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\def\makelabel##1{\upshape ##1}}% } {\end{list}\end{raggedright}} \newenvironment{specHead}[2]% {\vspace{20pt}\hrule\vspace{10pt}% \phantomsection\label{#1}\markright{#2}% \pdfbookmark[2]{#2}{#1}% \hspace{-0.75in}{\bfseries\fontsize{16pt}{18pt}\selectfont#2}% }{} \def\TheFullDate{1970-01-01 (revised: 01 January 1970)} \def\TheID{\makeatother } \def\TheDate{1970-01-01} \title{On Baysian Estimation of Loss of Estimators of Unknown Parameter of Binomial Distribution By Randhir Singh} \author{}\makeatletter \makeatletter \newcommand*{\cleartoleftpage}{% \clearpage \if@twoside \ifodd\c@page \hbox{}\newpage \if@twocolumn \hbox{}\newpage \fi \fi \fi } \makeatother \makeatletter \thispagestyle{empty} \markright{\@title}\markboth{\@title}{\@author} \renewcommand\small{\@setfontsize\small{9pt}{11pt}\abovedisplayskip 8.5\p@ plus3\p@ minus4\p@ \belowdisplayskip \abovedisplayskip \abovedisplayshortskip \z@ plus2\p@ \belowdisplayshortskip 4\p@ plus2\p@ minus2\p@ \def\@listi{\leftmargin\leftmargini 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\textcolor{GJMediumGrey}{\rule{\textwidth}{2pt}} \vskip16pt \textcolor{GJBlue}{\large\bfseries\abstractname\space} }{% \vskip8pt \textcolor{GJMediumGrey}{\rule{\textwidth}{2pt}} \vskip16pt } \usepackage[absolute,overlay]{textpos} \makeatother \usepackage{lineno} \linenumbers \begin{document} \author[1]{Randhir Singh} \renewcommand\Authands{ and } \date{\small \em Received: 1 January 1970 Accepted: 1 January 1970 Published: 1 January 1970} \maketitle \begin{abstract} This paper aims at the Bayesian estimation for the loss and risk functions of the unknown parameter of the binomial distribution under the loss function which is different from that given by Rukhin(1988). The estimation involves beta distribution, a natural conjugate prior density function for the unknown parameter. Estimators obtained are conservatively biased and have finite frequentist risk. \end{abstract} \keywords{Bayes Estimator, Loss Function, Risk Function, Binomial Distribution.} \begin{textblock*}{18cm}(1cm,1cm) % {block width} (coords) \textcolor{GJBlue}{\LARGE Global Journals \LaTeX\ JournalKaleidoscope\texttrademark} \end{textblock*} \begin{textblock*}{18cm}(1.4cm,1.5cm) % {block width} (coords) \textcolor{GJBlue}{\footnotesize \\ Artificial Intelligence formulated this projection for compatibility purposes from the original article published at Global Journals. However, this technology is currently in beta. \emph{Therefore, kindly ignore odd layouts, missed formulae, text, tables, or figures.}} \end{textblock*} \begin{textblock*}{10cm}(1.05cm,3cm) {{\textit{CrossRef DOI of original article:}} \underline{}} \end{textblock*}\let\tabcellsep& \section[{Issue ersion I}]{Issue ersion I}\par V IV ( F ) \hyperref[b4]{Rukhin(1988)} introduced a loss function given by, L(?, ?, ?) = w(?, ?)? ? 1 2 + ? 1 2\par (1.1)\par Where,? is an estimator of the loss function w(?, ?) ,which is non-negative. Guobing(2016) used this loss function and derived estimates of the loss and risk function of the parameter of Maxwell's distribution. \hyperref[b3]{Singh (2021)} took various forms of w(?, ?) and derived estimates of the loss and risk function of the parameter of a continuous distribution which gives Half-normal distribution,Rayleigh distribution and Maxwell's distribution as particular cases. \hyperref[b4]{Rukhin(1988)} considered the Bayesian estimation of the unknown parameter ? of the binomial distribution by takingw(?, ?) = (? ? ?) 2 (1.2)\par In this paper,Bayes estimate of the unknown parameter ? of the binomial distribution has been obtained by replacing w(?, ?) by w 1 (?, ?) given byw 1 (?, ?) = h(?)(? ? ?) 2 (1.3)\par Where,h(?) = 1 \{?(1 ? ?)\} (1.4) \section[{Notes}]{Notes}\par Summary-This paper aims at the Bayesian estimation for the loss and risk functions of the unknown parameter of the binomial distribution under the loss function which is different from that given by \hyperref[b4]{Rukhin(1988)}. The estimation involves beta distribution, a natural conjugate prior density function for the unknown parameter. Estimators obtained are conservatively biased and have finite frequentist risk. Let the random variable X follows binomial distribution with parameters n and ?.Where ? is unknown satisfying 0 ? ? ? 1.The prior p.d.f of ?,denoted by ? 1 (?) is as follows: \section[{Global}]{Global}? 1 (?) = ? ??1 (1??) ??1 B(?,?) if ? ? 0,? ? 0,0 < ? < 1 0 Otherwise (2.1)\par Under the assumption of prior probability density function (p.d.f.) for ? as above,Bayes estimates of ? derived by \hyperref[b4]{Rukhin (1988)} were as follows:For ? ? 0, ? ? 0 ? B (X) = (X + ?) (n + ? + ?) (2.2) ? B (X) = (X + ?)(n + ? ? X) (n + ? + ?) 2 (n + ? + ? + 1) (2.3)\par and for ? = 0, ? = 0? 0 (X) = X n (2.4) ? 0 (X) = X(n ? X) n 2 (n + 1) (2.5)\par It was shown thatE ? L(?, ? 0 , ? 0 ) = ? (2.6)\par Under,w 1 (?, ?) as above, the corresponding Bayes estimate is given by, For ? ? 0, ? ? 0? 1B (X) = E\{?h(?)/X\} E\{h(?)/X\} (2.7) Or, ? 1B (X) = (X + ? ? 1) A ? 2 (2.8)\par On simplification,provided,A = n + ? + ? > 2 and,? 1B (X) = E\{?h(?)/X\} ? \{? 1B (X)\} 2 E\{h(?)/X\} (2.9) \section[{Notes}]{Notes}\par Estimation of Loss and Risk of the Parameter of Binomial Distribution II. \section[{Issue ersion I V IV ( F )}]{Issue ersion I V IV ( F )}\par On Baysian Estimation of Loss of Estimators of Unknown Parameter of Binomial DistributionE ? L(?, ? 1B , ? 1B ) = E ? [h(?)(? ? (X + ? ? 1)(A ? 2) ?1 ) 2 ](A ? 2) 1/2 + (A ? 2) ?1/2 (2.11)\par Or,E ? L(?, ? 1B , ? 1B ) = [n+h(?)(1 ? ? + ?(? + ? ? 2)) 2 ](A?2) ?3/2 +(A?2) ?1/2 < ? (2.12) In this case, R(?, ? 1B ) = E ? \{h(?)(? ? ? 1B )\} 2 (2.13) Or, R(?, ? 1B ) = [n + h(?)\{1 ? ? + ?(? + ? ? 2)\} 2 ](A ? 2) ?2 (2.14)\par As mentioned by \hyperref[b2]{Keifer (1977)},an estimator ?(X)is said to be conservatively biased if,E ? \{?(X)\} ? R(?, ?) = E ? \{w(?, ?)\} (2.15)\par In the light of this condition,? 0 (X) as given by \hyperref[b4]{Rukhin (1988)} is not conservatively biased. In this case,E ? \{? 1B (X)\} = 1 A ? 2 (2.16)\par Let ? 0B (X) and ? 0B (X)be values of ? 1B (X) and ? 1B (X) ,respectively when,? = ? = 0.If possible let ,E ? \{? 0B (X)\} ? R(?, ? 0B (2.17) which holds if, ?2? 2 + 2? ? 1 ? 0 (2.18)\par which is a contradiction,since 0 < ? < 1 and maximum value of ?2? 2 + 2? ? 1 is? 1 2 which corresponds to ? = 1 2 .Moreover,?2? 2 + 2? ? 1 = ?1 for ? = 1 and ? = 0 Thus,? 0B (X) is not conservatively biased.\par When ? = ? = 1,we have,E ? \{? 1B (X)\} = R(?, ? 1B ) = 1 n (2.19) ) Or, ? 1B (X) = 1 A ? 2 (2.10) on simplification,provided,A = n + ? + ? > 2.\par We,see that, in this case ? 1B (X) does not depend upon X and is function of n,? and ? \section[{Notes}]{Notes}Issue ersion I V IV ( F )\par On Baysian Estimation of Loss of Estimators of Unknown Parameter of Binomial Distribution g(?) is a monotonically increasing function of ? over the set S = (0, 1) ? \{0.5\}.Hence, ? 1B (X) as above,presents a valid 'frequentist report' as mentioned by \hyperref[b0]{Berger(1985)}.\par The results are summerized in the following: THEOREM.Let (? 1B , ? 1B ) be Bayes estimators of the unknown parameter ? of the binomial distribution under the loss function L(?, ?, ?) =1 \{?(1??)\} (? ??) 2 ? ? 1 2 +? 1 2\par and beta prior density with known parameters ? and ?.Then,the frequentist risk E ? L(?, ? 1B , ? 1B ) is finite for all values of ? and ? provided 0 < ? < 1.For ? = ? = 0, ? 1B (X) is not conservatively biased. The estimator ? 1B (X) is conservatively biased for? = ? = 1 and for ? = ? > 1 satisfying ? ? 1 + 2?(1??) (2??1) 2 ,? = 0.5.However, if ? = ? > 1, ? = 0.5, ? 1B (X) is also conservatively biased.\par When,? = ? > 1, ? = 0. \begin{figure}[htbp] \noindent\textbf{}\includegraphics[]{image-2.png} \caption{\label{fig_0}}\end{figure} \begin{figure}[htbp] \noindent\textbf{} \par \begin{longtable}{P{0.17149122807017542\textwidth}P{0.3578947368421052\textwidth}P{0.14912280701754385\textwidth}P{0.13421052631578947\textwidth}P{0.03728070175438596\textwidth}} 5,\tabcellsep \tabcellsep \tabcellsep \\ which holds if\tabcellsep \multicolumn{2}{l}{E ? \{? 1B (X)\} ? R(?, ? 1B )}\tabcellsep (2.21)\tabcellsep Notes\\ \tabcellsep \multicolumn{2}{l}{? ? 1 + g(?)}\tabcellsep (2.22)\\ .Where,\tabcellsep \tabcellsep \tabcellsep \\ \tabcellsep g(?) =\tabcellsep 2?(1 ? ?) (2? ? 1) 2\tabcellsep (2.23)\end{longtable} \par \caption{\label{tab_0}}\end{figure} \footnote{© 2022 Global Journals} \backmatter \begin{bibitemlist}{1} \bibitem[Keifer ()]{b2}\label{b2} ‘Conditional confidence statements and con fidence estimators’. J Keifer . \textit{J. Am. Statist. Assoc} 1977. 72 p. . \bibitem[Rukhin ()]{b4}\label{b4} ‘Estimating the loss of estimators of a binomial parameter’. A L Rukhin . \textit{Biometrika} 1988. 75 p. . \bibitem[Fan ()]{b1}\label{b1} ‘Estimation of the Loss and Risk Functions of parameter of Maxwell's distribution’. Guobing Fan . \xref{http://dx.doi.org/10.11648/j.sjams.20160404.12}{10.11648/j.sjams.20160404.12}. \textit{Science Journal of Applied Mathematics and Statistics} 2016. 2016. 4 (4) p. . \bibitem[Singh ()]{b3}\label{b3} ‘On Bayesian Estimation of Loss and Risk Functions’. Randhir Singh . \xref{http://dx.doi.org/10.11648/j.sjams.20210903.11}{10.11648/j.sjams.20210903.11}. \textit{Science Journal of Applied Mathematics and Statistics} 2021. 2021. 9 (3) p. . \bibitem[Berger ()]{b0}\label{b0} ‘The frequentist viewpoint and conditioning’. J Berger . \textit{Proceedings of the Berkley Conference in Honor of Jerry Neyman and Jack Keifer}, L Lecam, R Olshen (ed.) (the Berkley Conference in Honor of Jerry Neyman and Jack KeiferBelmont, Cailf) 1985. Wadsworth. p. . \end{bibitemlist} \end{document}