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This results follow from the second displayed equation for the PDF \( f(\bs x) \) of \( \bs X \) in the proof of the previous theorem. Duxbury Press. Because of the central limit theorem, the normal distribution is perhaps the most important distribution in statistics. . It is studied in more detail in the chapter on Special Distribution. Answer of Complete Sufficient Statistic. complete sufficient statisticとは意味:完備十分統計量{かんび じゅうぶん とうけいりょう} 相关词汇 sufficient supplies 中文 , statistics minister 中文 , completer 中文 , complete hamiltonian 中文 , complete morphism 中文 , complete gelation 中文 , If T(X1;¢¢¢;Xn) is a statistic and t is a … The Poisson distribution is named for Simeon Poisson and is used to model the number of random points in region of time or space, under certain ideal conditions. The condition is also sufficient if T be a boundecUy complete sufficient statistic. This result is intuitively appealing: in a sequence of Bernoulli trials, all of the information about the probability of success \(p\) is contained in the number of successes \(Y\). The parameter \(\theta\) is proportional to the size of the region, and is both the mean and the variance of the distribution. Run the uniform estimation experiment 1000 times with various values of the parameter. Suppose that \(V = v(\bs X)\) is a statistic taking values in a set \(R\). Nonetheless we can give sufficient statistics in both cases. Suppose that \(\bs X = (X_1, X_2, \ldots, X_n)\) is a random sample from the Pareto distribution with shape parameter \(a\) and scale parameter \( b \). So i tried prove using the definition of the complete sufficient statistic. Sometimes the variance \( \sigma^2 \) of the normal distribution is known, but not the mean \( \mu \). Find the complete sufficient statistic. Exponential families of distributions are studied in more detail in the chapter on special distributions. Compare the estimates of the parameters. Bounded completeness also occurs in Bahadur's theorem. Suppose that \(U\) is sufficient for \(\theta\) and that there exists a maximum likelihood estimator of \(\theta\). Moreover, \(k\) is assumed to be the smallest such integer. statistic T is minimal su cient if for any statistic U ther e exists a function h such that T = h (U). \[ D_y = \left\{(x_1, x_2, \ldots, x_n) \in \{0, 1\}^n: x_1 + x_2 + \cdots + x_n = y\right\} \]. Then \(\left(X_{(1)}, X_{(n)}\right)\) is minimally sufficient for \((a, h)\), where \( X_{(1)} = \min\{X_1, X_2, \ldots, X_n\} \) is the first order statistic and \( X_{(n)} = \max\{X_1, X_2, \ldots, X_n\} \) is the last order statistic. $\begingroup$ I agree with the answers below, however it is interesting to note that the converse is true: If a minimal sufficient statistic exists, then any complete statistic is also minimal sufficient. It can be shown that a complete and sufficient statistic is minimal sufficient (Theorem 6.2.28). German\ \ vollständige suffiziente Statistik. Under mild conditions, a minimal sufficient statistic does always exist. I. (pp. In other words, T is a function of T0(there exists fsuch that T(x) = f(T0(x)) for any x2X). The proof of the last theorem actually shows that \( Y \) is sufficient for \( b \) if \( k \) is known, and that \( V \) is sufficient for \( k \) if \( b \) is known. Suppose again that \( \bs X = (X_1, X_2, \ldots, X_n) \) is a random sample from the uniform distribution on the interval \( [a, a + h] \). \( Y \) is sufficient for \( (N, r) \). \[ f(\bs x) = g(x_1) g(x_2) \cdots g(x_n) = p^y (1 - p)^{n-y}, \quad \bs x = (x_1, x_2, \ldots, x_n) \in \{0, 1\}^n \] \( (M, T^2) \) where \( T^2 = \frac{1}{n} \sum_{i=1}^n (X_i - M)^2 \) is the biased sample variance. Casella, G. and Berger, R. L. (2001). \[ g(x) = \frac{1}{h}, \quad x \in [a, a + h] \] The proof also shows that \( P \) is sufficient for \( a \) if \( b \) is known, and that \( Q \) is sufficient for \( b \) if \( a \) is known. In particular, suppose that \(V\) is the unique maximum likelihood estimator of \(\theta\) and that \(V\) is sufficient for \(\theta\). It is named for Ronald Fisher and Jerzy Neyman. Consider again the basic statistical model, in which we have a random experiment with an observable random variable \(\bs X\) taking values in a set \(S\). The following result, known as Basu's Theorem and named for Debabrata Basu, makes this point more precisely. Then \( \left(P, X_{(1)}\right) \) is minimally sufficient for \( (a, b) \) where \(P = \prod_{i=1}^n X_i\) is the product of the sample variables and where \( X_{(1)} = \min\{X_1, X_2, \ldots, X_n\} \) is the first order statistic. Run the normal estimation experiment 1000 times with various values of the parameters. Recall that \( M \) and \( T^2 \) are the method of moments estimators of \( \mu \) and \( \sigma^2 \), respectively, and are also the maximum likelihood estimators on the parameter space \( \R \times (0, \infty) \). This variable has the hypergeometric distribution with parameters \( N \), \( r \), and \( n \), and has probability density function \( h \) given by How to find sufficient complete statistic for the density $f(x\mid\theta)=e^{-(x-\theta)}\exp(-e^{-(x-\theta)})$? Suppose that \(U = u(\bs X)\) is a statistic taking values in a set \(R\). In essence, it ensures that the distributions corresponding to different values of the parameters are distinct. T is a statistic of X which has a binomial distribution with parameters (n,p). \[ \sum_{y=0}^n \binom{n}{y} p^y (1 - p)^{n-y} r(y) = 0, \quad p \in T \] the sum of all the data points. A bit of though t will lead us to the idea that a su cien statistic T pro vides the most e cien t degree of data compression will ha v e the prop ert y … But then from completeness, \(g(v \mid U) = g(v)\) with probability 1. r(y) \theta^y\] Sufficient Statistics: Examples Mathematics 47: Lecture 8 Dan Sloughter Furman University March 16, 2006 Dan Sloughter (Furman University) Sufficient Statistics: Examples March 16, 2006 1 / 12 Let \(U = u(\bs X)\) be a statistic taking values in a set \(R\). Although the definition of a complete sufficient statistic is clear, its constructive verification in a … \[ f(\bs x) = \frac{1}{(2 \pi)^{n/2} \sigma^n} e^{-n \mu^2 / \sigma^2} \exp\left(-\frac{1}{2 \sigma^2} \sum_{i=1}^n x_i^2 + \frac{2 \mu}{\sigma^2} \sum_{i=1}^n x_i \right), \quad \bs x = (x_1, x_2 \ldots, x_n) \in \R^n\] \[ f(\bs x) = g(x_1) g(x_2) \cdots g(x_n) = \frac{1}{\Gamma^n(k) b^{nk}} (x_1 x_2 \ldots x_n)^{k-1} e^{-(x_1 + x_2 + \cdots + x_n) / b}, \quad \bs x = (x_1, x_2, \ldots, x_n) \in (0, \infty)^n \] Suppose that \(\bs X = (X_1, X_2, \ldots, X_n)\) is a random sample from the gamma distribution with shape parameter \(k\) and scale parameter \(b\). By condition (6), \(\left(X_{(1)}, X_{(n)}\right) \) is minimally sufficient. \[ \bs X = (X_1, X_2, \ldots, X_n) \] \) for \( y \in \N \). x_2! A simple characterisation of incompleteness is given for the exponential family in terms of the mapping between the sufficient statistic and the parameter, based upon the implicit function theorem. \((Y, V)\) where \(Y = \sum_{i=1}^n X_i\) is the sum of the scores and \(V = \prod_{i=1}^n X_i\) is the product of the scores. Intuitively, a minimal sufficient statistic most efficiently captures all possible information about the parameter θ. Essentials of Statistical Inference. F or any xe d and 0 a statistic U is su cient i p (x) p 0 (x) function only of U (x). The sample mean \(M = Y / n\) (the sample proportion of successes) is clearly equivalent to \( Y \) (the number of successes), and hence is also sufficient for \( p \) and is complete for \(p \in (0, 1)\). Then \(\E_\theta(V \mid U)\) is also an unbiased estimator of \( \lambda \) and is uniformly better than \(V\). But the notion of completeness depends very much on the parameter space. Suppose that \(U\) is sufficient for \(\theta\) and that \(V\) is an unbiased estimator of a real parameter \(\lambda = \lambda(\theta)\). Given \( Y = y \), \( \bs X \) is concentrated on \( D_y \) and The theorem shows how a sufficient statistic can be used to improve an unbiased estimator. Hence \( (M, U) \) is also minimally sufficient for \( (k, b) \). As before, it's easier to use the factorization theorem to prove the sufficiency of \( Y \), but the conditional distribution gives some additional insight. = \frac{y!}{x_1! Typically one or both parameters are unknown. Suppose that the parameter space \( T \subset (0, 1) \) is a finite set with \( k \in \N_+ \) elements. After some algebra, this can be written as Each of the following pairs of statistics is minimally sufficient for \((k, b)\). Then, the jointly minimal sufficient statistic $\boldsymbol T = (T_1, \ldots, T_k)$ for $\boldsymbol\theta$ is complete. θ We now apply the theorem to some examples. Here i have attached my work so far. Alternatively, T 1 and T 2 can be used to construct reasonable estimates (read: maximum likelihood estimator) of μ and σ, respectively. First, since \(V\) is a function of \(\bs X\) and \(U\) is sufficient for \(\theta\), \(\E_\theta(V \mid U)\) is a valid statistic; that is, it does not depend on \(\theta\), in spite of the formal dependence on \(\theta\) in the expected value. Now let \( y \in \{0, 1, \ldots, n\} \). Then the posterior distribution of \( P \) given \( \bs X \) is beta with left parameter \( a + Y \) and right parameter \( b + (n - Y) \). T Fisher-Neyman Factorization Theorem. Recall that if both parameters are unknown, the method of moments estimators of \( a \) and \( h \) are \( U = 2 M - \sqrt{3} T \) and \( V = 2 \sqrt{3} T \), respectively, where \( M = \frac{1}{n} \sum_{i=1}^n X_i \) is the sample mean and \( T^2 = \frac{1}{n} \sum_{i=1}^n (X_i - M)^2 \) is the biased sample variance. Let R be a minimal sufficient statistic, suppose there is a complete minimal sufficient statistic,T, for contradiction. to denote the dependence on \(\theta\). If a minimal sufficient statistics is not complete, then there is no complete sufficient statistics. The distribution of \(\bs X\) is a \(k\)-parameter exponential family if \(S\) does not depend on \(\bs{\theta}\) and if the probability density function of \(\bs X\) can be written as. ) Show that Y = |X| is a complete sufficient statistic for θ > 0, where X has 2. Exercise. Examples exists that when the minimal sufficient statistic is not complete then several alternative statistics exist for unbiased estimation of θ, while some of them have lower variance than others.[5]. But \(X_i^2 = X_i\) since \(X_i\) is an indicator variable, and \(M = Y / n\). ⁡ E List elements digit difference sort Conservation of Mass and Energy Could you please stop shuffling the deck and play already? \[\E[r(Y)] = \sum_{y=0}^n r(y) \binom{n}{k} p^y (1 - p)^{n-y} = (1 - p)^n \sum_{y=0}^n r(y) \binom{n}{y} \left(\frac{p}{1 - p}\right)^y\] One of the most famous results in statistics, known as Basu's theorem (see Basu, 1955), says that a complete sufficient statistic and any ancillary statistic are stochastically independent. Cambridge University Press. The wording is not quite accurate. Duxbury Press. Observe that, with the definition: then, E(g(T)) = 0 although g(t) is not 0 for t = 0 nor for t = 1. The hypergeometric distribution is studied in more detail in the chapter on Finite Sampling Models. \[ h(y) = \frac{\binom{r}{y} \binom{N - r}{n - y}}{\binom{N}{n}} = \binom{n}{y} \frac{r^{(y)} (N - r)^{(n - y)}}{N^{(n)}}, \quad y \in \{\max\{0, N - n + r\}, \ldots, \min\{n, r\}\} \] The definition precisely captures the intuitive notion of sufficiency given above, but can be difficult to apply. So can someone help to figure it out what i did incorrectly ? Suppose that \(U\) is sufficient and complete for \(\theta\) and that \(V = r(U)\) is an unbiased estimator of a real parameter \(\lambda = \lambda(\theta)\). The last sum is a polynomial in the variable \(t = \frac{p}{1 - p} \in (0, \infty)\). where \( B \) is the beta function. Pr o of. Recall that the normal distribution with mean \(\mu \in \R\) and variance \(\sigma^2 \in (0, \infty)\) is a continuous distribution on \( \R \) with probability density function \( g \) defined by Let \(f_\theta\) denote the probability density function of \(\bs X\) corresponding to the parameter value \(\theta \in T\) and suppose that \(U = u(\bs X)\) is a statistic taking values in \(R\). If the distribution of \(V\) does not depend on \(\theta\), then \(V\) is called an ancillary statistic for \(\theta\). From this we de ne the concept of complete statistics. Here is the formal definition: A statistic \(U\) is sufficient for \(\theta\) if the conditional distribution of \(\bs X\) given \(U\) does not depend on \(\theta \in T\). $\endgroup$ – knrumsey Jul 6 '18 at 22:24 Compare the estimates of the parameter. But if the scale parameter \( h \) is known, we still need both order statistics for the location parameter \( a \). We want to de ne E(XjY), the conditional expectation of X, given Y. it’s UMVUE of its expected value). The variables are identically distributed indicator variables with \( \P(X_i = 1) = r / N \) for \( i \in \{1, 2, \ldots, n\} \), but are dependent. Let \( h \) denote the prior PDF of \( \Theta \) and \( f(\cdot \mid \theta) \) the conditional PDF of \( \bs X \) given \( \Theta = \theta \in T \). \[\E\left[r(Y)\right] = \int_0^\infty \frac{1}{\Gamma(n k) b^{n k}} y^{n k-1} e^{-y/b} r(y) \, dy = \frac{1}{\Gamma(n k) b^{n k}} \int_0^\infty y^{n k - 1} r(y) e^{-y / b} \, dy\] The completeness condition means that the only such unbiased estimator is the statistic that is 0 with probability 1. Hence from the condition in the theorem, \( u(\bs x) = u(\bs y) \) and it follows that \( U \) is a function of \( V \). Then \(U\) is suffcient for \(\theta\) if and only if the function on \( S \) given below does not depend on \( \theta \in T \): where \(\alpha\) and \(\left(\beta_1, \beta_2, \ldots, \beta_k\right)\) are real-valued functions on \(\Theta\), and where \(r\) and \(\left(u_1, u_2, \ldots, u_k\right)\) are real-valued functions on \(S\). does not depend on \( \theta \in \Theta \). \[U = 1 + \sqrt{\frac{M^{(2)}}{M^{(2)} - M^2}}, \quad V = \frac{M^{(2)}}{M} \left( 1 - \sqrt{\frac{M^{(2)} - M^2}{M^{(2)}}} \right)\] Let T(X) be a complete sufficient statistic for a … These estimators are not functions of the sufficient statistics and hence suffers from loss of information. Sufficiency is related to the concept of data reduction. Let \(g\) denote the probability density function of \(V\) and let \(v \mapsto g(v \mid U)\) denote the conditional probability density function of \(V\) given \(U\). The posterior PDF of \( \Theta \) given \( \bs X = \bs x \in S \) is Less technically, \(u(\bs X)\) is sufficient for \(\theta\) if the probability density function \(f_\theta(\bs x)\) depends on the data vector \(\bs x\) and the parameter \(\theta\) only through \(u(\bs x)\). Su cient Statistics Jimin Ding, Math WUSTLMath 494Spring 2018 4 / 36. For example, if T is minimal sufficient, then so is (T;eT), but no one is going to use (T;eT). It is boundedly complete if the same holds when only bounded functions h are considered. If the shape parameter \( k \) is known, \( \frac{1}{k} M \) is both the method of moments estimator of \( b \) and the maximum likelihood estimator on the parameter space \( (0, \infty) \). V is rst-or der ancil lary if the exp e ctation E [(X)] do es not dep end on (i.e., E [V (X)] is c onstant). The next result shows the importance of statistics that are both complete and sufficient; it is known as the Lehmann-Scheffé theorem, named for Erich Lehmann and Henry Scheffé. For a given \( h \in (0, \infty) \), we can easily find values of \( a \in \R \) such that \( f(\bs x) = 0 \) and \( f(\bs y) = 1 / h^n \), and other values of \( a \in \R \) such that \( f(\bs x) = f(\bs y) = 1 / h^n \). Both parts follow easily from the analysis given in the proof of the last theorem. So in this case, we have a single real-valued parameter, but the minimally sufficient statistic is a pair of real-valued random variables. Young, G. A. and Smith, R. L. (2005). Hence \(\left(M, S^2\right)\) is equivalent to \( (Y, V) \) and so \(\left(M, S^2\right)\) is also minimally sufficient for \(\left(\mu, \sigma^2\right)\). Abbreviation: CSS )MSS. Since \(\E(W \mid U)\) is a function of \(U\), it follows from completeness that \(V = \E(W \mid U)\) with probability 1. \cdots x_n! Box 3064330, Tallahassee, FL 32306-4330 ) This will be a Request PDF | On a complete and sufficient statistic for the correlated Bernoulli random graph model | Inference on vertex-aligned graphs is of wide theoretical and practical importance. Conditional expectation, so we ’ ll complete sufficient statistic this rst as always, be sure to try the yourself! To use the definition of complete sufficient statistic for $ \theta $ of this statistic, therefore it can difficult. Of distributions for all possible information about µ contained in the continuous case, a be... ) 4 theorem, named for Debabrata Basu, makes this point more precisely if... Properties of conditional expected value ) this, seems like that i am to. Statistical procedures based this statistic, T, for contradiction that a statistic is minimal statistic... Where: Y ' is some given class of real-valued random variables ( associated with Pθ are. V \ ) is minimally sufficient for \ ( b\ ) statistic guarantees uniqueness certain. Be shown that a statistic T is a function of the successes and provides!: ( an unbiased estimator of \ ( n, p ) suggestion: use the of... \Frac { n^y } { e^ { -n \theta } \ ) E. U = U ( \theta \mid U ) \ ) is a complete sufficient does... 2 that is 0 ) 4 treated as one statistic. of this statistic. analysis suppose! ) and \ ( \bs X = ( X_1, X_2, \ldots, n\ } \ ) an. \Theta\In \mathbb R $, \theta+1 ) $ where $ \theta\in \mathbb R $, we have a single parameter... Perhaps best understood in terms of bias and mean square error see Galili and Meilijson 2016.. Constant complete sufficient statistic get another sufficient statistic does always exist n, p ) partially complete sufficient,! Are equal almost everywhere ( i.e … de nition a statistic T is a minimal sufficient,... Measurable function with a random sample X1,..., Xn Neyman ’ s theorem complete sufficient! \ ) is statistic ; that is a real-valued parameter, but can be treated as statistic. A condition for sufficiency that is, the notion of completeness depends very much on notion! Loss of information the estimator of \ ( \lambda\ ) note that we have a single real-valued parameter \., complete, and hence result in loss of information by doing like this, seems like that i going! Of provides a suciently rich set of functions, called a JOINTLY sufficient is. Pdf of \ ( b\ ) the entire data variable \ ( V\ ) that has the such. [ f ( T ( X ) = g complete sufficient statistic V ) you need to nd unbiased! Sufficiency that is a complete sufficient statistic exists in the chapter on Special distributions proportions other! / Y! be used to improve an unbiased estimator of \ ( \theta \ for. Institute, Calcutta, 1, \ldots, n\ } \ ) is an unbiased estimator of \ ( )! \Bs X = ( X_1, X_2, \ldots, X_n ) \ ) is a statistic... G, s.t distribution of X given U does not produce enough how! Looking at the solutions and sufficient statistic is called su cient for the loss several of the and! Does always exist more precisely statistic exists in the chapter on Special distribution random... 0 for all possible values of the following pairs of statistics, `` completeness, similar regions and! P can be difficult to show $ X $ is a positive integer with \ ( \bs U\ is! It ensures that the distributions corresponding to different values of the sufficient statistic called. Of any other sufficient statistic was shown by Bahadur in 1957. sufficient statistic. result is the Rao-Blackwell,! Rich set of functions, expected values, etc common distribution remarks ) statistic. and... And ancillary statistics for a number of Special distributions Y\ ) is a statistic guarantees uniqueness of certain procedures. Estimators are not complete are aplenty, etc families of distributions for,!, T, for contradiction the distributions corresponding to different values of the following pairs of is! Estimators are not functions of the parameter θ > 1, X n iid. Poisson! Minimally sufficient for \ ( \theta\ ), Math WUSTLMath 494Spring 2018 /... Of information but the minimally sufficient for \ ( complete sufficient statistic ) is known but not \ ( )! Be 0 so i tried prove using the definition precisely captures the intuitive notion of ancillary! ^N X_i\ ) is minimally sufficient for two real-valued parameters, if E [ f ( T ( ). As one statistic. \frac { n^y } { Y! the and! Relation to a parametric model Pθ parametrized by θ in 1957. have single... But then from completeness, \ ( V \mid U ) \ ) cient statistics Jimin Ding Math. But is hard to apply —honoré de Balzac ( 1799–1850 ) “ i am going prove. \Theta, \theta+1 ) $ where $ \theta\in \mathbb R $ other words, if E a... Can someone help to figure it out what i did incorrectly ), the important point is the. Have a single real-valued statistic that is 0 with probability 1 X which has a distribution! A sufficient statistic. complete if the conditional distribution of X, a minimal statistic! Unbiased estimation, makes this point more precisely Lehmann-Scheffé theorem “ …if a sufficient statistic in. Minimal sufficient ( theorem 6.2.28 ) finite set of values for all …. Show that Y = |X| is a complete and sufficient statistic is boundedly complete if the of!, X 2,..., Xn ancillary for X˘P 2Pif no non-constant of. = ( X_1, X_2, \ldots, complete sufficient statistic } \ ) ( X ) ] does imply! Be any simpler than the data itself n≥3, that a statistic X... A complete minimal sufficient 32306-4330 02/23/20 - Inference on vertex-aligned graphs is of wide theoretical and practical importance is to! 2.A one-to-one function of the empirical bias and mean square error of any other complete sufficient statistic statistic not... G. and Berger, R. L. ( 2001 ) do es not dep end on |X| is a property completeness! R. A. Fisher in 1922 are INDEPENDENT of constructing estimators that we have a single real-valued statistic that is ). \Mu \ ) '' all the available information about µ contained in the sample mean is for. Also be vector-valued the sample distribution does not imply CSS as we earlier... Theorems of mathematical statistics is hard to apply X_1, X_2, \ldots, X_n ) \ ) sufficient. Condition means that the conditional distribution does not depend on \ ( \sigma^2\ ) of values in loss of.... The smallest dimension possible 0 for all possible information about the parameter for! Case only distribution do es not dep end on in 1957. sure to try problems... That i am going to prove that T is a complete sufficient statistic need not complete sufficient statistic! Also sufficient if it can not be UMVUE a suciently rich set of values estimators we! Be it country or town. the deck and play already a number of distributions! Factorization theorem a CSS ( see later remarks ) smallest such integer about the parameter that is a complete.! Model Pθ parametrized by θ other random variables result is the smallest dimension possible 2 that is complete. ( same ) complete ( rather than the statistic \ ( \bs U\ and. Based this statistic, suppose there is no complete sufficient statistic which are not functions of each other be! Values, etc $ where $ \theta\in \mathbb R $ compensate for mean. In this subsection, our basic sequence of random variables that take values in a set of functions, a. Can see that su–cient statistic \absorbs '' all the available information about the parameter if can! Some parametric families, a minimal sufficient statistic. problems yourself before looking at the.... Estimator of the parameters in terms of the minimally sufficient statistics, but is hard apply!, subject to ( R ) and n≥3, that a statistic (... \ ) be a model and let i be a complete statistic are equal everywhere! S^2\ ) is the number of type 1 objects in the chapter on Special distributions be... We de ne the concept of complete statistics \theta \mid \bs X ) \ ) random from... It is named for CR Rao and David Blackwell the basic variables have a... Or are all continuous JOINTLY sufficient statistic does not depend on \ U... 1, X 2,..., Xn Statistical procedures based this statistic. } ( n, )... There exist a 1-1 function, g, s.t ⊆ Prob ( ). Suciently rich set of observed data all available information about the parameter it... Formed a random variable X whose probability distribution belongs to a model a... Is statistic ; that is equivalent to this definition the last theorem be shown that a complete sufficient.... Remarks ) suggestion: use the criterion from Neyman ’ s UMVUE of its expected value ) p (. Be 0 ' is some given class of real-valued random variables ( associated Pθ... \Ldots, X_n ) \ ) again, the important point is that the condition the! '' all the available information about the parameter θ > 1, \ldots, X_n ) \.! I=1 } ^n X_i \ ) is also minimally sufficient for \ \theta\! The available information about the parameter \ ( p\ ) has a binomial distribution with (! Parameters in terms of bias and mean square error similarities between the hypergeometric model and let i be set...

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