minimal sufficient statistic for gamma distribution

For example, the sample mean, sample variance, the largest and smallest order statistics. PFAS have been associated with many adverse human health effects. These keywords were added by machine and not by the authors. In general, if Y is a sufficient statistic for a parameter , then every one-to-one function of Y not involving is also a sufficient statistic for . Let's take a look at another example. Suppose that X 1, , X n are iid N(, 2). Find a minimal sufficient statistic for 2 using the Lehmann-Scheff Theorem. d) Find the method of moment estimator of e. Example: (1) G(q,1): For the gamma distribution with shape parameter q and scale parameter 1 we have T(x)=Slogx i, E={x We can always achieve this by reparametrization. For any minimal s-dimensional exponential family the statistic (P i T 1(X i);:::; P i T s(X i)) is a minimal su cient statistic. T(x) is a function of T(x) means that if T(x) = T(y), then T(x) = T(y) . A su cient statistic T is minimal su cient if for any statistic U ther e exists a function h such that T = h (U). Basu, D. (1959), The Family of Ancillary Statistics, Sankhy, A, 21, 247256. I still could not see why the entire sample is NOT a minimal sufficient statistic;p Also, is minimal sufficient statistic UNIQUE? (1981), Order Statistics, Wiley, New York, NY. Show that T = Pn i=1 Xi is a su-cient statistic for . By Cramer-Rao this yields the eciency (minimal variance) of the unbiased estimator X. Boos, D.D., and Hughes-Oliver, J.M. f Let be given. Solved - Sufficient statistic for a Gamma distribution conditional probabilitygamma distributionself-studysufficient-statistics I am confused about the steps I need in order to solve the equation below. Lehmann, E.L., and Scheff, H. (1950), Completeness, Similar Regions, and Unbiased Estimation, Sankhy, 10, 305340. (A) Compared to a strain that expresses GFP::MEL-28 from the endogenous mel-28 locus after CRISPR/Cas9-mediated GFP knock-in (top panel), expression of GFP::MEL-28 full-length and mutant proteins from transgenes inserted by microparticle bombardment or MosSCI is either similar or lower, thus arguing against the possibility of artifacts induced by . Would also like to add that an important property of a minimal sufficient statistic is that it can be expressed as a function of another non-minimal sufficient statistic. Nobody wants to help me with this problem? Modified 7 years, 8 months ago. (1976), Sufficiency, Minimal Sufficiency, and the Lack Thereof, The American Statistician, 30, 3435. Provided by the Springer Nature SharedIt content-sharing initiative, Over 10 million scientific documents at your fingertips, Not logged in Mann, N.R., Schafer, R.E., and Singpurwalla, N.D. (1974), Methods for Statistical Analysis of Reliability and Life Data, Wiley, New York, NY. This is a preview of subscription content, access via your institution. How many rectangles can be observed in the grid? Ask Question Asked 7 years, 8 months ago. using the functions ? MathSciNet Datta, G.S., and Sarker, S.K. nm4d0H~qhN$MG>|*k\`P`z/S+RSQd70b -&nM*"x)xVl9wI1O1K/GN*xc|V8F7`=, AB*u*e9+,VRt,|{BCo,ZZ7#U%8XH-^78{]AUO kr C GH:HrChBI Number of unique permutations of a 3x3x3 cube. 4 which is of the desired form. The previous example suggests that there can be more than one sufficient statistic for a parameter . Rice, J. MathSciNet Viewed 10k times 4 $\begingroup$ I am confused about the steps I need in order to solve the equation below. Math Statistics Q&A Library (1) Find a minimal sufficient statistic for 0 based on an independent sample of size n from each of the following distributions, and indicate if the statistic is complete: (a) the Gamma distribution with density e f(x; a, B) = x > 0, T(a) with 0 = (a, 3), a > 0, 8 > 0; (b) the Uniform distribution on (0 - 1,0+1), with 0 E R; (c) the Geometric distribution with . Lehmann, E.L., and Scheff, H. (1950), Completeness, Similar Regions, and Unbiased Estimation, Sankhy, 10, 305340. Google Scholar. 2022 Springer Nature Switzerland AG. nuna mixx next bassinet caviar; sheet pan chicken fajitas ree drummond MathSciNet We review their content and use your feedback to keep the quality high. If the shape parameter k is held fixed, the resulting one-parameter family of distributions is a natural exponential family . First of all about the sufficient statistic, according to Wiki: If the probability density function is f ( x ), then T is sufficient for if and only if nonnegative functions g and h can be found such that f ( x ) = h ( x ) g ( T ( x )), (1981), Order Statistics, Wiley, New York, NY. As a result, T is a minimal su cient statistic. Suppose that the distribution of X is a k-parameter exponential family with natural sufficient statistic U=h(X). PubMedGoogle Scholar, 2014 Springer International Publishing Switzerland, Olive, D.J. Example 6.2.15. This is a preview of subscription content, access via your institution. This example can be generalized to higher dimensions, where the sucient statistics are cosines of general spherical coordinates. Proof: For every set of nonnegative integers x1;;xn, the joint probability mass function fn(xj) of X1;;Xn is as follows: fn(xj) = Koehn, U., and Thomas, D.L. Rice, J. These keywords were added by machine and not by the authors. Sampson, A., and Spencer, B. (See Keener Ex. Koehn, U., and Thomas, D.L. (1991), On Proving that $\overline{X}$ and S This leads to the notion of minimal su ciency De nition 2 (Minimal Su ciency). Covalent and Ionic bonds with Semi-metals, Is an athlete's heart rate after exercise greater than a non-athlete. This complements the idea of a minimal statistic being the coarsest description of a random sample! CrossRef Find the (minimal) sucient statistic for parameter . 2 are Independent, The American Statistician, 45, 121122. (1984), Kruskals Proof of the Joint Distribution of $\overline{X}$ and s - 119.195.102.159. Casella, G., and Berger, R.L. 14. CrossRef 1Find the complete minimal sufficient statistic for ? Cox, D.R., and Hinkley, D.V. ~{{LU}0#A (yD'&% iAl jK`M{W_[O ;D@1Xn ;U_~L0d$1ZRw| eD*(&yl. The parameters satisfy 1<3 and 2<4. +X n and let f be the joint density of X 1, X 2,., X n. Dan Sloughter (Furman University) Sucient Statistics: Examples March 16, 2006 2 / 12 To avoid confusing the distribution of . /Length 3827 David, H.A. The order statistic only contains the values. A useful characterization of minimal sufficiency is that when the density f exists, S ( X) is minimal sufficient if and only if is independent of : S ( x) = S ( y) 5Ah3e:3,U52.1N0:'a=ujuqqTLnA;qq@q;g1C?t4"3$/oM+jf6o#4*E(eiRC RD+Q@jF?Y $1-6L|J'79 pRz{NHDM&YV->X2*;f>u A{VO# R , J73zML('q- axzI#g"ruY. Anyone you share the following link with will be able to read this content: Sorry, a shareable link is not currently available for this article. distribution and the statistic is a random variable and hence also comes from some probability distribution. Minimum number of random moves needed to uniformly scramble a Rubik's cube? The resulting exponential family distribution is known as the Fisher-von Mises distribution. !. K{9\ YW/Hsg_R"BB:\..?f>[r-3A|[~\w{z7w|5ef Halmos, P.R., and Savage, L.J. (1958), Examples of Inconsistency of Maximum Likelihood Estimators, Sankhy, 20, 207210. Given $X_1, X_2,\ldots, X_n$ are i.i.d with $f(x) = \frac{1}{\pi}\frac{1}{1+x^2}$. https://doi.org/10.1007/978-3-319-04972-4_4, Shipping restrictions may apply, check to see if you are impacted, Tax calculation will be finalised during checkout. 3.12) Let (X 1, Y 1),,(X n,Y ) be a random sample from this pdf. subsonic spotify plugin; how to recover a deleted soundcloud account; easiest elden ring bosses. minimal sufficient statistic of Cauchy distribution. CrossRef You'll get a detailed solution from a subject matter expert that helps you learn core concepts. Let T(X) be a statistic. Anyone you share the following link with will be able to read this content: Sorry, a shareable link is not currently available for this article. MathSciNet Why is HIV associated with weight loss/being underweight? 2003-2022 Chegg Inc. All rights reserved. It is clear the this is just the special case Gamma(r=2;2). David, H.A. (a)The likelihood function is L( ) = Yn . Provided by the Springer Nature SharedIt content-sharing initiative, Over 10 million scientific documents at your fingertips, Not logged in Google Scholar. 2022 Springer Nature Switzerland AG. MATH CrossRef (1958), Examples of Inconsistency of Maximum Likelihood Estimators, Sankhy, 20, 207210. Google Scholar. Cox, D.R., and Hinkley, D.V. Boos, D.D., and Hughes-Oliver, J.M. |K iS{isgRnt9B97bSf+wU]HZm^%~/UQlU:&=Jr{r^Ipwem\1HRwZN/P+^1oTs)TF vI/!Q;~Zh o4E5jMz| f?^'E3 bW5v&Zf*ly oW{;\Hwc-`5ki (1976), Sufficiency, Minimal Sufficiency, and the Lack Thereof, The American Statistician, 30, 3435. Minimal sufficient statistics for 2-parameter exponential distribution. Minimal sufficient statistics for Cauchy distribution statistics polynomials statistical-inference order-statistics 4,187 It is trivial to see the order statistics T ( X) = ( X ( 1), , X ( n)) are sufficient, hence we only need to prove one direction: that if the ratio is constant as a function of , then T ( x) = T ( y). Intuitively, a minimal sufficient statistic most efficiently captures all possible information about the parameter . Sampson, A., and Spencer, B. CrossRef Johanson, S. (1979), Introduction to the Theory of Regular Exponential Families, Institute of Mathematical Statistics, University of Copenhagen, Copenhagen, Denmark. stream MATH 3 0 obj << Springer, Cham. How to go about finding a Thesis advisor for Master degree, Prove If a b (mod n) and c d (mod n), then a + c b + d (mod n). In: Statistical Theory and Inference. Example 4.2. Minimal sufficient statistic is definitely. ]38[M(P Intuitively, a minimal sufficient statistic most efficiently captures all possible information about the parameter . 2, The American Statistician, 38, 134135. For the gamma and beta functions, a set of random numbers si, which have a distribution f (x) is obtained by solving the following equation for si for each value of ri, which are random and rectangular in distribution: ls>(x) dy = ri (6) As part of the computational procedure the values of ri are ranked and replaced in their random positions . Datta, G.S., and Sarker, S.K. My question: Using Bahadur's Factorization and Lehman-Scheffe's theorems, I obtain the sufficient statistic as $T(X) = (x_1, \ldots, x_n)$. For example, if T is minimal sufcient, then so is (T;eT), but no one is going to use (T;eT). MATH a) Find a sufficient statistic for 0. MATH A useful characterization of minimal sufficiency is that when the density f exists, S ( X) is minimal sufficient if and only if is independent of : S ( x) = S ( y) - 46.175.224.27. . Statistics and Probability questions and answers. Google Scholar. Why plants and animals are so different even though they come from the same ancestors? A statistic is a function of the data that does not depend on any unknown parameters, and a statistic is a random variable that has a distribution called the sampling distribution. CrossRef ,Xn given and T does not depend on , statistician B knows this . Sufficient statistic can be thought as partition of sample space X. r{$ bF9. Google Scholar. (1984), Kruskals Proof of the Joint Distribution of $\overline{X}$ and s PubMedGoogle Scholar, 2014 Springer International Publishing Switzerland, Olive, D.J. Sufficient statistic for a Gamma distribution. a) Find a sufficient statistic for 0. Sol : MathSciNet Would also like to add that an important property of a minimal sufficient statistic is that it can be expressed as a function of another non-minimal sufficient statistic. the sum of all the data points. _u;j769 3hoBSASjerf E1& LG21"lJ>=.B`#9H@G@N&vpe&.PYVFY5 5WR"s}pxw&%RD#tj(6BC\^+6{Dhjvu_"~YC:dqG2V(D HD;qdcN5-W'ZPcJ6EIq)O|cfA`tl.cA1|c#->b #USHX Lehmann, E.L. (1983), Theory of Point Estimation, Wiley, New York, NY. S2 Fig: Analysis of MEL-28 expression levels. Example 2: Suppose that X1;;Xn form a random sample from a Poisson distribution for which the value of the mean is unknown ( > 0). MathSciNet 2 are Independent, The American Statistician, 45, 121122. MATH MATH ) be the bivariate pdf for the uniform distribution on the rectangle with lower left corner (1,2) and upper right corner (3,4) in 82. Google Scholar. Lehmann, E.L. (1983), Theory of Point Estimation, Wiley, New York, NY. What mathematical algebra explains sequence of circular shifts on rows and columns of a matrix? T(X)=X is a sufficient statistics but NOT minimal. jn#4R-nXNyqi ~"FKaIffU-Y}[rjv;+E:LpY?pTlI.SQswoXV + x n and the natural parameter () = ln(/(1)), the log-odds, Example 6 (Gamma random variables). Definition 5.1 (Minimal Sufficient Statistic) A sufficient statistic T(X) is called a minimal sufficient statistic if, for any other sufficent statistic T(X), T(x) is a function of T(x) . UW-Madison (Statistics) Stat 609 Lecture 22 2015 1 / 15 Google Scholar. Sufficient Statistics. (1991), On Proving that $\overline{X}$ and S n) is a sufcient statistic of dimensionn. I must use conditional distribution (and NOT the factorization theorem). 92f~W*e_a~Exwp 7|Mk)=Z[r4Vmn(\Uv_`>@Ish*y4`_'>) (1975), On Statistics Independent of a Sufficient Statistic: Basus Lemma, The American Statistician, 29, 4042. What are the best sites or free software for rephrasing sentences? The latter expression is the moment-generating function of a Gamma(P n i=1 a i;b) random variable and, by uniqueness of moment-generating functions, P n i=1 X i must have this distribution. (2008), A General Proof of Some Known Results of Independence Between Two Statistics, The American Statistician, 62, 141143. (1949), Applications of the Radon-Nikodym Theorem to the Theory of Sufficient Statistics, The Annals of Mathematical Statistics, 20, 225241. MathSciNet Q: $X_1, . Therefore, using the formal definition of sufficiency as a way of identifying a sufficient statistic for a parameter $\theta$ can often be a daunting road to . Per- and polyfluoroalkyl substances (PFAS) are a class of thousands of persistent, organic fluorinated chemicals added to materials and products mainly to repel stains and water. However, as noted above, there usually exists a statistic U that is sufficient for and has smaller dimension, so that we can achieve real data reduction. 4.25. distribution acts like a Gaussian distribution as a function of the angular variable x, with mean and inverse variance . Remark 1. (2002), Statistical Inference, 2nd ed., Duxbury, Belmont, CA. (1998), Applications of Basus Theorem, The American Statistician, 52, 218221. Lehmann, E.L. (1986), Testing Statistical Hypotheses, 2nd ed., Wiley, New York, NY. (1974), Theoretical Statistics, Chapman and Hall, London, UK. For T, if x 6=y but T(x) = T(y), then x and y provides the same information and can be treated as the same. (25 pts) b) Find the minimal sufficient statistic for 0. c) Find the maximum likelihood estimator of . 4.26. MATH For example X = (1,2,3,4,5) and X= (1,3,2,4,5) both has the same order statistic T(X) = (1,2,3,4,5). Bowman and L. R. Shenton 90. . The order statistic only contains the values. In . Zehna, P.W. (1988), Mathematical Statistics and Data Analysis, Wadsworth, Belmont, CA. data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAKAAAAB4CAYAAAB1ovlvAAADOUlEQVR4Xu3XQUpjYRCF0V9RcOIW3I8bEHSgBtyJ28kmsh5x4iQEB6/BWQ . Casella, G., and Berger, R.L. https://doi.org/10.1007/978-3-319-04972-4_4, DOI: https://doi.org/10.1007/978-3-319-04972-4_4, eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0). (1974), Theoretical Statistics, Chapman and Hall, London, UK. Let Y, Y ,Y, denote a random sample from Gamma distribution(a=2,0). As another example, if we take a normal distribution in which the mean and the variance are functionally related, e.g., the N(;2) distribution, then the distribution will be neither in minimal sufcient statistic is unique in the sense that two statistics that are functions of each other can be treated as one statistic. 3)Verify if the ancillary statistic formed in part (b)(ii) is independent of the complete sufficient statistic found in part (b)(i). d) Find the method of moment estimator of e. e) Find the minimum variance unbiased estimator of 0. Google Scholar. More than a million books are available now via BitTorrent. . Google Scholar. (2014). (2014). This video is a demonstration of how to find minimal sufficient statistics for the Beta distribution using the results of Fisher's factorisation theorem, and the derivation is valid when all of. . Consider a sample of size n from Gamma(,). T(X) = X contains more information than T(X) = (x(1),x(2),,x(n)). Construction of a minimal su cien t . Intuitively, a minimal sufficient statistic most efficiently captures all possible information about the parameter . citibank vision statement; geysermc missing profile public key; javascript wait for ajax call to return; sufficient cause examples in epidemiology October 26, 2022 A useful characterization of minimal sufficiency is that when the density f exists, S ( X) is minimal sufficient if and only if is independent of : S ( x) = S ( y) Hot Network Questions Stigler, S.M. T(X)=X is a sufficient statistics but NOT minimal. Springer, Cham. MATH Department of Mathematics, Southern Illinois University, Carbondale, IL, USA, You can also search for this author in Google Scholar. We aimed to determine whether buildings with "healthier" materialsdefined here as reportedly free of all PFASexhibit lower PFAS in dust. It is natural to ask if there is a statistic T that giv es the maximal degree of data compression vis- a-vis . Basu, D. (1959), The Family of Ancillary Statistics, Sankhy, A, 21, 247256. Statistical Theory and Inference pp 101128Cite as. cative constant, is called the natural sufcient statistic.Wesay that an exponential family is minimal if the functions c(q) and the statistics T(x) each are linearly independent. /Filter /FlateDecode Not to mention that we'd have to find the conditional distribution of $X_1, X_2, \ldots, X_n$ given $Y$ for every $Y$ that we'd want to consider a possible sufficient statistic! You can generate an ordered set from an unordered sample, but you cannot recover the original sample from an ordered sample. Find a minimal sufficient statistic for $\theta$. Google Scholar. x\[s~`%4MfZ:3Ih(R! Halmos, P.R., and Savage, L.J. Show that the sufficient statistics given above for the Bernoulli, Poisson, normal, gamma, and beta families are minimally sufficient for the given parameters. Bahadur, R.R. I intended to make this text accessible to researchers with a minimal amount of theoretical statistical background, . How many axis of symmetry of the cube are there? How many ways are there to solve a Rubiks cube? Bahadur, R.R. and ?. This process is experimental and the keywords may be updated as the learning algorithm improves. Stigler, S.M. (1975), On Statistics Independent of a Sufficient Statistic: Basus Lemma, The American Statistician, 29, 4042. Google Scholar. For more information about this format, please see the Archive Torrents collection. (2008), A General Proof of Some Known Results of Independence Between Two Statistics, The American Statistician, 62, 141143. https://doi.org/10.1007/978-3-319-04972-4_4, Shipping restrictions may apply, check to see if you are impacted, Tax calculation will be finalised during checkout. https://doi.org/10.1007/978-3-319-04972-4_4, DOI: https://doi.org/10.1007/978-3-319-04972-4_4, eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0). T(X) = X contains more information than T(X) = (x(1),x(2),,x(n)). Heuristically, a minimal sufcient statistic is a sufcient statistic with the smallest dimensionk, where 1k n.Ifkis small and does not depend onn, then there is considerable dimension reduction. 2, The American Statistician, 38, 134135. Example 24-3 Department of Mathematics, Southern Illinois University, Carbondale, IL, USA, You can also search for this author in (2002), Statistical Inference, 2nd ed., Duxbury, Belmont, CA. (1949), Applications of the Radon-Nikodym Theorem to the Theory of Sufficient Statistics, The Annals of Mathematical Statistics, 20, 225241. Let X 1, , X n be iid exponential () random variables. Show that U is a minimally sufficient for . (1998), Applications of Basus Theorem, The American Statistician, 52, 218221. (f)The ChiSq(r) PDF is given on page 152 of the book. 16. Minimal Sufficient Statistics The entire data variable X is trivially sufficient for . Rubik's Cube Stage 6 -- show bottom two layers are preserved by $ R^{-1}FR^{-1}BBRF^{-1}R^{-1}BBRRU^{-1} $. Let us do this: In such a case, the sufficient statistic may be a set of functions . Statistical Theory and Inference pp 101128Cite as. T(X) = X contains RANK while the order statistic does not. For example X = (1,2,3,4,5) and X= (1,3,2,4,5) both has the same order statistic T(X) = (1,2,3,4,5). 2Form an ancillary statisticfor ? However, the correct answer turned out to be $T(X) = (x_{(1)}, x_{(2)},\ldots, x_{(n)})$. Johanson, S. (1979), Introduction to the Theory of Regular Exponential Families, Institute of Mathematical Statistics, University of Copenhagen, Copenhagen, Denmark. A bivariate normal distribution with all parameters unknown is in the ve parameter Exponential family. 5. The sample minimum min(n) = Y(1) (4.6) and the observed min y(1) is the smallest value of the observed data. , X_n$ is a random sample from a Gamma(2, ) distribution. Find the minimal sufficient statistic for $\frac{1}{\pi^{n}(\prod_{i=1}^{n} 1+(x_i-\theta)^2)}$. 15. 2. What is the probability of genetic reincarnation? The Factorization Theorem is used to nd a sufcient statistic. More generally, the "unknown parameter" may represent a vector of unknown quantities or may represent everything about the model that is unknown or not fully specified. Find a four-dimensional sufficient statistic for (1,2,3,4). the information by using a few key features (statistics). Mann, N.R., Schafer, R.E., and Singpurwalla, N.D. (1974), Methods for Statistical Analysis of Reliability and Life Data, Wiley, New York, NY. % Statistical analyses are performed using Easyfit tool. Sufficient Statistics. It allows to find best fitting probability density function of traffic flows, approximation toward selected distributions as Pareto and Gamma and random number generation with selected distribution. For a multidimensional parameter space, the exponential family is dened with the product in the exponential replaced by the inner product. Properties of Estimators for the Gamma Distribution, K. O. This process is experimental and the keywords may be updated as the learning algorithm improves. Typically, the sufficient statistic is a simple function of the data, e.g. Well, there is only one way to solve the problem write down the joint pdf and apply the Factorization Theorem. T(X) = X contains RANK while the order statistic does not. Now, I can find a sufficient statistic using the factorisation theorem ($\sum X_i$), but I don't think that this statistic is in fact minimal sufficient. Find a minimal sufficient statistic for using the fact that the exponential distribution is a 1P-REF. The gamma distribution is a two-parameter exponential family with natural parameters k 1 and 1/ (equivalently, 1 and ), and natural statistics X and ln ( X ). The question seems to imply that there exists a minimal sufficient statistic, but I'm not even sure that there is one. Part of Springer Nature. Usually the term "observed" is dropped. Part of Springer Nature. Interestingly, minimal su cient statistics are quite easy to nd when working with min-imal exponential families. In: Statistical Theory and Inference. How can I calculate the number of permutations of an irregular rubik's cube? If the range of X is Rk, then there exists a minimal sufcient statistic. Experts are tested by Chegg as specialists in their subject area. Z ~ 0.2123 is a sufficient approximation to a standard normal random variable. Naturally, we would like to find the statistic U that has the smallest dimension possible. Zehna, P.W. (1988), Mathematical Statistics and Data Analysis, Wadsworth, Belmont, CA. aJLa75(lL# =xcB"@=il8uEnDP:Cl 6*1u0J,KLn]E m5)5ud,7) 4 But I cannot see why the first answer is incorrect, because if we are given the entire sample set, we could easily obtain the order statistic of that sample, so how could $T(X) = (x_1, \ldots, x_n)$ is wrong? MathSciNet A statistic is a function of the data that does not depend on any unknown parameters, and a statistic is a random variable that has a distribution called the sampling distribution. This is a demonstration of how to find the minimal sufficient statistics of the parameters of an Inverse Normal (Inverse Gaussian) distribution.These short v. You can generate an ordered set from an unordered sample, but you cannot recover the original sample from an ordered sample. (25 pts) b) Find the minimal sufficient statistic for 0. c) Find the maximum likelihood estimator of . Lehmann, E.L. (1986), Testing Statistical Hypotheses, 2nd ed., Wiley, New York, NY. Statistics and Probability Statistics and Probability questions and answers Let Y, Y ,.Y, denote a random sample from Gamma distribution (a=2,0). >> %PDF-1.5
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