Remarks on the generalized doubly truncated gamma distribution

A C T A U N I V E R S I T A T I S L O D Z I E N S I S

FO L IA O EC O N O M IC A 141, 1997

Tadeusz Gerstenkorn* Joanna Gerstenkorn**

R E M A R K S ON T H E G E N E R A L IZ E D D O U B LY T R U N C A T E D G A M M A D IST R IB U T IO N

Abstract. In the present paper we discuss a doubly truncated generalized gam m a

distribution and give form ulae for the m om ents o f this distribution and special cases together with examples o f calculations.

Key words: truncated probability distributions, m om ents, generalized gam m a distribution.

1. IN T R O D U C T IO N

After introducing the generalized gam m a distribution to the literature by S t a c y (1962), the investigations of this distribution were carried out by m any authors such as Š r ó d k a (1964, 1966, 1967, 1970), S t a c y and M i h r a n (1965), M a l i k (1967), W a s i l e w s k i (1967), H a r t e r ( 1967), L i e n h a r d and M e y e r (1967), R o s ł o n e k (1968), H a g e r and B a i n (1970), P o d o l s k i (1972), K r ó l i k o w s k a (1973), J a k u s z e n k o w (1973, 1974, 1976), L a j k ó (1977), L a w l e s s (1980), A c h e a r and B o l - f a r i n e (1986), M a s w a d a h (1991).

The im portance o f this distribution lies in the fact it is a generalization o f the gamma, Weibull, Rayleigh and X distributions significant in m any technical and economic applications.

In the present paper we deal with a generalized doubly truncated gam m a distribution and give form ulae for the m om ents o f this distribution together with examples of calculations. This will enable us to apply the distribution and its special cases whenever the range o f the variable is essentially bounded to some interval o f values. In the practice of operation researches, these m ay be the cases quite frequently encountered.

* U niversity o f Łódź, Faculty o f M athem atics. ** U niversity o f Łódź, C hair o f Statistical M ethods.

2. T H E R ESU LTS

The density function of random variable X with the generalized gam m a distribution is given by

A part from m any articles in journals, distribution (1) is also easy to find in the lexicon M ü l l e r et al. 1975 and handbook G e r s t e n k o r n , Ś r o d k a (1983) listed in the references.

As some special cases of (1) we have: 1) gam m a distribution if a = 1, 2) Weibull distribution if a = p, 3) Rayleigh distribution if a = p — 2, 4) x distribution if a = 2, p = n,

(most frequently, a = 1/2, see M ü l l e r et al. 1975, p. 34),

a) a = 2, p = n = 2 - the Maxwell distribution with 2 com ponents (most frequently, X = 1/2a 2).

b) a = 2, p = n = 3 - the Maxwell distribution with 3 com ponents (most frequently, X = 1/2<т2).

We assume th at f ( x ) is a positive density function o f random variable

X with P ( X e ( c , í/)) = 1. The density function f ^ u ) o f random variable U,

given by

is called the density of a doubly truncated distribution of random variable

X. We consider a truncated distribution if the values of random variable

have to be limited to interval [a, b].

D istributions 1-4 are well know n and have been investigated in m any papers.

F o r distribution (1), we have from (2)

for jc > 0; a, p, X > 0, for a: < 0.

(

)

(

2

)

D enote

h

W„ = f vp~ i+ne ~ iy'dv, и = 0, 1 , 2 , ... (4)

a

Lemma. F o r (4), we have

W„ = j - [ a p+n~ae ~ Álŕ - b p+n- ae - Xa* + (p + n - a)ITn_ J . (5)

Proof. We can write 1 b

W„= - - r f-A<xva- 1e - iv"v',+n- adv. Aoc a

A fter integration by parts, we have

1 b

W„ = - T [(vp+n- ae - Xv’) l ba- ( p + n - a ) f v p- 1+n- ae - Xv‘dv] = (5)

A C C a

We denote by m n the m om ent of order n o f random variable U with density function (3), th at is,

m n = E i l ß ) = J unJ\{u)du. (6)

a

From (6) and (4) we immediately have

Theorem. If random variable U has density function (3) o f the truncated generalized gam m a distribution (1), then, for и > 1,

m n = W„/W0 (7)

where W„ is given by (4). Examples.

We calculate m 2 in the ease of truncated generalized gam m a distribution (3). F rom (7) and (5) we have

m 2 = T [ар+2- ае ~ Ха’ - b p+2- * e - Xb‘ + (p + 2 - a) W2- e]. Aoc VYq

We consider the following special cases: 1. Let a = 1, Я = 1; then (gam m a distribution)

m 2 = (ap+íe~a - b p+1e~b + (p + 1 ) fV1) / W 0 =

W e notice that

W Q - J vp_ 1e ~ vdv = Г (p, b) - Г(/7, a) (8)

a

where Г (p, x ) represents the so-called incomplete gam m a function

Г(р, a) = J- vp~ i e~'dv,

о

which means that one can calculate (8) with the aid e.g. K . Pearson’s Tables. 2. If p = 1 = a (special case of the W eibull distribution), then

W 0 = e - a - e ~ b,

therefore

m 2 = {a2e ~ 2 — b2e~b + 2[(a + \)e~a - (b + 1 )е “ ь]}/(<Га - e " b) =

= {(a2 + 2a + 2)e~a - (b2 + 2b + 2) e - b}l(e~a - e~b). 3. Let a = p = 2 (Rayleigh distribution). In this case

m 2 = W 2I W 0 = 2^ p r ( a 2e _A“J - b2e ~Xb’ + 2 ^ 0),

however,

Ж

0 =

= - ^

“AvJ)la =

therefore

(a2 + ^ ) e - Aal- ( f c 2 + ^ ) e - A'’1

ffl2 = g-й * _ g-Abä •

4. Let а = 2, p = 3 (Maxwell distribution with 3 com ponents, then

т г= ~ b 3 e ~ Xb' + 3 W o ) where, in this case

W 0 = J v2e ~ Xv'dv.

a

A fter integration by parts, we have

and, further,

ь I JlXb [Z __ __

J e ~ Xv'dv = J е" ,2/2Л = / - [Ф(72Я6) - Ф(^2Яа)]. я \/2Я ,/2Аа V ^

Using the tables o f the norm al distribution Ф(/), we obtain the required result.

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Tadeusz Gerstenkorn, Joanna Gerstenkorn

U W AG I O U O G Ó L N IO NY M PO D W Ó JN IE U CIĘTY M R O Z K Ł A D Z IE G A M M A

W pracy przedstaw iony jest podwójnie ucięty uogólniony rozkład gam m a i podane są wzory na m om enty tego rozkładu oraz jego przypadków szczególnych w raz z przykładam i ich wyliczeń.