From e1ceb2b7b46ed58e9acf76cb943d4e3ae2711d74 Mon Sep 17 00:00:00 2001 From: Mikis Stasinopoulos Date: Tue, 11 Jun 2024 19:49:04 +0100 Subject: [PATCH] continuous --- Continuous_Distributions.qmd | 2 +- docs/Continuous_Distributions.html | 2022 ++++++++++++++-------------- docs/search.json | 6 +- 3 files changed, 1018 insertions(+), 1012 deletions(-) diff --git a/Continuous_Distributions.qmd b/Continuous_Distributions.qmd index 1441c5d..704e7b3 100644 --- a/Continuous_Distributions.qmd +++ b/Continuous_Distributions.qmd @@ -616,7 +616,7 @@ m3 <- histDist(Rdax, family=GT, nbins=30, line.col="black") ## END -[back to the index](https://mstasinopoulos.github.io/ShortCourse/) +[back to the index](https://mstasinopoulos.github.io/Porto_short_course/) ::: {layout-ncol="3," layout-nrow="1"} ![](book-1.png){width="300"} ![](BOOK-2.png){width="323"} ![](book3.png){width="333"} The Books diff --git a/docs/Continuous_Distributions.html b/docs/Continuous_Distributions.html index 177170e..bdc7caa 100644 --- a/docs/Continuous_Distributions.html +++ b/docs/Continuous_Distributions.html @@ -1,327 +1,465 @@ - - - - - - - - - - - - - -GAMLSS Short Course - Continuous Distributions - - - - - + - - - - - + - - - - - - - - - - - - - + .callout.callout-style-simple { + padding: 0em 0.5em; + border-left: solid #acacac .3rem; + border-right: solid 1px silver; + border-top: solid 1px silver; + border-bottom: solid 1px silver; + display: flex; + } - + .callout.callout-style-default { + border-left: solid #acacac .3rem; + border-right: solid 1px silver; + border-top: solid 1px silver; + border-bottom: solid 1px silver; + } -
-
- -
- -
- - - - -
+ .callout .callout-body-container { + flex-grow: 1; + } -
-
-

Continuous Distributions

-
+ .callout.callout-style-simple .callout-body { + font-size: 1rem; + font-weight: 400; + } + + .callout.callout-style-default .callout-body { + font-size: 0.9rem; + font-weight: 400; + } + .callout.callout-titled.callout-style-simple .callout-body { + margin-top: 0.2em; + } + .callout:not(.callout-titled) .callout-body { + display: flex; + } -
+ .callout:not(.no-icon).callout-titled.callout-style-simple .callout-content { + padding-left: 1.6em; + } -
-
Authors
-
-

Mikis Stasinopoulos

-

Bob Rigby

-

Gillian Heller

-

Fernanda De Bastiani

-

Niki Umlauf

-
-
+ .callout.callout-titled .callout-header { + padding-top: 0.2em; + margin-bottom: -0.2em; + } + + .callout.callout-titled .callout-title p { + margin-top: 0.5em; + margin-bottom: 0.5em; + } - + .callout.callout-titled.callout-style-simple .callout-content p { + margin-top: 0; + } + + .callout.callout-titled.callout-style-default .callout-content p { + margin-top: 0.7em; + } + + .callout.callout-style-simple div.callout-title { + border-bottom: none; + font-size: .9rem; + font-weight: 600; + opacity: 75%; + } + + .callout.callout-style-default div.callout-title { + border-bottom: none; + font-weight: 600; + opacity: 85%; + font-size: 0.9rem; + padding-left: 0.5em; + padding-right: 0.5em; + } + + .callout.callout-style-default div.callout-content { + padding-left: 0.5em; + padding-right: 0.5em; + } + + .callout.callout-style-simple .callout-icon::before { + height: 1rem; + width: 1rem; + display: inline-block; + content: ""; + background-repeat: no-repeat; + background-size: 1rem 1rem; + } + + .callout.callout-style-default .callout-icon::before { + height: 0.9rem; + width: 0.9rem; + display: inline-block; + content: ""; + background-repeat: no-repeat; + background-size: 0.9rem 0.9rem; + } + + .callout-title { + display: flex + } -
- + .callout-icon::before { + margin-top: 1rem; + padding-right: .5rem; + } + + .callout.no-icon::before { + display: none !important; + } + + .callout.callout-titled .callout-body > .callout-content > :last-child { + padding-bottom: 0.5rem; + margin-bottom: 0; + } + + .callout.callout-titled .callout-icon::before { + margin-top: .5rem; + padding-right: .5rem; + } + + .callout:not(.callout-titled) .callout-icon::before { + margin-top: 1rem; + padding-right: .5rem; + } + + /* Callout Types */ + + div.callout-note { + border-left-color: #4582ec !important; + } + + div.callout-note .callout-icon::before { + background-image: url('data:image/png;base64,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'); + } + + div.callout-note.callout-style-default .callout-title { + background-color: #dae6fb + } + + div.callout-important { + border-left-color: #d9534f !important; + } + + div.callout-important .callout-icon::before { + background-image: url('data:image/png;base64,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'); + } + + div.callout-important.callout-style-default .callout-title { + background-color: #f7dddc + } + div.callout-warning { + border-left-color: #f0ad4e !important; + } + + div.callout-warning .callout-icon::before { + background-image: url('data:image/png;base64,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'); + } + + div.callout-warning.callout-style-default .callout-title { + background-color: #fcefdc + } + + div.callout-tip { + border-left-color: #02b875 !important; + } -
+ div.callout-tip .callout-icon::before { + background-image: url('data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACAAAAAgCAYAAABzenr0AAAAAXNSR0IArs4c6QAAAERlWElmTU0AKgAAAAgAAYdpAAQAAAABAAAAGgAAAAAAA6ABAAMAAAABAAEAAKACAAQAAAABAAAAIKADAAQAAAABAAAAIAAAAACshmLzAAADr0lEQVRYCe1XTWgTQRj9ZjZV8a9SPIkKgj8I1bMHsUWrqYLVg4Ue6v9BwZOxSYsIerFao7UiUryIqJcqgtpimhbBXoSCVxUFe9CTiogUrUp2Pt+3aUI2u5vdNh4dmMzOzHvvezuz8xNFM0mjnbXaNu1MvFWRXkXEyE6aYOYJpdW4IXuA4r0fo8qqSMDBU0v1HJUgVieAXxzCsdE/YJTdFcVIZQNMyhruOMJKXYFoLfIfIvVIMWdsrd+Rpd86ZmyzzjJmLStqRn0v8lzkb4rVIXvnpScOJuAn2ACC65FkPzEdEy4TPWRLJ2h7z4cArXzzaOdKlbOvKKX25Wl00jSnrwVxAg3o4dRxhO13RBSdNvH0xSARv3adTXbBdTf64IWO2vH0LT+cv4GR1DJt+DUItaQogeBX/chhbTBxEiZ6gftlDNXTrvT7co4ub5A6gp9HIcHvzTa46OS5fBeP87Qm0fQkr4FsYgVQ7Qg+ZayaDg9jhg1GkWj8RG6lkeSacrrHgDaxdoBiZPg+NXV/KifMuB6//JmYH4CntVEHy/keA6x4h4CU5oFy8GzrBS18cLJMXcljAKB6INjWsRcuZBWVaS3GDrqB7rdapVIeA+isQ57Eev9eCqzqOa81CY05VLd6SamW2wA2H3SiTbnbSxmzfp7WtKZkqy4mdyAlGx7ennghYf8voqp9cLSgKdqNfa6RdRsAAkPwRuJZNbpByn+RrJi1RXTwdi8RQF6ymDwGMAtZ6TVE+4uoKh+MYkcLsT0Hk8eAienbiGdjJHZTpmNjlbFJNKDVAp2fJlYju6IreQxQ08UJDNYdoLSl6AadO+fFuCQqVMB1NJwPm69T04Wv5WhfcWyfXQB+wXRs1pt+nCknRa0LVzSA/2B+a9+zQJadb7IyyV24YAxKp2Jqs3emZTuNnKxsah+uabKbMk7CbTgJx/zIgQYErIeTKRQ9yD9wxVof5YolPHqaWo7TD6tJlh7jQnK5z2n3+fGdggIOx2kaa2YI9QWarc5Ce1ipNWMKeSG4DysFF52KBmTNMmn5HqCFkwy34rDg05gDwgH3bBi+sgFhN/e8QvRn8kbamCOhgrZ9GJhFDgfcMHzFb6BAtjKpFhzTjwv1KCVuxHvCbsSiEz4CANnj84cwHdFXAbAOJ4LTSAawGWFn5tDhLMYz6nWeU2wJfIhmIJBefcd/A5FWQWGgrWzyORZ3Q6HuV+Jf0Bj+BTX69fm1zWgK7By1YTXchFDORywnfQ7GpzOo6S+qECrsx2ifVQAAAABJRU5ErkJggg=='); + } + + div.callout-tip.callout-style-default .callout-title { + background-color: #ccf1e3 + } + + div.callout-caution { + border-left-color: #fd7e14 !important; + } + + div.callout-caution .callout-icon::before { + background-image: url('data:image/png;base64,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'); + } + div.callout-caution.callout-style-default .callout-title { + background-color: #ffe5d0 + } + + + + + +
+
+ +
+

Continuous Distributions

+ +
+
+
+Mikis Stasinopoulos +
+
+
+
+Bob Rigby +
+
+
+
+Gillian Heller +
+
+
+
+Fernanda De Bastiani +
+
+
+
+Niki Umlauf +
+
+
-
-

Types

+
+
+

Types

  • \((-\infty, \infty)\) real line \(\Re\) [Chapter 4 of Rigby et al (2019)]

  • \((0, \infty )\) positive real line \(\Re^{+}\) [Chapter 5]

  • \((0,1)\) real line on interval \(\Re_{(0,1)}\) (not containing zero or one) [Chapter 6]

-
+
+

Explicit distributions in real line, \(\Re\).

-
-

Location and Scale family

+ +
+
+

Location and Scale family

if \[Y\sim {D}(\mu,\sigma,\nu,\tau)\] then \[\varepsilon=(Y-\mu)/\sigma\sim {D}(0,1,\nu,\tau),\]

i.e. \(Y=\mu+\sigma\varepsilon\), so \(Y\) is a scaled and shifted version of the random variable \(\varepsilon\).

-
+
-
-

location

-
-
-
-
-

-
-
-
-
-
-
-

scale

-
-
-
-
-

-
-
-
-
-
-
-

2 parameter in \(\Re\)

+
+
+

location

+ +
+
+

scale

+ +
+
+

2 parameter in \(\Re\)

  • Gumbel, GU(\(\mu\), \(\sigma\)), left skew

  • Logistic, LO(\(\mu\), \(\sigma\)), lepto

    • @@ -329,38 +467,22 @@

    2 parameter in

    Reverse Gumbel RG(\(\mu\), \(\sigma\)), right skew

-
-

2 parameter in \(\Re\) (con.)

+
+

2 parameter in \(\Re\) (con.)

  • Normal against Logistic
-
-
-
-
-

-
-
-
-
-
-
-

2 parameter in \(\Re\) (con.)

+ +
+
+

2 parameter in \(\Re\) (con.)

  • Normal against Gumbel and reverse Gumbel
-
-
-
-
-

-
-
-
-
-
-
-

3 parameter in \(\Re\)

+ +
+
+

3 parameter in \(\Re\)

  • exponential Gaussian: exGAUS\((\mu, \sigma, \nu)\) for modelling right skew data,

  • normal family: NOF\((\mu, \sigma, \nu)\) for modelling mean and variance relationships following the power law;

    • @@ -369,104 +491,48 @@

    3 parameter in

    skew normal: SN1\((\mu, \sigma, \nu)\) and SK2\((\mu, \sigma, \nu)\) for modellinng skewness in data.

-
-

3 parameter in \(\Re\) (con.)

+
+

3 parameter in \(\Re\) (con.)

  • Skew Normal type 1
-
-
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-

-
-
-
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3 parameter in \(\Re\) (con.)

+ +
+
+

3 parameter in \(\Re\) (con.)

  • Skew Normal type 1
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3 parameter in \(\Re\) (TEST 1)

-
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-
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3 parameter in \(\Re\) PE

+ +
+
+

3 parameter in \(\Re\) (TEST 1)

+ +
+
+

3 parameter in \(\Re\) PE

  • Power Exponential
-
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3 parameter in \(\Re\) PE (test)

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3 parameter in \(\Re\) TF

+ +
+
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3 parameter in \(\Re\) PE (test)

+ +
+
+

3 parameter in \(\Re\) TF

  • t family
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3 parameter in \(\Re\) TF (test)

-
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4 parameter in \(\Re\)

+ +
+
+

3 parameter in \(\Re\) TF (test)

+ +
+
+

4 parameter in \(\Re\)

  • exponential generalised beta type 2,
    \(EGB2(\mu,\sigma,\nu,\tau)\) skewness and leptokurtosis;

    • @@ -476,88 +542,56 @@

    4 parameter in

    skew exponential power \(\texttt{SEP1}(\mu,\sigma, \nu, \tau)\), \(\texttt{SEP2}(\mu,\sigma, \nu, \tau)\), \(\texttt{SEP3}(\mu,\sigma, \nu, \tau)\) and \(\texttt{SEP4}(\mu,\sigma, \nu, \tau)\) skewness and lepto-platy;

-
-

4 parameter in \(\Re\) (con.)

+
+

4 parameter in \(\Re\) (con.)

  • sinh-arcsinh, \(\texttt{SHASH}(\mu,\sigma, \nu, \tau)\), \(\texttt{SHASHo}(\mu,\sigma, \nu, \tau)\) and \(\texttt{SHASHo2}(\mu,\sigma, \nu, \tau)\) skewness and lepto-platy;

  • skew t, \(\texttt{ST1}(\mu,\sigma, \nu, \tau)\), \(\texttt{ST2}(\mu,\sigma, \nu, \tau)\), \(\texttt{ST3}(\mu,\sigma, \nu, \tau)\), \(\texttt{ST4}(\mu,\sigma, \nu, \tau)\), \(\texttt{ST5}(\mu,\sigma, \nu, \tau)\) and \(\texttt{SST}(\mu,\sigma, \nu, \tau)\) skewness and leptokurtosis.

-
-

4 parameter in \(\Re\) SEP1

+
+

4 parameter in \(\Re\) SEP1

  • SEP1\(\mu=0, \sigma=1, \nu=0,1,2, \tau=1\)
-
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-
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4 parameter in \(\Re\) SEP1

+ +
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4 parameter in \(\Re\) SEP1

  • SEP1\(\mu=0, \sigma=1, \nu=0,1,2, \tau=2\)
-
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4 parameter in \(\Re\) SEP1

+ +
+
+

4 parameter in \(\Re\) SEP1

  • SEP1\(\mu=0, \sigma=1, \nu=0,1,2, \tau=5\)
+ +
+
+

4 parameter in \(\Re\) SEP1

-
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4 parameter in \(\Re\) SEP1

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4 parameter in \(\Re\) NET

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Summary

- + +
+

4 parameter in \(\Re\) NET

+ +
+
+

Summary

+
@@ -682,51 +716,53 @@

Summary

-
-
-
+
+
+

Explicit Distributions in positive real line \(\Re^+\)

-
-

the scale family

+ +
+
+

the scale family

If a random variable is distributed as \[Y\sim D(\mu,\sigma,\nu,\tau)\]

and
\[\varepsilon=(Y/\mu) \sim D(1,\sigma,\nu,\tau)\] then \(Y\) has a scale family

-
+
-
-

the weibull 3

+
+
+

the weibull 3

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the weibull 3 (con.)

+
+
+

the weibull 3 (con.)

-
-

+
+

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+
-
-

1 and 2 parameter positive real line, \(\Re^+\).

+
+
+

1 and 2 parameter positive real line, \(\Re^+\).

-
-

Weibull survival and hazard

+
+

Weibull survival and hazard

-
-

3 parameter positive real line \(\Re^+\)

+
+

3 parameter positive real line \(\Re^+\)

  • Box-Cox Cole and Green, \(\texttt{BCCG}(\mu, \sigma, \nu)\) known also as the LMS method in centile estimation.

  • generalised gamma, \(\texttt{GG}(\mu, \sigma, \nu)\)

  • generalised inverse Gaussian, \(\texttt{GIG}(\mu, \sigma, \nu)\)

  • log-normal family, \(\texttt{LNO}(\mu, \sigma, \nu)\) based on the standard Box-Cox transformation

-
-
-
+
+

the BCCG

-
-

+
+

-
+
-
+
+ +
+

the BCCG (con.)

-
-

+
+

-
+
-

4 parameter positive real line \(\Re^+\)

+
+
+

4 parameter positive real line \(\Re^+\)

  • Box-Cox power exponential, \(\texttt{BCPE}(\mu,\sigma,\nu,\tau)\) and \(\texttt{BCPEo}(\mu,\sigma,\nu,\tau)\) skewness and platy-lepto;

  • Box-Cox t, \(\texttt{BCT}(\mu,\sigma,\nu,\tau)\) and \(\texttt{BCTo}(\mu,\sigma,\nu,\tau)\) skewness and leptokurtosis;

  • generalised beta type 2, \(\texttt{GB2}(\mu,\sigma,\nu,\tau)\) skewness and platy-lepto.

-
-
-
+
+

the BCT

-
-

+
+

-
+
-
+
+ +
+

the BCT (con.)

-
-

+
+

-
+
-

the BCPE

+
+
+

the BCPE

-
-

+
+

-
+
-
-

Summary

- + +
+

Summary

+
@@ -946,13 +986,15 @@

Summary

{.striped .hover}

-
-
-
+
+
+

Explicit Distributions on \(\Re_{(0,1)}\)

-
-

Distributions on \(\Re_{(0,1)}\)

- + + +
+

Distributions on \(\Re_{(0,1)}\)

+
@@ -995,26 +1037,28 @@

Distributions on

Distributions
-
-

the BE

+
+

the BE

-
-

+
+

-
+
-
-
+
+
+

Choosing distributions

-
-

Choosing distribution

+ +
+
+

Choosing distribution

global deviance

\[ \begin{split} @@ -1028,8 +1072,8 @@

Choosing dis {\rm SBC} &= \texttt{GDEV}+ \log n \cdot df\ . \end{split}\]

-
-

Choosing distribution

+
+

Choosing distribution

prediction global deviance

\[\begin{split} TDEV=& -2, \ell (\boldsymbol{\widetilde{\theta}}) \\ @@ -1037,8 +1081,8 @@

Choosing d \end{split}\]

\({\widetilde{ y}}\) indicates the response variable values at the validation (or test) sample,

-
-

How to fit distributions in R

+
+

How to fit distributions in R

  • optim() or mle() R functions requiring initial parameter values

  • gamlsssML() fits a distribution (using MLE) on the response with no explanatory variables

    • @@ -1048,43 +1092,31 @@

    How

  • chooseDist() fits a set of distributions on a fitted model and chooses the one with the smallest GAIC

-
-

Example: DAX returns data

+
+

Example: DAX returns data

-
data(EuStockMarkets)
-dax <-EuStockMarkets[,"DAX"]
-Rdax <- diff(log(dax))
-plot(Rdax)
-
-
-
-

-
-
-
+
data(EuStockMarkets)
+dax <-EuStockMarkets[,"DAX"]
+Rdax <- diff(log(dax))
+plot(Rdax)
+
-
-
-

Example: DAX returns data (con.)

+
+
+

Example: DAX returns data (con.)

-
library(gamlss.ggplots)
-y_hist(Rdax)
-
-
-
-

-
-
-
+
library(gamlss.ggplots)
+y_hist(Rdax)
+
-
-
-

fitDist()

+
+
+

fitDist()

-
f1 <- fitDist(Rdax)
+
f1 <- fitDist(Rdax)
-
f1$fits
+
f1$fits
        GT       SEP2         PE        PE2        JSU       JSUo       SEP1 
 -11967.683 -11965.533 -11962.464 -11962.464 -11961.477 -11961.477 -11961.305 
@@ -1101,8 +1133,8 @@ 
fitDist()
-
-

chooseDist()

+
+

chooseDist()

GAMLSS-RS iteration  1: Global Deviance = -11737.208 eps = 0.000000
@@ -1121,491 +1153,465 @@

chooseDist()

-
-

chooseDist()(con.)

+
+

chooseDist()(con.)

-
m2 <- gamlss2(Rdax~1, family=GT, trace=FALSE)
-fitted(m2, parameter="mu", type="parameter")[1,1]
+
m2 <- gamlss2(Rdax~1, family=GT, trace=FALSE)
+fitted(m2, parameter="mu", type="parameter")[1,1]
[1] 0.0007286232
-
fitted(m2, parameter="sigma", type="parameter")[1,2]
+
fitted(m2, parameter="sigma", type="parameter")[1,2]
[1] 0.01005041
-
fitted(m2, parameter="nu", type="parameter")[1,3]
+
fitted(m2, parameter="nu", type="parameter")[1,3]
[1] 3.633405
-
fitted(m2, parameter="tau", type="parameter")[1,4]
+
fitted(m2, parameter="tau", type="parameter")[1,4]
[1] 1.603723
-
-

histDist()

+
+

histDist()

-
m3 <- histDist(Rdax, family=GT, nbins=30, line.col="black")
-
-
-
-

-
-
-
+
m3 <- histDist(Rdax, family=GT, nbins=30, line.col="black")
+
-
-
-
+
+
+

practical 2

-
-

END

-

back to the index

-
-
-
+
+
+

END

+

back to the index

+
-

The Books

+

The Books

+
+

+
+

www.gamlss.com

+
+
+
+ + -
-
+ + + + + + + + + + + + - - - + + + + - function fireSlideChanged(previousSlide, currentSlide) { - fireSlideEnter(); + - - + + + + - - - - + } else { + return undefined; + } + }; + var bibliorefs = window.document.querySelectorAll('a[role="doc-biblioref"]'); + for (var i=0; i + \ No newline at end of file diff --git a/docs/search.json b/docs/search.json index b1cc892..e6cfe41 100644 --- a/docs/search.json +++ b/docs/search.json @@ -290,8 +290,8 @@ "objectID": "Continuous_Distributions.html#types", "href": "Continuous_Distributions.html#types", "title": "Continuous Distributions", - "section": "", - "text": "\\((-\\infty, \\infty)\\) real line \\(\\Re\\) [Chapter 4 of Rigby et al (2019)]\n\\((0, \\infty )\\) positive real line \\(\\Re^{+}\\) [Chapter 5]\n\\((0,1)\\) real line on interval \\(\\Re_{(0,1)}\\) (not containing zero or one) [Chapter 6]" + "section": "Types", + "text": "Types\n\n\\((-\\infty, \\infty)\\) real line \\(\\Re\\) [Chapter 4 of Rigby et al (2019)]\n\\((0, \\infty )\\) positive real line \\(\\Re^{+}\\) [Chapter 5]\n\\((0,1)\\) real line on interval \\(\\Re_{(0,1)}\\) (not containing zero or one) [Chapter 6]" }, { "objectID": "Continuous_Distributions.html#location-and-scale-family", @@ -592,7 +592,7 @@ "href": "Continuous_Distributions.html#end", "title": "Continuous Distributions", "section": "END", - "text": "END\nback to the index\n\n\n\n\n\n\n The Books" + "text": "END\nback to the index\n\n\n\n The Books\n\n\n\n\n\n\n\nwww.gamlss.com" }, { "objectID": "diagnostics.html",