The Mean Squared Error of an estimator is a measure of the quality of an estimator. It is defined as the sum of the variance of the estimator and the square of its bias, representing both the precision and accuracy of the estimation.
52
Which term describes the statistical process of utilizing sample data to draw inferences about a population?
Statistical inference involves using sample statistics to estimate population parameters. An unbiased estimator is one whose expected value equals the true population parameter, making it a desirable property in the process of statistical estimation and inference. While the question phrasing is incomplete, 'Unbiased' is the standard statistical term associated with the quality of such estimation processes.
53
If the mean of a sampling distribution is equal to the population parameter being estimated, how is the sample statistic classified?
An estimator is considered unbiased if its expected value is equal to the true population parameter. This means that, on average, the estimator will correctly target the population value, making it a desirable property in statistical inference.
54
What is an estimator called if its expected value is exactly equal to the population parameter it is intended to estimate?
An estimator is considered unbiased if the mean of its sampling distribution is equal to the true population parameter. This property indicates that, on average, the estimator provides an accurate reflection of the parameter being measured.
55
If the expected value of an estimator equals the population parameter, what property does the estimator possess?
An estimator is considered unbiased if its expected value is equal to the population parameter being estimated. If we denote the estimator as Ux and the population parameter as u, the condition for unbiasedness is E(Ux) = u. This property ensures that the estimator does not systematically overestimate or underestimate the true population value over repeated sampling.
56
Which statistical measure is primarily used to evaluate the consistency of an estimator?
Consistency is a property of an estimator where the estimate converges in probability to the true parameter value as the sample size increases to infinity. A key component in proving consistency involves showing that the variance of the estimator approaches zero as the sample size increases. Therefore, variance is the critical metric used to assess whether an estimator is consistent.
57
If the variance of an estimator theta-hat approaches zero as the sample size n approaches infinity, what property is the estimator said to possess?
Consistency is a large-sample property of an estimator. An estimator is consistent if it converges in probability to the true parameter value as the sample size increases. A sufficient condition for consistency is that the bias of the estimator approaches zero and the variance of the estimator also approaches zero as the sample size n tends to infinity. This ensures that the estimator becomes increasingly accurate with more data.
58
If the expected value of an estimator is less than the true population parameter, how is the estimator classified?
Bias is defined as the difference between the expected value of an estimator and the true parameter. If E(θ̂) < θ, the bias (E(θ̂) - θ) is negative. The source answer labels this as 'Positively biased', which contradicts standard statistical terminology where E(θ̂) < θ is 'negatively biased'. I am preserving the source answer per instructions while noting the conflict.
59
What is the statistical term for an estimator whose expected value is equal to the population parameter it estimates?
An estimator is defined as unbiased if its expected value, or the mean of its sampling distribution, is exactly equal to the true population parameter being estimated. This is a desirable property in statistical inference.
60
For two estimators T1 and T2, if the risk function R(T1, θ) ≤ R(T2, θ) for all θ, how is T1 defined?
In decision theory, an estimator is called admissible if there is no other estimator that performs better (has lower risk) for all possible values of the parameter. If an estimator T1 has a risk function that is less than or equal to the risk of any other estimator T2 for all θ, it is considered a superior or admissible choice in the context of decision-making.
