0}$.
Then the result follows from
the fact that we have
$E_{\theta}$ $\left[U(X)\mid H(X)\right]$ $\in$ $\mathcal{U}_{0}$ $\left(\mathscr{H}_{\mathfrak{a}}\right)$
and the following identities hold
\begin{equation*}
E_{\theta}\left[T(X)U(X)\right] =
E_{\theta} \left\{ E_{\theta} \left[g\left(H(X)\right)U(X)\mid H(X)\right]\right\} =
E_{\theta} \left\{g\left(H(X)\right)E_{\theta}\left[U(X)\mid H(X)\right]\right\},
\end{equation*}
where $U(X)$ is an unbiased estimator of 0.
Note that $E_{\theta}\left[ U(X)\mid H(X) \right]$
is a statistic since $E_{\theta}$ $\left\{ T(X) - \left[T(X) - U(X) \right]\mid H(X)\right\}$
dose not depend on $\theta$.
The converse is obvious.
\end{proof}



\begin{theorem}
\label{thm:6}
Let $X = \left[X_{1}, \ldots, X_{n}\right]$
and suppose $X_1, \ldots, X_n$
are random variables from an
unknown population $P_{\theta}$, $\theta \in \Theta$.
Let $H(X)$ be a $\mathscr{H}$-sufficient
statistic for $ \mathfrak{a}$.
In addition, suppose for every unbiased estimator
$T(X)$ for $\mathfrak{a}$ there is a measurable
function $g$ such that $T(X)=g\left(H(X)\right)$.
Then $T(X)$ is a UMVUE if $E_{\theta} \left[ U(X)\mid H(X) \right] = 0$ almost surely $P_{\theta}$
for every $U(X) \in \mathcal{U}_{0}$ and every $\theta \in \Theta$.
\end{theorem}


\begin{proof}
For $U(X) \in \mathcal{U}_{0}$, we have
$E_{\theta} \left[ T(X)U(X) \right]$ $=$
$E_{\theta} \left\{ g\left(H(X)\right) E\left[ U(X)\mid H(X) \right] \right\}$ $=$ $0$
since $E_{\theta}$ $\left[ U(X)\mid H(X) \right]$ $=$ $0$ almost surely $P_{\theta}$.
So, $T(X)$ is a UMVUE.
\end{proof}



\section{Complete \texorpdfstring{$\mathscr{H}$}{H}-sufficient statistic}


We are interested in finding a $\mathscr{H}$-sufficient
statistic with the simplest structure.
Therefore, we
define a minimal $\mathscr{H}$-sufficient statistic as a
$\mathscr{H}$-sufficient statistic which is a function of any other
$\mathscr{H}$-sufficient statistic.


\begin{definition}
[Minimal $\mathscr{H}$-sufficient statistics]
Let $X = \left[X_{1}, \ldots, X_{n}\right]$
and suppose $X_1, \ldots, X_n$
are random variables from an
unknown population $ \mathcal{P}=\left\{ P_{\theta}:\theta \in \Theta \right\}$.
Let $T(X)$ be a $\mathscr{H}$-sufficient statistic for $\mathfrak{a}$.
A statistic $T(X)$ is called a minimal
$\mathscr{H}$-sufficient statistic for $\mathfrak{a}$ if and only
if, for any other statistic $S(X)$ that is a $\mathscr{H}$-sufficient
for $\mathfrak{a}$, there exists a measurable function $\psi$ such that
$T(X) = \psi \left(S(X)\right)$ almost surely $\mathcal{P}$.
\end{definition}


\begin{theorem}
Let $X = \left[X_{1}, \ldots, X_{n}\right]$
and suppose $X_1, \ldots, X_n$
are random variables from an
unknown population $\mathcal{P}=\left\{ P_{\theta}:\theta \in \Theta \right\}$.
Let $T(X)$ be a complete sufficient statistic for $\mathcal{P}$ (or
$\theta$) such that $T(X)$, $T : \mathscr{X} \to \mathbb{R}$, has mean $\mathfrak{a}$.
Then any
$\mathscr{H}$-sufficient statistic for $\mathfrak{a}$ is a
sufficient statistic for $ \mathcal{P}$ (or $\theta$).
\end{theorem}

\begin{proof}
Let $H(X)$ be a $\mathscr{H}$-sufficient statistic for
$\mathfrak{a}$, then var $\left\{ E \left[ (T(X)\mid H(X) \right] \right\}$ $\leq$ var$\left[ T(X) \right]$.
Since $T(X)$ is a UMVUE, $T(X)=E \left[ T(X)\mid H(X) \right]$ almost surely $\mathcal{P}$.
So, there is a measurable function $g$ such that $T(X)=g\circ H(X)$ almost surely $\mathcal{P}$,
and thus $H(X)$ is a sufficient statistic.
\end{proof}

Thus, we can apply $\mathscr{H}$-sufficient statistics for
$\mathfrak{a}$ in case complete sufficient statistics do not exist.
Intuitively, a $\mathscr{H}$-sufficient statistic with the complete property will
be a minimal $\mathscr{H}$-sufficient statistic.
The following
theorem, a version of the main theorem (Bahadur's
theorem), see~\cite{Ba}, states an important property of minimal $\mathscr{H}$-sufficient statistics.


\begin{theorem}
\label{thm:5}
Let $X = \left[X_{1}, \ldots, X_{n}\right]$
and suppose $X_1, \ldots, X_n$
are random variables from an
unknown population $\mathcal{P}=\left\{ P_{\theta}:\theta \in \Theta \right\}$.
If $T(X)$, $T : \mathscr{X} \to \mathbb{R}$, is a complete
$\mathscr{H}$-sufficient statistic for $\mathfrak{a}$ then $T(X)$
is a minimal $\mathscr{H}$-sufficient statistic for $\mathfrak{a}$.
\end{theorem}

\begin{proof}
Let $S(X)$ be a $\mathscr{H}$-sufficient statistic for
$\mathfrak{a}$.
Then $T(X)= E\left[ T(X)\mid S(X) \right]$ almost surely
$\mathcal{P}$ since $T(X)$ is a UMVUE.
\end{proof}



We illustrate by an example that the complete
$\mathscr{H}$-sufficient statistic may not exist.


\begin{example}[Complete $\mathscr{H}$-sufficient
statistics may not exist]
\label{ex:3}
Let $X$ be a random variable with
$\mathcal{P}=\left\{\textup{Bin}(\theta,0.5) : \theta \in \{1,2,\ldots\}\right\}$.
Then $X$ is a $\mathscr{H}$-sufficient
statistic for $\theta$.
But a complete $\mathscr{H}$-sufficient
statistic for $\theta$ dose not exist.
Otherwise, for every $k \in \mathbb{R}$ and some $k_{0} \in\mathbb{R}$,
we would have $E\left[ 2X+k(-1)^{X+1}\mid g(X) \right] = 2X+k_{0}(-1)^{X+1}$
almost surely $\mathcal{P}$,
where $g(X)$ is assumed to be a complete $\mathscr{H}$-sufficient
statistic for $\theta$; but this is not true.
We also note that there does not exist any UMVUE for $\theta$.
\end{example}


\section{Some applications}

In this section, some examples are presented for which
Theorem~\ref{thm:1} and Theorem~\ref{thm:2} are \hbox{applicable.}




\subsection{When the minimal sufficient statistic is not complete}

Consider a case where UMVUE exists, but the minimal
sufficient statistics are not complete.
LST cannot be
used to obtain UMVUEs.
We illustrate through some examples that
we can find a UMVUE without having complete
sufficiency.
Therefore, some worries in the literature on the
inadequacy of the LST and RBT for obtaining UMVUEs can be removed,
and the seemingly unbeatable obstacles can be overcome by using $\mathscr{H}$-sufficient statistics.



\begin{example}[Example of~\cite{LS1}]
\label{ex:4}
Let $X$ be a discrete random variable from
$P_{\theta}$ with the probability mass function
$P_{\theta}(X=-1)=\theta$, $P_{\theta} (X=k)=(1-\theta)^2\theta^k$, $k=0,1,2,\ldots$,
where $\theta \in (0, 1)$ is unknown.
We note that $I_{\{0\}}(X)$ is a complete and minimal
$\mathscr{H}$-sufficient statistic for $(1-\theta)^2$
since, for
every $\theta \in (0, 1)$ and every $\alpha\in\mathbb{R}$, we have
$E_{\theta}\left[ I_{\{0\}}(X)+\alpha X\mid I_{\{0\}}(X)\right] =I_{\{0\}}(X)$
almost surely $P_{\theta}$.
Hence, by Theorem~\ref{thm:2}, $I_{\{0\}}(X)$ is a UMVUE for $(1-\theta)^2$
and thus $AI_{\{0\}}(X)+B$ is a UMVUE for $A(1-\theta)^2+B$.

For an alternative, note that, for every
$\theta \in (0, 1)$ and $\alpha\in\mathbb{R}$, we have
$E_{\theta} \left[ \alpha X\mid I_{\{0\}}(X)\right] =0$ almost surely $P_{\theta}$
and thus the same
result can be obtained by using Theorem~\ref{thm:6}.
\end{example}


So far, Examples~\ref{ex:1} and~\ref{ex:2} have shown usefulness of $\mathscr{H}$-sufficiency.
However, in both cases, the considered estimation problem is a rather esoteric one.
The following examples seem more reasonable.


\begin{example}
\label{ex:5}
Let $X_1, \ldots, X_n$
be independent and identical random variables from an
unknown population
$P_{\theta}$ with the probability density function
\begin{equation*}
f(x;\mu,\sigma)= \frac {x-\mu}{\sigma^{2}} \expe^{-\frac {x-\mu}{\sigma}}I_{(\mu,\infty)}(x),
\end{equation*}
where $\theta=( \mu,\sigma)\in \mathbb{R}\times \mathbb{R}^{+}$ is
an unknown parameter.

Suppose now that $\mu$ is known.
So, $\bar{X}$ is
a complete sufficient statistic for $\sigma$.
By using the RBT, we can see that $\bar{X}$ is a $\mathscr{H}$-sufficient
statistic for $\mu+2\sigma$
since $E_{\theta} \big[ \delta(X) \mid \bar{X} \big] = \bar{X}$
almost surely $P_{\theta}$ for every
$\delta(X) \in \mathcal{U}_{\mu+2\sigma}$.
We note that
$\bar{X}$ is a complete and minimal $\mathscr{H}$-sufficient
statistic for $\mu+2\sigma$.
Hence, from Theorem~\ref{thm:2},
$\bar{X}$ is a UMVUE for $\mu+2\sigma$.
Then any function of
$\bar{X}$ is a UMVUE.
\end{example}

\begin{example}
Let $X_1, \ldots, X_n$
be independent and identical random variables from an
unknown population
$P_{\theta}$ with the probability density function
\begin{equation*}
f(x;\mu,\sigma)=2 \frac {x-\mu}{\sigma^{2}} I_{(\mu,\mu+\sigma)}(x),
\end{equation*}
where $\theta = (\mu,\sigma)\in \mathbb{R}\times \mathbb{R}^{+}$ is
an unknown parameter.
By the same argument as in Example~\ref{ex:5},
we can see that $\max\left(X_1, \ldots, X_n\right)$ is a complete
and minimal $\mathscr{H}$-sufficient statistic for
$\mu+\frac {2n}{2n+1}\sigma$.
Hence, by Theorem~\ref{thm:2}, $\max\left(X_1, \ldots, X_n\right)$
is a UMVUE for $\mu+\frac {2n}{2n+1}\sigma$.
Then any function of $\max\left(X_1, \ldots, X_n\right)$ is a UMVUE.
\end{example}


\subsection{When a complete and sufficient statistic is not available}


Even though, there exist complete sufficient statistics in the
following examples, namely
$\max$ $\left[ 1, \max\left(X_1, \ldots, X_n\right) \right]$ and
$X$ $I_{\mathbb{N}\setminus \left\{ m, m+1\right\}}$ $(X)$,
we can apply Theorems~\ref{thm:1} and~\ref{thm:2} for obtaining their UMVUEs.



\begin{example}[Example of~\cite{Sh}]
Let $X_1, \ldots, X_n$
be independent and identical random variables from $P_{\theta}$, the uniform distribution on the
interval $(0, \theta)$ with $\Theta = [1, \infty)$.
Then $X_{(n)}$ is
not complete, although it is still sufficient for $\theta$.
Thus, the RBT and LST are not applicable.
We now illustrate how to
use Theorem~\ref{thm:1} to find a UMVUE of $\theta$.
Let $U\left(X_{(n)}\right)$ be an unbiased estimator of $0$ in
$\mathcal{U}_{0} \left(X_{(n)}\right)$.

We can show that $H \left(X_{(n)}\right) =I_{[0, 1]} \left(X_{(n)}\right) + \frac {n+1}{n}X_{(n)}I_{(1, \infty)}\left(X_{(n)}\right)$
is a complete and $\mathscr{H}$-sufficient statistic for $\theta$,
though we need only
\begin{equation*}
E_{\theta}\left[ I_{[0, 1]}\left(X_{(n)}\right) +
\frac {n+1}{n}X_{(n)}I_{(1, \infty)}\left(X_{(n)}\right) + U\left(X_{(n)}\right)
\;\middle|\; H\left(X_{n}\right)\right] =H\left(X_{(n)}\right)
\end{equation*}
almost surely $P_{\theta}$ for every $\theta\in \Theta$.
Hence, $I_{[0, 1]} \left(X_{(n)}\right) + \frac {n+1}{n}X_{(n)}I_{(1, \infty)}\left(X_{(n)}\right)$ is a UMVUE for $\theta$.
\end{example}



\begin{example}[Example of~\cite{St}]
Let $X$ be a random variable having the discrete uniform distribution
with the probability mass function given by
\begin{equation*}
P_{N}(x)=\begin{cases}
N^{-1}, & \textrm{ if } x=1,\ldots,N,
\\
0, & \text{ otherwise.}
\end{cases}
\end{equation*}
We have excluded the value $N = m$ for some fixed $m \geq 1$ from $\left\{P_{N} : N \geq 1\right\}$.
Let $P$ $=$ $\left\{P_{N} : N \geq 1, N \neq m \right\}$.
We can see that
\begin{equation*}
H(X)=\begin{cases}
2X-1, & \textrm{ if } X\neq m, X\neq m+1,
\\
2m, & \textrm{ if } X= m, X= m+1
\end{cases}
\end{equation*}
is a complete and
$\mathscr{H}$-sufficient statistic for $N$, and also it is a UMVUE for $N$,
which can be proved similarly to the above example.
\end{example}






\subsection{A note on the structure of UMVUE}

We now show that the structure of UMVUE depends on a
$\mathscr{H}$-sufficient statistic for $E \left( \textrm{UMVUE} \right)$.



\begin{theorem}
\label{thm:9}
Let $\mathcal{P} = \left\{ P_{\theta}:\theta \in \Theta \right\}$
be a family of distributions.
Suppose that there is a sufficient
statistic $S(X)$ for $\mathcal{P}$ and a $\mathscr{H}$-sufficient
statistic $H(X)$ for $\mathfrak{a}$.
For any function such as $\alpha \left(S(X)\right)$
which is a \textup{UMVUE} there
exists a function $\beta \left(H(X)\right)$ so that
$\alpha \left( S(X)\right)= \beta\left(H(X)\right)$ almost surely $\mathcal{P}$.
\end{theorem}

\begin{proof}
The proof is an easy consequence of Theorem~\ref{thm:1}, where
$E_{\theta}$ $\left[\textrm{UMVUE}\mid S(X)\right]$ and
$E_{\theta}$ $\left[\textrm{UMVUE}\mid H(X) \right]$ are UMVUEs.
The proof follows by the uniqueness of UMVUEs.
\end{proof}



\section{Conclusions}

Sufficient statistics are of central concern for statisticians.
They play a fundamental role in the theorems of Rao-Blackwell and
Lemann-Scheff\'e.
By Theorem~\ref{thm:1}, every sufficient statistic
is a $\mathscr{H}$-sufficient statistic.
The class of $\mathscr{H}$-sufficient statistics contains all of the sufficient
statistics and also some statistics that are not necessarily
sufficient.
So, the factorization theorem, and its corollaries,
should not hold generally for $\mathscr{H}$-sufficient statistics.
The concepts closest to
$\mathscr{H}$-sufficient statistics are those of ``partial
sufficient'' and ``sufficient subspace''.
But they are slightly different.


More research based on the concept of
$\mathscr{H}$-sufficiency are under investigation.
They are

\begin{itemize}

\item
Generalizing $\mathscr{H}$-sufficiency to multi-parameter cases.

\item
How to find $\mathscr{H}$-sufficient statistics.

\end{itemize}

\section*{Declaration of interests}
The authors do not work for, advise, own shares in, or receive funds from
any organization that could benefit from this article, and have declared no affiliations other than their research organizations.

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