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In statistics, a consistent estimator or asymptotically consistent estimator is an estimator—a rule for computing estimates of a parameter θ 0 —having the property that as the number of data points used increases indefinitely, the resulting sequence of estimates converges in probability to θ 0.
The term consistent estimator is short for “consistent sequence of estimators,” an idea found in convergence in probability. The basic idea is that you repeat the estimator’s results over and over again, with steadily increasing sample sizes.
Searches related to define consistency of an estimator equation
Definition 3.7 (Consistency in squared mean) A sequence of estimators \(\{\hat{\theta}_n:n\in\mathbb{N}\}\) is consistent in squared mean for \(\theta\) if \[\begin{align*} \lim_{n\to\infty}\mathrm{MSE}\big[\hat{\theta}_n\big]=0, \end{align*}\]
Consistency is a relatively weak property and is considered necessary of all reasonable estimators. This is in contrast to optimality properties such as efficiency which state that the estimator is “best”. Consistency of ˆθ can be shown in several ways which we describe below.
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- n is F^n (t) = P ^ n t
- 7.7.3.2 The Cramer-Rao Lower Bound (CRLB) and E ciency
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In this case, we cannot use Chebyshev's inequality unfortunately, because the maximum likelihood estimator is not unbiased. The CDF for ^
which is the probability that each individual sample is less than t because only in that case will the max be less than t, and we have independence so we can say P ^ n
Why did we de ne that nasty Fisher information? (Actually, it's much worse when is a vector instead of a single number, as the second derivative becomes a matrix of second partial derivatives). It would be great if the mean squared error of an estimator ^ was a low as possible. The Cramer-Rao Lower Bound actually gives a lower bound on the variance...
First, you have to check that it's unbiased, as the CRLB only holds for unbiased estimators... " n #
n Thus, we've shown that, since our e ciency is 1, our estimator is e cient. That is, it has the best pos-sible variance among all unbiased estimators of . This, again, is a really good property that we want to have. To reiterate, this means we cannot possibly do better in terms of mean squared error. Our bias is 0, and our variance is as low as it...
Consistency of an estimator means that as the sample size gets large the estimate gets closer and closer to the true value of the parameter. Unbiasedness is a finite sample property that is not affected by increasing sample size.
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 ≤ t), F θ(t) = P θ{X ≤ t}.
