Questions tagged [logistic-distribution]

A symmetric distribution which CDF is the logistic function.

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Logistic regression - Confidence interval for x axis

My question is very similar to the post Confidence interval for x-values given a probability in a logistic regression where ultimately no answer was given. I have also posted a similar version on the ...
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Where does the Logistic Distribution get its name?

Having read around on the topic I understand its application as a close approximation of the normal distribution with a nicer mathematical form, but where does its name come from? Is it associated ...
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Logistic Regression vs Random Forest

I have a dataset with a binary dependent variable and 6 dependent variables. I performed logistic regression and random forest and the AUC is higher for Logistic Regression. I would like though to ...
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Multinomial Logistic Regression as a latent variable model

I was reading the wiki entry for multinomial logistic regression https://en.wikipedia.org/wiki/Multinomial_logistic_regression#As_a_latent-variable_model and it states that we can view the multinomial ...
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1 answer
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Logistic distribution's minimal sufficient statistics

Distribution function(pdf) is If x is sample from population, f(x|θ) is I calculated f(x|theta)/f(y|theta), but I don't know how can I find minimal sufficient statistics.
3 votes
2 answers
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Why does the glm function converge and not give an error when all y's are equal to the same value?

I need to fit a univariate logistic model with few observations (between 10 and 20). In some cases, y is equal to the same value (example 1) for all observations. Theoretically, the model should not ...
2 votes
2 answers
136 views

Inverse Mills Ratio for Logit?

For $X\sim N(\mu,\sigma^2)$ , $$E[X|X>\alpha] = \mu +\sigma \frac{\phi\left(\frac{\alpha-\mu}{\sigma}\right)}{1-\Phi\left(\frac{\alpha-\mu}{\sigma}\right)} $$ Is there an analogous expression for ...
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1 answer
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Is the Hodges-Lehmann estimator 'optimal' for estimating the location parameter of Logistic distribution?

Is the Hodges-Lehmann estimator $\hat\theta_{HL}=\operatorname{median}\limits_{1\le i\le j\le n}\left\{\frac{X_i+X_j}{2}\right\}$ in some sense 'optimal' for estimating the location parameter $\theta$ ...
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Are their feature selection always go by linear model?

A generalized linear model maps a linear transformation of features to some response through monotonic function, does GLM feature selection always go by analyzing the coefs of this linear ...
2 votes
1 answer
83 views

Trade-off between omitting variables or dropping observations in multivariate logistic regression

Say you are selecting $n$ observations from a complex survey of $N$ individuals to create an analytical sample of relevant observations; and that you intend to fit a binomial multivariate logistic ...
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What are the preliminary analysis before running a logistic regression?

I have a dichotomous variable which represents if a student is accepted or not in a University. In order to do this I have about 60 variables (information of the students: gender, age, etc; their ...
3 votes
2 answers
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Regression with Logistic-Distribution errors (NOT Logistic Regression)

I was wondering if anyone ever tried to do a regression where the errors, instead of normal, would be assumed to be from the Logistic Distribution. I don't mean Logistic Regression, as I don't assume ...
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1 answer
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Method of moments and MLE

Suppose a random variable follows the logistic distribution, $X ∼ Logistic (\mu, \sigma)$ and we restrict our attention to random samples drawn from this random variable $X$. What would be the MoM and ...
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1 answer
264 views

When should you not use logistic regression GLM with a binary response variable?

I came across a set of data that does not seem to "flip" from 0 to 1 (binary response variable) when the predictor variable increase. Thus, this is the plot I got after plotting the ...
4 votes
1 answer
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What is $\int_{c}^{\infty}\Phi(a+bx) \phi( x ) dx$ for $c\in\mathbb{R}$

It is straightforward to show $$\int_{-\infty}^{\infty}\Phi(a+bx) \phi( x ) dx = \Phi\left(\frac{a}{\sqrt{1+b^2}}\right)$$ but what value does $$\int_{c}^{\infty}\Phi(a+bx) \phi( x ) dx$$ have for $c&...
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Relation between logistic regression and logistic distribution [duplicate]

When we are using logistic regression, we can get the probability that $y$ belongs to class $1$ as follows: $P(y=1|x;\theta)=\frac{1}{1+\exp(-\theta^Tx)}$. PDF of a logistic distribution is given as ...
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A Gaussian scale mixture representation of the logistic distribution

Can the logistic distribution with density function $$f(x) = \frac{e^{-x}}{\left(1 + e^{-x}\right)^2}$$ be represented as a Gaussian scale mixture? In other words, if \begin{align*} X|V &\sim N(0, ...
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What is (are) scenerios and practical settings that can possibly lead to the weibull-log-Logistic mixture distribution?

In my paper I studied Weibull-loglogistic mixture distributions in reliability and life testing, some structural properties of the model are presented including moments, reliability, hazard rate ...
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What are assumptions for logistic binomial regression if all independent and independent variables are dichotomous?

I have three independent and one dependent variables of the dichotomous type and I am trying to use logistic binomial regression. I have 117 observations in total. When I read the literature, some of ...
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logistic distribution

Is it true that almost every discrete data set containing positive values follow logistic distribution if the data set is log-transformed and standardised(subtracting mean and dividing by standard ...
1 vote
1 answer
959 views

What form of conjugate prior best fits this likelihood distribution? [closed]

Joint likelihood of a two part model consisting of logistic regression and log-normal model:
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1 answer
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Change of variable for logistic distribution to loglogistic - where do I go wrong?

Similar to what I described in this post I would like to make the following change of variable $$Y := \exp( \eta + \sigma e ), \sigma>0 $$ where $e$ has the standard logistic density $f_E(e) = \exp(...
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Comparison of logit and probit estimations

There are a lot of questions concerning logit and probit relations (led by 20523), but I'm still confused with a seemingly simple issue. On the one hand, often we see that for 'rule-of-thumb' ...
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1 answer
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How to invert a mixture of log-logistic distributions? [duplicate]

Say I have a CDF of the form $$F(x) = \sum_{i=1}^n \frac{ k_i }{ 1 + (x/\alpha_i)^{ - \beta_i } }$$ $$\sum_{i=1}^n k_i = 1$$ How do I find the quantile function, i.e. how do I invert F for n>1? I ...
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How to specify a risk model?

I'm reading this article. The authors indicated that for a random sample of 12 characteristics and 3600 patients affected in two arms (control and treatment arm), they fitted a risk model consisting ...
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2 answers
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What is the probability distribution used in logistic regression called?

In logistic regression, we set the probability of predicting a target $y$ given a data $x$ as, $\Pr(Y = 1|X;w) = \dfrac{\exp(w^TX)}{(1+\exp(w^TX))}$ What is exactly this probability distribution (or ...
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Independent variables should have exponential distribution in logistic regression

As per my understanding of logistic regression, a log of odds of the desired value of “y” should be in linear relation with the log (x). Does that mean that independent variables should have ...
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Location and scale estimation for log-logistic in scipy

I have to estimate the location and scale parameters for a log-logistic or fisk distribution using scipy. The shape parameter is already defined and I would like to estimate loc and scale parameter ...
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How can I calculate $\int^{\infty}_{-\infty}\Phi\left(\frac{w-a}{b}\right)\phi(w)\,\mathrm dw$ for the logistic distribution?

Take the cumulative distribution function (cdf) $\Phi(z) = \frac{1}{1+\exp(-z)}$ and probability density function (pdf) $\phi(z)=\Phi(z)(1-\Phi(z))$ of the logistic distribution. How can one calculate ...
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Logistic Regression For Classification [duplicate]

The origin of logistic regression is actually logistic curve which varies from the value 0 to the value 1. It looks like the letter S, and it specifies the growth of species. If our data distribution ...
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How is Logistic Regression related to Logistic Distribution?

We all know that logistic regression is used to calculate probabilities through the logistic function. For a dependent categorical random variable $y$ and a set of $n$ predictors $\textbf{X} = [X_1 \...
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Generalized logistic distribution

I saw on wikipedia https://en.wikipedia.org/wiki/Generalized_logistic_distribution that when $\alpha<\beta$, generalized type IV logistic distribution can be written as: $\frac{\exp(-\alpha x)}{(\...
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3 votes
2 answers
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Is there a connection between the normal and the logistic distribution?

Regarding Bayesian statistics I found in a script that there is such link, and the logistic arises in context of a normal distribution and a "binary state". However, I have no idea what is the meaning ...
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1 answer
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Infinite continuous response variable for logistic distribution? GLMM

When I would like to use the generalized linear mixed model (GLMM) for my data analysis, I would have to check the distribution of the response variable so as to decide the link-function. The result ...
4 votes
3 answers
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How to visualise coefficients of a Binomial Logistic Regression?

Hello all! Do you have an idea how best visualise the data from this table knowing they are coefficients of binomial logistic regressions? What I would like to visualise is a confrontation between ...
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Logistic Regression Models Without Main Effects?

I am building logistic regression models measuring human behaviour, which consist of categorical variables: demographics, conditions, and interactions between the demographics and the condition ...
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Getting logistic distribution from student's t-distribution

How can I get the probability density function (PDF) of standard logistic distribution from the PDF of symmetric student's t-distribution with location and scale parameters. In the $(\beta_1, \beta_2)$...
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Use of KS statistics to choose the probability cut-off in logistic regression model [closed]

I was doing some reading on choosing the score cut-off of a logistic regression model using the KS-Stats. Suppose I fitted a logistic regression model on the train data and now want to decide the ...
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1 answer
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interpretation of logistic coefficient

I have some toy data for an experiment where subjects are shown pictures A,B and C and then they are given a choice between choice A or choice B. I am interested in determining the effect of the ...
2 votes
1 answer
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does logistic distribution belongs to exponential family

Let $X$ have the logistic distribution with the PDF $$f(x) = \frac{\exp(-x-θ)}{(1+\exp(-x-θ))^{2}}$$ Does $f(x)$ belong to the exponential family? My solution is $\exp[(-2)\cdot \ln(1+\exp\{-x-θ\})-x-...
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Calculate sample mean confidence interval of noisy logistical distribution

I have $n$ samples which follow a logistic distribution with unknown $u$ and $s$; it is affected by a Gaussian noise with 0 mean. I would like to estimate its average $u$ with a confidence interval (...
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2 answers
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t-test with logistic and Gumbel distributions

I know that one of the basic assumptions of a t-test is that the data is drawn from a Gaussian distribution. Using an Anderson-Darling test, I've found that the datasets I am working with are either ...
26 votes
3 answers
3k views

Why is the logistic distribution called "logistic"?

What is "logistic" about the logistic distribution, in a common sense way? What is the etymology of and the lexical rationale for the name, not just pure math definition?
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Find an unbiased estimator from a random sample from the logistic distribution

So I'm not sure where to start on this one, so any hints will help me because I'm a little lost on finding unbiased estimators. So suppose $X_1, X_2,... X_n$ is a random sample from the logistic ...
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1 answer
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Obtaining the Log-logistic distribution from a truncated logistic distribution

Let $$f(x) = \frac{e^x}{(1+e^x)^2}~,~ -\infty \lt x \lt \infty~~~~~(1)$$ be the standard logistic pdf of a random variable $X$. Then one can obtain the pdf of the log-logistic distribution via the ...
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60 votes
4 answers
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Logistic Regression - Error Term and its Distribution

On whether an error term exists in logistic regression (and its assumed distribution), I have read in various places that: no error term exists the error term has a binomial distribution (in ...
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2 votes
1 answer
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How to compute for Bivariate Logistic Distribution

This is the logistic distribution of single random variable (taken from Wikipedia). $x$ = random variable $\mu$ = mean of all random variables $s$ = variance. Now, I want to do a Bivariate logistic ...
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3 answers
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Multivariate logistic distribution

The normal distribution can be generalized into the multivariate normal distribution. Can the logistic distribution also be generalized into a similar multivariate distribution? Is there a ...
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2 votes
1 answer
274 views

What is this distribution? Inverted S / cursive N

I have come across a graph pattern in two basically unrelated experiments, and I want to understand where it is coming from, or at least how to handle it statistically. I work in computational ...
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2 votes
1 answer
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Test for a logistic distribution in R

I have a set of data and I'd like to know whether this data set has a logistic distribution. When I made a histogram of my data set it seems to have a logistic distribution, but to be sure I'd like to ...