unbiased estimator for geometric distribution

Since the mse of any unbiased estimator is its variance, a UMVUE is ℑ-optimal in mse with ℑ being the class of all unbiased estimators. The function 1/I(θ) is often referred to as the Cram´er-Rao bound (CRB) on the variance of an unbiased estimator of θ. In particular, the arithmetic mean of the observations, $ \overline{X}\; = ( X _ {1} + \dots + X _ {n} ) / n $, Thus, the arithmetic mean is an unbiased estimate of the short-term expected return and the compounded geometric mean an unbiased estimate of the long-term expected return. The sample variance of a random variable demonstrates two aspects of estimator bias: firstly, the naive estimator is biased, which can be corrected by a scale factor; second, the unbiased estimator is not optimal in terms of mean squared error (MSE), which can be minimized by using a different scale factor, resulting in a biased estimator with lower MSE than the unbiased estimator. If $ T = T ( X) $ is a linear function. T = a _ {0} + $$, then it follows from (2) that the statistic, $$ k endstream endobj startxref The practical value of the Rao–Blackwell–Kolmogorov theorem lies in the fact that it gives a recipe for constructing best unbiased estimators, namely: One has to construct an arbitrary unbiased estimator and then average it over a sufficient statistic. Nevertheless, if $ \theta $ $$. Since the expected value of the statistic matches the parameter that it estimated, this means that the sample mean is an unbiased estimator for the population mean. T ( k) This article was adapted from an original article by M.S. Unbiased Estimation Binomial problem shows general phenomenon. There is also a modification of this definition (see [3]). The European Mathematical Society, A statistical estimator whose expectation is that of the quantity to be estimated. \sum _ { k= } 1 ^ { m } a _ {k} T _ {k} ( X) $$, $$ This short video presents a derivation showing that the sample mean is an unbiased estimator of the population mean. �R��!%R+\��g6�._�R-&��:�+̺�2Ө��I��0"Sq��Rs�TN( ��%ZQb��K�ژ�dgh��. 205. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. https://encyclopediaofmath.org/index.php?title=Unbiased_estimator&oldid=49645, E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1959), L.B. \sum _ { r= } 1 ^ \infty \theta ^ {k} ( 1 - \theta ) ^ {n-} k ,\ 0 < \theta < 1 . is such that, $$ The uniformly minimum variance unbiased estimator of the probability in the geometric distribution with unknown truncation parameter is constructed. 192 {\mathsf E} [ X ( X - 1 ) \dots ( X - k + 1 ) ] = {\mathsf E} This theorem asserts that if the family $ \{ {\mathsf P} _ \theta \} $ Kolmogorov [1] has shown that this only happens for polynomials of degree $ m \leq n $. Normally we also require that the inequality be strict for at least one . This page was last edited on 7 June 2020, at 14:59. \frac{1}{n} 1 & \textrm{ if } X = 1 , \\ the observation of $ X $ is the best point estimator of $ \theta $ f ^ { \prime } ( \theta ) ^ {2} , Derive an unbiased estimator of θ that depends on the data only through the suﬃcient statistic Pn i=1 Xi. \end{array} then the statistic $ T ^ {*} = {\mathsf E} _ \theta \{ T \mid \psi \} $ Klebanov, Yu.V. is an arbitrary unbiased estimator of a function $ f ( \theta ) $, This fact implies, in particular, that the statistic. $ 0 < \theta < 1 $. it follows that $ T _ {k} ( X) $ A proof that the sample variance (with n-1 in the denominator) is an unbiased estimator of the population variance. $ 0 \leq \theta \leq 1 $); Let us obtain an unbiased estimator of \theta. Approximate 100(1 = )% CI for : ^ pz 2 nI(^ ) Example (exponential model) Lifetimes of ve batteries measured in hours x $$. Regarding the mention of the log-normal distribution in a comment, what holds is that the geometric mean (G M) of the sample from a log-normal distribution is a biased but asymptotically consistent estimator of the median. Q ( z) = {\mathsf E} \{ z ^ {X} \} = \ Show that 2Y3 is an unbiased estimator of θ. Founded in 2005, Math Help Forum is dedicated to free math help and math discussions, and our math community welcomes students, teachers, educators, professors, mathematicians, engineers, and scientists. Suppose p(x;θ) satisﬁes, in addition to (i) and (ii), the following … that is, $ {\mathsf E} \{ F _ {n} ( x) \} = F ( x) $, {\mathsf E} _ {0} \{ T ( X) \} = g _ {z} ( \theta ) = \ is complete on $ [ 0 , 1 ] $, k = r , r + 1 ,\dots . in Example 5 is an efficient unbiased estimator of the parameter $ \theta $ "Note on the Unbiased Estimation of a Function of the Parameter of the Geometric Distribution" by Tamas Lengyel and since $ T _ {k} ( X) $ The geometric distribution of the number Y of failures before the first success is infinitely divisible, i.e., for any positive integer n, there exist independent identically distributed random variables Y 1, ... which yields the bias-corrected maximum likelihood estimator A more general definition of an unbiased estimator is due to E. Lehmann [2], according to whom a statistical estimator $ T = T ( X) $ \theta ^ {r} ( 1 - \theta ) ^ {k} ,\ \ i = 1 \dots n . and the function $ f ( \theta ) $, k $$, where $ I ( \theta ) $ Suppose that a random variable $ X $ Umvu estimator of the probability in the geometric distribution with unknown truncation parameter: Communications in Statistics - Theory and Methods: Vol 15, No 8 is an unbiased estimator of $ \theta $. . k $$. Mean square error is our measure of the quality of unbiased estimators, so the following definitions are natural. \right \} = \theta ^ {k} , n ^ {[} k] ( z \theta + q ) ^ {n - k } \theta ^ {k} . $$, $$ This is because, for the lognormal distribution it holds that E (X s) = … {\mathsf D} \{ T \} = Yu.V. It only gives an approximate value for the true value of the quantity to be estimated; this quantity was not known before the experiment and remains unknown after it has been performed. Geometric distribution Geom(p): ... asymptotically unbiased, consistent, and asymptotically e cient (has minimal variance), ... Cramer-Rao inequality: if is an unbiased estimator of , then Var( ) 1 nI( ). T ( k) \theta ( 1 - \theta ) ^ {k-} 1 = \theta . In other words, d(X) has ﬁnite variance for every value of the parameter and for any other unbiased estimator d~, Var d(X) Var d~(X): The efﬁciency of unbiased estimator d~, e(d~) = Var d(X) Var d~(X): Thus, the efﬁciency is between 0 and 1. {\mathsf P} \{ X = k \mid n , \theta \} = \ in the sense of minimum quadratic risk in the class of all unbiased estimators. Evidently, $ T $ Complement to Lecture 7: "Comparison of Maximum likelihood (MLE) and Bayesian Parameter Estimation" A modification of the MLE estimator (modified MLE) has been derivedin which case the bias is reduced. By definition, an estimator is a function t mapping the possible outcomes N + = { 1, 2, 3, … } to the reals. ( z \theta + q ) ^ {n} ,\ \ is complete, the statistic $ T ^ {*} $ (14.1) If b. d( )=0for all values of the parameter, then d(X) is called an unbiased estimator. %%EOF Klebanov, "A general definition of unbiasedness", L.B. \left ( \frac{1}{n ^ {[} k] } of the parameter $ \theta $ More generally, the statistic, $$ Suppose that a random variable $ X $ $$. $$, $$ \theta ( 1 - \theta ) \frac{1}{n ^ {[} k] } \geq {\mathsf E} _ \theta \{ L ( \theta , T ( X) ) \} \ \ = \ f ( \theta ) = a _ {0} + By saying “unbiased”, it means the expectation of the estimator equals to the true value, e.g. h�bbd``b`�$. The geometric distribution is a common discrete distribution in modeling the life time of a device in reliability theory. Any estimator that is not unbiased is called biased. $$. {\mathsf E} _ \theta \{ L ( \theta ^ \prime , T( X) ) \} Naturally, an experimenter is interested in the case when the class of unbiased estimators is rich enough to allow the choice of the best unbiased estimator in some sense. {\mathsf E} _ \theta \{ T \} = \ Hence, we take \hat\theta=X_ { (n)} as an estimator of \theta and check whether it is unbiased. The maximum likelihood estimator (MLE) and uniformly minimum variance unbiased estimator (UMVUE) for the parameters of a multivariate geometric distribution (MGD) have been derived. 12, 2019. that is, for any $ k = 0 \dots n $, $$ hold and T(X) is an unbiased estimator of ψ(θ) = θ. \left ( \begin{array}{c} More generally, the statistic. In this case a sufficient statistic is $ X = X _ {1} + {} \dots + X _ {n} $, be a random variable subject to the geometric distribution with parameter of success $ \theta $, �߅�|��6H4��V��G��6�֓'PW��aѺ2[�Ni�V�Y=؄^�-:B�[��Dc��);zf�b_��u�$U Q ^ {(} k) ( 1) = \ An unbiased estimator is frequently called free of systematic errors. This page describes the definition, expectation value, variance, and specific examples of the geometric distribution. taking values in a probability space $ ( \mathfrak X , \mathfrak B , {\mathsf P} _ \theta ) $, that is, for any integer $ k = 0 , 1 \dots $, $$ �xDo��Geb��K F�A��x,�x�;z"Ja��b��3� �d, �t��I\�M`pa�{�m��&��6��l|%��6A�gL�DV��_M�K�Ht /F�� T = c _ {1} X _ {1} + \dots + c _ {n} X _ {n} ,\ \ + E [Xn])/n = (nE [X1])/n = E [X1] = μ. Thus, if under the conditions of Example 5 one takes as the function to be estimated $ f ( \theta ) = 1 / \theta $, Determine the joint pdf of Y3 and the sufficient statistic Y5 for θ. Kolmogorov [1] has considered the problem of constructing unbiased estimators, in particular, for the distribution function of a normal law with unknown parameters. %PDF-1.5 %�� www.springer.com then $ T $ ��m�k��M��ǽ*Y��ڣ��i#��ߊ7_|ډ3/p V��Y��1��兂Jv��yL�f�]}Bȷ@��(��6�:��/WVa,-) �J��k endstream endobj 61 0 obj <>>>/Filter/Standard/Length 128/O(%ƻ_*�&KŮA�XenMR.��T=q�x�6�#)/P -1340/R 4/StmF/StdCF/StrF/StdCF/U(�u��0�s�iZJ` )/V 4>> endobj 62 0 obj <> endobj 63 0 obj <> endobj 64 0 obj <>stream {\mathsf P} \{ X = k \mid \theta \} = \theta ( 1 - \theta ) relative to any convex loss function for all $ \theta \in \Theta $. for $ 1 / \theta $. c _ {1} + \dots + c _ {n} = 1 , Lehmann-Sche e now clari es everything. is an unbiased estimator of $ \theta $. 2 Biased/Unbiased Estimation In statistics, we evaluate the “goodness” of the estimation by checking if the estimation is “unbi-ased”. 98 0 obj <>/Encrypt 61 0 R/Filter/FlateDecode/ID[<360643C4CBBDDCE537B2AF07AF860660><55FF3F73DBB44849A4C81300CF0D9128>]/Index[60 81]/Info 59 0 R/Length 149/Prev 91012/Root 62 0 R/Size 141/Type/XRef/W[1 2 1]>>stream if E[x] = then the mean estimator is unbiased. We introduce different types of estimators such as the maximum likelihood, method of moments, modified moments, L -moments, ordinary and weighted least squares, percentile, maximum product of spacings, and minimum distance estimators. In connection with this example the following question arises: What functions $ f ( \theta ) $ Hint: If U and V are i.i.d. \\ $$. e ^ {\theta ( z- 1) } , If varθ(U) ≤ varθ(V) for all θ ∈ Θ then U is a uniformly better estimator than V. {\mathsf P} \{ X = k \mid \theta \} = \ . is chosen. geometric(θ) random variables, then (U − 1) + (V − 1) has the negative binomial(2,θ) distribution. T ( X) = \ of the binomial law, since, $$ is the only, hence the best, unbiased estimator of $ \theta ^ {k} $. ( z \theta + q ) ^ {n - k } \theta ^ {k\ } = $$, A statistical estimator for which equality is attained in the Rao–Cramér inequality is called efficient (cf. \right ] ^ {2} \right \} = \ If this is to be unbiased, then--writing q = 1 − p --the expectation must equal 1 − q for all q in the interval [ 0, 1]. 1 \leq m \leq n , Efficient estimator). If the family $ \{ {\mathsf P} _ \theta \} $ Recall that if U is an unbiased estimator of λ, then varθ(U) is the mean square error. a function $ f : \Theta \rightarrow \Omega $ carries no useful information on $ \theta $. has to be estimated, mapping the parameter set $ \Theta $ If $ T ( X) $ Let $ X $ \left \{ The generating function $ Q( z) $ In that case the statistic $ a T + b $ . is called unbiased relative to a loss function $ L ( \theta , T ) $ Linnik and his students (see [4]) have established that under fairly wide assumptions the best unbiased estimator is independent of the loss function. and that as an estimator of $ f ( \theta ) $ into a certain set $ \Omega $, ��N�@B�OG��"��%��%1I 5��8-*��p� R9�B̓�s��q�&��8��5yJ��OQd(��f��|��$�T��X�y�6C�'��S��f� \right. Since \theta is the upper bound for the sample realization, the value from the sample that is closer to \theta is X_ { (n)}, the maximum of the sample. g _ {z} ( \theta ) = \mathop{\rm exp} \{ \theta ( z - 1 ) \} , Let $ X _ {1} \dots X _ {n} $ \theta , \theta ^ \prime \in \Theta . is uniquely determined. The geometric distribution on $ \N $ with success parameter $ p \in (0, 1) $ has probability density function \[ g(x) = p (1 - p)^x, \quad x \in \N \] This version of the geometric distribution governs the number of failures before the first success in a sequence of Bernoulli trials. {\mathsf D} \{ T \} \geq T _ {k} ( X) = \ This is reasonable as one may think of the compounded geometric mean … and $ 1 / n $. {\mathsf P} \{ X = k \mid r , \theta \} = \ \begin{array}{ll} We have considered different estimation procedures for the unknown parameters of the extended exponential geometric distribution. \right ) is expressed in terms of the sufficient statistic $ X $ For the geometric distribution Geometric[p], we prove that exactly the functions that are analytic at p = 1 have unbiased estimators and present the best estimators. is an unbiased estimator for a function $ f ( \theta ) $, that is, $$ That is, the Rao–Blackwell–Kolmogorov theorem implies that unbiased estimators must be looked for in terms of sufficient statistics, if they exist. r + k - 1 \\ Examples of Parameter Estimation based on Maximum Likelihood (MLE): the exponential distribution and the geometric distribution. Let $ X $ T ( X) = \ Since X = Y=nis an unbiased function of Y, this is the unique MVUE; there is no other unbiased estimator that achieves the same variance. and $ \theta $( \sum _ { k= } 0 ^ \infty th derivative, $$ Suppose that U and V are unbiased estimators of λ. the Rao–Cramér inequality implies that, $$ \tag{1 } have the same Poisson law with parameter $ \theta $, Suppose that the independent random variables $ X _ {1} \dots X _ {n} $ In this case the empirical distribution function $ F _ {n} ( x) $ has the binomial law with parameters $ n $ An unbiased estimator T(X) of ϑ is called the uniformly minimum variance unbiased estimator (UMVUE) if and only if Var(T(X)) ≤ Var(U(X)) for any P ∈ P and any other unbiased estimator U(X) of ϑ. \left ( \begin{array}{c} must hold for it, which is equivalent to, $$ 60 0 obj <> endobj 0 . X ^ {[} k] To compare ^and ~ , two estimators of : Say ^ is better than ~ if it has uniformly smaller MSE: MSE^ ( ) MSE ~( ) for all . \mathop{\rm log} [ \theta ^ {X} Proposition. Now … $$, $$ and the system of functions $ 1 , x , x ^ {2} \dots $ a _ {1} \theta + \dots + a _ {m} \theta ^ {m} ,\ \ by itself is an unbiased estimator of its mathematical expectation $ \theta $. is an unbiased estimator of $ \theta $. E [ (X1 + X2 + . it must satisfy the unbiasedness equation $ {\mathsf E} \{ T \} = \theta $, 0 & \textrm{ if } X \geq 2 . The generating function of this law, which can be expressed by the formula, $$ is very close to 1 or 0, otherwise $ T $ X ^ {[} k] n) based on a distribution having parameter value , and for d(X) an estimator for h( ), the bias is the mean of the difference d(X)h( ), i.e., b. d( )=E. Assume an i.i.d. n \\ $$. Applying the definition of expectation to the formula for the probabilities of a … $$, $$ Moreover, an unbiased estimator, like every point estimator, also has the following deficiency. for ECE662: Decision Theory. X ^ {[} r] = X ( X - 1 ) \dots ( X - r + 1 ) ,\ r = 1 , 2 \dots $ | x | < \infty $. that is, $$ $$. Let $ X _ {1} \dots X _ {n} $ Rukhin, "Unbiased estimation and matrix loss functions", S. Zacks, "The theory of statistical inference" , Wiley (1971). \frac{1}{I ( \theta ) } �]��(�!I�Uww��g� j4 [�gR]�� iG/3n��iK�(�l�}P�ט;�c�BĻ; ��b��P�t��H�@��p�$m��82WT��!^�C��B䑕�Vr)��g6��KtQ�� 3xUՓ1*��-=ى�+�F�Zї.�#�&�3�6]��Z^��`,�D~i5;2J��#�F�8��l�4�d�)�x�1(��}Md%67�{弱p/x�G�}x�L�z�t#�,�%�� y�2�-��+92w4��H�l��7R;�h"*:��:�E�y}��mq��ܵ��r\�_��>�"�4�U��DS��x/��܊�pA��}G�{�0�倐��V{��v�s be random variables having the same expectation $ \theta $, (c) Is the estimator … Linnik, A.L. is called an unbiased estimator of $ f ( \theta ) $. For the geometric distribution Geometric[p], we prove that exactly the functions that are analytic at p= 1 have unbiased estimators and present the best estimators. Namely, if $ T = T ( X) $ obtained by averaging $ T $ $$, $$ \tag{2 } $$. {\mathsf E} \{ X _ {1} \} = \dots = {\mathsf E} \{ X _ {n} \} = \theta . \frac{n}{\theta ( 1 - \theta ) } {\mathsf E} \{ | T - f ( \theta ) | ^ {2} \} ( 1 - \theta ) ^ {n-} X $$, $$ \frac \partial {\partial \theta } I ( \theta ) = {\mathsf E} In turn, an unbiased estimator of, say, f ( \theta ) = \theta ^ {2} is X ( X - 1 ) . is an unbiased estimator of $ F ( x) $, n ( n - 1 ) \dots ( n - k + 1 ) $ \theta \in \Theta $, For example, the Rao–Cramér inequality has a simple form for unbiased estimators. admit an unbiased estimator? \textrm{ for all } \ and assume that $ f ( \theta ) = a \theta + b $ If $ T $ The next example shows that there are cases in which unbiased estimators exist and are even unique, but they may turn out to be useless. In turn, an unbiased estimator of, say, $ f ( \theta ) = \theta ^ {2} $ In particular, Xis the only e cient estimator. \frac{1}{n} and $ \theta $, {\mathsf E} \left \{ is the Fisher amount of information for $ \theta $. is $ X ( X - 1 ) $. \begin{array}{ll} \right .$$. that is, $ {\mathsf E} \{ T \} = \theta $, constructed from the observations $ X _ {1} \dots X _ {n} $ Suppose that in the realization of a random variable $ X $ . \theta > 0 . Let $ T = T ( X) $ {\mathsf D} \{ T \} = \ \end{array} In this context an important role is played by the Rao–Blackwell–Kolmogorov theorem, which allows one to construct an unbiased estimator of minimal variance. It is known that the best unbiased estimator of the parameter $ \theta $( be a random variable having the binomial law with parameters $ n $ that admits a power series expansion in its domain of definition $ \Theta \subset \mathbf R _ {1} ^ {+} $. q = 1 - \theta , of this law can be expressed by the formula, $$ is expressed in terms of the sufficient statistic $ X $ admits an unbiased estimator, then the unbiasedness equation $ {\mathsf E} \{ T ( X) \} = f ( \theta ) $ $$, which implies that for any integer $ k = 1 \dots n $, Thus, there is a lower bound for the variance of an unbiased estimator of $ f ( \theta ) $, X ^ { [} r] = X ( X - 1 ) \dots ( X - r + 1 ) ,\ r = 1 , 2 \dots. is good only when $ \theta $ more precise goal would be to ﬁnd an unbiased estimator dthat has uniform minimum variance. then (see Example 6) there is no unbiased estimator $ T ( X) $ $$, is an unbiased estimator of $ f ( \theta ) = \theta ^ {r} $. = f ( \theta ) . e ^ {- \theta } ,\ \ has a risk not exceeding that of $ T $ that is, for any natural number $ k $, $$ $ \theta > 0 $. $$. 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