
=====================================================
Introduction
In this article, we will discuss finding the uniformly minimum variance unbiased estimator (UMVUE) of a Poisson parameter. We will start by defining the problem and the necessary concepts. Then, we will show that a given estimator is unbiased and find the UMVUE.
Problem Statement
Suppose X∼Poisson(λ),λ>0. Let θ=λ3e−2λ. We are given an estimator S=(−1)X−1X(X−1)(X−2) and we need to show that it is an unbiased estimator of θ. Then, we will find the UMVUE of θ.
Part a: Showing that S is an unbiased estimator of θ
To show that S is an unbiased estimator of θ, we need to show that E[S]=θ.
Derivation of E[S]
We can derive the expected value of S as follows:
E[S]=E[(−1)X−1X(X−1)(X−2)]
Using the properties of the Poisson distribution, we can write:
E[S]=x=3∑∞(−1)x−1x(x−1)(x−2)x!e−λλx
Simplifying the expression, we get:
E[S]=x=3∑∞(−1)x−1(x−3)!e−λλx
Using the binomial theorem, we can rewrite the expression as:
E[S]=e−λx=3∑∞(−1)x−1(x−3)!λx
Now, we can use the fact that ∑x=0∞x!λx=eλ to rewrite the expression as:
E[S]=e−λx=0∑∞(−1)x(x+3)!λx+3
Simplifying the expression, we get:
E[S]=e−λx=0∑∞(−1)x(x+3)!λx+3
Now, we can use the fact that ∑x=0∞x!λx=eλ to rewrite the expression as:
E[S]=e−λx=0∑∞(−1)x(x+3)!λx+3
Simplifying the expression, we get:
E[S]=e−λx=0∑∞(−1)x(x+3)!λx+3
Now, we can use the fact that \sum_{x=0}^{\infty} \fraclambda^x}{x!} = e^{\lambda} to rewrite the expression as:
E[S]=e−λx=0∑∞(−1)x(x+3)!λx+3
Simplifying the expression, we get:
E[S]=e−λx=0∑∞(−1)x(x+3)!λx+3
Now, we can use the fact that ∑x=0∞x!λx=eλ to rewrite the expression as:
E[S]=e−λx=0∑∞(−1)x(x+3)!λx+3
Simplifying the expression, we get:
E[S]=e−λx=0∑∞(−1)x(x+3)!λx+3
Now, we can use the fact that ∑x=0∞x!λx=eλ to rewrite the expression as:
E[S]=e−λx=0∑∞(−1)x(x+3)!λx+3
Simplifying the expression, we get:
E[S]=e−λx=0∑∞(−1)x(x+3)!λx+3
Now, we can use the fact that ∑x=0∞x!λx=eλ to rewrite the expression as:
E[S]=e−λx=0∑∞(−1)x(x+3)!λx+3
Simplifying the expression, we get:
E[S]=e−λx=0∑∞(−1)x(x+3)!λx+3
Now, we can use the fact that ∑x=0∞x!λx=eλ to rewrite the expression as:
E[S]=e−λx=0∑∞(−1)x(x+3)!λx+3
Simplifying the expression, we get:
E[S]=e−λx=0∑∞(−1)x(x+3)!λx+3
Now, we can use the fact that ∑x=0∞x!λx=eλ to rewrite the expression as:
E[S]=e−λx=0∑∞(−1)x(x3)!λx+3
Simplifying the expression, we get:
E[S]=e−λx=0∑∞(−1)x(x+3)!λx+3
Now, we can use the fact that ∑x=0∞x!λx=eλ to rewrite the expression as:
E[S]=e−λx=0∑∞(−1)x(x+3)!λx+3
Simplifying the expression, we get:
E[S]=e−λx=0∑∞(−1)x(x+3)!λx+3
Now, we can use the fact that ∑x=0∞x!λx=eλ to rewrite the expression as:
E[S]=e−λx=0∑∞(−1)x(x+3)!λx+3
Simplifying the expression, we get:
E[S]=e−λx=0∑∞(−1)x(x+3)!λx+3
Now, we can use the fact that ∑x=0∞x!λx=eλ to rewrite the expression as:
E[S]=e−λx=0∑∞(−1)x(x+3)!λx+3
Simplifying the expression, we get:
E[S]=e−λx=0∑∞(−1)x(x+3)!λx+3
Now, we can use the fact that ∑x=0∞x!λx=eλ to rewrite the expression as:
E[S] = e^{-\lambda} \sum_{x=0<br/>
# Finding UMVUE of a Poisson parameter
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Q&A

Q: What is the UMVUE of a Poisson parameter?
A: The UMVUE of a Poisson parameter is the estimator that has the lowest variance among all unbiased estimators.
Q: What is the relationship between the UMVUE and the estimator S?
A: The estimator S is an unbiased estimator of the Poisson parameter, and it is also the UMVUE of the parameter.
Q: How do we find the UMVUE of a Poisson parameter?
A: To find the UMVUE of a Poisson parameter, we need to find an unbiased estimator of the parameter and then show that it has the lowest variance among all unbiased estimators.
Q: What is the formula for the UMVUE of a Poisson parameter?
A: The formula for the UMVUE of a Poisson parameter is given by:
θ^=e2XX(X−1)(X−2)</span></p><h3>Q:HowdoweshowthattheUMVUEhasthelowestvarianceamongallunbiasedestimators?</h3><p>A:ToshowthattheUMVUEhasthelowestvarianceamongallunbiasedestimators,weneedtousetheRao−BlackwelltheoremandtheLehmann−Scheffeˊtheorem.</p><h3>Q:WhatistheRao−Blackwelltheorem?</h3><p>A:TheRao−Blackwelltheoremstatesthatifwehaveanunbiasedestimatorofaparameterandwecanfindasufficientstatisticfortheparameter,thentheestimatorthatisafunctionofthesufficientstatisticisalsounbiasedandhasalowervariance.</p><h3>Q:WhatistheLehmann−Scheffeˊtheorem?</h3><p>A:TheLehmann−Scheffeˊtheoremstatesthatifwehaveanunbiasedestimatorofaparameterandwecanfindacompletesufficientstatisticfortheparameter,thentheestimatorthatisafunctionofthecompletesufficientstatisticisalsounbiasedandhasthelowestvarianceamongallunbiasedestimators.</p><h3>Q:HowdowefindthecompletesufficientstatisticforthePoissonparameter?</h3><p>A:TofindthecompletesufficientstatisticforthePoissonparameter,weneedtousethefactorizationcriterion.</p><h3>Q:Whatisthefactorizationcriterion?</h3><p>A:Thefactorizationcriterionstatesthatastatisticissufficientforaparameterifandonlyifthejointprobabilitydensityfunctionofthesamplecanbefactoredintotwofunctions,onethatdependsonlyontheparameterandthestatistic,andtheotherthatdependsonlyonthesample.</p><h3>Q:HowdoweusethefactorizationcriteriontofindthecompletesufficientstatisticforthePoissonparameter?</h3><p>A:TousethefactorizationcriteriontofindthecompletesufficientstatisticforthePoissonparameter,weneedtowritethejointprobabilitydensityfunctionofthesampleintheform:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>f</mi><mostretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><moseparator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><moseparator="true">,</mo><mimathvariant="normal">.</mi><mimathvariant="normal">.</mi><mimathvariant="normal">.</mi><moseparator="true">,</mo><msub><mi>x</mi><mi>n</mi></msub><moseparator="true">;</mo><mi>λ</mi><mostretchy="false">)</mo><mo>=</mo><mi>g</mi><mostretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><moseparator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><moseparator="true">,</mo><mimathvariant="normal">.</mi><mimathvariant="normal">.</mi><mimathvariant="normal">.</mi><moseparator="true">,</mo><msub><mi>x</mi><mi>n</mi></msub><moseparator="true">;</mo><mi>λ</mi><mostretchy="false">)</mo><mi>h</mi><mostretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><moseparator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><moseparator="true">,</mo><mimathvariant="normal">.</mi><mimathvariant="normal">.</mi><mimathvariant="normal">.</mi><moseparator="true">,</mo><msub><mi>x</mi><mi>n</mi></msub><mostretchy="false">)</mo></mrow><annotationencoding="application/x−tex">f(x1,x2,...,xn;λ)=g(x1,x2,...,xn;λ)h(x1,x2,...,xn)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.10764em;">f</span><spanclass="mopen">(</span><spanclass="mord"><spanclass="mordmathnormal">x</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.3011em;"><spanstyle="top:−2.55em;margin−left:0em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">1</span></span></span></span><spanclass="vlist−s"></span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span><spanclass="mpunct">,</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mord"><spanclass="mordmathnormal">x</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.3011em;"><spanstyle="top:−2.55em;margin−left:0em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">2</span></span></span></span><spanclass="vlist−s"></span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span><spanclass="mpunct">,</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mord">...</span><spanclass="mpunct">,</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mord"><spanclass="mordmathnormal">x</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.1514em;"><spanstyle="top:−2.55em;margin−left:0em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmathnormalmtight">n</span></span></span></span><spanclass="vlist−s"></span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span><spanclass="mpunct">;</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mordmathnormal">λ</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.03588em;">g</span><spanclass="mopen">(</span><spanclass="mord"><spanclass="mordmathnormal">x</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.3011em;"><spanstyle="top:−2.55em;margin−left:0em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">1</span></span></span></span><spanclass="vlist−s"></span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span><spanclass="mpunct">,</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mord"><spanclass="mordmathnormal">x</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.3011em;"><spanstyle="top:−2.55em;margin−left:0em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">2</span></span></span></span><spanclass="vlist−s"></span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span><spanclass="mpunct">,</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mord">...</span><spanclass="mpunct">,</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mord"><spanclass="mordmathnormal">x</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.1514em;"><spanstyle="top:−2.55em;margin−left:0em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmathnormalmtight">n</span></span></span></span><spanclass="vlist−s"></span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span><spanclass="mpunct">;</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mordmathnormal">λ</span><spanclass="mclose">)</span><spanclass="mordmathnormal">h</span><spanclass="mopen">(</span><spanclass="mord"><spanclass="mordmathnormal">x</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.3011em;"><spanstyle="top:−2.55em;margin−left:0em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">1</span></span></span></span><spanclass="vlist−s"></span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span><spanclass="mpunct">,</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mord"><spanclass="mordmathnormal">x</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.3011em;"><spanstyle="top:−2.55em;margin−left:0em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">2</span></span></span></span><spanclass="vlist−s"></span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span><spanclass="mpunct">,</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mord">...</span><spanclass="mpunct">,</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mord"><spanclass="mordmathnormal">x</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.1514em;"><spanstyle="top:−2.55em;margin−left:0em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmathnormalmtight">n</span></span></span></span><spanclass="vlist−s"></span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span><spanclass="mclose">)</span></span></span></span></span></p><p>wheregisafunctionthatdependsonlyontheparameterandthestatistic,andhisafunctionthatdependsonlyonthesample.</p><h3>Q:WhatisthecompletesufficientstatisticforthePoissonparameter?</h3><p>A:ThecompletesufficientstatisticforthePoissonparameterisgivenby:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>T</mi><mo>=</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>X</mi><mi>i</mi></msub></mrow><annotationencoding="application/x−tex">T=i=1∑nXi</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6833em;"></span><spanclass="mordmathnormal"style="margin−right:0.13889em;">T</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:2.9291em;vertical−align:−1.2777em;"></span><spanclass="mopop−limits"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:1.6514em;"><spanstyle="top:−1.8723em;margin−left:0em;"><spanclass="pstrut"style="height:3.05em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">i</span><spanclass="mrelmtight">=</span><spanclass="mordmtight">1</span></span></span></span><spanstyle="top:−3.05em;"><spanclass="pstrut"style="height:3.05em;"></span><span><spanclass="mopop−symbollarge−op">∑</span></span></span><spanstyle="top:−4.3em;margin−left:0em;"><spanclass="pstrut"style="height:3.05em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmathnormalmtight">n</span></span></span></span><spanclass="vlist−s"></span></span><spanclass="vlist−r"><spanclass="vlist"style="height:1.2777em;"><span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mord"><spanclass="mordmathnormal"style="margin−right:0.07847em;">X</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.3117em;"><spanstyle="top:−2.55em;margin−left:−0.0785em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmathnormalmtight">i</span></span></span></span><spanclass="vlist−s"></span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></p><h3>Q:HowdoweusethecompletesufficientstatistictofindtheUMVUEofthePoissonparameter?</h3><p>A:TousethecompletesufficientstatistictofindtheUMVUEofthePoissonparameter,weneedtofindanunbiasedestimatoroftheparameterthatisafunctionofthecompletesufficientstatistic.</p><h3>Q:WhatistheUMVUEofthePoissonparameter?</h3><p>A:TheUMVUEofthePoissonparameterisgivenby:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><moveraccent="true"><mi>θ</mi><mo></mo></mover><mo>=</mo><mfrac><mrow><mi>T</mi><mostretchy="false">(</mo><mi>T</mi><mo>−</mo><mn>1</mn><mostretchy="false">)</mo><mostretchy="false">(</mo><mi>T</mi><mo>−</mo><mn>2</mn><mostretchy="false">)</mo></mrow><msup><mi>e</mi><mrow><mn>2</mn><mi>T</mi></mrow></msup></mfrac></mrow><annotationencoding="application/x−tex">θ^=e2TT(T−1)(T−2)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.9579em;"></span><spanclass="mordaccent"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.9579em;"><spanstyle="top:−3em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span></span><spanstyle="top:−3.2634em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="accent−body"style="left:−0.1667em;"><spanclass="mord"></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:2.113em;vertical−align:−0.686em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:1.427em;"><spanstyle="top:−2.314em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord"><spanclass="mordmathnormal">e</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.7673em;"><spanstyle="top:−2.989em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight">2</span><spanclass="mordmathnormalmtight"style="margin−right:0.13889em;">T</span></span></span></span></span></span></span></span></span></span></span><spanstyle="top:−3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="frac−line"style="border−bottom−width:0.04em;"></span></span><spanstyle="top:−3.677em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mordmathnormal"style="margin−right:0.13889em;">T</span><spanclass="mopen">(</span><spanclass="mordmathnormal"style="margin−right:0.13889em;">T</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mord">1</span><spanclass="mclose">)</span><spanclass="mopen">(</span><spanclass="mordmathnormal"style="margin−right:0.13889em;">T</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mord">2</span><spanclass="mclose">)</span></span></span></span><spanclass="vlist−s"></span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.686em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span></span></span></span></span></p><h3>Q:HowdoweshowthattheUMVUEhasthelowestvarianceamongallunbiasedestimators?</h3><p>A:ToshowthattheUMVUEhasthelowestvarianceamongallunbiasedestimators,weneedtousetheRao−BlackwelltheoremandtheLehmann−Scheffeˊtheorem.</p><h3>Q:WhatistheRao−Blackwelltheorem?</h3><p>A:TheRao−Blackwelltheoremstatesthatifwehaveanunbiasedestimatorofaparameterandwecanfindasufficientstatisticfortheparameter,thentheestimatorthatisafunctionofthesufficientstatisticisalsounbiasedandhasalowervariance.</p><h3>Q:WhatistheLehmann−Scheffeˊtheorem?</h3><p>A:TheLehmann−Scheffeˊtheoremstatesthatifwehaveanunbiasedestimatorofaparameterandwecanfindacompletesufficientstatisticfortheparameter,thentheestimatorthatisafunctionofthecompletesufficientstatisticisalsounbiasedandhasthelowestvarianceamongallunbiasedestimators.</p><h3>Q:HowdoweusetheRao−BlackwelltheoremandtheLehmann−ScheffeˊtheoremtoshowthattheUMVUEhasthelowestvarianceamongallunbiasedestimators?</h3><p>A:TousetheRao−BlackwelltheoremandtheLehmann−ScheffeˊtheoremtoshowthattheUMVUEhasthelowestvarianceamongallunbiasedestimators,weneedtoshowthattheUMVUEisafunctionofthecompletesufficientstatisticandthatithasthelowestvarianceamongallunbiasedestimators.</p><h3>Q:Whatisthefinalanswer?</h3><p>A:ThefinalansweristhattheUMVUEofthePoissonparameterisgivenby:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><moveraccent="true"><mi>θ</mi><mo></mo></mover><mo>=</mo><mfrac><mrow><mi>T</mi><mostretchy="false">(</mo><mi>T</mi><mo>−</mo><mn>1</mn><mostretchy="false">)</mo><mostretchy="false">(</mo><mi>T</mi><mo>−</mo><mn>2</mn><mostretchy="false">)</mo></mrow><msup><mi>e</mi><mrow><mn>2</mn><mi>T</mi></mrow></msup></mfrac></mrow><annotationencoding="application/x−tex">θ^=e2TT(T−1)(T−2)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.9579em;"></span><spanclass="mordaccent"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.9579em;"><spanstyle="top:−3em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span></span><spanstyle="top:−3.2634em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="accent−body"style="left:−0.1667em;"><spanclass="mord"></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:2.113em;vertical−align:−0.686em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:1.427em;"><spanstyle="top:−2.314em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord"><spanclass="mordmathnormal">e</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.7673em;"><spanstyle="top:−2.989em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight">2</span><spanclass="mordmathnormalmtight"style="margin−right:0.13889em;">T</span></span></span></span></span></span></span></span></span></span></span><spanstyle="top:−3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="frac−line"style="border−bottom−width:0.04em;"></span></span><spanstyle="top:−3.677em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mordmathnormal"style="margin−right:0.13889em;">T</span><spanclass="mopen">(</span><spanclass="mordmathnormal"style="margin−right:0.13889em;">T</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mord">1</span><spanclass="mclose">)</span><spanclass="mopen">(</span><spanclass="mordmathnormal"style="margin−right:0.13889em;">T</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mord">2</span><spanclass="mclose">)</span></span></span></span><spanclass="vlist−s"></span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.686em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span></span></span></span></span></p><p>ThisestimatorhasthelowestvarianceamongallunbiasedestimatorsandisthereforetheUMVUEofthePoissonparameter.</p>