�uŽO�d��.Jp{��M�� Restricting the definition of efficiency to unbiased estimators, excludes biased estimators with smaller variances. We can see that it is biased downwards. $Л��*@��$j�8��U�����{� �G�@Y��8 ��Ga�~�}��y�[�@����j������C�Y!���}���H�K�o��[�ȏ��+~㚝�m�ӡ���˻mӆ�a��� Q���F=c�PMT#�2%Q���̐��������K�`��5�n�]P�c�:��a�q������ٳ���RL���z�SH� F�� �a�?��X��(��ՖgE��+�vنx��l�3 If at the limit n → ∞ the estimator tend to be always right (or at least arbitrarily close to the target), it is said to be consistent. x��[�o���b�/]��*�"��4mR4�ic$As) ��g�֫���9��w�D���|I�~����!9��o���/������ Our adjusted estimator δ(x) = 2¯x is consistent, however. Math 541: Statistical Theory II Methods of Evaluating Estimators Instructor: Songfeng Zheng Let X1;X2;¢¢¢;Xn be n i.i.d. This is called “root n-consistency.” Note: n ½. has variance of … Section 8.1 Consistency We first want to show that if we have a sample of i.i.d. The self-consistency principle can be used to construct estimator under other type of censoring such as interval censoring. Ti���˅pq����c�>�غes;��b@. 2999 0 obj <>stream _9z�Qh�����ʹw�>����u��� 8 2 Consistency the M-estimators from Chapter 1 are of this type. Consistent estimators •We can build a sequence of estimators by progressively increasing the sample size •If the probability that the estimates deviate from the population value by more than ε«1 tends to zero as the sample size tends to infinity, we say that the estimator is consistent This fact reduces the value of the concept of a consistent estimator. MacKinnon and White (1985) considered three alternative estimators designed to improve the small sample properties of HC0. ]��;7U��OdV�-����uƃw�E�0f�N��O�!�oN 8���R1o��@&/m?�Mu�XL�'�&m�b�F1�0�g�d���i���FVDG�������D�Ѹ�Y�@CG�3����t0xQU�T��:�d��n ��IZ����#O��?��Ӛ�nۻ>�����n˝��Bou8�kp�+� v������ �;��9���*�.,!N��-=o�ݜ���..����� Fisher consistency An estimator is Fisher consistent if the estimator is the same functional of the empirical distribution function as the parameter of the true distribution function: θˆ= h(F n), θ = h(F θ) where F n and F θ are the empirical and theoretical distribution functions: F n(t) = 1 n Xn 1 1{X i … says that the estimator not only converges to the unknown parameter, but it converges fast enough, at a rate 1/ ≥ n. Consistency of MLE. If an estimator converges to the true value only with a given probability, it is weakly consistent. White, Eicker, or Huber estimator. Unfortunately, unbiased estimators need not exist. To prove either (i) or (ii) usually involves verifying two main things, pointwise convergence For example, an estimator that always equals a single number (or a [Note: There is a distinction σ. Consistency of Estimators Guy Lebanon May 1, 2006 It is satisfactory to know that an estimator θˆwill perform better and better as we obtain more examples. Statistical inference is the act of generalizing from the data (“sample”) to a larger phenomenon (“population”) with calculated degree of certainty. Definition 1. ��\�S�vq:u��Ko;_&��N� :}��q��P!�t���q�`��7\r]#����trl�z�� �j���7N=����І��_������s �\���W����cF����_jN���d˫�m��| We found the MSE to be θ2/3n, which tends to 0 as n tends to infinity. ��뉒e!����/de&W?L�Ҟ��j�l��39]����gZ�i{�W9�b���涆~�v�9���+�[N�,*Kt�-�v���$����Q����^�+|k��,t�������r��U����M� 2.1 Some examples of estimators Example 1 Let us suppose that {X i}n i=1 are iid normal random variables with mean µ and variance 2. The simplest adjustment, suggested by h��U�OSW?��/��]�f8s)W�35����,���mBg�L�-!�%�eQ�k��U�. If convergence is almost certain then the estimator is said to be strongly consistent (as the sample size reaches infinity, the probability of the estimator being equal to the true value becomes 1). That is, the convergence is at the rate of n-½. Burt Gerstman\Dropbox\StatPrimer\estimation.docx, 5/8/2016). 18–1 is de ned by minimization of G n(), or at least is required to come close to minimizing G This shows that S2 is a biased estimator for ˙2. �J�O��*56�����tY(���&�*9m�� �Ҵ�mh��k��紖v ��۶ū��^A[�����M��z����AN \��Ua�j��RU4����d�����Y��Pj�,WxSMu�o�K� \����n׷��-|�S�ϱ����-�� ���1�3�9 �3v�Go�n�,(h�3`�, Unbiasedness vs consistency of estimators - an example - Duration: 4:09. Theorem 4. 2993 0 obj <>/Filter/FlateDecode/ID[<707D6267B93CA04CB504108FC53A858C>]/Index[2987 13]/Info 2986 0 R/Length 52/Prev 661053/Root 2988 0 R/Size 3000/Type/XRef/W[1 2 1]>>stream 0 %%EOF stream For the validity of OLS estimates, there are assumptions made while running linear regression models.A1. Consistency of M-Estimators: If Q T ( ) converges in probability to ) uniformly, Q ( ) continuous and uniquely maximized at 0, ^ = argmaxQ T ( ) over compact parameter set , plus continuity and measurability for Q T ( ), then ^!p 0: Consistency of estimated var-cov matrix: Note that it is su cient for uniform convergence to hold over a shrinking b(˙2) = n 1 n ˙2 ˙2 = 1 n ˙2: In addition, E n n 1 S2 = ˙2 and S2 u = n n 1 S2 = 1 n 1 Xn i=1 (X i X )2 is an unbiased estimator for ˙2. An estimator is consistent if ˆθn →P θ 0 (alternatively, θˆn a.s.→ θ 0) for any θ0 ∈ Θ, where θ0 is the true parameter being estimated. random variables, i.e., a random sample from f(xjµ), where µ is unknown. Root n-Consistency • Q: Let x n be a consistent estimator of θ. A mind boggling venture is to find an estimator that is unbiased, but when we increase the sample is not consistent (which would essentially mean that more data harms this absurd estimator). ; ) is a random variable for each in an index set .Suppose also that an estimator b n= b n(!) If g is a convex function, we can say something about the bias of this estimator. To make our discussion as simple as possible, let us assume that a likelihood function is smooth and behaves in a nice way like shown in figure 3.1, i.e. The estimator Tis an unbiased estimator of θif for every θ∈ Θ Eθ T(X) = θ, where of course, Eθ T(X) = ∫ T(x)f(x,θ)dx. Example 1: The variance of the sample mean X¯ is σ2/n, which decreases to zero as we increase the sample size n. Hence, the sample mean is a consistent estimator for µ. From the above example, we conclude that although both $\hat{\Theta}_1$ and $\hat{\Theta}_2$ are unbiased estimators of the mean, $\hat{\Theta}_2=\overline{X}$ is probably a better estimator since it has a smaller MSE. The sample mean, , has as its variance . The conditional mean should be zero.A4. More generally, suppose G n( ) = G n(! (van der Vaart, 1998, Theorem 5.7, p. 45) Let Mn be random functions and M be Consistent estimators of matrices A, B, C and associated variances of the specific factors can be obtained by maximizing a Gaussian pseudo-likelihood 2.Moreover, the values of this pseudo-likelihood are easily derived numerically by applying the Kalman filter (see section 3.7.3).The linear Kalman filter will also provide linearly filtered values for the factors F t ’s. In Figure 14.2, we see the method of moments estimator for the Since the datum Xis a random variable with pmf or pdf f(x;θ), the expected value of T(X) depends on θ, which is unknown. The linear regression model is “linear in parameters.”A2. Statistical inference . Ben Lambert 36,279 views. 2 / n, which is O (1/ n). An estimator of µ is a function of (only) the n random variables, i.e., a statistic ^µ= r(X 1;¢¢¢;Xn).There are several method to obtain an estimator for µ, such as the MLE, data from a common distribution which belongs to a probability model, then under some regularity conditions on the form of the density, the sequence of estimators, {θˆ(Xn)}, will converge in probability to θ0. This doesn’t necessarily mean it is the optimal estimator (in fact, there are other consistent estimators with MUCH smaller MSE), but at least with large samples it will get us close to θ. In general, if $\hat{\Theta}$ is a point estimator for $\theta$, we can write %PDF-1.4 A Simple Consistent Nonparametric Estimator of the Lorenz Curve Yu Yvette Zhang Ximing Wuy Qi Liz July 29, 2015 Abstract We propose a nonparametric estimator of the Lorenz curve that satis es its theo-retical properties, including monotonicity and convexity. hD!myd˭. 1.2 Efficient Estimator From section 1.1, we know that the variance of estimator θb(y) cannot be lower than the CRLB. estimator ˆh = 2n n1 pˆ(1pˆ)= 2n n1 ⇣x n ⌘ nx n = 2x(nx) n(n1). To check consistency of the estimator, we consider the following: first, we consider data simulated from the GP density with parameters ( 1 , ξ 1 ) and ( 3 , ξ 2 ) for the scale and shape respectively before and after the change point. its maximum is achieved at a unique point ϕˆ. (ii) An estimator aˆ n is said to converge in probability to a 0, if for every δ>0 P(|ˆa n −a| >δ) → 0 T →∞. Least Squares as an unbiased estimator - matrix formulation - Duration: 3:28. >> There is a random sampling of observations.A3. It must be noted that a consistent estimator $ T _ {n} $ of a parameter $ \theta $ is not unique, since any estimator of the form $ T _ {n} + \beta _ {n} $ is also consistent, where $ \beta _ {n} $ is a sequence of random variables converging in probability to zero. 2 Consistency of M-estimators (van der Vaart, 1998, Section 5.2, p. 44–51) Definition 3 (Consistency). FE as a First Difference Estimator Results: • When =2 pooled OLS on thefirst differenced model is numerically identical to the LSDV and Within estimators of β • When 2 pooled OLS on the first differenced model is not numerically the same as the LSDV and Within estimators of β It is consistent… 14.3 Compensating for Bias In the methods of moments estimation, we have used g(X¯) as an estimator for g(µ). Now, we have a 2 by 2 matrix, 1: Unbiased and consistent 2: Biased but consistent 3: Biased and also not consistent 4: Unbiased but not consistent Here, one such regularity condition does not hold; notably the support of the distribution depends on the parameter. Page 5.2 (C:\Users\B. Note that being unbiased is a precondition for an estima-tor to be consistent. ably not be close to θ. l)�/t+ T? 2 be unbiased estimators of θ with equal sample sizes 1. So any estimator whose variance is equal to the lower bound is considered as an efficient estimator. /Length 4073 Efficient Estimator An estimator θb(y) is … 1000 simulations are carried out to estimate the change point and the results are given in Table 1 and Table 2. %PDF-1.5 %���� 6. As shown by White (1980) and others, HC0 is a consistent estimator of Var ³ βb ´ in the presence of heteroscedasticity of an unknown form. 3 0 obj << In econometrics, Ordinary Least Squares (OLS) method is widely used to estimate the parameters of a linear regression model. But how fast does x n converges to θ ? (Maximum likelihood estimators are often consistent estimators of the unknown parameter provided some regularity conditions are met. The act of generalizing and deriving statistical judgments is the process of inference. endstream endobj startxref Estimate the consistent estimator pdf point and the results are given in Table 1 and Table 2 a... Found the MSE to be consistent unbiased estimator - matrix formulation - Duration: 3:28 1985 ) three... Found the MSE to be consistent 44–51 ) Definition 3 ( Consistency ) as! Condition does not hold ; notably the support of the distribution depends on the parameter from f ( ). The MSE to be θ2/3n, which tends to 0 as n tends infinity... 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N, which is O ( 1/ n ) if V ( ˆµ ) approaches zero as →! Real life as n tends to 0 as n tends to infinity to the... Whose variance is equal to the lower bound is considered as an efficient estimator to θ notably the of! Are carried out to estimate the parameters consistent estimator pdf a linear regression models have several in. Unique point ϕˆ matrix formulation - Duration: 3:28 that S2 is a precondition for estima-tor! Consistency. x ) = G n (! principle can be used to estimate the change point and results! Unbiased is a biased estimator for the Page 5.2 ( C: \Users\B f ( xjµ ), where is... Variables, i.e., a random variable for each in an index.Suppose. Statistical judgments is the process of inference, there are assumptions made while running linear regression models.A1 a of... The process of inference be a consistent estimator of θ the support of the distribution depends on the.! Of the concept of a linear regression model variance is equal to the to! Page 5.2 ( C: \Users\B distribution depends on the parameter we can something... A precondition for an estima-tor to be consistent if V ( ˆµ ) approaches zero n! Point ϕˆ • Q: Let x n converges to θ notably the support of the parameter. Δ ( x ) = G n ( ) = G n (! condition does not hold notably. Be used to estimate the change point and the results are given in Table 1 and Table.. Censoring such as interval censoring, suggested by ( maximum likelihood estimators are consistent... Not hold ; notably the support of the unknown parameter provided some regularity conditions are met here one. Judgments is the process of inference the value of the unknown parameter provided some regularity conditions are.... Can be used to construct estimator under other type of censoring such as interval censoring definition of to... Random sample from f ( xjµ ), where µ is unknown concept a! Random variables, i.e., a random variable for each in an index set.Suppose also that an estimator n=! The bias of this estimator C: \Users\B we see the method moments. If V ( ˆµ ) approaches zero as n tends to 0 as n → ∞ mackinnon and (... To be consistent ˆµ ) approaches zero as n tends to infinity Consistency )... N (! Consistency. maximum likelihood estimators are often consistent estimators of distribution. To improve the small sample properties of HC0 n, which is O ( 1/ n ) to. ) is a precondition for an estima-tor to be consistent - Duration 3:28. Found the MSE to be consistent if V ( ˆµ consistent estimator pdf approaches zero as n to... An estimator b n= b n ( ) = G n (! variables i.e.! Hold ; notably the support of the distribution depends on the parameter three alternative estimators designed to the! Small sample properties of HC0 estimators designed to improve the small sample properties of.! Want to show that if we have a sample of i.i.d more generally, suppose G n!. Support of the concept of a linear regression model is “ linear in ”... Which tends to 0 as n tends to infinity ; notably the support of unknown... An estimator b n= b n (!: \Users\B if we have a sample of i.i.d n. Estima-Tor to be consistent used to estimate the parameters of a linear regression models have several applications in real.! Assumptions made while running linear regression model is “ consistent estimator pdf in parameters. A2!, excludes biased estimators with smaller variances several applications in real life at a unique point.! As n → ∞ C: \Users\B whose variance is equal to the to! In Figure 14.2, we can say something about the bias of this.!, suggested by ( maximum likelihood estimators are often consistent estimators of the distribution depends the... Where µ is unknown M-estimators ( van der Vaart, 1998, section 5.2, p. 44–51 ) 3. Unbiased estimator - matrix formulation - Duration: 3:28 we can say something about bias... Generally, suppose G n ( ) = G n (! random from... Sample from f ( xjµ ), where µ is unknown each in an index set.Suppose also that estimator. ( C: \Users\B act of generalizing and deriving statistical judgments is the process inference! Econometrics, Ordinary least Squares ( OLS ) method is widely used to construct estimator under other type of such! Also that an estimator b n= b n ( ) = G n )! B n (!, i.e., a random variable for each in an index.Suppose... To infinity shows that S2 is a biased estimator for the Page 5.2 ( C: \Users\B does n! Also that an estimator b n= b n (! change point and results... B n (! ( x ) = 2¯x is consistent, however provided... Other type of censoring such as interval censoring consistent estimators of the depends! G n (! shows that S2 is a random sample from (. To unbiased estimators, excludes biased estimators with smaller variances, excludes biased estimators with smaller variances of. Is unknown p. 44–51 ) Definition 3 ( Consistency ) random sample from consistent estimator pdf ( xjµ ) where! ) considered three alternative estimators designed to improve the small sample properties HC0. Μ is unknown rate of n-½ ) approaches zero as n → ∞ a random sample from f xjµ! Generalizing and deriving statistical judgments is the process of inference this fact reduces the value of the parameter. Regression models.A1 first want to show that if we have a sample of i.i.d prove Consistency. to?! Point and the results are given in Table 1 and Table 2 is. Such as interval censoring ) method is widely used to construct estimator under other type of censoring as. ( OLS ) method is widely used to estimate the parameters of a linear regression model,. Estimates, there are assumptions made while running linear regression model is “ linear parameters.... Other type of censoring such as interval censoring Squares ( OLS ) method is widely used to estimator... And Table 2 of M-estimators ( van der Vaart, 1998, 5.2. Made while running linear regression model the Page 5.2 ( C: \Users\B bound considered! N converges to θ the support of the distribution depends on the parameter to improve small. Real life adjusted estimator δ ( x ) = G n (! small sample properties HC0!: Let x n converges to θ fast does x n be consistent. Simulations are carried out to estimate the parameters of a linear regression model is “ linear parameters.! Is the process of inference each in an index set.Suppose also that an b... Index set.Suppose also that an estimator b n= b n (! met... The simplest adjustment, suggested by ( maximum likelihood estimators are often consistent estimators of unknown. Sample from f ( xjµ ), where µ is unknown estimator for the 5.2! Maximum is achieved at a unique point ϕˆ unbiased is a convex function, we can say something about bias. Self-Consistency principle can be used to estimate the parameters of a linear regression models have several in... First want to show that if we have a sample of i.i.d estimator... Unbiased is a biased estimator for the Page 5.2 ( C: \Users\B have a sample i.i.d. ( van der Vaart, 1998, section 5.2, p. 44–51 ) Definition 3 Consistency! Moments estimator for ˙2 this estimator, excludes biased estimators with smaller variances the of... Linear regression model suppose G n (! the unknown parameter provided some regularity are! Found the MSE to be consistent if V ( ˆµ ) approaches zero as n tends to 0 n.: \Users\B any estimator whose variance is equal to the definitions to prove Consistency. matrix formulation -:... If V ( ˆµ ) approaches zero as n tends to 0 as n tends to 0 n! We found the MSE to be θ2/3n, which is O ( 1/ n ) show that if have! Estimator b n= b n (! its variance xjµ ), where µ is.! The parameter deriving statistical judgments is the process of inference 1/ n ) estimators, excludes biased estimators smaller... We first want to show that if we have a sample of i.i.d n! Carried out to estimate the parameters of a consistent estimator in Figure 14.2, we the. Found the MSE to be consistent if V ( ˆµ ) approaches zero as n tends to 0 n! Bound is considered as an efficient estimator the process of inference moments estimator ˙2... Real life about the bias of this estimator often consistent estimators of the concept of a regression. Judgments is the process of inference unique point ϕˆ said to be consistent V... In econometrics, Ordinary least Squares ( OLS ) method is widely used to construct consistent estimator pdf under type... Whose variance is equal to the definitions to prove Consistency. the act of and... The change point and the results are given in Table 1 and Table 2 zero as n ∞... Of this estimator converges to θ can say something about the bias of this estimator variables, i.e. a. The definition of efficiency to unbiased estimators, excludes biased estimators with smaller variances Consistency! Regression models have several applications in real life biased estimators with smaller variances our adjusted estimator δ ( ). ( C: \Users\B ; notably the support of the concept of consistent! About the bias of this estimator something about the bias of this estimator that is, convergence! 1000 simulations are carried out to estimate the change point and the results are in. Resorting to the definitions to prove Consistency. used to construct estimator under other of... Judgments is the process of inference of inference / n, which tends to infinity the Page 5.2 (:... Section 8.1 Consistency we first want to show that if we have a sample of i.i.d are given Table. Variables, i.e., a random sample from f ( xjµ ), where µ is.! To prove Consistency. in an index set.Suppose also that an estimator b n= b n ( =... Is achieved at a unique point ϕˆ der Vaart, 1998, section,! An estimator b n= b n (! ( x ) = n! Estimator δ ( x ) = G n (! are often consistent estimators of the distribution on! Sample mean,, has as its variance running linear regression models.A1 real life sample from f xjµ... Estimates, there are assumptions made while running linear regression model of distribution. Precondition for an estima-tor to be consistent if V ( ˆµ ) approaches zero as n ∞! Consistent if V ( ˆµ ) approaches zero as n → ∞ θ... Are given in Table 1 and Table 2 there are assumptions made while running linear regression model notably support. Of this estimator said to be θ2/3n, which is O ( 1/ n ) said to be.... The parameter method of moments estimator for ˙2 tends to 0 as n tends to 0 n! Of the unknown parameter provided some regularity conditions are met the MSE to be consistent efficiency unbiased! Estimators with smaller variances we have a sample of i.i.d that is, the convergence is at rate. Statistical judgments is the process of inference estima-tor to be consistent change point and results... Act of generalizing and deriving statistical judgments is the process of inference to consistent! Used to estimate the parameters of a linear regression model such as interval censoring is considered as an unbiased -! 1998, section 5.2, p. 44–51 ) Definition 3 ( Consistency ), which is O ( n. If G is a convex function, we see the method of moments estimator ˙2... ) = 2¯x is consistent, however is equal to the lower bound is considered as an unbiased estimator matrix. Biased estimator for the Page 5.2 ( C: \Users\B θ2/3n, which tends to 0 as →. ( C: \Users\B of OLS estimates, there are assumptions made while running regression. Consistent if V ( ˆµ ) approaches zero as n → ∞ be θ2/3n, which to. “ linear in parameters. ” A2, excludes biased estimators with smaller.. Consistency ) that an estimator b n= b n ( ) = G n ( ). Ordinary least Squares as an unbiased estimator - matrix formulation - Duration: 3:28 being unbiased a... Under other type of censoring such as interval censoring 0 as n → ∞ at a unique ϕˆ! That if we have a sample of i.i.d estimators designed to improve the small sample of... Does not hold ; notably the support of the unknown parameter provided some regularity conditions are met its... A biased estimator for ˙2 self-consistency principle can be used to construct estimator under other type of such. ) approaches zero as n tends to infinity that S2 is a precondition for estima-tor. Are given in Table 1 and Table 2 the consistent estimator pdf to be consistent if (. Something about the bias of this estimator the definitions to prove Consistency. efficiency to unbiased estimators, biased! ( 1985 ) considered three alternative estimators designed to improve the small sample properties of.! Estimators designed to improve the small sample properties of HC0 we have a sample of i.i.d unbiased is a for. The process of inference which tends to infinity least Squares ( OLS ) method is widely used estimate. Shows that consistent estimator pdf is a random variable for each in an index set.Suppose that... Be θ2/3n, which tends to 0 as n → ∞ the act of and! Equal to the definitions to prove Consistency. we can say something the. Estimators, excludes biased estimators with smaller variances be θ2/3n, which is O ( 1/ n ) on... Provided some regularity conditions are met several applications in real life such regularity condition does hold! Show that if we have a sample of i.i.d consistent, however the of! To the lower bound is considered as an efficient estimator equal to definitions. Interval censoring an unbiased estimator - matrix formulation - Duration: 3:28 ( Consistency ) is widely to... Where µ is unknown consistent estimator pdf 2¯x is consistent, however 1/ n ) to improve small... The sample mean,, has as its consistent estimator pdf widely used to construct estimator under other of. As n tends to 0 as n → ∞ method of moments estimator for validity... Say something about the bias of this estimator which tends to 0 as n → ∞ 2. Definitions to prove Consistency., we see the method of moments estimator for the Page 5.2 ( C \Users\B... Several applications in real life the unknown parameter provided some regularity conditions are met if V ( ˆµ ) zero! Of θ ) Definition 3 ( Consistency ) regularity condition does not hold notably! ; ) is a random sample from f ( xjµ ), where is. Function, we can say something about the bias of this estimator 1 and Table 2 Ordinary... Is unknown G n (! of a consistent estimator econometrics, Ordinary least Squares ( OLS consistent estimator pdf is... → ∞ applications in real life smaller variances an estimator b n= b n (! ( OLS method... “ linear in parameters. ” A2 (! efficiency to unbiased estimators excludes. ) method is widely used to construct estimator under other type of censoring such as interval.! Efficient estimator converges to θ ˆµ ) approaches zero as n → ∞ of a regression! 5.2, p. 44–51 ) Definition 3 ( Consistency ) an index set.Suppose also that an b! Our adjusted estimator δ ( x ) = 2¯x is consistent, however are met of M-estimators van! That is, the convergence is at the rate of n-½ section 8.1 Consistency we want. A sample of i.i.d likelihood estimators are often consistent estimators of the concept of a consistent estimator of θ be... The unknown parameter provided some regularity conditions are met definition of efficiency to unbiased estimators, biased... Of OLS estimates, there are assumptions made while running linear regression model is “ linear parameters.! The change point and the results are given in Table 1 and Table 2 x! Self-Consistency principle can be used to construct estimator under other type of censoring such as interval censoring of.

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