A MODIFIED WEIBULL HAZARD RATE AS GENERATOR OF A GENERALIZED MAXWELL DISTRIBUTION

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AS GENERATOR OF A GENERALIZED MAXWELL DISTRIBUTION

VIOREL GH. VOD ˘A

We first introduce a slightly modified Weibull hazard rate, namely, h(t) = (k/a²)t^2k−1, t > 0, a, k > 0, which fork = 1/2 and k = 1 provide the exponential and Rayleigh densities, respectively. Using the transformation f1(t) = t·f(t)/E(T),where f(t) is the density function of the variableT havingh(t) as hazard rate, a generalized variant of Maxwell density function is obtained, namely f1(t) =C·t^2kexp(−t^2k/2a²),whereCis a norming constant. Some inferences on f1 are made and the special casek= 1/2 is also discussed.

AMS 2000 Subject Classification: 62P05.

Key words: p.d.f.-probability density function, c.d.f.-cumulative distribution function, modified Weibull p.d.f., generalized Maxwell, unbiasedness, hazard rate, SPRT-Sequential Probability Ratio Test.

1. INTRODUCTION

As is well-known, a reliability model is usually represented by the so- called hazard (or failure) rate associated with the failure behaviour of the entity under consideration (see Gnedenkoet al. [7] or Barlow and Proschan [2]).

IfT is a positive continuous random variable with p.d.f. f(t) and c.d.f.

F(t) = Prob{T 6t},withF(t)<1 for every t, then

(1.1) h(t) = f(t)

1−F(t), t>0,

is called hazard or failure rate. As Gertsbakh [6, pages 24–25] pointed out, this term is borrowed from demography or survival analysis (the statistical study of lifetime for living beings, where the concept of mortality rate, instead of failure rate is exclusively used).

The variableT is the lifetime (life span) of a given object or system, that is, the time from its “birth” (starting operating moment) until its “death” (ir- reversible out-of-working state). If the entity is non-renewable (non-restoring), then the average value E(T) ofT is just the mean durability calculated up to

MATH. REPORTS11(61),2 (2009), 171–179

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its “first” failure which is in fact the last one, too. We refer here to an entire class of similar objects for which an indicator as the average life makes sense.

Taking into account that the reliability (or “survivor”) function of T is R(t) = Prob{T > t} = 1−F(t), it follows immediately from (1.1) that h(t) =−R⁰(t)/R(t),hence

(1.2) R(t) = exp

− Z t

0

h(u)du

and f(t) =h(t) exp

− Z t

0

h(u)du

. Therefore, the second equation (1.2) will provide p.d.f.(s) if one chooses specific forms for the hazard rate function, h. A list of the most used p.d.f.(s) in reliability theory is given by Blischke and Murthy [4, pages 128–129].

For instance, if we take h(u) = θ k u^k−1, u > 0, θ, k > 0, we obtain the Weibull p.d.f. f(t) = θ·k·t^k−1exp (−θk), t > 0, θ, k > 0. Details on Weibull model may be found in Abernethy et al. [1], Isaic-Maniu [8], Lawless [10], Johnson et al. [9].

In the present paper we shall show that a modified Weibull hazard rate, namely,

(1.3) h(t) = k

a²t^2k−1, t>0, a, k >0,

can generate by a suitable transformation on its corresponding p.d.f., a generalized variant of the Maxwell distribution (notice that the Maxwell p.d.f. can- not be derived from the Weibull one by particularization of the shape/power parameter kas is the case of exponential or Rayleigh p.d.f.(s).

2. THE MODIFIED WEIBULL P.D.F.

Equation (1.3) yields immediately via (1.2) the p.d.f.

(2.1) T :f(t;θ, k) = k

a²t^2k−1exp

−t^2k 2a²

, t>0, a, k >0,

which is a Weibull p.d.f. with 2kas shape parameter (ifkis a natural number, then the shape parameter is always an even number, which is not the case of the usual Weibull p.d.f).

It is easy to see that ifk= 1/2,then we one obtain the exponential p.d.f.

f(t;a,1/2) = 1/2a²

exp −t/2a²

, while if k = 1 then we get a Rayleigh p.d.f., f(t;a,1) = 1/a²

·t·exp −t²/2a² .

The c.d.f. and the reliability function ofT are (2.2) F(t;a, k) = 1−exp

−t^2k 2a²

and R(t;a, k) = exp

−t^2k 2a²

, respectively.

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The form ofF(t;a, k) can be generalized by “exponentiation” as Mud- holkar and Srivastava [11] originally proposed for the classical Weibull form.

In our case (2.2) we should have (2.3) F_{EM W}(t;b, a, k) =

1−e^t^2k^/2a²b

, t>0, a, b, k >0,

where using the above authors’ vocabulary we must read EMW as “exponentiated modified Weibull” distribution.

The form (2.3) is not our concern here. A special case (for k = 1, a= 1/λ√

2 and arbitraryb >0),that is, (2.4) F(t;b, λ) =

h

1−e^−(λt)² ib

, t>0, λ, b >0, has been studied by Raqab and Kundu [13].

For our stated purpose, we now only need the mean-value of T. After some simple algebra we get

(2.5) E(T) = Z ∞

0

t·f(t;a, k) dt= 2a²1/2k

·Γ (1 + 1/2k), where Γ (·) is the well-known Gamma function

(2.6) Γ (x) =

Z ∞

0

u^x−1e^−udu.

Remember that Γ (x+ 1) =x! ifxis natural, Γ (1) = Γ (2) = 1, Γ (1/2) =√ π and since Γ (x+ 1) = x·Γ (x), we have Γ (1 + 1/2) = √

π/2; details on the Gamma function are given in Dorin et al. [5, pages 240–244)].

In (2.5) one easily recognizes the mean-value E(T) = 2a² if k = 1/2, that is, the case of an exponential variable.

3. THE GENERALIZED MAXWELL P.D.F.

Let T be a positive continuous random variable with finite (nonzero) mean valueE(T) and p.d.f.f(t).The functionf₁(t) =t·f(t)/E(t) is obviously the p.d.f. of a new variableT1 (see [18] and [19]). In our case, if T is given by (2.1), we have

(3.1) T1: f1(t;a, k) = k

2^1/2k·a^2+1/k·Γ (1 + 1/2k)t^2k·exp

−t^2k 2a²

,

where t>0,a, k >0. If k= 1, we have the classical Maxwell p.d.f., that is, (3.2) T_M : f₁(t;a,1) =

r2 π · 1

a³ ·t²·exp

− t² 2a²

,

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hence (3.1) can be regarded as a generalized Maxwell p.d.f. It is interesting to notice that if we consider T with p.d.f.

(3.3) T : f(t;θ, k) = 2θ^k+1

Γ (k+ 1)·t^2k+1·exp −θt²

, t>0, θ, k >0, that is the so-called generalized Rayleigh p.d.f. (see Johnsonet al. [9, page 479]

or [17]), then by our transformation we get (3.3) T1: f1(t;θ, k) = 2θ^k+3/2

Γ (k+ 3/2)t^2k+2·exp −θt²

, t>0, θ, k >0, which provides the Maxwell p.d.f. for k= 0. Anyway, the form (3.1) is more general since its exponential factor depends on the shape parameter k.

3.1. ESTIMATION PROCEDURES

In this subsection we shall deal with the estimation of the scale parameter aassuming thatkis known. The log-likelihood function associated with (3.1) is

lnL=nlnk−(n/2k) ln 2−n(2 + 1/k) lna−nln Γ(1 + 1/2k)+

(3.5)

+2k

n

X

1

lnti− 1/2a²

·

n

X

1

t^2k_i ,

where t1, t2, . . . , tn is an independent random sample on T1 (details on maximum likelihood estimation theory can be found in Blischke and Murthy [4, pages 139–148]). It follows that

(3.6) ∂lnL

∂a =−n(2 + 1/k) + 1/a²

·

n

X

1

t^2k_i ,

which provides the MLE (Maximum Likelihood Estimator) of a² as

(3.7) ˆa² = 1

n(2 + 1/k) ·

n

X

1

t^2k_i . We now remark that

1. Ifk= 1 (the classical Maxwell case), then ˆa²= (1/3n)·

n

P

1

t²_i and the statistic

(3.8) S= 3π−8

3nπ ·

n

X

1

t²_i

is an unbiased estimate of the variance of the classical Maxwell variable T_M, since Var(TM) = (3−8/π)·a² and, consequently,E(S) =S.

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2. The variable T_M² (whose density is given by (3.2) has a chi-square p.d.f. with 3 degrees of freedom and scale parameter a. Indeed,

(3.9) F(y) = Prob{T_M 6√

y}= 1 a³ ·

r2 π ·

Z ^√y

0

t²exp −t²/2a² dt and if we differentiate, we obtain the p.d.f. of T_M² as

(3.10) T_M² :f(t;a) = 1 a²√

2π ·t^1/2·exp

− t 2a²

. The mean-value of T_M² is E T_M²

= 3a² since for a natural integer n≥1 we have

(3.11)

Z

uⁿe^−u²du=−1

2uⁿ⁻¹e^−u²+n−1 2

Z

uⁿ⁻²e^−u²du

(see Smoleanski [14, page 122, formula 41.22]). In (3.10) we took t=v² and u=v/a√

2 in order to compute E T_M²

. Hence (3.12) Var (T_M) =E T_M²

−E²(T_M) = (3−8/π)a².

3. It is also interesting to mention that fT_M (t;a) has the modal value tmo =a√

2 since

(3.13) f⁰

TM (t;a) =f_TM (t;a)·a⁻²·2a²−t²

t ,

which provides

f_TM(t;a)

max = q2

π · _ae². The second derivative yields two inflection points, which is not the usual case in the p.d.f.(s) families related to failure phenomena. Isaic-Maniu [8, pages 21–24] showed that the reduced Weibull (RW) p.d.f., that is,

(3.14) f_RW(t;b) =b·t^b−1exp −t^b

, t≥0, b >0,

has two inflection points only if b > 2. If b= 2 (Rayleigh case) there is only one inflection point situated on the right-side of (f_RW)max.

4. TESTING A SIMPLE HYPOTHESIS ON THE SCALE PAREMETER

In this paragraph we shall test the statistical hypothesis (4.1) H₀ :a=a₀ versusH₁:a=a₁ (a₀ < a₁)

assuming that the shape parameterkis known. Here,H0 andH1are straight- forward suppositions about the theoretical average value (or mean-lifetime, in a reliability context) of the T1 variable.

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We shall consider the SPRT (Sequential Probability Ratio Test) proposed by Wald [20, §3, pages 37–52]. The log-likelihood ratio is

(4.2) r_n=

n

Q

1

f1(ti;a1, k) Qn

1

f₁(t_i;a₀, k)

= a₀

a1

n(1+2k)

·exp

"

−1 2

1 a²₁ − 1

a²₀ n

X

1

t^2k_i

# ,

where {t_i}_1≤i≤n is a sequential sample on T1. Taking natural logarithms, we have

(4.3) lnr_n=n(2 + 1/k) lna₀ a₁ −1

2 1

a²₁ − 1 a²₀

·

n

X

1

t^2k_i ,

which may be written as

(4.4) lnrn=−n(2 + 1/k) lna1

a₀ +1 2

1 a²₀ − 1

a²₁

·

n

X

1

t^2k_i .

DenotingA=β/(1−α) and B=β/(1−β)/α,where α and β are the usual risks in hypothesis testing theory (see Wald [20, §3.2, pages 40–42]), we get the decision rule below.

a. If Pn

1

t^2k_i ≤ ^ln^B

1 2

1 a2

0

−¹

a2 1

+n(2 + 1/k)· ^ln

a1 a0

1 2

1 a2 0

−¹

a2 1

accept the null-hypothesis H₀ and reject H₁.

b. If

n

P

1

t^2k_i ≥ ^ln^A

1 2

1 a2

0

−¹

a2 1

+n(2 + 1/k)· ^ln

a1 a0

1 2

1 a2

0

−¹

a2 1

reject H0 and accept the alternativeH₁.

c. If

n

P

1

t^2k_i is strictly greater then the right-hand side of a. and strictly smaller then the right-hand side of b., the experiment has to continue by taking a new measurement/observation on T₁.

The SPRT will be completely constructed if we also provide the OC- function (Operative Characteristic) L(a), which gives the power of the test and the so-called ASN (Average Sample Number) E_a(n) needed to perform the sequential procedure. These are given as

(4.5) L(a) = A^H(a)−1

A^H(a)−B^H(a) and E_a(n) = L(a) lnB+ [1−L(a)] lnA

Ea(z) ,

with H(a) given by Wald’s equation

(4.6) E

e^ZH(a)

= 1,

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where z = ln [f1(t;a1, k)/f1(t;a0, k)] and H(a) 6= 0. In Wald [20], the no- tation is h(a) instead of H(a): we made this change since we used h for the hazard rate. We easily obtain

(4.7) z= ln

a₀ a₁

2+1/k

−1 2

1 a²₁ − 1

a²₀

·t^2k

and, after some algebra, (4.8) E e^ZH

= (constant)· Z ∞

0

t^2kexp

−1 2

H a²₁ −H

a²₀ + 1 a²

t^2k

dt= 1 with the restriction

(4.9) H

a²₁ − H a²₀ + 1

a² >0 or H < (a1a0)² a² a²₁−a²₀.

The integral in (4.8) can be evaluated (using again Smoleanski [14]) by denoting

(4.10) t·

1 2

H a²₁ −H

a²₀ + 1 a²

1/2k

=u.

For instance, if k= 1 (the Maxwell case), we obtain

(4.11) 1

a³ · a0

a1

3H

· H

a²₁ − H a²₀ + 1

a² −3/2

= 1 and the denominator in ASN [see the second formula of (4.5)] is (4.12) E_a(z) = 3 lna₀

a1

−3 2

1 a²₁ − 1

a²₀ a²

·a². Ifk6= 1, the Gamma function will be involved.

One can also perform a sequential comparison of two generalized Maxwell distributions having the same shape parameter, amounting to a comparison between two average lifetimes. The procedure is also a SPRT in Girshick’s variant (presented in Wald [20, §4.2.4, pages 84–86]), his method being improved by V˘aduva [15] and Obreja [12]: their results allow a quicker computation of the OC-function and ASN.

5. FINAL COMMENTS: A CURIOUS CONNECTION

If in (3.1) we takek= 1/2,we obtain (5.1) f₁(t;a,1/2) = 1/4a⁴

t·exp −t/2a²

, t≥0, a >0, and denoting λ= 1/2a² we have a p.d.f.

(5.2) f(t;λ) =λ²t·exp (−λt), t≥0, λ >0,

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which can be generated by a homographic hazard rate (HHR) of the form

(5.3) h(t;λ) = λ²t

1 +λ t, t≥0, λ >0, which has been studied by Bˆarsan-Pipuet al. [3].

Such a peculiar variable (sayT₀) with the p.d.f. given by (5.2) has some interesting properties: the coefficient of variation CV (T0) = 1/√

2 (that is a constant), skewness β1 = √

2, kurtosis β2 = 6. These values are easily obtained since themth raw moment isE(T₀^m) =λ^−m·Γ (2 +m), m= 1,2, . . ..

Noticing that (skewness) = 2·(coefficient of variation), a Gamma variable is suggested, namely, a particular form of the reparametrizedGamma p.d.f.

(5.4) f(t;λ, θ) =α θ

α

· 1

Γ (α)t^α−1exp

−α θt

, t≥0, α, θ >0, see [16] or Johnson et al. [9]. If we takeα= 2 and λ= 2/θ, we obtain (5.2).

To conclude, the generalized Maxwell p.d.f. can be also regarded as a generalized form of a homographic hazard rate p.d.f. provided by (5.3).

REFERENCES

[1] R.B. Abernethy, J.E. Breneman, C.H. Medlin and G.L. Reinman, Weibull Analysis Handbook. Air Force Wright Aeronautical Labs., Technical Report AFWAL-TR-83-2079, Ohio, 1983.

[2] R.E. Barlow and F. Proschan, Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York, 1975.

[3] N. Bˆarsan-Pipu, Al. Isaic-Maniu and V.Gh. Vod˘a,An extensive study of the homographic hazard rate variable. Econ. Comput. Econ. Cyb. Stud. Res. (Bucharest) XXXIII (1999),1–4, 5–14.

[4] W.R. Blischke and D.N.P. Murthy,Reliability. Modeling, Prediction and Optimization.

Wiley, New York, 2000.

[5] Al.C. Dorin, Al. Isaic-Maniu and V.Gh. Vod˘a,Statistical Problems of Reliability.Ed.

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[6] I. Gertsbakh,Reliability Theory with Applications to Preventive Maintenance.Springer, Berlin, 2000.

[7] B.V. Gnedenko, Yu.K. Belyaev and A.D. Solovyiev,Mathematical Methods in Reliability Theory.Academic Press, New York, 1969.

[8] Al. Isaic-Maniu,The Weibull Method and Applications.Ed. Academiei, Bucharest, 1983 (Romanian).

[9] N.L. Johnson, S. Kotz and N. Balakrishnan, Continuous Univariate Distributions, Vol. 1, 2nd Edition. Wiley, New York, 1994.

[10] J.F. Lawless,Statistical methods in reliability.Technometrics25, 305–316.

[11] G.S. Mudholkar and D.K. Srivastava,Exponentiated Weibull family for analyzing bathtub failure data.IEEE Trans. ReliabilityR-42(1993), 299–302.

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[12] G. Obreja, On the OC-function and ASN for some sequential tests. Stud. Cerc. Mat.

21(1969),3, 451–461 (Romanian).

[13] M.Z. Raqab and D. Kundu, Burr type X distribution revisited. http://home.

iitk.ac.in/~kundu(2003).

[14] M.L. Smoleanski, Tables of Indefinite Integrals.Ed. Tehnic˘a, Bucharest, 1972. (Roma- nian translation of the Russian original “Tablitzy Neopredelyonnyh Integralov”, Nauka, Moskva, 1967.)

[15] I. V˘aduva, Sequential tests for exponential type distributions. Rev. Roumaine Math.

Pures Appl.12(1962),4, 705–716 (Russian).

[16] V.Gh. Vod˘a, The study of a certain Gamma-type distribution. Stud. Cerc. Mat. 26 (1974),2, 311–322 (Romanian).

[17] V.Gh. Vod˘a, Inferential procedures on a generalized Rayleigh variate (I). Aplikace Matematiky (Praha)21(1976),6, 395–412.

[18] V.Gh. Vod˘a,New models in durability tool-testing:Pseudo-Weibull distribution.Kyber- netika (Praha)25(1989),3, 209–215.

[19] V.Gh. Vod˘a,A new generalization of Rayleigh distribution. Reliability: Theory and Applications2(2007),2, 47–57. (www.gnedenko-forum.org/Journal/index.htm) [20] A. Wald,Sequential Analysis.Dover, New York, 1973.

Received 6 March 2008 Romanian Academy

Gheorghe Mihoc-Caius Iacob Institute of Mathematical Statistics and Applied Mathematics

Casa Academiei Romˆane Calea 13 Septembrie nr. 13 050711 Bucharest 5, Romania

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