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The bootstrap distribution of a parameter-estimator is often used to calculate confidence intervals for its population-parameter. [2] A variety of methods for constructing the confidence intervals have been proposed, although there is disagreement which method is the best.
The confidence interval can be expressed in terms of statistical significance, e.g.: "The 95% confidence interval represents values that are not statistically significantly different from the point estimate at the .05 level." [20] Interpretation of the 95% confidence interval in terms of statistical significance.
The best example of the plug-in principle, the bootstrapping method. Bootstrapping is a statistical method for estimating the sampling distribution of an estimator by sampling with replacement from the original sample, most often with the purpose of deriving robust estimates of standard errors and confidence intervals of a population parameter like a mean, median, proportion, odds ratio ...
Classically, a confidence distribution is defined by inverting the upper limits of a series of lower-sided confidence intervals. [15] [16] [page needed] In particular, For every α in (0, 1), let (−∞, ξ n (α)] be a 100α% lower-side confidence interval for θ, where ξ n (α) = ξ n (X n,α) is continuous and increasing in α for each sample X n.
when the probability distribution of the value is known, it can be used to calculate an exact confidence interval; when the probability distribution is unknown, Chebyshev's or the Vysochanskiï–Petunin inequalities can be used to calculate a conservative confidence interval; and
Confidence bands can be constructed around estimates of the empirical distribution function.Simple theory allows the construction of point-wise confidence intervals, but it is also possible to construct a simultaneous confidence band for the cumulative distribution function as a whole by inverting the Kolmogorov-Smirnov test, or by using non-parametric likelihood methods.
a) The expression inside the square root has to be positive, or else the resulting interval will be imaginary. b) When g is very close to 1, the confidence interval is infinite. c) When g is greater than 1, the overall divisor outside the square brackets is negative and the confidence interval is exclusive.
The jackknife pre-dates other common resampling methods such as the bootstrap. Given a sample of size n {\displaystyle n} , a jackknife estimator can be built by aggregating the parameter estimates from each subsample of size ( n − 1 ) {\displaystyle (n-1)} obtained by omitting one observation.