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Log-logistic distribution
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<h2 id="characterization">Characterization</h2> <p>There are several different parameterizations of the distribution in use. The one shown here gives reasonably interpretable parameters and a simple form for the <a href="/facts/Cumulative_distribution_function/WaKU8tp4">cumulative distribution function</a>.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a><a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> The parameter α > 0 {\displaystyle \alpha >0} is a <a href="/facts/Scale_parameter/6lxcyKUf">scale parameter</a> and is also the <a href="/facts/Median/YwXGWTyr">median</a> of the distribution. The parameter β > 0 {\displaystyle \beta >0} is a <a href="/facts/Shape_parameter/7jaG8RDp">shape parameter</a>. The distribution is <a href="/facts/Unimodal/Xo3bhgeE">unimodal</a> when β > 1 {\displaystyle \beta >1} and its <a href="/facts/Statistical_dispersion/xqevRHmz">dispersion</a> decreases as β {\displaystyle \beta } increases. </p><p>The <a href="/facts/Cumulative_distribution_function/WaKU8tp4">cumulative distribution function</a> is </p> F ( x ; α , β ) = 1 1 + ( x / α ) − β = ( x / α ) β 1 + ( x / α ) β = x β α β + x β {\displaystyle {\begin{aligned}F(x;\alpha ,\beta )&={1 \over 1+(x/\alpha )^{-\beta }}\\[5pt]&={(x/\alpha )^{\beta } \over 1+(x/\alpha )^{\beta }}\\[5pt]&={x^{\beta } \over \alpha ^{\beta }+x^{\beta }}\end{aligned}}} <p>where x > 0 {\displaystyle x>0} , α > 0 {\displaystyle \alpha >0} , β > 0. {\displaystyle \beta >0.} </p><p>The <a href="/facts/Probability_density_function/zvfybna4">probability density function</a> is </p> f ( x ; α , β ) = ( β / α ) ( x / α ) β − 1 ( 1 + ( x / α ) β ) 2 {\displaystyle f(x;\alpha ,\beta )={\frac {(\beta /\alpha )(x/\alpha )^{\beta -1}}{\left(1+(x/\alpha )^{\beta }\right)^{2}}}} <h3>Alternative parameterization</h3> <p>An alternative parametrization is given by the pair μ , s {\displaystyle \mu ,s} in analogy with the logistic distribution: </p> μ = ln ( α ) {\displaystyle \mu =\ln(\alpha )} s = 1 / β {\displaystyle s=1/\beta } <h2 id="properties">Properties</h2> <h3>Moments</h3> <p>The k {\displaystyle k} th raw <a href="/facts/Moment_(mathematics)/GL7vJ1nb">moment</a> exists only when k < β , {\displaystyle k<\beta ,} when it is given by<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a><a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p> E ( X k ) = α k B ( 1 − k / β , 1 + k / β ) = α k k π / β sin ( k π / β ) {\displaystyle {\begin{aligned}\operatorname {E} (X^{k})&=\alpha ^{k}\operatorname {B} (1-k/\beta ,1+k/\beta )\\[5pt]&=\alpha ^{k}\,{k\pi /\beta \over \sin(k\pi /\beta )}\end{aligned}}} <p>where B is the <a href="/facts/Beta_function/Dcm1JAxb">beta function</a>. Expressions for the <a href="/facts/Expected_value/1XV0JKL8">mean</a>, <a href="/facts/Variance/ULBJKXD1">variance</a>, <a href="/facts/Skewness/rdS8luFa">skewness</a> and <a href="/facts/Kurtosis/yqGgry89">kurtosis</a> can be derived from this. Writing b = π / β {\displaystyle b=\pi /\beta } for convenience, the mean is </p> E ( X ) = α b / sin b , β > 1 , {\displaystyle \operatorname {E} (X)=\alpha b/\sin b,\quad \beta >1,} <p>and the variance is </p> Var ( X ) = α 2 ( 2 b / sin 2 b − b 2 / sin 2 b ) , β > 2. {\displaystyle \operatorname {Var} (X)=\alpha ^{2}\left(2b/\sin 2b-b^{2}/\sin ^{2}b\right),\quad \beta >2.} <p>Explicit expressions for the skewness and kurtosis are lengthy.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> As β {\displaystyle \beta } tends to infinity the mean tends to α {\displaystyle \alpha } , the variance and skewness tend to zero and the excess kurtosis tends to 6/5 (see also related distributions below). </p> <h3>Quantiles</h3> <p>The <a href="/facts/Quantile_function/sf0jbCXh">quantile function</a> (inverse cumulative distribution function) is : </p> F − 1 ( p ; α , β ) = α ( p 1 − p ) 1 / β . {\displaystyle F^{-1}(p;\alpha ,\beta )=\alpha \left({\frac {p}{1-p}}\right)^{1/\beta }.} <p>It follows that the <a href="/facts/Median/YwXGWTyr">median</a> is α {\displaystyle \alpha } , the lower <a href="/facts/Quartile/XLlNsqEP">quartile</a> is 3 − 1 / β α {\displaystyle 3^{-1/\beta }\alpha } and the upper quartile is 3 1 / β α {\displaystyle 3^{1/\beta }\alpha } . </p> <h2 id="applications">Applications</h2> <h3>Survival analysis</h3> <p>The log-logistic distribution provides one <a href="/facts/Parametric_model/FeynAa0f">parametric model</a> for <a href="/facts/Survival_analysis/q8w13cjl">survival analysis</a>. Unlike the more commonly used <a href="/facts/Weibull_distribution/1fEWxbKl">Weibull distribution</a>, it can have a non-<a href="/facts/Monotonic/MjQK0YL2">monotonic</a> <a href="/facts/Hazard_function/shlTIKKg">hazard function</a>: when β > 1 , {\displaystyle \beta >1,} the hazard function is <a href="/facts/Unimodal/Xo3bhgeE">unimodal</a> (when β {\displaystyle \beta } ≤ 1, the hazard decreases monotonically). The fact that the cumulative distribution function can be written in closed form is particularly useful for analysis of survival data with <a href="/facts/Censoring_(statistics)/aa8HkK1a">censoring</a>.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> The log-logistic distribution can be used as the basis of an <a href="/facts/Accelerated_failure_time_model/JlbMvvuj">accelerated failure time model</a> by allowing α {\displaystyle \alpha } to differ between groups, or more generally by introducing covariates that affect α {\displaystyle \alpha } but not β {\displaystyle \beta } by modelling log ( α ) {\displaystyle \log(\alpha )} as a linear function of the covariates.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> </p><p>The <a href="/facts/Survival_function/jkdCmDKq">survival function</a> is </p> S ( t ) = 1 − F ( t ) = [ 1 + ( t / α ) β ] − 1 , {\displaystyle S(t)=1-F(t)=[1+(t/\alpha )^{\beta }]^{-1},\,} <p>and so the <a href="/facts/Hazard_function/shlTIKKg">hazard function</a> is </p> h ( t ) = f ( t ) S ( t ) = ( β / α ) ( t / α ) β − 1 1 + ( t / α ) β . {\displaystyle h(t)={\frac {f(t)}{S(t)}}={\frac {(\beta /\alpha )(t/\alpha )^{\beta -1}}{1+(t/\alpha )^{\beta }}}.} <p>The log-logistic distribution with shape parameter β = 1 {\displaystyle \beta =1} is the marginal distribution of the inter-times in a geometric-distributed <a href="/facts/Counting_process/4pAaYF5U">counting process</a>.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> </p> <h3>Hydrology</h3> <p>The log-logistic distribution has been used in hydrology for modelling stream flow rates and precipitation.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a><a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> </p><p>Extreme values like maximum one-day rainfall and river discharge per month or per year often follow a <a href="/facts/Log-normal_distribution/V2RXmmlE">log-normal distribution</a>.<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> The log-normal distribution, however, needs a numeric approximation. As the log-logistic distribution, which can be solved analytically, is similar to the log-normal distribution, it can be used instead. </p><p>The blue picture illustrates an example of fitting the log-logistic distribution to ranked maximum one-day October rainfalls and it shows the 90% <a href="/facts/Confidence_belt/NS8mG5UE">confidence belt</a> based on the <a href="/facts/Binomial_distribution/UMoFMjDj">binomial distribution</a>. The rainfall data are represented by the <a href="/facts/Plotting_position/h2jKBjYm">plotting position</a> <i>r</i>/(<i>n</i>+1) as part of the <a href="/facts/Cumulative_frequency_analysis/NtFhHr9h">cumulative frequency analysis</a>. </p> <h3>Economics</h3> <p>The log-logistic has been used as a simple model of the <a href="/facts/Distribution_of_wealth/8k7fg8rU">distribution of wealth</a> or <a href="/facts/Income_distribution/5ewT5Q6S">income</a> in <a href="/facts/Economics/HYctA2Dn">economics</a>, where it is known as the Fisk distribution.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> Its <a href="/facts/Gini_coefficient/HledImpA">Gini coefficient</a> is 1 / β {\displaystyle 1/\beta } .<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> </p> <table><tbody><tr><th>Derivation of Gini coefficient</th></tr><tr><td><p>The Gini coefficient for a continuous probability distribution takes the form:</p> G = 1 μ ∫ 0 ∞ F ( 1 − F ) d x {\displaystyle G={1 \over {\mu }}\int _{0}^{\infty }F(1-F)dx} <p>where F {\displaystyle F} is the CDF of the distribution and μ {\displaystyle \mu } is the expected value. For the log-logistic distribution, the formula for the Gini coefficient becomes:</p> G = sin ( π / β ) α π / β ∫ 0 ∞ d x [ 1 + ( x / α ) − β ] [ 1 + ( x / α ) β ] {\displaystyle G={\sin(\pi /\beta ) \over {\alpha \pi /\beta }}\int _{0}^{\infty }{dx \over {[1+(x/\alpha )^{-\beta }][1+(x/\alpha )^{\beta }]}}} <p>Defining the substitution z = x / α {\displaystyle z=x/\alpha } leads to the simpler equation:</p> G = sin ( π / β ) π / β ∫ 0 ∞ d z ( 1 + z − β ) ( 1 + z β ) {\displaystyle G={\sin(\pi /\beta ) \over {\pi /\beta }}\int _{0}^{\infty }{dz \over {(1+z^{-\beta })(1+z^{\beta })}}} <p>And making the substitution u = 1 / ( 1 + z β ) {\displaystyle u=1/(1+z^{\beta })} further simplifies the Gini coefficient formula to:</p> G = sin ( π / β ) π ∫ 0 1 u − 1 / β ( 1 − u ) 1 / β d u {\displaystyle G={\sin(\pi /\beta ) \over {\pi }}\int _{0}^{1}u^{-1/\beta }(1-u)^{1/\beta }du} <p>The integral component is equivalent to the standard <a href="/facts/Beta_function/Dcm1JAxb">beta function</a> B ( 1 − 1 / β , 1 + 1 / β ) {\displaystyle {\text{B}}(1-1/\beta ,1+1/\beta )} . The beta function may also be written as:</p> B ( x , y ) = Γ ( x ) Γ ( y ) Γ ( x + y ) {\displaystyle {\text{B}}(x,y)={\Gamma (x)\Gamma (y) \over {\Gamma (x+y)}}} <p>where Γ ( ⋅ ) {\displaystyle \Gamma (\cdot )} is the <a href="/facts/Gamma_function/SjGQcnyw">gamma function</a>. Using the properties of the gamma function, it can be shown that:</p> B ( 1 − 1 / β , 1 + 1 / β ) = 1 β Γ ( 1 − 1 / β ) Γ ( 1 / β ) {\displaystyle {\text{B}}(1-1/\beta ,1+1/\beta )={1 \over {\beta }}\Gamma (1-1/\beta )\Gamma (1/\beta )} <p>From <a href="/facts/Euler%27s_reflection_formula/aKUhTbH0">Euler's reflection formula</a>, the expression can be simplified further:</p> B ( 1 − 1 / β , 1 + 1 / β ) = 1 β π sin ( π / β ) {\displaystyle {\text{B}}(1-1/\beta ,1+1/\beta )={1 \over {\beta }}{\pi \over {\sin(\pi /\beta )}}} <p>Finally, we may conclude that the Gini coefficient for the log-logistic distribution G = 1 / β {\displaystyle G=1/\beta } .</p></td></tr></tbody></table> <h3>Networking</h3> <p>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 <a href="/facts/Real-time_computing/p9HxgL4L">real-time</a> guarantees (for example, when an application is displaying data coming from a remote <a href="/facts/Sensor/RN8SXbjs">sensor</a> connected to the Internet). It has been shown to be a more accurate probabilistic model for that than the <a href="/facts/Log-normal_distribution/V2RXmmlE">log-normal distribution</a> or others, as long as abrupt changes of regime in the sequences of those times are properly detected.<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> </p> <h2 id="related-distributions">Related distributions</h2> <ul><li>If X ∼ LL ( α , β ) {\displaystyle X\sim \operatorname {LL} (\alpha ,\beta )} then k X ∼ LL ( k α , β ) . {\displaystyle kX\sim \operatorname {LL} (k\alpha ,\beta ).} </li> <li>If X ∼ LL ( α , β ) {\displaystyle X\sim \operatorname {LL} (\alpha ,\beta )} then X k ∼ LL ( α k , β / | k | ) . {\displaystyle X^{k}\sim \operatorname {LL} (\alpha ^{k},\beta /|k|).} </li> <li> LL ( α , β ) ∼ Dagum ( 1 , α , β ) {\displaystyle \operatorname {LL} (\alpha ,\beta )\sim {\textrm {Dagum}}(1,\alpha ,\beta )} (<a href="/facts/Dagum_distribution/rmgoUXvM">Dagum distribution</a>).</li> <li> LL ( α , β ) ∼ SinghMaddala ( 1 , α , β ) {\displaystyle \operatorname {LL} (\alpha ,\beta )\sim {\textrm {SinghMaddala}}(1,\alpha ,\beta )} (<a href="/facts/Singh%E2%80%93Maddala_distribution/75pvKu6P">Singh–Maddala distribution</a>).</li> <li> LL ( γ , σ ) ∼ β ′ ( 1 , 1 , γ , σ ) {\displaystyle {\textrm {LL}}(\gamma ,\sigma )\sim \beta '(1,1,\gamma ,\sigma )} (<a href="/facts/Beta_prime_distribution/b0a1jjJm">Beta prime distribution</a>).</li> <li>If <i>X</i> has a log-logistic distribution with scale parameter α {\displaystyle \alpha } and shape parameter β {\displaystyle \beta } then <i>Y</i> = log(<i>X</i>) has a <a href="/facts/Logistic_distribution/QlQNt0uC">logistic distribution</a> with location parameter log ( α ) {\displaystyle \log(\alpha )} and scale parameter 1 / β . {\displaystyle 1/\beta .} </li> <li>As the shape parameter β {\displaystyle \beta } of the log-logistic distribution increases, its shape increasingly resembles that of a (very narrow) <a href="/facts/Logistic_distribution/QlQNt0uC">logistic distribution</a>. Informally:</li></ul> LL ( α , β ) → L ( α , α / β ) as β → ∞ . {\displaystyle \operatorname {LL} (\alpha ,\beta )\to L(\alpha ,\alpha /\beta )\quad {\text{as}}\quad \beta \to \infty .} <ul><li>The log-logistic distribution with shape parameter β = 1 {\displaystyle \beta =1} and scale parameter α {\displaystyle \alpha } is the same as the <a href="/facts/Pareto_distribution/PgMDbLrR">generalized Pareto distribution</a> with location parameter μ = 0 {\displaystyle \mu =0} , shape parameter ξ = 1 {\displaystyle \xi =1} and scale parameter α : {\displaystyle \alpha :} </li></ul> LL ( α , 1 ) = GPD ( 1 , α , 1 ) . {\displaystyle \operatorname {LL} (\alpha ,1)=\operatorname {GPD} (1,\alpha ,1).} <ul><li>The addition of another parameter (a shift parameter) formally results in a <a href="/facts/Shifted_log-logistic_distribution/EBUz1Yab">shifted log-logistic distribution</a>, but this is usually considered in a different parameterization so that the distribution can be bounded above or bounded below.</li></ul> <h3>Generalizations</h3> <p>Several different distributions are sometimes referred to as the generalized log-logistic distribution, as they contain the log-logistic as a special case.<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> These include the <a href="/facts/Burr_Type_XII_distribution/75pvKu6P">Burr Type XII distribution</a> (also known as the <i>Singh–Maddala distribution</i>) and the <a href="/facts/Dagum_distribution/rmgoUXvM">Dagum distribution</a>, both of which include a second shape parameter. Both are in turn special cases of the even more general <i>generalized beta distribution of the second kind</i>. Another more straightforward generalization of the log-logistic is the <a href="/facts/Shifted_log-logistic_distribution/EBUz1Yab">shifted log-logistic distribution</a>. </p><p>Another generalized log-logistic distribution is the log-transform of the <a href="/facts/Metalog_distribution/MWJJ3edV">metalog distribution</a>, in which power series expansions in terms of p {\displaystyle p} are substituted for <a href="/facts/Logistic_distribution/QlQNt0uC">logistic distribution</a> parameters μ {\displaystyle \mu } and σ {\displaystyle \sigma } . The resulting <a href="/facts/Metalog_distribution/MWJJ3edV">log-metalog distribution</a> is highly shape flexible, has simple closed form <a href="/facts/Probability_density_function/zvfybna4">PDF</a> and <a href="/facts/Quantile_function/sf0jbCXh">quantile function</a>, can be fit to data with linear least squares, and subsumes the log-logistic distribution is special case. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/List_of_probability_distributions/qrDTbPkd">Probability distributions: List of important distributions supported on semi-infinite intervals</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Shoukri, M.M.; Mian, I.U.M.; Tracy, D.S. (1988), "Sampling Properties of Estimators of the Log-Logistic Distribution with Application to Canadian Precipitation Data", The Canadian Journal of Statistics, 16 (3): 223–236, doi:10.2307/3314729, JSTOR 3314729 <a href="/wiki/The_Canadian_Journal_of_Statistics" target="_blank">/wiki/The_Canadian_Journal_of_Statistics</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Ashkar, Fahim; Mahdi, Smail (2006), "Fitting the log-logistic distribution by generalized moments", Journal of Hydrology, 328 (3–4): 694–703, Bibcode:2006JHyd..328..694A, doi:10.1016/j.jhydrol.2006.01.014 <a href="/wiki/Journal_of_Hydrology" target="_blank">/wiki/Journal_of_Hydrology</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Tadikamalla, Pandu R.; Johnson, Norman L. (1982), "Systems of Frequency Curves Generated by Transformations of Logistic Variables", Biometrika, 69 (2): 461–465, CiteSeerX 10.1.1.153.9487, doi:10.1093/biomet/69.2.461, JSTOR 2335422 <a href="/wiki/CiteSeerX_(identifier)" target="_blank">/wiki/CiteSeerX_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Tadikamalla, Pandu R. (1980), "A Look at the Burr and Related Distributions", International Statistical Review, 48 (3): 337–344, doi:10.2307/1402945, JSTOR 1402945 <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>McLaughlin, Michael P. (2001), A Compendium of Common Probability Distributions (PDF), p. A–37, retrieved 2008-02-15 <a href="http://www.causascientia.org/math_stat/Dists/Compendium.pdf" target="_blank">http://www.causascientia.org/math_stat/Dists/Compendium.pdf</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Bennett, Steve (1983), "Log-Logistic Regression Models for Survival Data", Journal of the Royal Statistical Society, Series C, 32 (2): 165–171, doi:10.2307/2347295, JSTOR 2347295 <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Collett, Dave (2003), Modelling Survival Data in Medical Research (2nd ed.), CRC press, ISBN 978-1-58488-325-8 <a href="978-1-58488-325-8" target="_blank">978-1-58488-325-8</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Di Crescenzo, Antonio; Pellerey, Franco (2019), "Some results and applications of geometric counting processes", Methodology and Computing in Applied Probability, 21 (1): 203–233, doi:10.1007/s11009-018-9649-9, S2CID 254793416 <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Shoukri, M.M.; Mian, I.U.M.; Tracy, D.S. (1988), "Sampling Properties of Estimators of the Log-Logistic Distribution with Application to Canadian Precipitation Data", The Canadian Journal of Statistics, 16 (3): 223–236, doi:10.2307/3314729, JSTOR 3314729 <a href="/wiki/The_Canadian_Journal_of_Statistics" target="_blank">/wiki/The_Canadian_Journal_of_Statistics</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Ashkar, Fahim; Mahdi, Smail (2006), "Fitting the log-logistic distribution by generalized moments", Journal of Hydrology, 328 (3–4): 694–703, Bibcode:2006JHyd..328..694A, doi:10.1016/j.jhydrol.2006.01.014 <a href="/wiki/Journal_of_Hydrology" target="_blank">/wiki/Journal_of_Hydrology</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Ritzema, H.P., ed. (1994), Frequency and Regression Analysis, Chapter 6 in: Drainage Principles and Applications, Publication 16, International Institute for Land Reclamation and Improvement (ILRI), Wageningen, The Netherlands, pp. 175–224, ISBN 978-90-70754-33-4 <a href="978-90-70754-33-4" target="_blank">978-90-70754-33-4</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Fisk, P.R. (1961), "The Graduation of Income Distributions", Econometrica, 29 (2): 171–185, doi:10.2307/1909287, JSTOR 1909287 <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Kleiber, C.; Kotz, S (2003), Statistical Size Distributions in Economics and Actuarial Sciences, Wiley, ISBN 978-0-471-15064-0 <a href="978-0-471-15064-0" target="_blank">978-0-471-15064-0</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Gago-Benítez, A.; Fernández-Madrigal J.-A., Cruz-Martín, A. (2013), "Log-Logistic Modeling of Sensory Flow Delays in Networked Telerobots", IEEE Sensors Journal, 13 (8), IEEE Sensors 13(8): 2944–2953, Bibcode:2013ISenJ..13.2944G, doi:10.1109/JSEN.2013.2263381, S2CID 47511693{{citation}}: CS1 maint: multiple names: authors list (link) <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>Kleiber, C.; Kotz, S (2003), Statistical Size Distributions in Economics and Actuarial Sciences, Wiley, ISBN 978-0-471-15064-0 <a href="978-0-471-15064-0" target="_blank">978-0-471-15064-0</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> </ol>