Pedometric mapping: Difference between revisions

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Template:Primary sources In probability and statistics, the generalized beta distribution^[1] is a continuous probability distribution with five parameters, including more than thirty named distributions as limiting or special cases. It has been used in the modeling of income distribution, stock returns, as well as in regression analysis. The exponential generalized Beta (EGB) distribution follows directly from the GB and generalizes other common distributions.

Definition

A generalized beta random variable, Y, is defined by the following probability density function:

G B (y; a, b, c, p, q) = \frac{| a | y^{a p - 1} (1 - (1 - c) (y / b)^{a})^{q - 1}}{b^{a p} B (p, q) (1 + c (y / b)^{a})^{p + q}} for 0 < y^{a} < \frac{b^{a}}{1 - c},

and zero otherwise. Here the parameters satisfy $0 \leq c \leq 1$ and $b$ , $p$ , and $q$ positive. The function B(p,q) is the beta function.

File:GBtree.jpg

GB distribution tree

Properties

Moments

It can be shown that the hth moment can be expressed as follows:

E_{G B} (Y^{h}) = \frac{b^{h} B (p + h / a, q)}{B (p, q)}_{2} F_{1} [\begin{matrix} p + h / a, h / a; c \\ p + q + h / a; \end{matrix}],

where $_{2} F_{1}$ denotes the hypergeometric series (which converges for all h if c<1, or for all h/a<q if c=1 ).

Related distributions

The generalized beta encompasses a number of distributions in its family as special cases. Listed below are its three direct descendants, or sub-families.

Generalized beta of first kind (GB1)

The generalized beta of the first kind is defined by the following pdf:

G B 1 (y; a, b, p, q) = \frac{| a | y^{a p - 1} (1 - (y / b)^{a})^{q - 1}}{b^{a p} B (p, q)}

for $0 < y^{a} < b^{a}$ where $b$ , $p$ , and $q$ are positive. It is easily verified that

G B 1 (y; a, b, p, q) = G B (y; a, b, c = 0, p, q) .

The moments of the GB1 are given by

E_{G B 1} (Y^{h}) = \frac{b^{h} B (p + h / a, q)}{B (p, q)} .

The GB1 includes the beta of the first kind (B1), generalized gamma(GG), and Pareto as special cases:

B 1 (y; b, p, q) = G B 1 (y; a = 1, b, p, q),

G G (y; a, β, p) = \lim_{q \to \infty} G B 1 (y; a, b = q^{1 / a} β, p, q),

P A R E T O (y; b, p) = G B 1 (y; a = - 1, b, p, q = 1) .

Generalized beta of the second kind (GB2)

The GB2 (also known as the Generalized Beta Prime) is defined by the following pdf:

G B 2 (y; a, b, p, q) = \frac{| a | y^{a p - 1}}{b^{a p} B (p, q) (1 + (y / b)^{a})^{p + q}}

for $0 < y < \infty$ and zero otherwise. One can verify that

G B 2 (y; a, b, p, q) = G B (y; a, b, c = 1, p, q) .

The moments of the GB2 are given by

E_{G B 2} (Y^{h}) = \frac{b^{h} B (p + h / a, q - h / a)}{B (p, q)} .

The GB2 nests common distributions such as the generalized gamma (GG), Burr type 3, Burr type 12, lognormal, Weibull, gamma, Lomax, F statistic, Fisk or Rayleigh, chi-square, half-normal, half-Student's, exponential, and the log-logistic.^[2]

Beta

The beta distribution (B) is defined by:Template:Cn

B (y; b, c, p, q) = \frac{y^{p - 1} (1 - (1 - c) (y / b))^{q - 1}}{b^{p} B (p, q) (1 + c (y / b))^{p + q}}

for $0 < y < b / (1 - c)$ and zero otherwise. Its relation to the GB is seen below:

B (y; b, c, p, q) = G B (y; a = 1, b, c, p, q) .

The beta family includes the betas of the first and second kind^[3] (B1 and B2, where the B2 is also referred to as the Beta prime), which correspond to c = 0 and c = 1, respectively.

A figure showing the relationship between the GB and its special and limiting cases is included above (see McDonald and Xu (1995) ).

Exponential generalized beta distribution

Letting $Y \sim G B (y; a, b, c, p, q)$ , the random variable $Z = \ln (Y)$ , with re-parametrization, is distributed as an exponential generalized beta (EGB), with the following pdf:

E G B (z; δ, σ, c, p, q) = \frac{e^{p (z - δ) / σ} (1 - (1 - c) e^{(z - δ) / σ})^{q - 1}}{| σ | B (p, q) (1 + c e^{(z - δ) / σ})^{p + q}}

for $- \infty < \frac{z - δ}{σ} < \ln (\frac{1}{1 - c})$ , and zero otherwise. The EGB includes generalizations of the Gompertz, Gumbell, extreme value type I, logistic, Burr-2, exponential, and normal distributions.

Included is a figure showing the relationship between the EGB and its special and limiting cases (see McDonald and Xu (1995) ).

File:EGBtree.jpg

EGB distribution tree

Moment generating function

Using similar notation as above, the moment-generating function of the EGB can be expressed as follows:

M_{E G B} (Z) = \frac{e^{δ t} B (p + t σ, q)}{B (p, q)}_{2} F_{1} [\begin{matrix} p + t σ, t σ; c \\ p + q + t σ; \end{matrix}] .

Uses

The flexibility provided by the GB family is used in modeling the distribution of:Template:Cn

family income
stock returns
insurance losses

Applications involving members of the EGB family include:Template:Cn

partially adaptive estimation of regression
time series models

References

↑ McDonald, James B. & Xu, Yexiao J. (1995) "A generalization of the beta distribution with applications," Journal of Econometrics, 66(1–2), 133–152 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park.
↑ McDonald, J.B. (1984) "Some generalized functions for the size distributions of income", Econometrica 52, 647–663.
↑ Stuart, A. and Ord, J.K. (1987): Kendall's Advanced Theory of Statistics, New York: Oxford University Press.

Bibliography

C. Kleiber and S. Kotz (2003) Statistical Size Distributions in Economics and Actuarial Sciences. New York: Wiley
Johnson, N. L., S. Kotz, and N. Balakrishnan (1994) Continuous Univariate Distributions. Vol. 2, Hoboken, NJ: Wiley-Interscience.

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[1] McDonald, James B. & Xu, Yexiao J. (1995) "A generalization of the beta distribution with applications," Journal of Econometrics, 66(1–2), 133–152 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park.

[2] McDonald, J.B. (1984) "Some generalized functions for the size distributions of income", Econometrica 52, 647–663.

[3] Stuart, A. and Ord, J.K. (1987): Kendall's Advanced Theory of Statistics, New York: Oxford University Press.

[1]

[2]

[3]

@@ Line 1: / Line 1: @@
+{{primary sources|date=April 2013}}
+In [[probability and statistics]], the '''generalized beta distribution'''<ref>McDonald, James B. & Xu, Yexiao J. (1995) "A generalization of the beta distribution with applications," ''Journal of Econometrics'', 66(1–2), 133–152 {{doi|10.1016/0304-4076(94)01612-4}}</ref> is a [[continuous probability distribution]] with five parameters, including more than thirty named distributions as [[Limiting case|limiting]] or [[special case]]s. It has been used in the modeling of [[income distribution]], stock returns, as well as in [[regression analysis]]. The '''exponential generalized Beta (EGB) distribution''' follows directly from the GB and generalizes other common distributions.
+== Definition ==
+A generalized beta random variable, ''Y'', is defined by the following probability density function:
+:<math> GB(y;a,b,c,p,q) = \frac{|a|y^{ap-1}(1-(1-c)(y/b)^{a})^{q-1}}{b^{ap}B(p,q)(1+c(y/b)^{a})^{p+q}} \quad \quad \text{  for } 0<y^{a}< \frac{b^a}{1-c} , </math>
+and zero otherwise. Here the parameters satisfy <math> 0 \le c \le 1 </math> and <math> b </math>, <math> p </math>, and <math> q </math> positive. The function ''B''(''p,q'') is the [[beta function]].
-Shares in LVMH (LVMH.PA), the world's biggest luxury goods group, fell sharply on Wednesday after an unexpected slowdown in sales growth at its fashion and leather business, which includes the Louis Vuitton, [http://www.pcs-systems.co.uk/Images/celinebag.aspx Celine] and Dior brands.<br><br>The share price was down 6.4 percent at 135.50 euros by 1140 GMT, a six-week low and wiping around 4.8 billion euros ($6.5 billion)off the market value of France's fourth-biggest listed company.<br><br>The group, which also owns Ruinart champagne and Hennessy cognac, saw sales growth at the fashion and leather division slide to 3 percent in the third quarter, against expectations of 7 to 8 percent.<br>In a conference call, Chief Financial Officer Jean-Jacques Guiony blamed price increases in Japan for the slowdown, as well as softer demand for some brands.<br>However, one London-based analyst noted that Japan accounted for only around 15 percent of LVMH's fashion and leather sales, "so we did not get a full explanation".<br><br>LVMH has been trying to stem a decline in Louis Vuitton's sales growth by introducing new and pricier leather bags, which analysts expected would lead to short-term losses in sales.<br>"I understand that the repositioning of Louis Vuitton takes time and may be a bumpy ride," said Exane BNP Paribas analyst Luca Solca.<br>Before the results were announced, LVMH shares were up 4.4 percent since January 1, underperforming the overall luxury sector, which was up more than 20 percent.<br><br>Analysts said concerns about the future growth of Louis [http://www.dailymail.co.uk/home/search.html?sel=site&searchPhrase=Vuitton Vuitton] had been exacerbated by the announcement this month that it was parting company with its star designer, Marc Jacobs.<br>SUPPLY CONSTRAINTS<br>Guiony said the recent launch of new leather collections had not come in time to have a meaningful impact on sales, and acknowledged that production was constrained by a lack of high-quality leather.<br><br>"Without these supply constraints, we would produce more than what we do," Guiony said.<br>Louis Vuitton shop assistants polled by Reuters last month said they had been provided with only a small number of new handbags, such as the Capucines model, priced at 3,500 euros, which had flown off the shelves.<br>LVMH has been buying tanneries to secure supplies but experts say the market is under pressure partly because the number of calves raised and slaughtered is driven more by demand for meat -- which has been in decline -- than by demand for quality hides.<br><br>In addition, China, the luxury industry's main driver since the late 2000s, has started to run out of steam in the last year due to an economic slowdown and a government crackdown on gift-giving.<br>Guiony said Vuitton's sales in mainland China were "flattish" but, thanks to sales to Chinese tourists, overall sales growth to Chinese customers was in the "mid-single digits plus".<br>Guiony said trends in watches and jewelry had slightly improved in China, but not in fashion and leather.<br><br>He said trading remained difficult in Europe, particularly for perfume and cosmetics, where sales were "flattish".<br>LVMH's overall sales grew 2 percent in the third quarter. Organic growth was 8 percent, of which 6 percentage points were accounted for by the relative weakness of the U.S. dollar and the Japanese yen against the euro.<br>($1=0.7406 euros)<br>(Additional reporting by James Regan; Editing by James Jukwey and Kevin Liffey)
+[[File:GBtree.jpg|thumb|GB distribution tree]]
+== Properties ==
+=== Moments ===
+It can be shown that the ''h''th moment can be expressed as follows:
+:<math> \operatorname{E}_{GB}(Y^{h})=\frac{b^{h}B(p+h/a,q)}{B(p,q)}{}_{2}F_{1} \begin{bmatrix}
+ p + h/a,h/a;c \\
+ p + q +h/a;
+\end{bmatrix},
+</math>
+where <math>{}_{2}F_{1}</math> denotes the [[hypergeometric series]] (which converges for all ''h'' if ''c''<1, or for all ''h''/''a''<''q'' if ''c''=1 ).
+== Related distributions ==
+The generalized beta encompasses a number of distributions in its family as special cases. Listed below are its three direct descendants, or sub-families.
+=== Generalized beta of first kind (GB1) ===
+The generalized beta of the first kind is defined by the following pdf:
+:<math> GB1(y;a,b,p,q) = \frac{|a|y^{ap-1}(1-(y/b)^{a})^{q-1}}{b^{ap}B(p,q)} </math>
+for <math> 0< y^{a}<b^{a} </math> where <math> b </math>, <math> p </math>, and <math> q </math> are positive. It is easily verified that
+:<math> GB1(y;a,b,p,q) = GB(y;a,b,c=0,p,q). </math>
+The moments of the GB1 are given by
+:<math> \operatorname{E}_{GB1}(Y^{h}) = \frac{b^{h}B(p+h/a,q)}{B(p,q)}. </math>
+The GB1 includes the [[Beta distribution|beta of the first kind]] (B1), [[Generalized gamma distribution|generalized gamma]](GG), and [[Pareto distribution|Pareto]] as special cases:
+:<math> B1(y;b,p,q) = GB1(y;a=1,b,p,q) ,</math>
+:<math> GG(y;a,\beta,p) = \lim_{q \to \infty}
+GB1(y;a,b=q^{1/a}\beta,p,q) ,</math>
+:<math> PARETO(y;b,p) = GB1(y;a=-1,b,p,q=1) . </math>
+=== Generalized beta of the second kind (GB2) ===
+The GB2 (also known as the [[Generalized_beta_prime_distribution#Generalization|Generalized Beta Prime]]) is defined by the following pdf:
+:<math> GB2(y;a,b,p,q) = \frac{|a|y^{ap-1}}{b^{ap}B(p,q)(1+(y/b)^a)^{p+q}} </math>
+for <math> 0< y < \infty </math> and zero otherwise. One can verify that
+:<math> GB2(y;a,b,p,q) = GB(y;a,b,c=1,p,q). </math>
+The moments of the GB2 are given by
+:<math> \operatorname{E}_{GB2}(Y^h) = \frac{b^h B(p+h/a,q-h/a)}{B(p,q)}. </math>
+The GB2 nests common distributions such as the generalized gamma (GG), Burr type 3, Burr type 12, [[lognormal]], [[Weibull distribution|Weibull]], [[Gamma distribution|gamma]], [[Lomax distribution|Lomax]], [[F-distribution|F statistic]], Fisk or [[Rayleigh distribution|Rayleigh]], [[Chi-squared distribution|chi-square]], [[Half-normal distribution|half-normal]], half-Student's, [[Exponential distribution|exponential]], and the [[Log-logistic distribution|log-logistic]].<ref>McDonald, J.B. (1984) "Some generalized functions for the size distributions of income", ''Econometrica'' 52, 647–663.</ref>
+=== Beta ===
+The [[beta distribution]] (B) is defined by:{{cn|date=April 2013}}
+:<math> B(y;b,c,p,q) = \frac{y^{p-1}(1-(1-c)(y/b))^{q-1}}{b^{p}B(p,q)(1+c(y/b))^{p+q}} </math>
+for <math> 0<y<b/(1-c) </math> and zero otherwise. Its relation to the GB is seen below:
+:<math> B(y;b,c,p,q) = GB(y;a=1,b,c,p,q). </math>
+The beta family includes the betas of the first and second kind<ref>Stuart, A. and Ord, J.K. (1987): Kendall's Advanced Theory of Statistics, New York: Oxford University Press.</ref> (B1 and B2, where the B2 is also referred to as the [[Beta prime distribution|Beta prime]]), which correspond to ''c'' = 0 and ''c'' = 1, respectively.
+A figure showing the relationship between the GB and its special and limiting cases is included above (see McDonald and Xu (1995) ).
+== Exponential generalized beta distribution ==
+Letting <math> Y \sim GB(y;a,b,c,p,q) </math>, the random variable <math> Z = \ln(Y) </math>, with re-parametrization, is distributed as an exponential generalized beta (EGB), with the following pdf:
+:<math> EGB(z;\delta,\sigma,c,p,q) = \frac{e^{p(z-\delta)/\sigma}(1-(1-c)e^{(z-\delta)/\sigma})^{q-1}}{|\sigma|B(p,q)(1+ce^{(z-\delta)/\sigma})^{p+q}}</math>
+for <math> -\infty < \frac{z-\delta}{\sigma}<\ln(\frac{1}{1-c}) </math>, and zero otherwise.
+The EGB includes generalizations of the [[Gompertz distribution|Gompertz]], [[Gumbel distribution|Gumbell]], [[Type I extreme value distribution|extreme value type I]], [[Logistic distribution|logistic]], Burr-2, [[Exponential distribution|exponential]], and [[Normal distribution|normal]] distributions.
+Included is a figure showing the relationship between the EGB and its special and limiting cases (see McDonald and Xu (1995) ).
+[[File:EGBtree.jpg|thumb|EGB distribution tree]]
+=== Moment generating function ===
+Using similar notation as above, the [[moment-generating function]] of the EGB can be expressed as follows:
+:<math> M_{EGB}(Z)=\frac{e^{\delta t}B(p+t\sigma,q)}{B(p,q)}{}_{2}F_{1} \begin{bmatrix}
+ p + t\sigma,t\sigma;c \\
+ p + q +t\sigma;
+\end{bmatrix}.
+</math>
+== Uses ==
+The flexibility provided by the GB family is used in modeling the distribution of:{{cn|date=April 2013}}
+* family income
+* stock returns
+*insurance losses
+Applications involving members of the EGB family include:{{cn|date=April 2013}}
+* partially adaptive estimation of regression
+* time series models
+==References==
+<references />
+==Bibliography==
+* C. Kleiber and S. Kotz (2003) ''Statistical Size Distributions in Economics and Actuarial Sciences''. New York: Wiley
+* Johnson, N. L., S. Kotz, and N. Balakrishnan (1994) ''Continuous Univariate Distributions''. Vol. 2, Hoboken, NJ: Wiley-Interscience.
+{{ProbDistributions|continuous-bounded}}
+[[Category:Continuous distributions]]
+[[Category:Probability distributions]]

Pedometric mapping: Difference between revisions

Revision as of 18:23, 9 September 2013

Contents

Definition

Properties

Moments

Related distributions

Generalized beta of first kind (GB1)

Generalized beta of the second kind (GB2)

Beta

Exponential generalized beta distribution

Moment generating function

Uses

References

Bibliography

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Pedometric mapping: Difference between revisions

Revision as of 18:23, 9 September 2013

Definition

Properties

Moments

Related distributions

Generalized beta of first kind (GB1)

Generalized beta of the second kind (GB2)

Beta

Exponential generalized beta distribution

Moment generating function

Uses

References

Bibliography

Navigation menu

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