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Statistical function that defines the quantiles of a probability distribution
statistics, the quantile function of a probability distribution is the inverse of its cumulative distribution function. That is, the quantile function of a distribution
Quantile_function
Statistical method of dividing data into equal-sized intervals for analysis
distribution function of a random variable is known, the q-quantiles are the application of the quantile function (the inverse function of the cumulative
Quantile
Probability distribution
approximation 1−p≈2nπen2 The quantile function of a distribution is the inverse of the cumulative distribution function. The quantile function of the standard normal
Normal_distribution
Statistical modeling technique
However, the main attraction of quantile regression goes beyond this and is advantageous when conditional quantile functions are of interest. Different measures
Quantile_regression
Comparison of two distributions
plot (quantile–quantile plot) is a probability plot, a graphical method for comparing two probability distributions by plotting their quantiles against
Q–Q_plot
Continuous probability distribution
distribution function (quantile function) of the logistic distribution is a generalization of the logit function. Its derivative is called the quantile density
Logistic_distribution
Function in statistics
In statistics, the logit (logistic unit) or log-odds function is the quantile function associated with the standard logistic distribution. It has many
Logit
Probability that random variable X is less than or equal to x
{\displaystyle F(x)=p} . This defines the inverse distribution function or quantile function. Some distributions do not have a unique inverse (for example
Cumulative distribution function
Cumulative_distribution_function
Continuous probability distribution
of fitting to data with linear least squares; simple, closed-form quantile function (inverse CDF) equations that facilitate simulation; a simple, closed-form
Metalog_distribution
Family of continuous probability distributions
simulation studies since its probability density function, cumulative distribution function and quantile functions can be expressed in closed form. This distribution
Kumaraswamy_distribution
Statistic which divides data into four same-sized parts for analysis
In statistics, quartiles are a type of quantiles which divide the number of data points into four parts, or quarters, of more-or-less equal size. The
Quartile
Description of continuous random distribution
probability density function (PDF), density function, or simply density of an absolutely continuous random variable, is a function whose value at any given
Probability_density_function
Φ is the cumulative distribution function of the standard normal distribution. The formula for the quantile function is G ( p ) = 1 4 [ γ Φ − 1 ( p )
Birnbaum–Saunders distribution
Birnbaum–Saunders_distribution
Basic method for pseudo-random number sampling
involves computing the quantile function of the distribution — in other words, computing the cumulative distribution function (CDF) of the distribution
Inverse_transform_sampling
Discrete-variable probability distribution
a probability mass function (PMF), discrete probability density function, probability function, or frequency function is a function that gives the probability
Probability_mass_function
Fourier transform of the probability density function
distribution function F X {\displaystyle F_{X}} , the inverse cumulative distribution function Q X {\displaystyle Q_{X}} (also called the quantile function), and
Characteristic function (probability theory)
Characteristic_function_(probability_theory)
Concept in probability theory and statistics
theory and statistics, the moment generating function of a real-valued random variable is a generating function that provides an alternative specification
Moment_generating_function
Statistical function that converts a probability to a standard normal score
distributed. Mathematically, the probit function is the quantile function (the inverse of the cumulative distribution function (CDF)) associated with the standard
Probit
Probability distribution
distribution that has a constant failure rate. The quantile function (inverse cumulative distribution function) for Exp(λ) is F − 1 ( p ; λ ) = − ln ( 1 −
Exponential_distribution
Probability distribution
instance of the hypergeometric function. For information on its inverse cumulative distribution function, see quantile function § Student's t-distribution
Student's_t-distribution
Sigmoid shape special function
Φ is known as the normal quantile function, or probit function and may be expressed in terms of the inverse error function as probit ( p ) = Φ − 1
Error_function
Symmetric probability distribution
continuous, symmetric probability distribution defined in terms of its quantile function. It is typically used to identify an appropriate distribution (see
Tukey_lambda_distribution
Set of quantities in probability theory
cumulant generating function (CGF) K(t), which is a generating function that is the natural logarithm of the moment generating function: K ( t ) = log
Cumulant
quantile function F − 1 ( y ) {\displaystyle F^{-1}(y)} . These distributions are called quantile-parameterized because for a given set of quantile pairs
Quantile-parameterized distribution
Quantile-parameterized_distribution
Discrete probability distribution
the quantile function (corresponding to a lower tail area p) of the chi-squared distribution with n degrees of freedom and F−1(p;n,1) is the quantile function
Poisson_distribution
Probability distribution
dx\\[6pt]&=\log(4\pi \gamma )\end{aligned}}} The derivative of the quantile function, the quantile density function, for the Cauchy distribution is: Q ′ ( p ; γ ) = γ
Cauchy_distribution
Financial mathematical measure
representations of the inverse pricing function in terms of established probability distribution quantile functions, including the formula below. For k>0
Implied_volatility
Continuous probability distribution
Having a closed form expression for the quantile function, may make it a more flexible alternative for a quantile regression model against the classical
Unit_Weibull_distribution
Graphical technique in statistics
computed in exactly the same way. The normal quantile function Φ−1 is simply replaced by the quantile function of the desired distribution. In this way,
Normal_probability_plot
Graph of wealth or income distribution
theory. Alternatively, for a cumulative distribution function F(x) with its inverse, the quantile function x(F), the Lorenz curve L(F) is directly given by:
Lorenz_curve
Continuous probability distribution
>0} are shape parameters. The inverse cumulative distribution function (quantile function) is Q X ( u ; θ ) = α α − log ( 1 − ( 1 − u ) 1 / β ) {\displaystyle
Modified Kumaraswamy distribution
Modified_Kumaraswamy_distribution
Continuous probability distribution
{e} ^{x^{\beta }}\right)}} where x > 0 {\displaystyle x>0} . The quantile function of the Chen distribution is given by: Q ( p ; λ , β ) = [ ln { 1
Chen_distribution
Average value of a random variable
Expectile – related to expectations in a way analogous to that in which quantiles are related to medians Law of total expectation – the expected value of
Expected_value
Power series derived from a discrete probability distribution
generating function of a discrete random variable is a power series representation (the generating function) of the probability mass function of the random
Probability generating function
Probability_generating_function
Measure for evaluating probabilistic forecasts
y)=(x-y)^{2}} The following scoring functions are strictly consistent for the α {\displaystyle \alpha } -quantile, i.e. T ( F ) = q α {\displaystyle T(F)=q_{\alpha
Scoring_rule
Continuous probability distribution
fitting skewed data. A quantile regression model has also been derived from the distribution. Its probability density function is defined as: f ( x ∣
Unit_Gompertz_distribution
Probability distribution and special case of gamma distribution
degrees of freedom. These values can be calculated evaluating the quantile function (also known as "inverse CDF" or "ICDF") of the chi-squared distribution;
Chi-squared_distribution
Continuous probability distribution for a non-negative random variable
(see also related distributions below). The quantile function (inverse cumulative distribution function) is F − 1 ( p ; α , β ) = α ( p 1 − p ) 1 / β
Log-logistic_distribution
Measure of the asymmetry of random variables
} where Q is the quantile function (i.e., the inverse of the cumulative distribution function). The numerator is difference between
Skewness
Mathematical function for the probability a given outcome occurs in an experiment
variable, a location at which the probability density function has a local peak. Quantile: the q-quantile is the value x {\displaystyle x} such that P ( X
Probability_distribution
Functional relationship between two quantities
generation function using random samples, the bundle methodology is based on residual quantile functions (RQFs), also called residual percentile functions, which
Power_law
Probability distribution
gamma function, and P ( ⋅ , ⋅ ) {\displaystyle P(\cdot ,\cdot )} denotes the regularized lower incomplete gamma function. The quantile function can be
Generalized gamma distribution
Generalized_gamma_distribution
Variable representing a random phenomenon
{\displaystyle \operatorname {D} } can be generated by calculating the quantile function of D {\displaystyle \operatorname {D} } on a randomly-generated number
Random_variable
Technique to make two distributions statistically identical
from the cumulative distribution function of the distribution. However, any reference distribution can be used. Quantile normalization is frequently used
Quantile_normalization
and d is a parameter. From these relationships, the associated RMM quantile function is (Shore, 2011): w = log ( y ) = μ + ( α λ ) [ ( η + c z ) λ −
Response_modeling_methodology
Continuous probability distribution
distribution function is F ( x ; k , β ) = 1 − e − ( β x ) k , {\displaystyle F(x;k,\beta )=1-e^{-(\beta x)^{k}},} the quantile function is Q ( p ; k
Weibull_distribution
Particular case of the generalized extreme value distribution
{\displaystyle b} addition algorithms. Since the quantile function (inverse cumulative distribution function), Q ( p ) {\displaystyle Q(p)} , of a Gumbel
Gumbel_distribution
Measure of inequality of a statistical distribution
coefficient may be expressed in terms of the quantile function Q(F) (inverse of the cumulative distribution function: Q(F(x)) = x) G=12μ∫01∫01|Q(F1)−Q(F2)|dF1dF2
Gini_coefficient
Probability distribution
}}\right),} where erf is the error function, a standard function in many mathematical software packages. The quantile function (or inverse CDF) is written:
Half-normal_distribution
Statistical characterization of distribution functions
dispersion function of order p is defined as the L p {\displaystyle L_{p}} -distance between the quantile function Q X {\displaystyle Q_{X}} and the quantile function
Dispersion_function
Largest and oldest high-IQ society in the world
us.mensa.org. Retrieved 22 February 2023. See Normal distribution#Quantile function. American Mensa. "Take the Mensa Admission Test". www.us.mensa.org
Mensa_International
Measure of variation in statistics
}{2}}}\right)=1-\alpha ,} where q p {\displaystyle q_{p}} is the p-th quantile of the chi-square distribution with k degrees of freedom, and 1 − α is
Standard_deviation
Estimate of an interval in which future observations will fall
prediction is to estimate the parameters and then use the associated quantile function – for example, one could use the sample mean x‾ as an estimate for
Prediction_interval
Measure of statistical dispersion
defined as follows: Q1=CDF−1(0.25), Q3=CDF−1(0.75), where CDF−1 is the quantile function. The interquartile range and median of some common distributions are
Interquartile_range
Middle quantile of a data set or probability distribution
the median is of central importance in robust statistics. Median is a 2-quantile; it is the value that partitions a set into two equal parts. The median
Median
Mathematical equation related to human death rate
useful to work with the quantile function Q ( u ) {\displaystyle Q(u)} , defined as the inverse of the cumulative distribution function. A closed-form expression
Gompertz–Makeham law of mortality
Gompertz–Makeham_law_of_mortality
Concept in probability theory
(a)}}} The result that, for a nonnegative random variable X, the quantile function of X satisfies: Q X ( 1 − p ) ≤ E ( X ) p , {\displaystyle Q_{X}(1-p)\leq
Markov's_inequality
Kth smallest value in a statistical sample
median is some function of the two (usually the average) and hence not an order statistic. Similar remarks apply to all sample quantiles. Given any random
Order_statistic
Statistical test for multiple comparisons
same α . In addition, R offers a cumulative distribution function (ptukey) and a quantile function (qtukey) for q . The Tukey confidence limits for all pairwise
Tukey's_range_test
Concept in information theory
distributions which do not have an explicit density function expression, but have an explicit quantile function expression, Q(p), then h(Q) can be defined in
Differential_entropy
Statistical measure
{\displaystyle Q_{3}={\text{CDF}}^{-1}(0.75),} where CDF−1 is the quantile function. When normalizing by the mean value of the measurements, the term
Root_mean_square_deviation
Probability distribution in economics
{x}{b}})^{-a-1}}{\left(({\tfrac {x}{b}})^{-a}+1\right)^{p+1}}}\right).} The quantile function is given by Q ( u ; a , b , p ) = b ( u − 1 / p − 1 ) − 1 / a = b
Dagum_distribution
Method for estimating the unknown parameters in a linear regression model
{\bigg ]}} at the 1 − α confidence level, where q denotes the quantile function of standard normal distribution, and [·]jj is the j-th diagonal element
Ordinary_least_squares
Generalization of the one-dimensional normal distribution to higher dimensions
covariance matrix and χ k 2 ( p ) {\displaystyle \chi _{k}^{2}(p)} is the quantile function for probability p {\displaystyle p} of the chi-squared distribution
Multivariate normal distribution
Multivariate_normal_distribution
Statistical distribution
probability distribution or cumulative distribution function. Rather, it is a discrete form of a quantile function (inverse cumulative distribution) in reverse
Rank–size_distribution
Calculation of complex statistical distributions
specific quantile of interest within a desired margin of error. Let q {\displaystyle q} denote the desired quantile (e.g., 0.025) of a real-valued function g
Markov_chain_Monte_Carlo
Statistical measure of variability
reciprocal of the quantile function Φ − 1 {\displaystyle \Phi ^{-1}} (also known as the inverse of the cumulative distribution function) for the standard
Median_absolute_deviation
Probability distribution
ordinary meaning of 'the xth quantile of the standard normal distribution', rather than being a shorthand for 'the (1 − x)th quantile'. Secondly, this formula
Binomial_distribution
graphical representation of income inequality because it is a form of quantile function and it is considered useful when comparing two different areas or
Pen's_parade
Algorithm that generates an approximation of a random number sequence
is a pseudo-random number generator for P, where F∗:(0,1)→ℝ is the quantile function of P, i.e. F∗(x):=inf{t∈ℝ:x≤F(t)}. Intuitively, an arbitrary distribution
Pseudorandom_number_generator
Statistical measure of how far values spread from their average
random variable X {\displaystyle X} is discrete with probability mass function x 1 ↦ p 1 , x 2 ↦ p 2 , … , x n ↦ p n {\displaystyle x_{1}\mapsto p_{1}
Variance
Function of the observed sample results
statistic for given fixed p-values; this corresponds to computing the quantile function (inverse CDF). As an example of a statistical test, an experiment
P-value
Diagnostic plot of binary classifier ability
non-linearly transformed x- and y-axes. The transformation function is the quantile function of the normal distribution, i.e., the inverse of the cumulative
Receiver operating characteristic
Receiver_operating_characteristic
Regression analysis technique
the cumulative distribution function (CDF) of e {\displaystyle e} as F e , {\displaystyle F_{e},} and the quantile function (inverse CDF) of e {\displaystyle
Binomial_regression
Statistic expressing the amount of random sampling error in a survey's results
values of z 1 − α {\displaystyle z_{1-\alpha }} are given by the quantile function of the normal distribution (which the 68–95–99.7 rule approximates)
Margin_of_error
Field of machine learning
Will; Ostrovski, Georg; Silver, David; Munos, Remi (2018-07-03). "Implicit Quantile Networks for Distributional Reinforcement Learning". Proceedings of the
Reinforcement_learning
Family of probability distributions
whole real line. Since the cumulative distribution function is invertible, the quantile function for the GEV distribution has an explicit expression
Generalized extreme value distribution
Generalized_extreme_value_distribution
Probability distribution
the paper, which is available for the computing of the cdf, pmf, quantile function, and random number generation of the Poisson binomial distribution
Poisson_binomial_distribution
Statistical sequence characterizing probability distributions
This integral can often be made more tractable by introducing the quantile function Q X {\displaystyle Q_{X}} via the change of variables y = F X ( x
L-moment
Utility-representation theorem in Decision Theory
over X {\displaystyle X} , with the naturally-induced mixture function. Quantile functions: for any CDF F : R → [ 0 , 1 ] {\displaystyle F:\mathbb {R} \to
Mixture-space_theorem
Motion of a body subject only to gravity
\beta )} is the quantile function of the beta distribution; also known as the inverse function of the regularized incomplete beta function I x ( α , β )
Free_fall
Term in statistical hypothesis testing
(thus no longer involving n) and so through use of the corresponding quantile function Φ−1, we obtain that the null should be rejected if Tn>tα≈Φ−1(0.95)≈1
Power_(statistics)
Fourth standardized moment in statistics
version of the fourth L-moment; measures based on four population or sample quantiles. These are analogous to the alternative measures of skewness that are
Kurtosis
Probability distribution
Y=\ln X} has a normal distribution, and quantiles are preserved under monotonic transformations, the quantiles of X {\displaystyle X} are q X ( α ) = exp
Log-normal_distribution
Method of estimating the parameters of a statistical model, given observations
probability density function, cumulative distribution function, or quantile function, to generate predictions of probabilities or quantiles of out-of-sample
Maximum_likelihood_estimation
Error rate graph
deviate mapping (or normal quantile function, or inverse normal cumulative distribution) is given by the probit function, so that the horizontal axis
Detection_error_tradeoff
Set of statistical processes for estimating the relationships among variables
different procedures to estimate alternative location parameters (e.g., quantile regression or Necessary Condition Analysis) or estimate the conditional
Regression_analysis
Dark matter halo formation model
\beta )} is the quantile function of the Beta distribution, also known as the inverse function of the regularized incomplete beta function I x ( α , β )
Spherical_collapse_model
Probability distribution of the test statistic under the null hypothesis
to genomics. 2008." Van Der Laan, Mark J., and Alan E. Hubbard. "Quantile-function based null distribution in resampling based multiple testing." Statistical
Null_distribution
Continuous probability distribution
Champernowne distribution, which has exponential tails. The inverse cdf (or quantile function) for a uniform variate 0 ≤ p < 1 is F − 1 ( p ) = − 2 π arsinh [ cot
Hyperbolic secant distribution
Hyperbolic_secant_distribution
Distance function defined between probability distributions
{\displaystyle F_{1}^{-1}} and F 2 − 1 {\displaystyle F_{2}^{-1}} are the quantile functions (inverse CDFs). In the case of p = 1 {\displaystyle p=1} , a change
Wasserstein_metric
Collective term for blood tests used to check the function of the thyroid
different equation. The Thyroid Feedback Quantile-based Index (TFQI) is another parameter for thyrotropic pituitary function. It was defined to be more robust
Thyroid_function_tests
Statistical test
D_{\mathrm {max} }} by plugging P z {\displaystyle P_{z}} into the Quantile Function. D m a x = Q ( P z ) ≈ 1.7317 {\displaystyle D_{\mathrm {max} }=Q(P_{z})\approx
Chauvenet's_criterion
Chart of correlation statistics
{z_{1-\alpha /2}}{\sqrt {N}}}} where N is the sample size, z is the quantile function of the standard normal distribution and α is the significance level
Correlogram
Continuous probability distribution
inverse probability transform method. For the two-parameter model, the quantile function (inverse CDF) is F − 1 ( u ) = σ exp − 1 β log ( 1 − u ) , 0 <
Benini_distribution
Moment of a random variable minus its mean
continuous univariate probability distribution with probability density function f(x), the n-th moment about the mean μ is μ n = E [ ( X − E [ X ] )
Central_moment
Probability distribution
(1976). Alternative forms to this distribution, with the corresponding quantile function, have been given by Ashour and Abdel-Hamid and by Mudholkar and Hutson
Skew_normal_distribution
Class of statistical models
response variable via a link function and by allowing the magnitude of the variance of each measurement to be a function of its predicted value. Generalized
Generalized_linear_model
Statistic which divides a data set into 100 parts and analyzes it as a percentage
that 97% of the data points are less than it. Percentiles are a type of quantiles, obtained by a subdivision into 100 groups. The 25th percentile (P25)
Percentile
Multi-dimensional version of a confidence interval
{\varepsilon ^{\operatorname {T} }\varepsilon }{n-p}}.} Further, F is the quantile function of the F-distribution, with p and ν = n − p {\displaystyle \nu =n-p}
Confidence_region
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