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The cumulative distribution function (shown as F(x)) gives the p values as a function of the q values. The quantile function does the opposite: it gives the q values as a function of the p values. Note that the portion of F(x) in red is a horizontal line segment.
In statistics, a Q–Q plot (quantile–quantile plot) is a probability plot, a graphical method for comparing two probability distributions by plotting their quantiles against each other. [1] A point ( x , y ) on the plot corresponds to one of the quantiles of the second distribution ( y -coordinate) plotted against the same quantile of the ...
For a population, of discrete values or for a continuous population density, the k-th q-quantile is the data value where the cumulative distribution function crosses k/q. That is, x is a k-th q-quantile for a variable X if Pr[X < x] ≤ k/q or, equivalently, Pr[X ≥ x] ≥ 1 − k/q. and Pr[X ≤ x] ≥ k/q.
In particular, the quantile is 1.96; therefore a normal random variable will lie outside the interval in only 5% of cases. The following table gives the quantile z p {\textstyle z_{p}} such that X {\textstyle X} will lie in the range μ ± z p σ {\textstyle \mu \pm z_{p}\sigma } with a specified probability p {\textstyle p} .
Here F X is the cumulative distribution function of X, f X is the corresponding probability density function, Q X (p) is the corresponding inverse cumulative distribution function also called the quantile function, [2] and the integrals are of the Riemann–Stieltjes kind.
A quantile-based credible interval, which is computed by taking the inter-quantile interval [, +] for some predefined [,]. For instance, the median credible interval (MCI) of probability γ {\displaystyle \gamma } is the interval where the probability of being below the interval is as likely as being above it, that is to say the interval [ q ...
The individual point forecasts are used as independent variables and the corresponding observed target variable as the dependent variable in a standard quantile regression setting. [8] The Quantile Regression Averaging method yields an interval forecast of the target variable, but does not use the prediction intervals of the individual methods.
QPD transformations are governed by a general property of quantile functions: for any quantile function = and increasing function (), = (()) is a quantile function. [8] For example, the quantile function of the normal distribution , x = μ + σ Φ − 1 ( y ) {\displaystyle x=\mu +\sigma \Phi ^{-1}(y)} , is a QPD by the Keelin and Powley ...