@Fawkes4494d3 kindly have a look at this problem. Definition: First-Order Ancillary Statistic. If the range of X is Rk, then there exists a minimal sufcient statistic. You are correct. Is it possible to make a high-side PNP switch circuit active-low with less than 3 BJTs? With new ARBURG injection machines, we can guarantee top-notch plastic manufacturing. This is equivalent to the requirement that the conditional distribution, given $ X= x $, Check for more Examples in complete sufficient statistics : https://youtu.be/pW0TkAzxP4gLearn the correct way to use the definition of complete sufficient st. is said to be sufficient for $ \{ P _ \theta \} $ : 95%) by the [Q] Are there terms to describe a data point that is [Q] Is there any test equivalent to two-way ANOVA that [Q] Why isn't there a significance level of .02, .03, or [Q] Why is it more statistically accurate to round down [Q] / [D] People's silly ideas on statistics - how to [Q] If you had 3-5 years to prep for a PhD Stats, what [Q] Why do Errors Not Need to be Normal in Logistic [Q] In Bayesian statistics, what does the posterior [Q] I'm trying to fit a "Buy Till You Die" LTV model [C] At the job interview, is it bad to say you're looking [D] Which is the best book to Master statistics for a Press J to jump to the feed. Light bulb as limit, to what is current limited to? Statistics are defined as a field that involves tabulating, depicting, and describing data sets. A simple instance is X U ( , + 1) where R. It is not difficult to show X is a minimal sufficient statistic for . In the case where there exists at least one minimal sufficient statistic, a statistic which is sufficient and boundedly complete, is necessarily minimal sufficient. https://stats.stackexchange.com/questions/187869/are-complete-statistics-always-sufficient._3K2ydhts9_ES4s9UpcXqBi{display:block;padding:0 16px;width:100%} 2022, OReilly Media, Inc. All trademarks and registered trademarks appearing on oreilly.com are the property of their respective owners. in relation (*) has the meaning of the probability of the elementary event $ \{ \omega \} $. Connect and share knowledge within a single location that is structured and easy to search. on $ \mathbf R ^ {d} $. Can lead-acid batteries be stored by removing the liquid from them? Is this minimal sufficient complete or not? Example 6.2.15. ErnLTPiI wB!]NW2\ s[/'\EM9G5W4xQ %>{ `:q` sg=:Cp'`!\B*DF(Kz:R Zn}{+g3+:\s4 Does the following in the condition, I dont give an answer because im not sure, but it could be that you are right, and theres a mistake in Casella Berger. Although T = [X (1), X (N)] T is minimal sufficient for of the uniform distribution in Example 9.12, and together they contain as much information about as the original samples, it turns out they also contain information that is not relevant for estimating . A statistic Tis complete for XP 2Pif no non-constant function of T is rst-order ancillary. This change repeals the individual cosmetology law (Occupations Code 1602) and expands the " Barbering and Cosmetology " law in Occupations Code 1603. Statistical Inference-Notes-Part4-Lehmann Scheffe Theorem If this minimal sufficient statistic is not complete then there exists a function $h(T)$ of the minimal sufficient statistic such that. Since complete sufcient statistics are particularly effective in reducing the data, it is not surprising that a complete sufcient statistic is always minimal. MIT, Apache, GNU, etc.) E.g., let $ X _ {1} \dots X _ {n} $ For the uniform pdf , observe that the statistic V = X(N)X(1) is first-order ancillary because its mean does not depend on : Get Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications now with the OReilly learning platform. De nition 4. Reddit and its partners use cookies and similar technologies to provide you with a better experience. \sum _ {i = 1 } ^ { n } ( x _ {i} - \mu ) ^ {2} \right ] = Use MathJax to format equations. Y.1:2h$>RYzJj01PX7cBao TSqC\A%+!pKL'.X^0hMXJ@#/R} ^2MD/:^~dRL |q]!G?]A%);R+E[C.<2,GE#:`Q1oA4 j"#L/pv )a 3 ;8=JUl}K@UI+RB;`pU=)tzj{1d 3gxL|N,Wo`a. $$. p _ {\mu , \sigma ^ {2} } ( x _ {1} \dots x _ {n} ) = @keyframes _1tIZttmhLdrIGrB-6VvZcT{0%{opacity:0}to{opacity:1}}._3uK2I0hi3JFTKnMUFHD2Pd,.HQ2VJViRjokXpRbJzPvvc{--infoTextTooltip-overflow-left:0px;font-size:12px;font-weight:500;line-height:16px;padding:3px 9px;position:absolute;border-radius:4px;margin-top:-6px;background:#000;color:#fff;animation:_1tIZttmhLdrIGrB-6VvZcT .5s step-end;z-index:100;white-space:pre-wrap}._3uK2I0hi3JFTKnMUFHD2Pd:after,.HQ2VJViRjokXpRbJzPvvc:after{content:"";position:absolute;top:100%;left:calc(50% - 4px - var(--infoTextTooltip-overflow-left));width:0;height:0;border-top:3px solid #000;border-left:4px solid transparent;border-right:4px solid transparent}._3uK2I0hi3JFTKnMUFHD2Pd{margin-top:6px}._3uK2I0hi3JFTKnMUFHD2Pd:after{border-bottom:3px solid #000;border-top:none;bottom:100%;top:auto} and the value zero with probability $ 1 - \nu $( Solved: Let X ~ Ber(n1; p), Y ~ Ber(n2; p^2), where X and Y are independent. rev2022.11.7.43014. Let X ~ Ber(n1; p), Y ~ Ber(n2; p^2), where X and Y are independent yields exhaustive material for statistical inferences about the parameter $ \theta $, Definition: First-Order Ancillary Statistic Statistic V is first-order ancillary if is independent of . ) ,\ \theta \in \Theta , Homeuse 308nm Excimer System for Vitiligo and Psoriasis Quick Detail 308nm Excimer Light Machine deliver only monochromatic 308nm, which has been widely regarded as the gold standard for treatment vitiligo and psoriasis with high efficiency, short.Laser treatments must be performed by a physician, a PA under supervision of a physician or a NP working under a protocol signed by a physician. and unknown variance $ \sigma ^ {2} $. $$. PDF Lecture 23: Minimal sufciency - University of Wisconsin-Madison Get full access to Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications and 60K+ other titles, with free 10-day trial of O'Reilly. SOME ASPECTS OF EDUCATION IN AMERICA (W. Gray, M.A., Wanganui). The pretreatment NLR (2.2/ Introduction Pancreatic cancer has the poorest prognosis of any major malignancy (5-year survival rate, 6%) (1). ._9ZuQyDXhFth1qKJF4KNm8{padding:12px 12px 40px}._2iNJX36LR2tMHx_unzEkVM,._1JmnMJclrTwTPpAip5U_Hm{font-size:16px;font-weight:500;line-height:20px;color:var(--newCommunityTheme-bodyText);margin-bottom:40px;padding-top:4px;text-align:left;margin-right:28px}._2iNJX36LR2tMHx_unzEkVM{-ms-flex-align:center;align-items:center;display:-ms-flexbox;display:flex}._2iNJX36LR2tMHx_unzEkVM ._24r4TaTKqNLBGA3VgswFrN{margin-left:6px}._306gA2lxjCHX44ssikUp3O{margin-bottom:32px}._1Omf6afKRpv3RKNCWjIyJ4{font-size:18px;font-weight:500;line-height:22px;border-bottom:2px solid var(--newCommunityTheme-line);color:var(--newCommunityTheme-bodyText);margin-bottom:8px;padding-bottom:8px}._2Ss7VGMX-UPKt9NhFRtgTz{margin-bottom:24px}._3vWu4F9B4X4Yc-Gm86-FMP{border-bottom:1px solid var(--newCommunityTheme-line);margin-bottom:8px;padding-bottom:2px}._3vWu4F9B4X4Yc-Gm86-FMP:last-of-type{border-bottom-width:0}._2qAEe8HGjtHsuKsHqNCa9u{font-size:14px;font-weight:500;line-height:18px;color:var(--newCommunityTheme-bodyText);padding-bottom:8px;padding-top:8px}.c5RWd-O3CYE-XSLdTyjtI{padding:8px 0}._3whORKuQps-WQpSceAyHuF{font-size:12px;font-weight:400;line-height:16px;color:var(--newCommunityTheme-actionIcon);margin-bottom:8px}._1Qk-ka6_CJz1fU3OUfeznu{margin-bottom:8px}._3ds8Wk2l32hr3hLddQshhG{font-weight:500}._1h0r6vtgOzgWtu-GNBO6Yb,._3ds8Wk2l32hr3hLddQshhG{font-size:12px;line-height:16px;color:var(--newCommunityTheme-actionIcon)}._1h0r6vtgOzgWtu-GNBO6Yb{font-weight:400}.horIoLCod23xkzt7MmTpC{font-size:12px;font-weight:400;line-height:16px;color:#ea0027}._33Iw1wpNZ-uhC05tWsB9xi{margin-top:24px}._2M7LQbQxH40ingJ9h9RslL{font-size:12px;font-weight:400;line-height:16px;color:var(--newCommunityTheme-actionIcon);margin-bottom:8px} \widehat \nu = ,Tk are jointly sucient statistics if the statistician who Removing repeating rows and columns from 2d array. Is a minimal sufficient statistic also a complete statistic Steady state heat equation/Laplace's equation special geometry. which is independent of $ \theta $. Laws and - tihn.mybiwag.de with respect to the measure $ \mu $. Scilit | Article - Pre-treatment neutrophil to lymphocyte ratio as a For tumble dryers: T1 and T1 Classic. This irrelevant information is called ancillary, and any component of a minimal sufficient statistic that . of its expectation $ e ( \theta ) = {\mathsf E} _ \theta X $. Complete Sufficient Statistic - an overview | ScienceDirect Topics ._3bX7W3J0lU78fp7cayvNxx{max-width:208px;text-align:center} Can FOSS software licenses (e.g. A complete and minimal sufficient statistic is a statistic that contains all the information in the sample about the parameter (s) of interest, and no additional information. $$, $$ The knowledge of the sufficient statistic $ X $ can be expressed in terms of $ \widehat \mu $ Sufficient statistic - Encyclopedia of Mathematics And$ h(T) $ is not identically zero. be a sequence of independent random variables which assume the value one with an unknown probability $ \nu $ \frac \mu {\sigma ^ {2} 3 0 obj << almost surely with respect to $ P _ \theta $, $$, $$ \frac{1}{2 \sigma ^ {2} } We welcome all researchers, students, professionals, and enthusiasts looking to be a part of an online statistics community. of the normal distribution. At present, surgical resection represents the only potentially curative treatment strategy in these patients, however, the 5-year survival rate following surgical resection remains low, at 5.5-21% (2,3). Complete Statistics February 4, 2016 Debdeep Pati 1 Complete Statistics Suppose XP ; 2. = 1 } ^ { n } x _ {i} ^ {2} + \sum _ {i = 1 } ^ { n } X _ {i} $$. Any kind of help is thanked in advanced. Su-ciency was introduced into the statistical literature by Sir Ronald A. Fisher (Fisher (1922)). Although T = [X(1), X(N)]T is minimal sufficient for of the uniform distribution in Example 9.12, and together they contain as much information about as the original samples, it turns out they also contain information that is not relevant for estimating . /*# sourceMappingURL=https://www.redditstatic.com/desktop2x/chunkCSS/TopicLinksContainer.d421885364b06dce936a_.css.map*/, Thanks. Basu's theorem is useful in proving the . which is independent of $ \theta $. Lesson 24: Sufficient Statistics Overview In the lesson on Point Estimation, we derived estimators of various parameters using two methods, namely, the method of maximum likelihood and the method of moments. My profession is written "Unemployed" on my passport. Thanks for contributing an answer to Mathematics Stack Exchange! The pound had an average inflation rate of 4.77% per year between 1930 and today, producing a cumulative price increase of 7,165.06%. By clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie policy. with respect to Lebesgue measure is given by the expression, $$ 17. From this we de ne the concept of complete statistics. there exists a version of the conditional probability $ P _ \theta ( A \mid X = x ) $ Built-under - lpxtju.mybiwag.de >> \sum _ {i = 1 } ^ { n } X _ {i} ,\ \sum _ {i = 1 } ^ { n } X _ {i} ^ {2} is a trivial sufficient statistic. Find a minimal sufficient statistic T and, using a nontrivial function, show that it is not complete. \frac{n \mu A sufficient statistic is minimal sufficient if it can be represented as a function of any other sufficient statistic. .c_dVyWK3BXRxSN3ULLJ_t{border-radius:4px 4px 0 0;height:34px;left:0;position:absolute;right:0;top:0}._1OQL3FCA9BfgI57ghHHgV3{-ms-flex-align:center;align-items:center;display:-ms-flexbox;display:flex;-ms-flex-pack:start;justify-content:flex-start;margin-top:32px}._1OQL3FCA9BfgI57ghHHgV3 ._33jgwegeMTJ-FJaaHMeOjV{border-radius:9001px;height:32px;width:32px}._1OQL3FCA9BfgI57ghHHgV3 ._1wQQNkVR4qNpQCzA19X4B6{height:16px;margin-left:8px;width:200px}._39IvqNe6cqNVXcMFxFWFxx{display:-ms-flexbox;display:flex;margin:12px 0}._39IvqNe6cqNVXcMFxFWFxx ._29TSdL_ZMpyzfQ_bfdcBSc{-ms-flex:1;flex:1}._39IvqNe6cqNVXcMFxFWFxx .JEV9fXVlt_7DgH-zLepBH{height:18px;width:50px}._39IvqNe6cqNVXcMFxFWFxx ._3YCOmnWpGeRBW_Psd5WMPR{height:12px;margin-top:4px;width:60px}._2iO5zt81CSiYhWRF9WylyN{height:18px;margin-bottom:4px}._2iO5zt81CSiYhWRF9WylyN._2E9u5XvlGwlpnzki78vasG{width:230px}._2iO5zt81CSiYhWRF9WylyN.fDElwzn43eJToKzSCkejE{width:100%}._2iO5zt81CSiYhWRF9WylyN._2kNB7LAYYqYdyS85f8pqfi{width:250px}._2iO5zt81CSiYhWRF9WylyN._1XmngqAPKZO_1lDBwcQrR7{width:120px}._3XbVvl-zJDbcDeEdSgxV4_{border-radius:4px;height:32px;margin-top:16px;width:100%}._2hgXdc8jVQaXYAXvnqEyED{animation:_3XkHjK4wMgxtjzC1TvoXrb 1.5s ease infinite;background:linear-gradient(90deg,var(--newCommunityTheme-field),var(--newCommunityTheme-inactive),var(--newCommunityTheme-field));background-size:200%}._1KWSZXqSM_BLhBzkPyJFGR{background-color:var(--newCommunityTheme-widgetColors-sidebarWidgetBackgroundColor);border-radius:4px;padding:12px;position:relative;width:auto} 1 day ago VEM Why are taxiway and runway centerline lights off center? Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Leaving this here for future reference, as Ive ran across a frustrating error in Casella and Berger before too: http://archived.stat.ufl.edu/casella/class/errata7.pdf. \frac{1}{n-} A corollary of the RaoBlackwellKolmogorov theorem states that if a complete sufficient statistic $ X $ The best answers are voted up and rise to the top, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, $T=\left( X_{(1)}, \sum_{i=1}^{n} (X_i - X_{(1)}) \right)$, You have found $E\left[X_{(1)}\right]=\frac{\theta^2}{n} + \theta$; it could also be worth finding $E\left[ \sum\limits_{i=1}^{n} \left(X_i - X_{(1)}\right)\right]=(n-1)\theta^2$. stream $$, $$ and if $ X _ {1} $ There's also live online events, interactive content, certification prep materials, and more. p _ \nu ( x _ {1} \dots x _ {n} ) = \prod _ {i = 1 } ^ { n } A sufficient statistic is minimal sufficient if it can be represented as a function of any other sufficient statistic. p _ \theta ( \omega ) = g _ \theta ( X ( \omega ) ) h ( \omega ) , $$, $$ When the migration is complete, you will access your Teams at stackoverflowteams.com, and they will no longer appear in the left sidebar on stackoverflow.com. Chapter 5 Sufficient Principle, Minimal Sufficient Statistics (Lecture 9.7 COMPLETE STATISTIC. Often the loss function $ g $ Sufficient, Complete and Ancillary Statistics - Random Services \sum _ {i = 1 } ^ { n } x _ {i} \right ) , Terms of service Privacy policy Editorial independence. Rao, "Linear statistical inference and its application" , Wiley (1973), E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1986), C.R. is taken to be a positive-definite quadratic form on $ \mathbf R ^ {d} $. parameter Exponential family. Salary: Depends on Qualifications Job Type: Job Number: 23-080 Location: Sonoma County, CA Department: Humanities and Religion Department Closing: Continuous Description [Q] Does a causal relationship translates as a [Q] Is there such a thing as a p-value that incorporates [Q] Does a high explained variance (e.g. in the Bernoulli scheme, while the sample mean $ \widehat \mu $ In other words, if E [f(T(X))] = 0 for all , then f(T(X)) = 0 with probability 1 for all . Can you say that you reject the null at the 95% level? [PDF] Moral Courage of the Nursing Students of Yazd University of OReilly members experience live online training, plus books, videos, and digital content from nearly 200 publishers. Homeuse 308nm Excimer System for Vitiligo and Psoriasis Quick Detail Complete statistics. However, of main interest are statistics which permit a real reduction of the statistical problem. I have found that the minimal sufficient statistic is $T=\left( X_{(1)}, \sum_{i=1}^{n} (X_i - X_{(1)}) \right)$. A sufficient statistic is known as minimal or necessary if it is a function of any other sufficient statistic. Minimal sufficient statistics are not unique Any two are in one to one from STAT 566 at University Of Arizona ._1LHxa-yaHJwrPK8kuyv_Y4{width:100%}._1LHxa-yaHJwrPK8kuyv_Y4:hover ._31L3r0EWsU0weoMZvEJcUA{display:none}._1LHxa-yaHJwrPK8kuyv_Y4 ._31L3r0EWsU0weoMZvEJcUA,._1LHxa-yaHJwrPK8kuyv_Y4:hover ._11Zy7Yp4S1ZArNqhUQ0jZW{display:block}._1LHxa-yaHJwrPK8kuyv_Y4 ._11Zy7Yp4S1ZArNqhUQ0jZW{display:none} An ancillary statistic does not contain any information about parameter . Now, completeness implies (g1 g2)(u) = 0 for all u, which implies that g1(u) = g2(u) for all u and therefore that g1(U) = g2(U). .FIYolDqalszTnjjNfThfT{max-width:256px;white-space:normal;text-align:center} only by means of the variables, $$ ( 2 \pi \sigma ^ {2} ) ^ {- n / 2 } \mathop{\rm exp} \left [ - Minimal Sufficient Statistic and Complete Sufficient Statistic . Sufficient statistic | Psychology Wiki | Fandom Solutions to Homework Assignment 9 - University of Hawaii Making statements based on opinion; back them up with references or personal experience. Then, $$ there exists a version of the conditional probability $ P _ \theta ( A \mid {\mathcal B} ) $ algebra $ {\mathcal A} = X ^ {-} 1 ( {\mathcal B} ) $ The European Mathematical Society, for a family of probability distributions $ \{ {P _ \theta } : {\theta \in \Theta } \} $ However, as noted above, there usually exists a statistic \(U\) that is sufficient for \(\theta\) and has smaller dimension, so that we can achieve real data reduction. Stack Overflow for Teams is moving to its own domain! @Henry right you are. If a minimal sufficient statistic exists, then any complete statistic is also a minimal sufficient statistic. Properties of Sufficient Statistics Sufficiency is related to several of the methods of constructing estimators that we have studied. 2022/2023 - FACULTY (ASSOCIATE) - RELIGIOUS STUDIES . and ?. Job Description Summary \frac{1}{n} To subscribe to this RSS feed, copy and paste this URL into your RSS reader. ._2ik4YxCeEmPotQkDrf9tT5{width:100%}._1DR1r7cWVoK2RVj_pKKyPF,._2ik4YxCeEmPotQkDrf9tT5{display:-ms-flexbox;display:flex;-ms-flex-align:center;align-items:center}._1DR1r7cWVoK2RVj_pKKyPF{-ms-flex-pack:center;justify-content:center;max-width:100%}._1CVe5UNoFFPNZQdcj1E7qb{-ms-flex-negative:0;flex-shrink:0;margin-right:4px}._2UOVKq8AASb4UjcU1wrCil{height:28px;width:28px;margin-top:6px}.FB0XngPKpgt3Ui354TbYQ{display:-ms-flexbox;display:flex;-ms-flex-align:start;align-items:flex-start;-ms-flex-direction:column;flex-direction:column;margin-left:8px;min-width:0}._3tIyrJzJQoNhuwDSYG5PGy{display:-ms-flexbox;display:flex;-ms-flex-align:center;align-items:center;width:100%}.TIveY2GD5UQpMI7hBO69I{font-size:12px;font-weight:500;line-height:16px;color:var(--newRedditTheme-titleText);white-space:nowrap;overflow:hidden;text-overflow:ellipsis}.e9ybGKB-qvCqbOOAHfFpF{display:-ms-flexbox;display:flex;-ms-flex-align:center;align-items:center;width:100%;max-width:100%;margin-top:2px}.y3jF8D--GYQUXbjpSOL5.y3jF8D--GYQUXbjpSOL5{font-weight:400;box-sizing:border-box}._28u73JpPTG4y_Vu5Qute7n{margin-left:4px} x[KsFWHy?!]8$RMR*~!HYlr 0uO/o0C8_MDp1Lwrq=*A-glUlxz'eRXofO^70-});fG8_k5&V46xSisgJM\-u"uIgylR_*nUt81L4qF%\IH'#l8$RI /.V B'm)"A%xW/%-C:5l&hWX#*= q2Y?. \sigma ^ {2} } Another form of Bahadur's theorem states that any sufficient and boundedly complete statistic over a finite-dimensional coordinate space is also minimal sufficient. This property is mathematically expressed as one of the results of the theory of statistical decision making which says that the set of decision rules based on a sufficient statistic forms an essentially complete class. PDF Completeness and sufficiency - University of Oklahoma Statistical Theory and Inference (gebundenes Buch) | Buchhandlung Schningh - Finding sufficient minimal sufficient and complete sufficient minimal sufcient statistic is unique in the sense that two statistics that are functions of each other can be treated as one statistic. A statistic $ X $ This is a subreddit for discussion on all things dealing with statistical theory, software, and application. An ancillary statistic is also first-order ancillary: if its distribution is independent of , then its expectation does not depend on . Statistics and Probability questions and answers. 2Form an ancillary statisticfor ? What do you call an episode that is not closely related to the main plot? $ \theta \in \Theta $. er"l#tI}!Q|4rXl4PPG5(.r,~=[3VtcNa@dnjmngJO}P/Vc'fS]D \sum _ {i = 1 } ^ { n } x _ {i} ,\ \sum _ {i = 1 } ^ { n } x _ {i} ^ {2} . @keyframes ibDwUVR1CAykturOgqOS5{0%{transform:rotate(0deg)}to{transform:rotate(1turn)}}._3LwT7hgGcSjmJ7ng7drAuq{--sizePx:0;font-size:4px;position:relative;text-indent:-9999em;border-radius:50%;border:4px solid var(--newCommunityTheme-bodyTextAlpha20);border-left-color:var(--newCommunityTheme-body);transform:translateZ(0);animation:ibDwUVR1CAykturOgqOS5 1.1s linear infinite}._3LwT7hgGcSjmJ7ng7drAuq,._3LwT7hgGcSjmJ7ng7drAuq:after{width:var(--sizePx);height:var(--sizePx)}._3LwT7hgGcSjmJ7ng7drAuq:after{border-radius:50%}._3LwT7hgGcSjmJ7ng7drAuq._2qr28EeyPvBWAsPKl-KuWN{margin:0 auto} Let ( X (1);:::;X . Let X ~ Ber(n1; p), Y ~ Ber(n2; p^2), where X and Y are independent We are currently having 14 injection machines here. (but MSS does not imply CSS as we saw earlier). By accepting all cookies, you agree to our use of cookies to deliver and maintain our services and site, improve the quality of Reddit, personalize Reddit content and advertising, and measure the effectiveness of advertising. then the inequality, $$ Clearly, we have failed to obtain any general result by conditioning on the minimal sufficient statistic since we cannot eliminate the possibility of improving on two "optimal" statistics by just averaging them. The joint density of the distributions of $ X _ {1} \dots X _ {n} $ is independent of $ \theta $. 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