# Log-logistic distribution parameter estimation

Chapter The Loglogistic Distribution. Generate Reference Book: File may be more up-to-date.

## [R] Log logistic Distribution - Parameter estimation

As may be indicated by the name, the loglogistic distribution has certain similarities to the logistic distribution. A random variable is loglogistically distributed if the logarithm of the random variable is logistically distributed.

Because of this, there are many mathematical similarities between the two distributions, as discussed in Meeker and Escobar . For example, the mathematical reasoning for the construction of the probability plotting scales is very similar for these two distributions.

The pdf for this distribution is given by:. The method used by the application in estimating the different types of confidence bounds for loglogistically distributed data is presented in this section. The complete derivations were presented in detail for a general function in Parameter Estimation.

The bounds around time for a given loglogistic percentile, or unreliability, are estimated by first solving the reliability equation with respect to time, as follows:. Set up the folio for times-to-failure data that includes interval and left censored data, then enter the data. The computed parameters for maximum likelihood are calculated to be:. Navigation menu Personal tools Log in. Namespaces Page Discussion.

Views Read View source View history. This page was last edited on 23 Decemberat Creative Commons Attribution. Index Chapter The Loglogistic Distribution.In probability and statisticsthe log-logistic distribution known as the Fisk distribution in economics is a continuous probability distribution for a non-negative random variable. It is used in survival analysis as a parametric model for events whose rate increases initially and decreases later, as, for example, mortality rate from cancer following diagnosis or treatment.

It has also been used in hydrology to model stream flow and precipitationin economics as a simple model of the distribution of wealth or incomeand in networking to model the transmission times of data considering both the network and the software.

The log-logistic distribution is the probability distribution of a random variable whose logarithm has a logistic distribution. It is similar in shape to the log-normal distribution but has heavier tails. Unlike the log-normal, its cumulative distribution function can be written in closed form. There are several different parameterizations of the distribution in use.

The one shown here gives reasonably interpretable parameters and a simple form for the cumulative distribution function. The cumulative distribution function is. The probability density function is. Expressions for the meanvarianceskewness and kurtosis can be derived from this. Explicit expressions for the skewness and kurtosis are lengthy. The log-logistic distribution provides one parametric model for survival analysis.

The fact that the cumulative distribution function can be written in closed form is particularly useful for analysis of survival data with censoring. The log-logistic distribution has been used in hydrology for modelling stream flow rates and precipitation.

Extreme values like maximum one-day rainfall and river discharge per month or per year often follow a log-normal distribution. As the log-logistic distribution, which can be solved analytically, is similar to the log-normal distribution, it can be used instead.

The log-logistic has been used as a simple model of the distribution of wealth or income in economicswhere it is known as the Fisk distribution. For the log-logistic distribution, the formula for the Gini coefficient becomes:. The beta function may also be written as:.

PB65: Maximum A Posteriori (MAP) Estimation

Using the properties of the gamma function, it can be shown that:. From Euler's reflection formulathe expression can be simplified further:. The log-logistic has been used as a model for the period of time beginning when some data leaves a software user application in a computer and the response is received by the same application after travelling through and being processed by other computers, applications, and network segments, most or all of them without hard real-time guarantees for example, when an application is displaying data coming from a remote sensor connected to the Internet.

It has been shown to be a more accurate probabilistic model for that than the log-normal distribution or others, as long as abrupt changes of regime in the sequences of those times are properly detected.

Several different distributions are sometimes referred to as the generalized log-logistic distributionas they contain the log-logistic as a special case. Both are in turn special cases of the even more general generalized beta distribution of the second kind. Another more straightforward generalization of the log-logistic is the shifted log-logistic distribution.

From Wikipedia, the free encyclopedia. Redirected from Loglogistic distribution. Log-logistic Probability density function. Derivation of Gini coefficient. Probability distributions List.In probability theorya log-normal or lognormal distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. A random variable which is log-normally distributed takes only positive real values.

It is a convenient and useful model for measurements in exact and engineering sciences, as well as medicineeconomics and other topics e. The distribution is occasionally referred to as the Galton distribution or Galton's distributionafter Francis Galton. A log-normal process is the statistical realization of the multiplicative product of many independent random variableseach of which is positive.

This is justified by considering the central limit theorem in the log domain sometimes call Gibrat's law. The log-normal distribution is the maximum entropy probability distribution for a random variate X —for which the mean and variance of ln X are specified.

Then, the distribution of the random variable. A positive random variable X is log-normally distributed i. The cumulative distribution function is.

This may also be expressed as follows: . Since the multivariate log-normal distribution is not widely used, the rest of this entry only deals with the univariate distribution.

### The RELIABILITY Procedure

However, the log-normal distribution is not determined by its moments. Consequently, the characteristic function of the log-normal distribution cannot be represented as an infinite convergent series. However, a number of alternative divergent series representations have been obtained. A relatively simple approximating formula is available in closed form, and is given by .

It equals the median. Note that the geometric mean is smaller than the arithmetic mean. This is due to the AM—GM inequalityand corresponds to the logarithm being convex down. In fact.

For any real or complex number nthe n -th moment of a log-normally distributed variable X is given by . Specifically, the arithmetic mean, expected square, arithmetic variance, and arithmetic standard deviation of a log-normally distributed variable X are respectively given by: .

For a log-normal distribution it is equal to . This estimate is sometimes referred to as the "geometric CV" GCV  due to its use of the geometric variance. Contrary to the arithmetic standard deviation, the arithmetic coefficient of variation is independent of the arithmetic mean.

That is, there exist other distributions with the same set of moments. The mode is the point of global maximum of the probability density function. Specifically, the median of a log-normal distribution is equal to its multiplicative mean, . For a log-normal random variable, the partial expectation is given by:.The principle of maximum entropy POME was employed to derive a new method of parameter estimation for the 3-parameter log-logistic distribution LLD3. Simulation results showed that POME's performance was superior in predicting quantiles of large recurrence intervals when population skew was greater than or equal to 2. In all other cases, POME's performance was comparable to other methods. This is a preview of subscription content, access via your institution.

Rent this article via DeepDyve. Ahmad, M. Journal of Hydrology 98, — Google Scholar. Bennett, S. Applied Statistics 32, — Burr, I. Guo, H. Haktanir, T. Hydrological Sciences Journal 36, No. Jaynes, E. Lawless, J. Biometrika 73, 2, — Levine, R. Massachusetts: MIT Press. Mielke, P. Monthly Weather Review— Shoukri, M. The Canadian Journal of Statistics 16, No. Singh, V. Hydrological Science and Technology 2, No. Tribus, M. New York: Pergamon Press. Download references. Reprints and Permissions. Parameter estimation for 3-parameter log-logistic distribution LLD3 by Pome. Stochastic Hydrol Hydraul 7, — Download citation.

Issue Date : September Search SpringerLink Search. Abstract The principle of maximum entropy POME was employed to derive a new method of parameter estimation for the 3-parameter log-logistic distribution LLD3. Immediate online access to all issues from Subscription will auto renew annually.

References Ahmad, M. Author information Affiliations Dept.Maximum likelihood estimation of the parameters of a statistical model involves maximizing the likelihood or, equivalently, the log likelihood with respect to the parameters. The parameter values at which the maximum occurs are the maximum likelihood estimates of the model parameters. The likelihood is a function of the parameters and of the data. Let be the observations in a random sample, including the failures and censoring times if the data are censored.

Let be the probability density of failure time, be the reliability function, and be the cumulative distribution function, where is the vector of parameters to be estimated. The probability density, reliability function, and CDF are determined by the specific distribution selected as a model for the data. The log likelihood is defined as.

Only the sums appropriate to the type of censoring in the data are included when the preceding equation is used. The Newton-Raphson algorithm is a recursive method for computing the maximum of a function. On the r th iteration, the algorithm updates the parameter vector with. That is. The default value of c is 0.

The default value of d is 0.

## Log-normal distribution

The relative Hessian criterion is useful in detecting the occasional case where no progress can be made in increasing the log likelihood, yet the gradient g is not zero. It is more convenient computationally to maximize log likelihoods that arise from location-scale models.

If you specify a distribution from Table If you specify the lognormal base 10 distribution, the logarithm base 10 of the response is used. The Weibull, lognormal, and log-logistic distributions in Table Table If the Weibull distribution is specified, the logarithms of the responses are used to obtain maximum likelihood estimates of the location and scale parameters of the extreme value distribution.

The maximum likelihood estimatesof the Weibull scale and shape parameters are computed as and. Maximum likelihood estimates for the Gompertz distributions are obtained by expressing the log-likelihood in terms of, and if applicable. After the log likelihood is maximized, parameter estimates and their standard errors are transformed from the logarithm metric to the standard metric by using the delta method.

The parameters of the three-parameter Weibull distribution are estimated by maximizing the log likelihood function. The threshold parameter must be less than the minimum failure timeunlessin which case, can be equal to.

If the shape parameter is less than one, then the density function in Table For any fixedmaximum likelihood estimates of the scale and shape parameters and exist. If in the iterative estimation procedure, the estimate of the threshold is set to the upper bound and maximum likelihood estimates of and are computed. The data set In specifies a lower bound for the threshold parameter of — for groups A, B, and D, and an upper bound of 3 for the threshold parameter for group D. For example, if you want to relate the lifetimes of electronic parts in a test to Arrhenius-transformed operating temperature, then an appropriate model might be.

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Here, [ 1 ]. There are two types of explanatory variables: continuous variables and classification variables. Continuous variables represent physical quantities, such as temperature or voltage, and they must be numeric. Continuous explanatory variables are sometimes called covariates. These are also referred to as categorical, dummy, qualitative, discrete, or nominal variables.

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Classification variables can be either character or numeric. The values of classification variables are called levels. An indicator variable is generated for each level of a classification variable and is used as an explanatory variable. See Nelsonp.Metrics details.

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It is presumed that samples for independent non informative set of priors for estimating LL parameters are drawn from posterior density function. It is shown that statistically consistent parameter estimates and their respective credible intervals can be constructed through the use of OpenBUGS.

Finally comparison of maximum likelihood estimate and Bayes estimates is carried out using three plots. Additively through this research it is established that computationally MCMC technique can be effortlessly put into practice. Elaborate procedure for applying MCMC, to estimate parameters of LL model, is demonstrated by making use of real survival data relating to bladder cancer patients.

The log-logistic LL distribution branded as the Fisk distribution in economics possesses a rather supple functional form. The LL distribution is among the class of survival time parametric models where the hazard rate initially increases and then decreases and at times can be hump-shaped. The LL distribution can be used as a suitable substitute for Weibull distribution.

It is in fact a mixture of Gompertz distribution and Gamma distribution with the value of the mean and the variance coincide—equal to one. The LL distribution as a life testing model has its own standing; it is an increasing failure rate IFR model and also is viewed as a weighted exponential distribution.

Scrolling through the literature on the subject distribution we see that Bain modeled LL distribution by a transformation of a well-known logistic variate. The properties of LL distribution have been deliberated upon by Ragab and Green who also worked on the order statistics for the said distribution.

Kantam et al. Kantam and Rao derived the modified maximum likelihood estimation MLE of this distribution.

Rosaiah et al. The properties, estimation and testing of linear failure rate using exponential and half-logistic distribution has been discussed thoroughly by Rao et al. The current research intends to use LL distribution for modeling the survival data and to obtain MLE utilizing associated probability intervals of the Bayes estimates.

It has been noticed that the Bayesian estimates may not be computed plainly under the assumption of independent uniform priors for the parameters. The authors will work under the assumption that both parameters—shape and scale, of the LL model are unknown. Bayesian estimates of parameters along with highest posterior density HPD credible intervals will be constructed.

Moreover, estimation of the reliability function will also be looked into.The shifted log-logistic distribution is a probability distribution also known as the generalized log-logistic or the three-parameter log-logistic distribution. The shift parameter adds a location parameter to the scale and shape parameters of the unshifted log-logistic. The properties of this distribution are straightforward to derive from those of the log-logistic distribution.

However, an alternative parameterisation, similar to that used for the generalized Pareto distribution and the generalized extreme value distributiongives more interpretable parameters and also aids their estimation. In this parameterisation, the cumulative distribution function CDF of the shifted log-logistic distribution is.

The probability density function PDF is. The three-parameter log-logistic distribution is used in hydrology for modelling flood frequency. From Wikipedia, the free encyclopedia. Shifted log-logistic Probability density function.

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Retrieved 1 October Probability distributions List. Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher soliton discrete uniform Zipf Zipf—Mandelbrot. Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S U Landau Laplace asymmetric Laplace logistic noncentral t normal Gaussian normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy—Widom variance-gamma Voigt.

Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate Laplace multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart.

Degenerate Dirac delta function Singular Cantor. Circular compound Poisson elliptical exponential natural exponential location—scale maximum entropy mixture Pearson Tweedie wrapped.

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