< List of probability distributions
Contents:
Logistic Distribution
The logistic distribution is used for modeling growth and logistic regression; it’s a symmetrical distribution with one peak that looks very similar to the normal distribution. In this blog post, we’ll break down the logistic distribution, its uses, and how it compares to the normal distribution.
What Is The Logistic Distribution?
The logistic distribution is a type of probability distribution often used in statistics and machine learning for modeling growth and logistic regression. It has a shape that is similar to the normal (or “Gaussian”) distribution—it’s symmetrical, unimodal (one peak), and has slightly fatter tails than the normal distribution. The differences between these two distributions are so small that they can be considered essentially identical for most applications.
The logistic distribution is often used because the curve has a relatively simple cumulative distribution formula to work with, compared to the normal distribution. The formula approximates the normal extremely well:
Where
- μ = the mean
- s = a scale parameter proportional to the standard deviation.
On the other hand, finding cumulative probabilities for the normal distribution usually involves looking up values in a z-table, rounding up or down to the nearest z-score. Exact values are usually found with statistical software, because the cumulative distribution function is difficult to work with, involving integration (from calculus):
The probability density function (PDF) for the logistic distribution can generally be expressed as:
Where
- μ = a location parameter.
- s = a scale parameter.
As the PDF can be expressed in terms of the square of the hyperbolic secant function (sech), it is occasionally called the sech-square(d) distribution [2].
Other logistic distribution properties:
Mean = μ
Variance (σ)2 = ⅓ π2 s2
Skewness (γ)1 = 0
Kurtosis (γ)2 = 6/5
Uses Of The Logistic Distribution
The logistic distribution is commonly used in various fields, including engineering, economics, finance, epidemiology, biostatistics, demography, psychology, sociology, computer science and other areas of applied mathematics. It can be used in many different scenarios such as:
- predicting future sales or customer demand over time;
- estimating customer attrition rates;
- measuring consumer preferences;
- forecasting population changes;
- predicting stock prices;
- analyzing health outcomes;
- estimating risk levels in financial investments;
- determining credit scores;
- forecasting marketing campaigns;
- developing pricing strategies;
- measuring customer satisfaction levels;
- predicting website traffic patterns;
- determining insurance premiums/claims costs/fraudulent claims rates etc.
How Does The Logistic Distribution Compare To Normal Distribution?
As we mentioned earlier, the logistic and normal distributions are extremely similar—the only notable difference between them being that the logistic tends to have slightly fatter tails than the normal distribution. This means that the area under each tail of the curve is larger than that of a normal curve at any given point on either side of its peak. This slight difference makes them useful for different purposes—whereas a normal curve may be better suited for some applications like estimating customer demand or forecasting sales trends over time (as we mentioned above), a logistic curve may be more appropriate when dealing with risk assessment or fraud detection due to its fatter tails which allow it to capture extreme values more accurately than its counterpart.
Stukel distribution
Stukel proposed a two-parameter class of generalized logistic distributions in 1988 [3], with the goal of extending the standard logistic model to improve fit in the noncentral probability regions and include asymmetric deviations in logistic models.
About the Stukel distribution
A Stukel distribution includes skewness into the link function and, according to Stukel, works well for many common applications. Stukel’s models are generalized, which means that many common link models (including symmetric and asymmetric models) can be included in family members. One disadvantage of any Stukel distribution is that when covariates are present, the models can result in improper posterior distributions for many types of noninformative improper priors, including the improper uniform prior for the regression coefficients [4].
Stukey’s generalized model is described as
Other logistic generalizations
Many other authors have proposed generalizations of the standard logistic distribution including:
- Aranda-Ordaz [5], who proposed two different one-parameter distributions — one for symmetric and one for asymmetric departures.
- Prentice [6], who modeled the expected probability curve with the cumulative distribution function (CDF) of the log F 1m1,2m2 distribution; the log F distributions includes as special cases the logistic distribution, the normal distribution, extreme minimum distribution, extreme maximum distribution, exponential distribution, LaPlace distribution and reflected exponential distribution. Thus, Prentice’s models can handle a range of nonstandard cases.
- Pregibon [7], who defined a family of link functions to include the logit link as a special case.
References
[1] No machine-readable author provided. Anarkman~commonswiki assumed (based on copyright claims)., CC BY-SA 3.0 http://creativecommons.org/licenses/by-sa/3.0/, via Wikimedia Commons
[2] Continuous univariate distributions: N.L. Johnson, S. Kotz & N. Balakrishnan (1995): Vol. 2, 2nd Edition. John Wiley, New York
[3] Stukel, T. A. (1988), “Generalized Logistic Models,” Journal of the American Statistical Association, 83, 426–431.
[4] CHEN, M.-H., DEY, D. K. and SHAO, Q.-M. (1999). A new skewed link model for dichotomous quantal response data. J. Amer. Statist. Assoc. 94 1172–1186. MR1731481
[5] Aranda-Ordaz, F. J. 1981. “On Two Families of Transformations to Additivity for Binary Response Data,”. Biometrika, 68: 357–363. [Google Scholar]
[6] Prentice, R. L. 1976. “Generalization of the Probit and Logit Methods for Dose Response Curves,”. Biometrics, 32: 761–768. [Google Scholar]
[7] Pregibon, D. 1980. “Goodness of Link Tests for Generalized Linear Models,”. Applied Statistics, 29: 15–24. [Google Scholar]