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

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1 A MODIFIED WEIBULL HAZARD RATE AS GENERATOR OF A GENERALIZED MAXWELL DISTRIBUTION VIOREL GH. VODĂ We first introduce a slightly modified Weibull hazard rate, namely, ht) = k/a )t k, t, a, k >, which for k = / and k = provide the exponential and Rayleigh densities, respectively. Using the transformation f t) = t ft)/et ), where ft) is the density function of the variable T having ht) as hazard rate, a generalized variant of Maxwell density function is obtained, namely f t) = C t k exp t k /a ), where C is a norming constant. Some inferences on f are made and the special case k = / is also discussed. AMS Subject Classification: 6P5. 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.. INTRODUCTION As is well-known, a reliability model is usually represented by the socalled hazard or failure) rate associated with the failure behaviour of the entity under consideration see Gnedenko et al. [7] or Barlow and Proschan []). If T is a positive continuous random variable with p.d.f. ft) and c.d.f. F t) = Prob{T t}, with F t) < for every t, then.) ht) = ft), t, F t) is called hazard or failure rate. As Gertsbakh [6, pages 4 5] 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 variable T is the lifetime life span) of a given object or system, that is, the time from its birth starting operating moment) until its death irreversible out-of-working state). If the entity is non-renewable non-restoring), then the average value ET ) of T is just the mean durability calculated up to MATH. REPORTS 6), 9), 7 79

2 7 Viorel Gh. Vodă 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 Rt) = Prob {T > t} = F t), it follows immediately from.) that ht) = R t)/rt), hence.) Rt) = exp { t } hu)du and { ft) = ht) exp t } hu)du. Therefore, the second equation.) 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 8 9]. For instance, if we take h u) = θ k u k, u, θ, k >, we obtain the Weibull p.d.f. ft) = θ k t k exp θk), t, θ, k >. Details on Weibull model may be found in Abernethy et al. [], Isaic-Maniu [8], Lawless [], Johnson et al. [9]. In the present paper we shall show that a modified Weibull hazard rate, namely,.3) ht) = k a tk, t, a, k >, 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. cannot be derived from the Weibull one by particularization of the shape/power parameter k as is the case of exponential or Rayleigh p.d.f.s).. THE MODIFIED WEIBULL P.D.F. Equation.3) yields immediately via.) the p.d.f..) T : f t; θ, k) = k ) a tk exp tk a, t, a, k >, which is a Weibull p.d.f. with k as shape parameter if k is 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 if k = /, then we one obtain the exponential p.d.f. f t; a, /) = /a ) exp t/a ), while if k = then we get a Rayleigh p.d.f., f t; a, ) = /a ) t exp t /a ). The c.d.f. and the reliability function of T are.) F t; a, k) = exp respectively. tk a ) and R t; a, k) = exp ) tk a,

3 3 A modified Weibull hazard rate as generator of a generalized Maxwell distribution 73 The form of F t; a, k) can be generalized by exponentiation as Mudholkar and Srivastava [] originally proposed for the classical Weibull form. In our case.) we should have.3) F EMW t; b, a, k) = e tk /a ) b, t, a, b, k >, where using the above authors vocabulary we must read EMW as exponentiated modified Weibull distribution. The form.3) is not our concern here. A special case for k =, a = /λ and arbitrary b > ), that is, [.4) F t; b, λ) = e λt)] b, t, λ, b >, has been studied by Raqab and Kundu [3]. For our stated purpose, we now only need the mean-value of T. After some simple algebra we get.5) ET ) = t f t; a, k) dt = a ) /k Γ + /k), where Γ ) is the well-known Gamma function.6) Γ x) = u x e u du. Remember that Γ x + ) = x! if x is natural, Γ ) = Γ ) =, Γ /) = π and since Γ x + ) = x Γ x), we have Γ + /) = π/; details on the Gamma function are given in Dorin et al. [5, pages 4 44)]. In.5) one easily recognizes the mean-value ET ) = a if k = /, 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 value ET ) and p.d.f. ft). The function f t) = t ft)/et) is obviously the p.d.f. of a new variable T see [8] and [9]). In our case, if T is given by.), we have ) k 3.) T : f t; a, k) = /k a +/k Γ + /k) tk exp tk a, where t, a, k >. If k =, we have the classical Maxwell p.d.f., that is, ) 3.) T M : f t; a, ) = π a 3 t exp t a,

4 74 Viorel Gh. Vodă 4 hence 3.) 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) = θk+ Γ k + ) tk+ exp θt ), t, θ, k >, that is the so-called generalized Rayleigh p.d.f. see Johnson et al. [9, page 479] or [7]), then by our transformation we get 3.3) T : f t; θ, k) = θk+3/ Γ k + 3/) tk+ exp θt ), t, θ, k >, which provides the Maxwell p.d.f. for k =. Anyway, the form 3.) is more general since its exponential factor depends on the shape parameter k. 3.. ESTIMATION PROCEDURES In this subsection we shall deal with the estimation of the scale parameter a assuming that k is known. The log-likelihood function associated with 3.) is 3.5) ln L = n ln k n/k) ln n + /k) ln a n ln Γ + /k)+ +k ln t i /a ) t k i, where t, t,..., t n is an independent random sample on T details on maximum likelihood estimation theory can be found in Blischke and Murthy [4, pages 39 48]). It follows that 3.6) ln L a = n + /k) + /a ) t k i, which provides the MLE Maximum Likelihood Estimator) of a as 3.7) â = n + /k) t k i. statistic We now remark that. If k = the classical Maxwell case), then â = /3n) 3.8) S = 3π 8 3nπ t i t i and the is an unbiased estimate of the variance of the classical Maxwell variable T M, since VarT M ) = 3 8/π) a and, consequently, ES) = S.

5 5 A modified Weibull hazard rate as generator of a generalized Maxwell distribution 75. The variable TM whose density is given by 3.) has a chi-square p.d.f. with 3 degrees of freedom and scale parameter a. Indeed, 3.9) F y) = Prob {T M y} = y a 3 π t exp t /a ) dt and if we differentiate, we obtain the p.d.f. of T M as 3.) T M : f t; a) = a π t/ exp t a The mean-value of TM is E TM) = 3a since for a natural integer n we have 3.) u n e u du = un e u + n u n e u du see Smoleanski [4, page, formula 4.]). In 3.) we took t = v and u = v/a in order to compute E T M). Hence ). 3.) Var T M ) = E T M) E T M ) = 3 8/π) a. 3. It is also interesting to mention that f TM t; a) has the modal value t mo = a since 3.3) f t; a) = f T TM t; a) a a t, M t which provides [ f TM t; a) ] max = π ae. 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 4] showed that the reduced Weibull RW) p.d.f., that is, 3.4) f RW t; b) = b t b exp t b), t, b >, has two inflection points only if b >. If b = 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.) H : a = a versus H : a = a a < a ) assuming that the shape parameter k is known. Here, H and H are straightforward suppositions about the theoretical average value or mean-lifetime, in a reliability context) of the T variable.

6 76 Viorel Gh. Vodă 6 We shall consider the SPRT Sequential Probability Ratio Test) proposed by Wald [, 3, pages 37 5]. The log-likelihood ratio is 4.) r n = n f t i ; a, k) = n f t i ; a, k) a a ) n+k) exp [ a a ) where {t i } i n is a sequential sample on T. Taking natural logarithms, we have 4.3) ln r n = n + /k) ln a a a ) a t k i, which may be written as 4.4) ln r n = n + /k) ln a a + a a Denoting A = β/ α) and B = β/ β) /α, where α and β are the usual risks in hypothesis testing theory see Wald [, 3., pages 4 4]), we get the decision rule below. a. If t k ln B i a a ) + n + /k) ) accept the null-hypothesis H and reject H. b. If t k ) + n + /k) ln A i a a ln a a a a ln a a a a ) t k i. t k i ] ) reject H and accept the alternative H. c. If is strictly greater then the right-hand side of a. and strictly t k i smaller then the right-hand side of b., the experiment has to continue by taking a new measurement/observation on T. The SPRT will be completely constructed if we also provide the OCfunction Operative Characteristic) La), 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) La) = AHa) A Ha) B Ha) and E a n) = with Ha) given by Wald s equation 4.6) E [ e ZHa)] =, La) ln B + [ La)] ln A, E a z),

7 7 A modified Weibull hazard rate as generator of a generalized Maxwell distribution 77 where z = ln [f t; a, k) /f t; a, k)] and Ha). In Wald [], the notation is ha) instead of Ha): we made this change since we used h for the hazard rate. We easily obtain ) +/k a 4.7) z = ln a a ) a t k and, after some algebra, 4.8) E e ZH) [ = constant) t k exp H a H a + ) ] a t k dt = with the restriction H 4.9) a H a + a > or H < a a ) a a ). a The integral in 4.8) can be evaluated using again Smoleanski [4]) by denoting [ H 4.) t a H a + )] /k a = u. For instance, if k = the Maxwell case), we obtain ) 3H 4.) a 3 a H a a H a + ) 3/ a = and the denominator in ASN [see the second formula of 4.5)] is 4.) E a z) = 3 ln a 3 a a ) a a a. If k, 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 [, 4..4, pages 84 86]), his method being improved by Văduva [5] and Obreja []: their results allow a quicker computation of the OC-function and ASN. 5. FINAL COMMENTS: A CURIOUS CONNECTION If in 3.) we take k = /, we obtain 5.) f t; a, /) = /4a 4) t exp t/a ), t, a >, and denoting λ = /a we have a p.d.f. 5.) f t;λ) = λ t exp λt), t, λ >,

8 78 Viorel Gh. Vodă 8 which can be generated by a homographic hazard rate HHR) of the form 5.3) h t; λ) = λ t, t, λ >, + λ t which has been studied by Bârsan-Pipu et al. [3]. Such a peculiar variable say T ) with the p.d.f. given by 5.) has some interesting properties: the coefficient of variation CV T ) = / that is a constant), skewness β =, kurtosis β = 6. These values are easily obtained since the mth raw moment is E T m) = λ m Γ + m), m =,,.... Noticing that skewness) = coefficient of variation), a Gamma variable is suggested, namely, a particular form of the reparametrized Gamma p.d.f. α ) α 5.4) f t; λ, θ) = θ Γ α) tα exp α θ t ), t, α, θ >, see [6] or Johnson et al. [9]. If we take α = and λ = /θ, we obtain 5.). 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 [] 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-83-79, Ohio, 983. [] R.E. Barlow and F. Proschan, Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York, 975. [3] N. Bârsan-Pipu, Al. Isaic-Maniu and V.Gh. Vodă, An extensive study of the homographic hazard rate variable. Econ. Comput. Econ. Cyb. Stud. Res. Bucharest) XXXIII 999), 4, 5 4. [4] W.R. Blischke and D.N.P. Murthy, Reliability. Modeling, Prediction and Optimization. Wiley, New York,. [5] Al.C. Dorin, Al. Isaic-Maniu and V.Gh. Vodă, Statistical Problems of Reliability. Ed. Economică, Bucharest, 994 Romanian). [6] I. Gertsbakh, Reliability Theory with Applications to Preventive Maintenance. Springer, Berlin,. [7] B.V. Gnedenko, Yu.K. Belyaev and A.D. Solovyiev, Mathematical Methods in Reliability Theory. Academic Press, New York, 969. [8] Al. Isaic-Maniu, The Weibull Method and Applications. Ed. Academiei, Bucharest, 983 Romanian). [9] N.L. Johnson, S. Kotz and N. Balakrishnan, Continuous Univariate Distributions, Vol., nd Edition. Wiley, New York, 994. [] J.F. Lawless, Statistical methods in reliability. Technometrics 5, [] G.S. Mudholkar and D.K. Srivastava, Exponentiated Weibull family for analyzing bathtub failure data. IEEE Trans. Reliability R-4 993), 99 3.

9 9 A modified Weibull hazard rate as generator of a generalized Maxwell distribution 79 [] G. Obreja, On the OC-function and ASN for some sequential tests. Stud. Cerc. Mat. 969), 3, Romanian). [3] M.Z. Raqab and D. Kundu, Burr type X distribution revisited. iitk.ac.in/~kundu 3). [4] M.L. Smoleanski, Tables of Indefinite Integrals. Ed. Tehnică, Bucharest, 97. Romanian translation of the Russian original Tablitzy Neopredelyonnyh Integralov, Nauka, Moskva, 967.) [5] I. Văduva, Sequential tests for exponential type distributions. Rev. Roumaine Math. Pures Appl. 96), 4, Russian). [6] V.Gh. Vodă, The study of a certain Gamma-type distribution. Stud. Cerc. Mat ),, 3 3 Romanian). [7] V.Gh. Vodă, Inferential procedures on a generalized Rayleigh variate I). Aplikace Matematiky Praha) 976), 6, [8] V.Gh. Vodă, New models in durability tool-testing:pseudo-weibull distribution. Kybernetika Praha) 5 989), 3, 9 5. [9] V.Gh. Vodă,A new generalization of Rayleigh distribution. Reliability: Theory and Applications 7),, [] A. Wald, Sequential Analysis. Dover, New York, 973. Received 6 March 8 Romanian Academy Gheorghe Mihoc-Caius Iacob Institute of Mathematical Statistics and Applied Mathematics Casa Academiei Române Calea 3 Septembrie nr Bucharest 5, Romania von voda@yahoo.com