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Prediction intervals are commonly used as definitions of reference ranges, such as reference ranges for blood tests to give an idea of whether a blood test is normal or not. For this purpose, the most commonly used prediction interval is the 95% prediction interval, and a reference range based on it can be called a standard reference range.
A weaker three-sigma rule can be derived from Chebyshev's inequality, stating that even for non-normally distributed variables, at least 88.8% of cases should fall within properly calculated three-sigma intervals. For unimodal distributions, the probability of being within the interval is at least 95% by the Vysochanskij–Petunin inequality ...
A prediction interval estimates the interval containing future samples with some confidence, γ. Prediction intervals can be used for both Bayesian and frequentist contexts. These intervals are typically used in regression data sets, but prediction intervals are not used for extrapolation beyond the previous data's experimentally controlled ...
For example, if n = 19, then [m,M] gives an 18/20 = 90% prediction interval – 90% of the time, the 20th observation falls between the smallest and largest observation seen heretofore. Likewise, n = 39 gives a 95% prediction interval, and n = 199 gives a 99% prediction interval.
In statistical prediction, the coverage probability is the probability that a prediction interval will include an out-of-sample value of the random variable. The coverage probability can be defined as the proportion of instances where the interval surrounds an out-of-sample value as assessed by long-run frequency. [2]
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.
The standard definition of a reference range for a particular measurement is defined as the interval between which 95% of values of a reference population fall into, in such a way that 2.5% of the time a value will be less than the lower limit of this interval, and 2.5% of the time it will be larger than the upper limit of this interval, whatever the distribution of these values.
In addition, 95% confidence intervals are also 83% prediction intervals: one (pre experimental) confidence interval has an 83% chance of covering any future experiment's mean. [3] As such, knowing a single experiment's 95% confidence intervals gives the analyst a reasonable range for the population mean.