Bir gama rasgele değişkeninin logaritmasının çarpıklığı

Gama rasgele değişkeni düşünün $X\sim\Gamma(\alpha, \theta)$ . Ortalama, varyans ve çarpıklık için düzgün formüller vardır:

\begin{aligned} E [X] & = α θ \\ Var [X] & = α θ^{2} = 1 / α \cdot E [X]^{2} \\ Skewness [X] & = 2 / \sqrt{α} \end{aligned}

$\begin{align} \mathbb E[X]&=\alpha\theta\\ \operatorname{Var}[X]&=\alpha\theta^2=1/\alpha\cdot\mathbb E[X]^2\\ \operatorname{Skewness}[X]&=2/\sqrt{\alpha} \end{align}$

Şimdi log dönüştürülmüş rasgele değişkeni düşünün . Wikipedia, ortalama ve varyans için formüller verir: $Y=\log(X)$

\begin{aligned} E [Y] & = ψ (α) + \log (θ) \\ Var [Y] & = ψ_{1} (α) \end{aligned}

$\begin{align} \mathbb E[Y]&=\psi(\alpha)+\log(\theta)\\ \operatorname{Var}[Y]&=\psi_1(\alpha)\\ \end{align}$

gama fonksiyonunun logaritmasının birinci ve ikinci türevleri olarak tanımlanan digamma ve trigamma fonksiyonları ile.

Çarpıklığın formülü nedir?

Tetragamma işlevi görünecek mi?

(Beni bu konuda merak eden şey, lognormal ve gama dağılımları arasında bir seçim, bkz. Gamma ve lognormal dağılımlar . Diğer şeylerin yanı sıra, çarpıklık özelliklerinde farklılık gösterirler. gama kütüğünün çarpıklığı negatif ama ne kadar olumsuz? ..)

gamma-distribution skewness logarithm

— amo diyor Reinstate Monica
kaynak

Does bu yardım? Yoksa bu mu?

— Stephan Kolassa

Log-gama dağılımının ne olduğundan tam olarak emin değilim. Lognormal normal ile ilişkili olduğu için gama ile ilgili ise, o zaman başka bir şey soruyorum (çünkü "lognormal" kafa karıştırıcı olarak, günlük (normal) değil exp (normal) dağılımıdır).

— amip Reinstate Monica diyor

@Glen_b: Dürüst olmak gerekirse, normalin üstelini "lognormal" olarak adlandırmanın çok daha tutarsız ve kafa karıştırıcı olduğunu söyleyebilirim. Rağmen, ne yazık ki, daha köklü.

— Stephan Kolassa

Ayrıca log-lojistik bkz @Stephan, log-Cauchy, log-Laplace vs vs. karşısında daha net bir şekilde kurulmuş kongresi

— Glen_b -Reinstate Monica

Evet; Bu nedenle bu dağılımla ilgili hiçbir yerde "log-gama" dememeye dikkat ettim. (Geçmişte log-normal ile tutarlı bir şekilde kullandım)

— Glen_b -Restate Monica

Yanıtlar:

Momenti üreten fonksiyon $M(t)$ arasında $Y=\ln X$ bir basit cebirsel formu vardır, çünkü bu durumda yararlıdır. Mgf tanımına göre,

\begin{aligned} M (t) & = E [e^{t \ln X}] = E [X^{t}] \\ = \frac{1}{Γ (α) θ^{α}} \int_{0}^{\infty} x^{α + t - 1} e^{- x / θ} d x \\ = \frac{θ^{t}}{Γ (α)} \int_{0}^{\infty} y^{α + t - 1} e^{- y} d y \\ = \frac{θ^{t} Γ (α + t)}{Γ (α)} . \end{aligned}

$\begin{aligned}M(t)&=\operatorname{E}[e^{t\ln X}]=\operatorname{E}[X^t]\\ &=\frac{1}{\Gamma(\alpha)\theta^\alpha}\int_0^\infty x^{\alpha+t-1}e^{-x/\theta}\,dx\\ &=\frac{\theta^{t}}{\Gamma(\alpha)}\int_0^\infty y^{\alpha+t-1}e^{-y}\,dy\\ &=\frac{\theta^t\Gamma(\alpha+t)}{\Gamma(\alpha)}.\end{aligned}$

Verdiğiniz beklentiyi ve varyansı doğrulayalım. Türevleri alırsak,

M^{'} (t) = \frac{Γ^{'} (α + t)}{Γ (α)} θ^{t} + \frac{Γ (α + t)}{Γ (α)} θ^{t} \ln (θ)

$M'(t)=\frac{\Gamma'(\alpha+t)}{\Gamma(\alpha)}\theta^t+\frac{\Gamma(\alpha+t)}{\Gamma(\alpha)}\theta^t\ln(\theta)$ ve

M^{″} (t) = \frac{Γ^{″} (α + t)}{Γ (α)} θ^{t} + \frac{2 Γ^{'} (α + t)}{Γ (α)} θ^{t} \ln (θ) + \frac{Γ (α + t)}{Γ (α)} θ^{t} \ln^{2} (θ) .

$M''(t)=\frac{\Gamma''(\alpha+t)}{\Gamma(\alpha)}\theta^t+\frac{2\Gamma'(\alpha+t)}{\Gamma(\alpha)}\theta^t\ln(\theta)+\frac{\Gamma(\alpha+t)}{\Gamma(\alpha)}\theta^t\ln^2(\theta).$ Bu nedenle,

Daha sonra, aşağıdaki

E [Y] = ψ^{(0)} (α) + \ln (θ), E [Y^{2}] = \frac{Γ^{″} (α)}{Γ (α)} + 2 ψ^{(0)} (α) \ln (θ) + \ln^{2} (θ) .

$\operatorname{E}[Y]=\psi^{(0)}(\alpha)+\ln(\theta),\qquad\operatorname{E}[Y^2]=\frac{\Gamma''(\alpha)}{\Gamma(\alpha)}+2\psi^{(0)}(\alpha)\ln(\theta)+\ln^2(\theta).$

Var (Y) = E [Y^{2}] - E [Y]^{2} = \frac{Γ^{″} (α)}{Γ (α)} - {(\frac{Γ^{'} (α)}{Γ (α)})}^{2} = ψ^{(1)} (α) .

$\operatorname{Var}(Y)=\operatorname{E}[Y^2]-\operatorname{E}[Y]^2=\frac{\Gamma''(\alpha)}{\Gamma(\alpha)}-\left(\frac{\Gamma'(\alpha)}{\Gamma(\alpha)}\right)^2=\psi^{(1)}(\alpha).$

Çarpıklığı bulmak için, kümülatör üretme fonksiyonuna dikkat edin (uç için olasılık @ olasılıktır) Birinci kümülatör basitçe . Hatırlamak

K (t) = \ln M (t) = t \ln θ + \ln Γ (α + t) - \ln Γ (α) .

$K(t)=\ln M(t)=t\ln\theta+\ln\Gamma(\alpha+t)-\ln\Gamma(\alpha).$

K^{'} (0) = ψ^{(0)} (α) + \ln (θ)

$K'(0)=\psi^{(0)}(\alpha)+\ln(\theta)$

, dolayısıyla sonraki kümülanlar

. Çarpıklık bu nedenle

ψ^{(n)} (x) = d^{n + 1} \ln Γ (x) / d x^{n + 1}

$\psi^{(n)}(x)=d^{n+1}\ln\Gamma(x)/dx^{n+1}$

K^{(n)} (0) = ψ^{(n - 1)} (α)

$K^{(n)}(0)=\psi^{(n-1)}(\alpha)$

n \geq 2

$n\geq2$

\frac{E [(Y - E [Y])^{3}]}{Var (Y)^{3 / 2}} = \frac{ψ^{(2)} (α)}{[ψ^{(1)} (α)]^{3 / 2}} .

$\frac{\operatorname{E}[(Y-\operatorname{E}[Y])^3]}{\operatorname{Var}(Y)^{3/2}}=\frac{\psi^{(2)}(\alpha)}{[\psi^{(1)}(\alpha)]^{3/2}}.$

Bir yan not olarak, bu özel dağıtım iyice onun AC Olshen tarafından incelenmiştir ortaya çıktı Pearson Tip III Dağıtım Dönüşümler , Johnson ve ark. En Sürekli Tekdeğişkenli Dağılımları da bu konuda küçük bir parça vardır. Thoseunlara bir bak.

— Francis
kaynak

Bu ayırt edilmelidir

yerine

, bu kümülant olarak üretim fonksiyonu - merkezi anlarla daha doğrudan ilgili -

K (t) = \log [M (t)] = t \log [θ] + \log [Γ (α + t)] - \log [Γ (α)]

$K (t)=\log [M (t)]=t\log [\theta]+\log [\Gamma (\alpha+t)]-\log [\Gamma (\alpha)]$

M (t)

$M (t)$

s k e w = K^{(3)} (0) = ψ^{(2)} (α)

$skew=K^{(3)}(0)=\psi^{(2)}(\alpha)$ where

ψ^{(n)} (z)

$\psi^{(n)}(z)$ is the polygamma function

— probabilityislogic

@probabilityislogic: very good call, changed my answer

— Francis

@probabilityislogic This is a great addition, thanks a lot. I just want to note, lest some readers be confused, that skewness is not directly given by the third cumulant: it's the third standardized moment, not the third central moment. Francis has it correct in his answer, but the last formula in your comment is not quite right.

— amoeba says Reinstate Monica

I. Direct computation

\int_{0}^{\infty} x^{ν - 1} e^{- μ x} (\ln x)^{p} d x

$\int_0^\infty x^{\nu-1}e^{-\mu x}(\ln x)^p dx$

p = 2, 3, 4

$p=2,3,4$

p = 1

$p=1$

Γ, ψ

$\Gamma, \psi$

ζ

$\zeta$

p

$p$ as a derivative of a gamma function so presumably it's feasible to go higher. So skewness is certainly doable but not especially "neat".

Details of the derivation of the formulas in 4.358 are in [2]. I'll quote the formulas given there since they're slightly more succinctly stated and put 4.352.1 in the same form.

Let $\delta= \psi(a)-\ln \mu$ . Then:

\begin{aligned} \int_{0}^{\infty} x^{a - 1} e^{- μ x} \ln x d x & = \frac{Γ (a)}{μ^{a}} {δ} \\ \int_{0}^{\infty} x^{a - 1} e^{- μ x} \ln^{2} x d x & = \frac{Γ (a)}{μ^{a}} {δ^{2} + ζ (2, a)} \\ \int_{0}^{\infty} x^{a - 1} e^{- μ x} \ln^{3} x d x & = \frac{Γ (a)}{μ^{a}} {δ^{3} + 3 ζ (2, a) δ - 2 ζ (3, a)} \\ \int_{0}^{\infty} x^{a - 1} e^{- μ x} \ln^{4} x d x & = \frac{Γ (a)}{μ^{a}} {δ^{4} + 6 ζ (2, a) δ^{2} - 8 ζ (3, a) δ + 3 ζ^{2} (2, a) + 6 ζ (4, a))} \end{aligned}

$\begin{align} \int_0^\infty x^{a-1} e^{-\mu x} \ln x \,dx &=\frac{\Gamma(a)}{\mu^a}\left\{ \delta \right\} \\ \int_0^\infty x^{a-1} e^{-\mu x} \ln^2\!x \,dx &=\frac{\Gamma(a)}{\mu^a}\left\{ \delta^2+\zeta(2,a) \right\} \\ \int_0^\infty x^{a-1} e^{-\mu x} \ln^3\!x \,dx &=\frac{\Gamma(a)}{\mu^a}\left\{ \delta^3+3\zeta(2,a)\delta-2\zeta(3,a) \right\} \\ \int_0^\infty x^{a-1} e^{-\mu x} \ln^4\!x \,dx &=\frac{\Gamma(a)}{\mu^a}\left\{ \delta^4+6\zeta(2,a)\delta^2-8\zeta(3,a)\delta + 3\zeta^2(2,a)+6\zeta(4,a)) \right\} \end{align}$

where $\zeta(z,q)=\sum_{n=0}^\infty \frac{1}{(n+q)^z}$ is the Hurwitz zeta function (the Riemann zeta function is the special case $q=1$ ).

Now on to the moments of the log of a gamma random variable.

Noting firstly that on the log scale the scale or rate parameter of the gamma density is merely a shift-parameter, so it has no impact on the central moments; we may take whichever one we're using to be 1.

If $X\sim \text{Gamma}(\alpha,1)$ then

E (\log^{p} X) = \frac{1}{Γ (α)} \int_{0}^{\infty} \log^{p} x x^{α - 1} e^{- x} d x .

$E(\log^{p}\!X) = \frac{1}{\Gamma(\alpha)}\int_0^\infty \log^{p}\!x\, x^{\alpha-1} e^{-x} \,dx.$

We can set $\mu=1$ in the above integral formulas, which gives us raw moments; we have $E(Y)$ , $E(Y^2)$ , $E(Y^3)$ , $E(Y^4)$ .

Since we have eliminated $\mu$ from the above, without fear of confusion we're now free to re-use $\mu_k$ to represent the $k$ -th central moment in the usual fashion. We may then obtain the central moments from the raw moments via the usual formulas.

Then we can obtain the skewness and kurtosis as $\frac{\mu_3}{\mu_2^{3/2}}$ and $\frac{\mu_4}{\mu_2^{2}}$ .

A note on terminology

It looks like Wolfram's reference pages write the moments of this distribution (they call it ExpGamma distribution) in terms of the polygamma function.

By contrast, Chan (see below) calls this the log-gamma distribution.

II. Chan's formulas via MGF

Chan (1993) [3] gives the mgf as the very neat $\Gamma(\alpha+t)/\Gamma(\alpha)$ .

(A very nice derivation for this is given in Francis' answer, using the simple fact that the mgf of $\log(X)$ is just $E(X^t)$ .)

Consequently the moments have fairly simple forms. Chan gives:

E (Y) = ψ (α)

$E(Y)=\psi(\alpha)$

and the central moments as

\begin{aligned} E (Y - μ_{Y})^{2} & = ψ^{'} (α) \\ E (Y - μ_{Y})^{3} & = ψ^{″} (α) \\ E (Y - μ_{Y})^{4} & = ψ^{‴} (α) \end{aligned}

$\begin{align} E(Y-\mu_Y)^2 &= \psi'(\alpha) \\ E(Y-\mu_Y)^3 &= \psi''(\alpha) \\ E(Y-\mu_Y)^4 &= \psi'''(\alpha) \end{align}$

and so the skewness is $\psi''(\alpha)/(\psi'(\alpha)^{3/2})$ and kurtosis is $\psi'''(\alpha)/(\psi'(\alpha)^{2})$ . Presumably the earlier formulas I have above should simplify to these.

Conveniently, R offers digamma ( $\psi$ ) and trigamma ( $\psi'$ ) functions as well as the more general polygamma function where you select the order of the derivative. (A number of other programs offer similarly convenient functions.)

Consequently we can compute the skewness and kurtosis quite directly in R:

skew.eg <- function(a) psigamma(a,2)/psigamma(a,1)^(3/2)
kurt.eg <- function(a) psigamma(a,3)/psigamma(a,1)^2

Trying a few values of a ( $\alpha$ in the above), we reproduce the first few rows of the table at the end of Sec 2.2 in Chan [3], except that the kurtosis values in that table are supposed to be excess kurtosis, but I just calculated kurtosis by the formulas given above by Chan; these should differ by 3.

(E.g. for the log of an exponential, the table says the excess kurtosis is 2.4, but the formula for $\beta_2$ is $\psi'''(1)/\psi'(1)^2$ ... and that is 2.4.)

Simulation confirms that as we increase sample size, the kurtosis of a log of an exponential is converging to around 5.4 not 2.4. It appears that the thesis possibly has an error.

Consequently, Chan's formulas for central moments appear to actually be the formulas for the cumulants (see the derivation in Francis' answer). This would then mean that the skewness formula was correct as is; because the second and third cumulants are equal to the second and third central moments.

Nevertheless these are particularly convenient formulas as long as we keep in mind that kurt.eg is giving excess kurtosis.

References

[1] Gradshteyn, I.S. & Ryzhik I.M. (2007), Table of Integrals, Series, and Products, 7th ed.
Academic Press, Inc.

[2] Victor H. Moll (2007)
The integrals in Gradshteyn and Ryzhik, Part 4: The gamma function
SCIENTIA Series A: Mathematical Sciences, Vol. 15, 37–46
Universidad Técnica Federico Santa María, Valparaíso, Chile
http://129.81.170.14/~vhm/FORM-PROOFS_html/final4.pdf

[3] Chan, P.S. (1993),
A statistical study of log-gamma distribution,
McMaster University (Ph.D. thesis)
https://macsphere.mcmaster.ca/bitstream/11375/6816/1/fulltext.pdf

— Glen_b -Reinstate Monica
kaynak

Cool. Thanks a lot! According to the encyclopedia entry that Stephan linked to above, the final answer for skewness is

ψ^{″} (α) / ψ^{'} (α)^{3 / 2}

$\psi''(\alpha)/\psi'(\alpha)^{3/2}$ (which almost qualifies as "neat"!). So it seems that all the scary zetas will have to cancel out.

— amoeba says Reinstate Monica

Sorry only just now saw your comment (I've been editing for about an hour or so); that's correct, though if the Encyclopedia gives kurtosis the way Chan gives it in his thesis, it seems that it's wrong (as given above), but readily corrected. The neat formulas appear to be for cumulants rather than standardized central moments.

— Glen_b -Reinstate Monica

Yes, the Encyclopedia does give the same formula for kurtosis.

— amoeba says Reinstate Monica

Hmm, I mean to refer to the things normally denoted

γ_{1}

$\gamma_1$ and

γ_{2}

$\gamma_2$ . I will fix.

— Glen_b -Reinstate Monica

I should probably add the note that the Hurwitz zeta function can be expressed in terms of the polygamma function, and vice versa:

ψ^{(n)} (z) = (- 1)^{n + 1} Γ (n + 1) ζ (n + 1, z)

$\psi^{(n)}(z)=(-1)^{n+1}\,\Gamma(n+1)\,\zeta(n+1,z)$ So, the answer to the @amoeba's question of "will the tetragamma function appear?" is YES.

— J. M. is not a statistician