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Continuous probability distribution for a non-negative random variable
and statistics, the log-logistic distribution (known as the Fisk distribution in economics) is a continuous probability distribution for a non-negative
Log-logistic_distribution
Continuous probability distribution
statistics, the logistic distribution is a continuous probability distribution. Its cumulative distribution function is the logistic function, which appears
Logistic_distribution
Probability distribution
log-logistic distribution is a probability distribution also known as the generalized log-logistic or the three-parameter log-logistic distribution.
Shifted log-logistic distribution
Shifted_log-logistic_distribution
Name for several different families of probability distributions
other families of distributions that have also been called generalized logistic distributions, see the shifted log-logistic distribution, which is a generalization
Generalized logistic distribution
Generalized_logistic_distribution
Risk measure estimating the average loss in the worst tail of the distribution
X} follows log-logistic distribution, i.e. the random variable ln ( 1 + X ) {\displaystyle \ln(1+X)} follows the logistic distribution with p.d.f.
Expected_shortfall
Function in statistics
statistics, the logit (logistic unit) or log-odds function is the quantile function associated with the standard logistic distribution. It has many uses in
Logit
Statistical model for a binary dependent variable
In statistics, a logistic model (or logit model) is a statistical model that models the log-odds of an event as a linear combination of one or more independent
Logistic_regression
Probability distribution used to model household income
Singh–Maddala distribution and is one of a number of different distributions sometimes called the "generalized log-logistic distribution". The Burr (Type
Burr_distribution
Probability distribution
In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally
Log-normal_distribution
the Wald distribution The Lévy distribution The log-Cauchy distribution The log-Laplace distribution The log-logistic distribution The log-metalog distribution
List of probability distributions
List_of_probability_distributions
2D graphic with logarithmic scales on both axes
model Log-normal distribution Log-logistic distribution Data transformation (statistics) Variance-stabilizing transformation Bourne, Murray. "7. Log-Log and
Log–log_plot
Probability distribution
distributed, then Y = logit(X)= log (X/(1-X)) is normally distributed. It is also known as the logistic normal distribution, which often refers to a multinomial
Logit-normal_distribution
Measure of inequality of a statistical distribution
Distribution". mathworld.wolfram.com. Retrieved 30 November 2022. "The Log-Logistic Distribution". randomservices.org. Retrieved 30 November 2022. Abdon, Mitch
Gini_coefficient
Probability distribution
log ( e − X 1 − e − X ) ∼ Logistic ( μ , β ) {\displaystyle \mu -\beta \log \left({\frac {e^{-X}}{1-e^{-X}}}\right)\sim \operatorname {Logistic}
Exponential_distribution
S-shaped curve
{p}{p+1-p}}=p} The conversion from the log-likelihood ratio of two alternatives also takes the form of a logistic curve. The logistic function is an offset and scaled
Logistic_function
Probability distribution
Weibull distribution with shape parameter greater than 0 but less than 1; the Burr distribution; the log-logistic distribution; the log-gamma distribution; the
Heavy-tailed_distribution
Conditional distribution in statistics
Weibull distribution and the left-truncated log-logistic distribution. The Tobit model employs truncated distributions. Other examples include truncated binomial
Truncated_distribution
Kind of numerical parameter of a parametric family of probability distributions
power distribution Fréchet distribution Gamma distribution Generalized extreme value distribution Log-logistic distribution Log-t distribution Inverse-gamma
Shape_parameter
Topic in probability theory and statistics
of the same family of distribution as X, in the following cases: Cauchy distribution, F distribution, log logistic distribution. Examples: If X is a Cauchy
Relationships among probability distributions
Relationships_among_probability_distributions
Particular case of the generalized extreme value distribution
generalized extreme value distribution (also known as the Fisher–Tippett distribution). It is also known as the log-Weibull distribution and the double exponential
Gumbel_distribution
Observation that in many real-life datasets, the leading digit is likely to be small
Muth distribution, Gompertz distribution, Weibull distribution, gamma distribution, log-logistic distribution and the exponential power distribution all
Benford's_law
Probability distribution
Singh–Maddala distribution. β ′ ( 1 , 1 , γ , σ ) = LL ( γ , σ ) {\displaystyle \beta '(1,1,\gamma ,\sigma )={\textrm {LL}}(\gamma ,\sigma )} the log logistic distribution
Beta_prime_distribution
Concept in statistics
half-logistic distribution is a continuous probability distribution—the distribution of the absolute value of a random variable following the logistic distribution
Half-logistic_distribution
Continuous probability distribution
a_{i}=0} otherwise. The log-logistic distribution, also known as the Fisk distribution in economics, is a special case of the log metalog where b l = 0
Metalog_distribution
Parametric model in survival analysis
somewhat similar in shape to the log-normal distribution but it has heavier tails. The log-logistic cumulative distribution function has a simple closed form
Accelerated failure time model
Accelerated_failure_time_model
Information-theoretic measure
{\displaystyle q} relative to a distribution p {\displaystyle p} over a given set is defined as follows: H ( p , q ) = − E p [ log q ] , {\displaystyle H(p
Cross-entropy
Family of probability distributions
\mathrm {Logistic} (2\alpha ,\beta )\ } (The sum is not a logistic distribution). Note that E { X + Y } = 2 α + 2 β γ ≠ 2 α = E { Logistic (
Generalized extreme value distribution
Generalized_extreme_value_distribution
Regression for more than two discrete outcomes
In statistics, multinomial logistic regression is a classification method that generalizes logistic regression to multiclass problems, i.e. with more than
Multinomial logistic regression
Multinomial_logistic_regression
Probability of survival beyond any specified time
distribution: survival function 1 is defined by an exponential distribution, 2 is defined by a Weibull distribution, 3 is defined by a log-logistic distribution
Survival_function
Mathematical concept
the log-logistic distribution (i.e. the log values of the data follow a logistic distribution), the Gumbel distribution, the exponential distribution, the
Probability distribution fitting
Probability_distribution_fitting
Probability distribution
\beta )} , then Y = log X 1 − X {\displaystyle Y=\log {\frac {X}{1-X}}} has a generalized logistic distribution, also called logistic-beta, with density
Beta_distribution
Class of statistical models
log-odds or logistic model. Generalized linear models cover all these situations by allowing for response variables that have arbitrary distributions
Generalized_linear_model
Mathematical model
fi(X) in the range −∞ to +∞. This may be contrasted to logistic models, similar to the logistic function, for which the output quantity lies in the range
Log-linear_model
Simple polynomial map exhibiting chaotic behavior
The logistic map is a discrete dynamical system defined by the quadratic difference equation It is a recurrence relation and a polynomial mapping of degree 2
Logistic_map
Probability distribution
the Cauchy, Student's t, and logistic distributions). (For other names, see Naming.) The univariate probability distribution is generalized for vectors
Normal_distribution
Probability distribution
logarithmic marginals, log x k 1 − x k {\displaystyle \log {\frac {x_{k}}{1-x_{k}}}} , which follow the logistic-beta distribution, B σ ( α k , ∑ i ≠ k
Dirichlet_distribution
Probability distribution
mode include the logistic distribution, hyperbolic secant distribution, and the Champernowne distribution. The Laplace distribution is easy to integrate
Laplace_distribution
regression Log-log plot Log-logistic distribution Logarithmic distribution Logarithmic mean Logistic distribution Logistic function Logistic regression
List_of_statistics_articles
Branch of statistics
distribution Hypertabastic distribution Lindley distribution Log-logistic distribution Weibull distribution Credit risk False conviction rate of inmates sentenced
Survival_analysis
Type of mathematical function
the binomial distribution, the logistic distribution, the extreme value distribution, the Laplace distribution, the chi distribution, the hyperbolic
Logarithmically concave function
Logarithmically_concave_function
Smooth approximation of one-hot arg max
probability distribution over K possible outcomes. It is a generalization of the logistic function to multiple dimensions, and is used in multinomial logistic regression
Softmax_function
Family of probability distributions related to the normal distribution
is called the skew-logistic distribution). The density can be rewritten as e − x 1 + e − x exp [ − θ log ( 1 + e − x ) + log ( θ ) ] {\displaystyle
Exponential_family
Statistical quantity
Kumaraswamy distribution Log-logistic distribution (Fisk distribution): Let β be the shape parameter. The variance and mean of this distribution are only
Nonparametric_skew
Continuous probability distribution
Weibull distribution Fisher–Tippett–Gnedenko theorem Logistic distribution Rosin–Rammler distribution for particle size analysis Rayleigh distribution Unit
Weibull_distribution
Indicator function of positive numbers
approximations are cumulative distribution functions of common probability distributions: the logistic, Cauchy and normal distributions, respectively. Approximations
Heaviside_step_function
System for rating game players
logistic distributions are in a way arbitrary points in a spectrum of distributions which would work well. In practice, both of these distributions work
Elo_rating_system
Concept in machine learning
{\displaystyle I[f]} for the logistic loss function can be directly found from equation (1) as f Logistic ∗ = log ( η 1 − η ) = log ( p ( 1 ∣ x ) 1 − p (
Loss functions for classification
Loss_functions_for_classification
Mathematical function, inverse of an exponential function
f(w) = wew, and of the logistic function, respectively. From the perspective of group theory, the identity log(cd) = log(c) + log(d) expresses a group isomorphism
Logarithm
Statistic quantifying the association between two events
{p}}_{01}}}\right)=\log \left({\dfrac {n_{11}n_{00}}{n_{10}n_{01}}}\right)}} . The distribution of the log odds ratio is approximately normal with: L ∼ N ( log (
Odds_ratio
Fourth standardized moment in statistics
Student's t-distribution, Rayleigh distribution, Laplace distribution, exponential distribution, Poisson distribution and the logistic distribution. Such distributions
Kurtosis
Technique used in statistics
would be best to use logistic regression. (Any data that is analysed with log-linear analysis can also be analysed with logistic regression. The technique
Log-linear_analysis
Mathematical model used for classification or regression
can be used to sample new data. Types of discriminative models include logistic regression (LR), conditional random fields (CRFs), decision trees among
Discriminative_model
Probabilistic classification algorithm
Bayes classifiers generally perform worse than more advanced models like logistic regressions, especially at quantifying uncertainty (with naive Bayes models
Naive_Bayes_classifier
Average uncertainty in variable's states
uniform probability distribution. That is, uncertainty is maximal when all possible events are equiprobable: H ( p 1 , … , p n ) ≤ log b n . {\displaystyle
Entropy_(information_theory)
Entropy of a process with only two probable values
the formula: H ( X ) = − p log p − ( 1 − p ) log ( 1 − p ) . {\displaystyle \operatorname {H} (X)=-p\log p-(1-p)\log(1-p).} The base of the logarithm
Binary_entropy_function
Regression model for ordinal dependent variables
In statistics, the ordered logit model or proportional odds logistic regression is an ordinal regression model—that is, a regression model for ordinal
Ordered_logit
Heavy-tail probability distribution
follows a logistic distribution with location log(λ) and scale 1.0. The Lomax distribution arises as a mixture of exponential distributions where the
Lomax_distribution
Law in seismology describing earthquake frequency and magnitude
given region and time period of at least that magnitude. log 10 N = a − b M {\displaystyle \log _{10}N=a-bM} or N = 10 a − b M {\displaystyle N=10^{a-bM}}
Gutenberg–Richter_law
Family of functions to transform data
logarithm log ( X i ) {\displaystyle \log(X_{i})} : X i log ( X i ) {\displaystyle X_{i}\log(X_{i})} This term is included in the logistic regression
Power_transform
Model for generating observable data in probability and statistics
Bayes. In this sense, Logistic Regression is often referred to as a discriminative classifier because we can view the distribution P ( Y ∣ X ) {\displaystyle
Generative_model
Number of occurrences in an experiment or study
such frequency distribution. The ideal number of classes may be determined or estimated by formula: number of classes = C = 1 + 3.3 log n {\displaystyle
Frequency_(statistics)
Mathematical function having a characteristic S-shaped curve or sigmoid curve
common in statistics as cumulative distribution functions (which go from 0 to 1), such as the integrals of the logistic density, the normal density, and
Sigmoid_function
Statistical modeling method
described using a skewed distribution such as the log-normal distribution or Poisson distribution (although GLMs are not used for log-normal data, instead
Linear_regression
Frequency with which an engineered system or component fails
combines both of these effects, as do the log-normal and hypertabastic distributions. After modelling a given distribution and parameters for h ( t ) {\displaystyle
Failure_rate
Mathematical function having a characteristic "bell"-shaped curve
Most of the window functions like the Kaiser window The derivative of the logistic function. This is a scaled version of the derivative of the hyperbolic
Bell-shaped_function
Measure of association used in epidemiology
since logistic regression, often associated with clinical trials, works with the log of the odds ratio, not relative risk. Because the (natural log of the)
Relative_risk
Statistical model for count data
Press. ISBN 978-0-521-63201-0. Christensen, Ronald (1997). Log-linear models and logistic regression. Springer Texts in Statistics (Second ed.). New York:
Poisson_regression
Regression analysis technique
inside the range 0 to 1. In the case of logistic regression, the link function is the log of the odds ratio or logistic function. In the case of probit, the
Binomial_regression
Distribution of an uncertain quantity
marginal distribution p ( x ) {\displaystyle p(x)} , so we have K L = ∫ p ( t ) ∫ p ( x ∣ t ) log [ p ( x ∣ t ) ] d x d t − ∫ p ( x ) log [ p ( x
Prior_probability
Function related to statistics and probability theory
the use of log-likelihoods (see Wilks' theorem), the test statistic is twice the difference in log-likelihoods and the probability distribution of the test
Likelihood_function
Statistical matching technique
versus control group—based on observed predictors, usually obtained from logistic regression to create a counterfactual group. Propensity scores may be used
Propensity_score_matching
Mathematical model
{IP}{I+E}}} Species–area relationship: log ( S ) = log ( c ) + z log ( A ) {\displaystyle \log(S)=\log(c)+z\log(A)\,} Population dynamics Population
Population_model
Probability distribution
triangle. Mathematics portal Logistic regression Multinomial distribution Negative binomial distribution Beta-binomial distribution Binomial measure, an example
Binomial_distribution
Probability distribution on a hyper-sphere of arbitrary dimension
r {\displaystyle r} is symmetric beta and the distribution for s {\displaystyle s} is symmetric logistic-beta: r ∼ Beta ( p − 1 2 , p − 1 2 ) , s ∼ B σ
Von_Mises–Fisher_distribution
Principle in Bayesian statistics
probability distribution would be uniform, and then the information entropy would be equal to its maximum possible value, log m {\displaystyle \log m} . The
Principle_of_maximum_entropy
Method used in statistics, pattern recognition, and other fields
variables and a categorical dependent variable (i.e. the class label). Logistic regression and probit regression are more similar to LDA than ANOVA is
Linear_discriminant_analysis
Non-parametric statistic used to estimate the survival function
the log likelihood will be: log ( L ) = ∑ j = 1 i ( d j log ( h j ) + ( n j − d j ) log ( 1 − h j ) + log ( n j d j ) ) {\displaystyle \log({\mathcal
Kaplan–Meier_estimator
Statistical measure of fit
example, for logistic regression, the upper bound is R M 2 ≤ 0.75 {\displaystyle R_{\text{M}}^{2}\leq 0.75} for a symmetric marginal distribution of events
Pseudo-R-squared
Probability distribution
Poisson regression and logistic regression. This takes advantage of the exponential family properties of the CMP distribution to obtain elegant model
Conway–Maxwell–Poisson distribution
Conway–Maxwell–Poisson_distribution
Measure of the asymmetry of random variables
theory and statistics is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. Similarly to kurtosis
Skewness
Method of estimating the parameters of a statistical model, given observations
parameters of the log-normal distribution are the same as those of the normal distribution fitted to the logarithm of the data. In fact, in the log-normal case
Maximum_likelihood_estimation
Statistical theorem
In statistics, Wilks' theorem offers an asymptotic distribution of the log-likelihood ratio statistic, which can be used to produce confidence intervals
Wilks'_theorem
Probability distribution
this case the distribution is a normal distribution, otherwise the distributions are shifted and possibly reversed log-normal distributions. Parameters
Generalized normal distribution
Generalized_normal_distribution
N-th root of the product of n numbers
standard deviation Harmonic mean Heronian mean Heteroscedasticity Log-normal distribution Muirhead's inequality Product Pythagorean means Quadratic mean
Geometric_mean
Unbiased statistical estimator minimizing variance
− x exp ( − θ log ( 1 + e − x ) + log ( θ ) ) {\displaystyle {\frac {e^{-x}}{1+e^{-x}}}\exp \left(-\theta \log(1+e^{-x})+\log(\theta )\right)}
Minimum-variance unbiased estimator
Minimum-variance_unbiased_estimator
Generalization of the one-dimensional normal distribution to higher dimensions
statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional
Multivariate normal distribution
Multivariate_normal_distribution
Probability distribution
statistics, Student's t distribution (or simply the t distribution) t ν {\displaystyle t_{\nu }} is a continuous probability distribution that generalizes the
Student's_t-distribution
Relative measure of dispersion expressed as the ratio of standard deviation to the mean
approximately log-normal distribution. In such cases, a more accurate estimate, derived from the properties of the log-normal distribution, is defined as:
Coefficient_of_variation
Probability distribution
statistics, the skew normal distribution is a continuous probability distribution that generalises the normal distribution to allow for non-zero skewness
Skew_normal_distribution
Statistical distribution for dependence between random variables
Mikhail, N.N.; Haq, M.S. (1978). "A class of bivariate distributions including the bivariate logistic". Journal of Multivariate Analysis. 8 (3): 405–412.
Copula_(statistics)
Hypothesis test to compare the survival distributions of two samples
The logrank test, or log-rank test, is a hypothesis test to compare the survival distributions of two samples. It is a nonparametric test and appropriate
Logrank_test
Statistical test comparing two probability distributions
Stephens, M. A. (1979). "Test of fit for the logistic distribution based on the empirical distribution function". Biometrika. 66 (3): 591–595. doi:10
Kolmogorov–Smirnov_test
Regression analysis for modeling ordinal data
(using the Iverson bracket [yi = k].) The log-likelihood of the ordered logit model is analogous, using the logistic function instead of Φ. In machine learning
Ordinal_regression
Fundamental theorem in probability theory and statistics
only positive values approaches a normal distribution, the product itself approaches a log-normal distribution. Many physical quantities (especially mass
Central_limit_theorem
Mathematical function for the probability a given outcome occurs in an experiment
distribution. Normal distribution (Gaussian distribution), for a single such quantity; the most commonly used absolutely continuous distribution Log-normal
Probability_distribution
Mathematical functions
type I generalizes the logistic function. If the parameters θ = 0 {\displaystyle \theta =0} , then it would become a logistic function. This function
Hyperbolastic_functions
Probability distribution used in multivariate hypothesis testing
a chi-squared distribution ( p − n + 1 2 − m ) log Λ ( p , m , n ) ∼ χ n p 2 . {\displaystyle \left({\frac {p-n+1}{2}}-m\right)\log \Lambda (p,m,n)\sim
Wilks's_lambda_distribution
Value that appears most often in a set of data
standard deviation σ = 0.25, the distribution of Y is weakly skewed. Using formulas for the log-normal distribution, we find: mean = e μ + σ 2 / 2 = e
Mode_(statistics)
{\displaystyle s} parameters of the logistic quantile function. The semi-bounded and bounded metalog distributions, which are the log and logit transforms, respectively
Quantile-parameterized distribution
Quantile-parameterized_distribution
Iterative method for finding maximum likelihood estimates in statistical models
maximizing the expected log-likelihood found on the E step. These parameter-estimates are then used to determine the distribution of the latent variables
Expectation–maximization algorithm
Expectation–maximization_algorithm
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LOG LOGISTIC-DISTRIBUTION
LOG LOGISTIC-DISTRIBUTION
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LOG LOGISTIC-DISTRIBUTION
LOG LOGISTIC-DISTRIBUTION
LOG LOGISTIC-DISTRIBUTION
LOG LOGISTIC-DISTRIBUTION
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