एक गामा यादृच्छिक चर के लघुगणक का तिरछापन

पर विचार करें गामा यादृच्छिक चर $X\sim\Gamma(\alpha, \theta)$ । माध्य, विचरण और तिरछापन के लिए स्वच्छ सूत्र हैं:

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

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

अब एक लॉग-परिवर्तित यादृच्छिक चर $Y=\log(X)$ । माध्य और विचरण के लिए विकिपीडिया सूत्र देता है:

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

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

डिगामा और ट्राइगम्मा फ़ंक्शंस के माध्यम से जो गामा फ़ंक्शन के लघुगणक के पहले और दूसरे डेरिवेटिव के रूप में परिभाषित किए गए हैं।

तिरछेपन का सूत्र क्या है?

Tetragamma फ़ंक्शन दिखाई देगा?

(इस बारे में मुझे जो अचरज हुआ वह यह है कि लोगनॉर्मल और गामा डिस्ट्रीब्यूशन के बीच एक विकल्प है । गामा बनाम लोगनॉर्मल डिस्ट्रीब्यूशन देखें । अन्य चीजों के अलावा, वे अपने तिरछे गुणों में भिन्न होते हैं। विशेष रूप से, लॉग्नोर्मल के लॉग का तिरछा होना शून्य के बराबर है।) गामा के लॉग का तिरछा होना नकारात्मक है। लेकिन कितना नकारात्मक? ..?

gamma-distribution skewness logarithm

— अमीबा का कहना है कि मोनिका को बहाल करो
स्रोत

क्या यह मदद करता है ? या ये ?

— Stephan Kolassa

मुझे यकीन नहीं है कि लॉग-गामा वितरण क्या है। यदि यह गामा से संबंधित है क्योंकि लॉगनॉर्मल सामान्य से संबंधित है, तो मैं कुछ और के बारे में पूछ रहा हूं (क्योंकि "लॉगनॉर्मल", भ्रामक रूप से, एक्स का वितरण है (सामान्य) लॉग (सामान्य) का नहीं।

— अमीबा का कहना है कि मोनिका

@Glen_b: ईमानदार होने के लिए, मैं कहूंगा कि सामान्य के घातांक को "लॉगनॉर्मल" कहना अधिक असंगत और भ्रमित करने वाला है। हालांकि, दुर्भाग्य से, अधिक स्थापित।

— स्टीफन कोलासा

@ स्टेफ़न लॉग-लॉजिस्टिक, लॉग-कौची, लॉग-लाप्लास आदि को भी देखते हैं। यह विपरीत की तुलना में अधिक स्पष्ट रूप से स्थापित सम्मेलन है

— Glen_b -Reate Monica

हां; मैं इस कारण से इस वितरण के संबंध में कहीं भी "लॉग-गामा" नहीं कहने के लिए सावधान हूं। (मैंने इसे अतीत में लॉग-सामान्य के अनुरूप तरीके से उपयोग किया है)

— Glen_b -Reinstate Monica

जवाबों:

पल पैदा समारोह के , इस मामले में उपयोगी है, क्योंकि यह एक सरल बीजीय रूप है। Mgf की परिभाषा से, हमारे पास $M(t)$ $Y=\ln X$

\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}

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

चलो अपेक्षा और आपके द्वारा दिए गए विचरण को सत्यापित करते हैं। डेरिवेटिव ले रहा है, हम और

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

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

इसलिए,

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) .

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

यह तो इस प्रकार

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) .

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

To find the skewness, note the cumulant generating function (thanks @probabilityislogic for the tip) is

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

$K(t)=\ln M(t)=t\ln\theta+\ln\Gamma(\alpha+t)-\ln\Gamma(\alpha).$ The first cumulant is thus simply

K^{'} (0) = ψ^{(0)} (α) + \ln (θ)

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

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

$n\geq2$ . The skewness is therefore

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

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

As a side note, this particular distribution appeared to have been thoroughly studied by A. C. Olshen in his Transformations of the Pearson Type III Distribution, Johnson et al.'s Continuous Univariate Distributions also has a small piece about it. Check those out.

— Francis
स्रोत

You should differentiate

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

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

M (t)

$M (t)$ as this is the cumulant generating function - more directly related to central moments -

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

Gradshteyn & Ryzhik [1] (sect 4.358, 7th ed) list explicit closed forms for

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

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

p = 2, 3, 4

$p=2,3,4$ while the

p = 1

$p=1$ case is done in 4.352 (assuming you regard expressions in

Γ, ψ

$\Gamma, \psi$ and

ζ

$\zeta$ functions as closed form) -- from which it is definitely doable up to kurtosis; they give the integral for all

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
स्रोत

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