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Questions and Answers
What is the goal of point estimation?
What is the goal of point estimation?
The goal of point estimation is to guess the true value of the parameter θ (or vector Θ), or more generally, to guess the true value of some function of say τ(θ), or for a vector of parameters Θ, some function, say τ(Θ).
What are the two main aspects of the point estimation problem?
What are the two main aspects of the point estimation problem?
The two main aspects are:
- Devising a means to obtain statistics that serve as point estimators with desirable properties.
- Establishing optimality criteria to determine and select the "best" point estimator.
What is an estimator?
What is an estimator?
An estimator is any statistic (known function of observable random variables) that itself is a random variable whose values are used to estimate τ(θ), where τ(·) is some function of the parameter θ. It is defined as an estimator of τ(θ).
Which of the following is NOT considered a desirable property of a point estimator?
Which of the following is NOT considered a desirable property of a point estimator?
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An estimator T' is considered more concentrated than T if the probability of T' falling within a certain distance λ from the true value τ(θ) is greater than or equal to the probability of T falling within the same distance.
An estimator T' is considered more concentrated than T if the probability of T' falling within a certain distance λ from the true value τ(θ) is greater than or equal to the probability of T falling within the same distance.
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An estimator T' is called Pitman-closer than T if the probability of T' being closer to the true value τ(θ) than T is greater than or equal to 1/2 for each θ in the parameter space.
An estimator T' is called Pitman-closer than T if the probability of T' being closer to the true value τ(θ) than T is greater than or equal to 1/2 for each θ in the parameter space.
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What does the Mean-Squared Error (MSE) of an estimator measure?
What does the Mean-Squared Error (MSE) of an estimator measure?
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An estimator T is considered unbiased if the expected value of T is equal to the true value τ(θ) for all θ in the parameter space.
An estimator T is considered unbiased if the expected value of T is equal to the true value τ(θ) for all θ in the parameter space.
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The bias of an unbiased estimator is equal to zero, and its MSE is equal to its variance.
The bias of an unbiased estimator is equal to zero, and its MSE is equal to its variance.
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What is the MVUE, and what is its significance?
What is the MVUE, and what is its significance?
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A sequence of estimators is considered mean-squared error consistent if the limit of the expected value of the squared error between the estimator and the true value τ(θ) approaches zero as the sample size n grows large.
A sequence of estimators is considered mean-squared error consistent if the limit of the expected value of the squared error between the estimator and the true value τ(θ) approaches zero as the sample size n grows large.
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A sequence of estimators is considered simple consistent if the probability of the estimator falling within a certain distance ε from the true value τ(θ) approaches 1 as the sample size n grows large.
A sequence of estimators is considered simple consistent if the probability of the estimator falling within a certain distance ε from the true value τ(θ) approaches 1 as the sample size n grows large.
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A sequence of estimators is considered best asymptotically normal (BAN) if it satisfies certain conditions, including that the distribution of the normalized estimator approaches a normal distribution with mean 0 and variance (σ*)2(θ) as the sample size n approaches infinity.
A sequence of estimators is considered best asymptotically normal (BAN) if it satisfies certain conditions, including that the distribution of the normalized estimator approaches a normal distribution with mean 0 and variance (σ*)2(θ) as the sample size n approaches infinity.
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What is the general approach for finding estimators using the method of moments?
What is the general approach for finding estimators using the method of moments?
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What is the rationale behind the method of maximum likelihood estimation (MLE)?
What is the rationale behind the method of maximum likelihood estimation (MLE)?
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How is the likelihood function defined in the context of MLE?
How is the likelihood function defined in the context of MLE?
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A maximum likelihood estimator (MLE) is defined as the value of 0 that maximizes the likelihood function.
A maximum likelihood estimator (MLE) is defined as the value of 0 that maximizes the likelihood function.
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Finding the maximum of the logarithm of the likelihood function is often easier than maximizing the likelihood function directly.
Finding the maximum of the logarithm of the likelihood function is often easier than maximizing the likelihood function directly.
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The invariance property of maximum-likelihood estimators states that if θ is the MLE for θ, and τ(·) is a function with a single-valued inverse, then τ(θ) is the MLE for τ(θ).
The invariance property of maximum-likelihood estimators states that if θ is the MLE for θ, and τ(·) is a function with a single-valued inverse, then τ(θ) is the MLE for τ(θ).
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An estimator T* is defined as a UMVUE if it is unbiased and if its variance is at least as small as the variance of any other unbiased estimator for the same parameter.
An estimator T* is defined as a UMVUE if it is unbiased and if its variance is at least as small as the variance of any other unbiased estimator for the same parameter.
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If T is an unbiased estimator for a parameter, then the MSE of T is simply equal to the variance of T.
If T is an unbiased estimator for a parameter, then the MSE of T is simply equal to the variance of T.
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The Cramer-Rao lower bound provides an upper bound for the variance of any unbiased estimator.
The Cramer-Rao lower bound provides an upper bound for the variance of any unbiased estimator.
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The lower bound provided by the Cramer-Rao inequality can be attained if and only if there exists a function that satisfies a specific relationship involving the estimator, the true parameter value, and the density function of the random variables.
The lower bound provided by the Cramer-Rao inequality can be attained if and only if there exists a function that satisfies a specific relationship involving the estimator, the true parameter value, and the density function of the random variables.
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What are the key uses of the Cramer-Rao lower bound?
What are the key uses of the Cramer-Rao lower bound?
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A one-parameter (θ is unidimensional) family of densities f(x; θ) that can be expressed in a specific form involving functions a(θ), b(x), c(θ), and d(x) is said to belong to the exponential family or exponential class.
A one-parameter (θ is unidimensional) family of densities f(x; θ) that can be expressed in a specific form involving functions a(θ), b(x), c(θ), and d(x) is said to belong to the exponential family or exponential class.
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A statistic is considered sufficient if it encapsulates all the information related to the parameter θ from a given random sample.
A statistic is considered sufficient if it encapsulates all the information related to the parameter θ from a given random sample.
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If a set of statistics is jointly sufficient, then any one-to-one function or transformation of that set is also jointly sufficient.
If a set of statistics is jointly sufficient, then any one-to-one function or transformation of that set is also jointly sufficient.
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The factorization theorem provides a criterion to determine if a statistic S is sufficient by examining whether the joint density of the random sample can be factored into two parts, one depending on the parameter θ and the statistic S, and the other not depending on θ.
The factorization theorem provides a criterion to determine if a statistic S is sufficient by examining whether the joint density of the random sample can be factored into two parts, one depending on the parameter θ and the statistic S, and the other not depending on θ.
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The main application of the exponential family in the context of sufficiency is to show that the sufficient statistics are complete.
The main application of the exponential family in the context of sufficiency is to show that the sufficient statistics are complete.
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When searching for UMVUEs, it is sufficient to consider only unbiased estimators that are functions of sufficient statistics.
When searching for UMVUEs, it is sufficient to consider only unbiased estimators that are functions of sufficient statistics.
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The Rao-Blackwell theorem states that, given an unbiased estimator, it is always possible to derive another unbiased estimator, which is a function of sufficient statistics, that has a smaller variance.
The Rao-Blackwell theorem states that, given an unbiased estimator, it is always possible to derive another unbiased estimator, which is a function of sufficient statistics, that has a smaller variance.
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The Lehmann-Scheffe theorem provides a sufficient condition for the existence of a UMVUE, requiring the existence of a complete sufficient statistic and an unbiased estimator for the desired parameter.
The Lehmann-Scheffe theorem provides a sufficient condition for the existence of a UMVUE, requiring the existence of a complete sufficient statistic and an unbiased estimator for the desired parameter.
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A Rao-Blackwellized estimator is the UMVUE if the sufficient statistic used is also complete.
A Rao-Blackwellized estimator is the UMVUE if the sufficient statistic used is also complete.
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How does Bayes estimation differ from other point estimation methods?
How does Bayes estimation differ from other point estimation methods?
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What are the prior and posterior distributions in Bayes estimation?
What are the prior and posterior distributions in Bayes estimation?
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What is the posterior Bayes estimator?
What is the posterior Bayes estimator?
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Bayes estimation provides an alternative to classical point estimation methods by incorporating prior belief while considering the observed data.
Bayes estimation provides an alternative to classical point estimation methods by incorporating prior belief while considering the observed data.
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For a single function of the parameter(s), say τ(θ), how many potential point estimators for τ(θ) exist?
For a single function of the parameter(s), say τ(θ), how many potential point estimators for τ(θ) exist?
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According to what criteria is the best estimator selected?
According to what criteria is the best estimator selected?
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What are two major aspects of the point estimation problem?
What are two major aspects of the point estimation problem?
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What is the object of point estimation?
What is the object of point estimation?
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What is the goal of the second aspect of the point estimation problem?
What is the goal of the second aspect of the point estimation problem?
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What is an estimator defined to be?
What is an estimator defined to be?
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What does X = (X1, X2 , ..., X)' represent?
What does X = (X1, X2 , ..., X)' represent?
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What does Θ = (θ1, θ2, ... θk)' represent?
What does Θ = (θ1, θ2, ... θk)' represent?
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What does Ωθ = θ represent?
What does Ωθ = θ represent?
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What does τ(θ) represent?
What does τ(θ) represent?
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What does X = (X1, X2, ..., Xn)' represent in example 1?
What does X = (X1, X2, ..., Xn)' represent in example 1?
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What does µ1 = X, µ2 = X, µ3 = mode, µ4 = X1, µ5 = (X(1) + X(n))/2 represent?
What does µ1 = X, µ2 = X, µ3 = mode, µ4 = X1, µ5 = (X(1) + X(n))/2 represent?
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What is the criterion used to determine the best estimate?
What is the criterion used to determine the best estimate?
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What is the key determining factor in assessing the closeness of an estimator to the true value?
What is the key determining factor in assessing the closeness of an estimator to the true value?
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What does the notation Po[τ(θ) -λ< T ≤ τ(θ) +λ] refer to?
What does the notation Po[τ(θ) -λ< T ≤ τ(θ) +λ] refer to?
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What does the notion of a more concentrated estimator imply?
What does the notion of a more concentrated estimator imply?
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What is an estimator called if it is more concentrated than any other estimator?
What is an estimator called if it is more concentrated than any other estimator?
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What is the notation Po[[T' - τ(θ)| < |T - τ(θ)|] ≥ 1/2 refer to?
What is the notation Po[[T' - τ(θ)| < |T - τ(θ)|] ≥ 1/2 refer to?
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What is an estimator called if it is Pitman-closer than any other estimator?
What is an estimator called if it is Pitman-closer than any other estimator?
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What is the MSE of an estimator with respect to τ(θ) defined as?
What is the MSE of an estimator with respect to τ(θ) defined as?
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What is the formula for calculating the MSE?
What is the formula for calculating the MSE?
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What does the MSE of an estimator for a certain target parameter represent?
What does the MSE of an estimator for a certain target parameter represent?
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What does the MSE of an estimator represent in terms of variability?
What does the MSE of an estimator represent in terms of variability?
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The smaller the MSE, the better the estimator.
The smaller the MSE, the better the estimator.
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What is the formula for calculating the MSE in terms of variance and bias?
What is the formula for calculating the MSE in terms of variance and bias?
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When does the MSE simplify to Var(T)?
When does the MSE simplify to Var(T)?
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Unbiased estimators, if they exist, are always unique.
Unbiased estimators, if they exist, are always unique.
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What is the bias of an unbiased estimator equal to?
What is the bias of an unbiased estimator equal to?
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The estimator T should have a minimum variance unbaised estimator (MVUE).
The estimator T should have a minimum variance unbaised estimator (MVUE).
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If E(T) = τ(θ), then MSE(T) = Var(T).
If E(T) = τ(θ), then MSE(T) = Var(T).
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Is MSE(T1) ≤ MSE(T2) if T1 is unbiased and T2 is biased?
Is MSE(T1) ≤ MSE(T2) if T1 is unbiased and T2 is biased?
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What are two common approaches for finding estimators?
What are two common approaches for finding estimators?
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What does µr = E[X^r] represent?
What does µr = E[X^r] represent?
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What does µ'r = E[(X1 - µx)^r] represent?
What does µ'r = E[(X1 - µx)^r] represent?
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What does Mr = 1/n ΣXi^r represent?
What does Mr = 1/n ΣXi^r represent?
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What does Mr' = 1/n Σ(Xi - X)^r represent?
What does Mr' = 1/n Σ(Xi - X)^r represent?
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What is the method of moments estimation defined as?
What is the method of moments estimation defined as?
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How are the method of moments estimators denoted?
How are the method of moments estimators denoted?
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MMEs are always unbiased and consistent.
MMEs are always unbiased and consistent.
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The estimators are found using the method of moments by equating the population moments with the sample moments and solving the solution of the equation Mr = μr(θ), for all r = 1, 2, ...
The estimators are found using the method of moments by equating the population moments with the sample moments and solving the solution of the equation Mr = μr(θ), for all r = 1, 2, ...
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Using central moments, rather than raw moments, can also be used to obtain equations.
Using central moments, rather than raw moments, can also be used to obtain equations.
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What is the procedure for obtaining MMEs?
What is the procedure for obtaining MMEs?
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What is the rationale behind maximum likelihood estimation?
What is the rationale behind maximum likelihood estimation?
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What does L(0; X1, X2, ..., Xn) represent?
What does L(0; X1, X2, ..., Xn) represent?
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The maximum likelihood estimator is the solution of the equation dL(θ)/dθ = 0.
The maximum likelihood estimator is the solution of the equation dL(θ)/dθ = 0.
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L(θ) and log L(θ) have their maxima at the same value of θ.
L(θ) and log L(θ) have their maxima at the same value of θ.
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What are the key steps in formulating the likelihood function?
What are the key steps in formulating the likelihood function?
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What is the MLE denoted as?
What is the MLE denoted as?
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What does the MLE of θ represent?
What does the MLE of θ represent?
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What is a specific value of the MLE called?
What is a specific value of the MLE called?
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The MLE method finds the value of θ that minimizes the probability of obtaining the observed random sample values.
The MLE method finds the value of θ that minimizes the probability of obtaining the observed random sample values.
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What is the maximum likelihood estimator defined to be?
What is the maximum likelihood estimator defined to be?
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What does the equation L(θ; X) = supee∈ΘL(θ; X) represent?
What does the equation L(θ; X) = supee∈ΘL(θ; X) represent?
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What is the significance of the invariance property of maximum likelihood estimators?
What is the significance of the invariance property of maximum likelihood estimators?
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What is an UMVUE defined to be?
What is an UMVUE defined to be?
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An UMVUE always exists.
An UMVUE always exists.
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There can be more than one UMVUE for a given parameter function.
There can be more than one UMVUE for a given parameter function.
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Why is the Cramer-Rao lower bound important?
Why is the Cramer-Rao lower bound important?
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What is the CRLB formula?
What is the CRLB formula?
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What does the Fisher's information measure (I(θ)) represent?
What does the Fisher's information measure (I(θ)) represent?
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Study Notes
Parametric Point Estimation
- Point estimation aims to guess the true value of a parameter (θ or vector Θ) or a function of that parameter (τ(θ)).
- Multiple potential estimators exist for τ(θ), but a primary goal is selecting the "best" or "optimal" one.
- The point estimation problem involves two aspects:
- Developing statistics suitable as estimators.
- Establishing criteria for determining the "best" estimator.
Illustration
- A statistic is a function of observable random variables.
- Estimators and estimation are part of the larger process of evaluating unknown parameters from data samples.
- Using a statistic to estimate a parameter is termed point estimation.
- Interval estimates provide a range of possible values for a parameter (as opposed to a single precise point estimate).
Definition and Notations
- An estimator is a statistic used to estimate a parameter.
- X = (X₁, X₂, ..., Xₙ) is a random sample.
- Θ = (θ₁, θ₂, ..., θₖ) is a vector of unknown parameters.
- Ω is the parameter space, containing all possible values of θ.
- τ(θ) is the target parameter.
- T(X) is a statistic used to estimate τ(θ).
Example 1 and Example 2
- Example 1: A random sample from a normal distribution (N(μ, σ²)).
- Example 2: A random sample from a normal distribution (N(μ, σ²)) where σ² is known.
Properties of Point Estimators
- Closeness: Estimators are evaluated at how close they typically are to the true parameter value.
- A more concentrated estimator is one likely to be closer to the parameter and thus better.
Definition. Pitman-Closer and Pitman-Closest
- Pitman-closer estimators are ranked based on closeness to the parameter being estimated, based on calculated probabilities.
- An estimator is considered Pitman-closest if it is closer than other estimators in terms of calculation probabilities.
Definition. Mean-Squared Error (MSE)
- MSE measures the average squared difference between the estimator and the parameter.
- Lower MSE values typically indicate better estimators.
Remarks
- Smaller MSE is generally preferred.
- Finding the estimator with the absolute smallest MSE is typically challenging.
Unbiasedness
- An estimator is unbiased if its expected value equals the true parameter value.
- Bias measures the difference between the estimator's expected value and the parameter.
Notes
- An unbiased estimator has a zero bias and the MSE equals the variance.
- Unbiased estimators are not necessarily unique.
Example 1 and Example 2 and Example 3
- Example 1: T₁ is unbiased while T₂ is biased.
- Example 2: Examples of unbiased/biased estimators, regarding a sample from a normally distributed data set.
- Example 3: Determine if a certain statistic is an unbiased estimator for a set parameter.
Asymptotic Properties
- Properties of estimators as the sample size increases. -Mean-Squared Error Consistency;
- A property that indicates how the estimator's expected square error approaches zero as sample size increases.
- Asymptotic behavior of bias and variance.
Examples 1, 2, 3
- Examples relating to the asymptotic properties of different estimators.
Methods of Finding Estimators
-
Method of Moments (MME):
- Equating population moments to sample moments to estimate parameters. -The number of equations depends on the unknown parameters.
- The estimator might not be unbiased or consistent in all situations.
Method of Maximum Likelihood Estimation (MLE)
- MLE selects parameter(s) that maximize the probability of obtaining the observed sample values.
Examples
- Illustrative examples of using the MLE approach.
Definition. Likelihood Function
- This is a function of the parameter(s) based on the joint probability density of sample data.
Definition. Maximum Likelihood Estimator (MLE)
- MLE is a parameter value that maximizes the likelihood function, usually calculated by taking the derivative of the likelihood function with respect to the parameters and setting it to zero.
Remarks
- The maximum likelihood estimator maximizes the probability of observed values.
- Finding maximum likelihood estimators involves finding solutions to equations where the likelihood function's derivatives equal zero.
Examples
- Problem sets showing different forms of maximum likelihood estimation calculations.
Theorem. Invariance Property of Maximum-Likelihood Estimators
- If a parameter has a single-valued inverse transformation, the maximum-likelihood estimator will change accordingly
Definition. Uniformly Minimum Variance Unbiased Estimator (UMVUE)
- A UMVUE is an unbiased estimator with the smallest variance among all unbiased estimators.
Remarks
- The UMVUE is the best possible unbiased estimator in terms of variance.
Theorem. Cramer-Rao Lower Bound (CRLB)
- Provides a lower bound for the variance of any unbiased estimator.
Examples
- Examples illustrating the use of the Cramer-Rao lower bound to find the UMVUE.
Definition. Exponential Family of Densities
- A family of probability density functions that can be written in a specific form.
- Most parametric families are within the exponential family.
Examples
- A series of examples illustrating and testing the exponential families of densities.
Definition. Sufficient Statistic
- A statistic (a function of the sample) that contains all of the information needed to estimate the parameter in question, and is not influenced by irrelevant information from the data.
Definition. Jointly Sufficient Statistics
- A set of sufficient statistics is one that contains all of the information from multiple different sample data points, and are not subject to unwanted properties inherent from the data being used.
Theorem. Factorization Theorem
- Defines a sufficient statistic with respect to its properties using mathematically proven relations.
Examples
- Examples of proving sufficient statistics regarding various distributions.
Theorem. Rao-Blackwell
- Used for finding the best unbiased estimator that takes into consideration sufficient statistics.
Theorem. Lehmann-Scheffe
- Conditions are given for finding the UMVUE of a parameter based on sufficient statistics
Remarks
- The criteria for finding the UMVUE using the Lehmann-Scheffe theorem.
Examples
- Example problems that determine UMVUES.
Bayes Estimation
- A method of estimation that considers prior information about the parameter being estimated.
Definition. Prior and Posterior Distributions
- Prior probability distribution, prior to obtaining data or outcomes.
- Conditional probability distribution, after observing additional data or outcomes.
Definition. Posterior Bayes Estimator
- The estimate considering both the prior and posterior distributions.
Examples
- Providing examples using Bernoulli and normal probability distributions.
End of Slide
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This quiz explores the concepts of point estimation, focusing on the selection of optimal estimators and the role of statistics in estimating unknown parameters. It delves into the definitions, notations, and differences between point and interval estimation. Test your understanding of this fundamental aspect of statistical analysis.
