THE EQUILIBRIUM RENEWAL PROCESS AS DENSITY GENERATOR

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AS DENSITY GENERATOR

VIOREL GH. VOD ˘A

We derive some new probability density functions – pdf(s) – starting from the pdf of the first variable in a so-called equilibrium renewal process. The Weibull pdf is used first and some properties of the variable derived are emphasized. Some other generators such as Gamma and power are also employed and the difficulties arising in the procedure are put into light.

AMS 2000 Subject Classification: 62P05.

Key words: equilibrium renewal process, reliability function, incomplete Gamma function, half-normal pdf, reliability context, MLE-maximum likelihood estimator.

1. A SHORT REVIEW OF SOME USUAL PROCEDURES FOR PDF GENERATION

There are several ways to construct a peculiar density function of a continuous random variable. One of them is to ask for a so-called “system of frequency curves” – the best known being probably that of Karl Pearson (1857–1936) derived in 1895 (see Rodriguez [15]). Rodriguez has presented a synthesis of five such systems, mentioned also the nature of their generating mechanism: 1) Pearson (differential equation); 2) Gram–Charlier–Edgeworth;

3) Burr (differential equation); 4) Johnson (transformation to normality);

5) Tukey’s lambda (a special transformation). The first four systems provide pdf(s) in an explicit form (see [15, p. 218]).

We may add to this list a less known system proposed by Rafaele D’Addario (1899–1974) in [5] improved by himself in [6]. Details are presented by Guerrieri [9]. It is noticeable also the “suprasystem of probability distributions” advanced by Savageau [16], as a set of simultaneous differential equations. Voit [19] proposed a somewhat related system which he baptized as “The S-distribution system” – a four parameter ordinary differential equation, considered to be “a good candidate for representing and analyzing failure data” (Yu and Voit [20, p. 596]).

A unified generalization of Burr-Hatke andS-system of distributions has been given recently by the present author (see [18]).

MATH. REPORTS10(60),3 (2008), 289–296

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In the reliability context, the usual way to derive densities or cdf (s)- cumulative distribution functions, is to employ the hazard (failure) rate function z(x) ≥ 0, x ≥ 0, associated to a random variable X representing the failure behavior of a given entity. The failure density f(x) is given then as (1.1) X :f(x) =z(x)·exp

− Z x

0

z(u)du

.

Several choices forz(u) – that is constant, increasing, decreasing or bath- tub shaped positive functions – will provide a wide range of such densities (see Blischke and Murthy, 2000 [3, p. 128–129]). In the same reliability framework, another procedure is to consider the form

(1.2) X⁽¹⁾ :g(x) = x

E(x) ·f(x), whereE(x) =R∞

0 x f(x) dxwithf the pdf of the original variableX. The new class of densities (g(x)) presents in some cases the so-called conservability property, that is, if for instanceXis Gamma distributed, thenX⁽¹⁾preserves the same class of distributions. It is not the case of the Weibull variable which furnishes X⁽¹⁾ as a Pseudo-Weibull one (see [17]).

2. DEFINITIONS

A less exploited way to construct pdf(s) related to failure phenomena is that to use some notions from renewal theory.

Definition 1. Anordinary renewal process is a sequence of nonnegative, independent and identically distributed random variables{X₁, X2, . . . , Xn, . . .}

where X_i is interpreted as the random lifetime of the ith item and S_r = X1+X2+· · ·+Xr(S0= 0) is then random time at which therth replacement takes place (Barlow and Proschan [2, p. 96]).

Definition 2. Let {X₁, X2, . . . , Xn, . . .} be an ordinary renewal process for which f is the probability density function ofX₂, X₃, . . . variables. If the pdf of X₁ is

(2.1) f_e(x) = [1−F(x)]/µ,

whereF(x) =Rx

0 f(u)duand µ=R∞

0 x·f(x)dxare the cdf ofX_i and the average lifetime, respectively, then the renewal process is called an equilibrium process (Cox [4, p. 28]).

Definition 3. A continuous positive random variable X has a two-parameter Weibull cdf – Fx – if it has the form

(2.2) X : FX(x;θ, k) = 1−exp

−(θx)^k

, x≥0, θ, k >0.

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Consequently, the reliability functionRX(x;θ, k) has the formRX(x;θ, k) = 1−F_X(x;θ, k) = exph

−(θx)^ki .

3. THE WEIBULL CDF AS GENERATOR

Using formula (2.1), the Weibull variable yields

X₁ : f₁(x;θ, k) = R_X(x;θ, k)

E(X) =

(3.1)

= θ

Γ(1 + 1/k)exp

−θ^kx^k

, x≥0, θ, k >0,

since

(3.2) E(X) = 1

θΓ (1 + 1/k), where Γ(x) =R∞

0 e^−tt^x−1dt.

Consequences.(1) The form (3.1) is a generalization of the classical exponential pdf: for k = 1 one obtainsf₁(x;θ) =θ·exp (−θx), x≥ 0, θ > 0 (some other generalizations have been given by Dob´o [7] and Khan [14]).

(2) The form (3.1) may be also regarded as a generalization of the half- normal pdf, namely, f(x;σ) =

2/πσ²1/2

·exp −x²/2σ²

, x ≥0, σ > 0, if in (3.1) one takes k = 2, θ^k = 1/2σ², since Γ (3/2) =√

π/2 (see Hahn and Shapiro [11, p. 77]).

3.1. SOME PROPERTIES OF THE NEW PDF

Property 1. The coefficient of variation of X1 does not depend on the scale parameter θ.

Proof. A straightforward computation of the rth noncentral moment of X₁ provides

E X₁^r

= Z ∞

0

x^rf₁(x;θ, k) dx= 1

k θ^rΓ (1 + 1/k)· Z ∞

0

y^r+1^k ⁻¹·e^−ydy= (3.3)

= 1

k θ^r · Γ ^r+1_k Γ (1 + 1/k)

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(after the change of variable θ^kx^k=y). Hence

(3.4)

E[X1] = Γ (2/k)

kθ Γ (1 + 1/k) and Var [X1] = 1

kθ² ·

Γ (3/k)

Γ (1 + 1/k)− Γ²(2/k) Γ²(1 + 1/k)·1

k

.

Since the variation coefficient is CV (X1) = p

Var (X1)/E(X1), it is obvious that it only depends on the shape parameter (k).

Remark. This fact may be used in parameter estimation of pdf(s) as follows: compute ¯x and s (sample average and sample standard deviation) from an independent sample onX₁, then equate the statistics/¯x toCV(X₁).

We shall obtain an estimate of k- say ˆk. Then from the equation ¯x=E(X1) one can extract ˆθ– an estimator ofθ. Thismoment-type estimation procedure has been suggested by Emil J. Gumbel (1891–1966) – see [10] and has been intensively employed by Dorin et al. [8].

Property 2. The distribution function of X₁ is

(3.5) F₁(x;θ, k) = Γ^1/k_x/θ(1/k) k·Γ (1 + 1/k),

where the numerator in (3.5)is the incomplete Gamma function(see Abramowitz and Stegun [1, p. 863]).

Proof. We can immediately write F₁(x;θ, k) = θ

Γ (1 + 1/k) · Z x

0

exp −θ^k·u^k du= (3.6)

= 1

k·Γ (1 + 1/k)

Z x^1/k/θ

0

y¹^k⁻¹e^−ydy,

where we made the change of variable u= θ⁻¹y^1/k, hence (3.6) is just (3.5).

Now, taking into account formula 6.5.1 from [1, p. 87], namely, P(a, x) = [Γ (a)]⁻¹Rx

0 t^a−1e^−tdt, we in fact have F1(x;θ, k) = P ¹_k, x^1/k/θ

while for k= 1/n,n∈N\{0}, we can use the approximation ([1, formula 6.5.1], p. 88) (3.7) P_n(n,x) = 1−en−1(x)·e^−x,

where e^−x =

n−1

P

j=0 x^j

j!. For anyk >0, we can use a continued fraction expansion ([1, formula 6.5.31])

(3.8) Γ (a, x) = e^−xx^a 1

x+·1−a 1+ · 1

x+·2−a 1+

2 x+ 1· · ·

,

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where Γ (a, x) = Γ (a)−P(a, x)·Γ (a), x >0, |a|<∞.

Property 3. If X₁ has the pdf given by (3.1), then the variable X₁ is Gamma distributed.

Proof. Let us write

(3.9) F(u) = Prob{X₁^k≤u}= θ Γ (1 + 1/k)

Z u^1/k

0

e^−θ^k^x^kdx which by taking the derivative provides

(3.10) dF(u)

du =f(u) = θ

kΓ (1 + 1/k) ·u¹^k⁻¹exp(−θ^ku),

that is, a gamma pdf (fork= 1 one has the exponential pdf – as is well-known).

As a consequence, we may state

Property 4. If in (3.1) the shape parameter is known, then the MLE of 1/θ^k is Gamma distributed, unbiased and of minimum variance.

Proof. Let x₁, x₁, . . . , x_n be an independent sample on X₁. The log- likelihood function associated with the pdf of X1 is

(3.11) lnL=nlnθ−nln Γ (1 + 1/k)−θ^k

n

X

1

x^k_i,

hence

(3.12) ∂lnL

∂θˆ = n

θˆ−k·θˆ^k−1·

n

X

1

x^k_i = 0, which gives

(3.13) ˆ1

θ^k

!

MLE

= k n·

n

X

1

x^k_i.

By the stability property of the Gamma distribution, the statistic (3.13) is Gamma distributed, too. Unbiasedness and minimum variance features result from the already known properties of this distribution (see Johnson et al. [13, p. 338–343]).

4. SOME OTHER PDF GENERATORS

Formula (2.1) can easily provide new pdf (s) if for the generating variable X there are explicit expressions of its reliability function as well as of its theoretical mean-value.

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Example 1. Consider X with the cdf

(4.1) X : F(x;θ) = 1−(1 +θx) exp (−θx), x≥0, θ >0, which is in fact obtained from a Gamma pdf

(4.2) f(x;θ, k) = θ^k

Γ (k)·x^k−1exp (−θx), x≥0, k >0,

by taking k = 2. Since R(x;θ) = (1+θx) e^−θx and E(X) = 2/θ, the new pdf is

(4.3) X₁ : f₁(x;θ) = 1

2 θ+θ²x

e^−θx, x≥0, θ >0.

Remark. One has to observe that f₁ is a mixture of two pdf(s): an exponential and a peculiar Gamma: f1 = p1g1+p2g2 where p1 =p2 = 1/2, g₁ =θe^−θx and g₂ =θ²xe^−θx.

Example 2. Let X be the generalized exponential variable proposed by Dob´o [7] with the cdf

(4.4) X : F(x;λ, θ,a) = 1−

λ+θ λ+θe^ax

1/aθ

, x≥0, a >0, θ, λ≥0.

If λis very small, thenE(X)≈θ, hence X₁ has the density (4.5) X1 : f1(x;λ, θ, a)≈ 1

θ ·

λ+θ λ+θe^ax

1/aθ

, x≥0, a >0, θ, λ≥0.

We note that forλ= 0 we obtain the exponential pdf.

Example 3. LetX be the power variable with cdf (4.6) X : F(x;δ,b) =

x b

δ

, 0≤x≤b, δ >0.

SinceE(X) =δb/(δ+ 1), the pdf of X₁ is (4.7) X1 : f1(x;δ,b) = δ+ 1

δb

1−x b

δ

, 0≤x≤b, δ >0.

5. FINAL COMMENTS

If the cdf of the initial variableXand its mean-value cannot be expressed explicitly, some difficulties will arise. A suggestive example is for instance the case of Hjorth’s reliability function (see [12]) defined as

(5.1) X : R(x;θ, δ) = (1 +x)^−θexp −δx²/2

, x≥0, θ, δ >0, (if θ= 0 we recover the Rayleigh reliability function).

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Unfortunately, the moments of X are difficult to evaluate. Indeed, if we denote

(5.2) I(u, v) =

Z ∞

0

e^−ut²^/2 (1+t)^v dt, after some algebra we get

(5.3) I(u, v+ 1) = 1 +I(u, v)−u·I(u, v−1)

which allows (by numerical methods) the computation ofE(X), which in fact is I(δ, θ).

REFERENCES

[1] M. Abramowitz and A. Irene Stegun, Handbook of Mathematical Functions. Moscow, Nauka, 1979. (Russian translation of 1964 NBS original, Washington, D.C.)

[2] R.E. Barlow and F. Proschan, Mathematical Theory of Reliability.Wiley, New York, 1965.

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

Wiley, New York, 2000.

[4] D.R. Cox,Renewal Theory.Wiley, New York, 1963.

[5] R. D’Addario,Un metodo per la rappresentazione analitica delle distribuzioni statistiche.

Annali dell’Istituto di Statistica dell’Universit`a di BariXVI(1939), 36–45.

[6] R. D’Addario, Sulle repartizioni la cui media superiormente o inferiormente “in- completa” cresce linearmente col crescere della variabile distributiva. Giornale degli Economisti ed Annali di Economia (Roma)11-12(1969), 20–28.

[7] A. Dob´o,Reliability of ageing components. Quality and Reliability (Budapest), Special Edition, pp. 53–56, 1976.

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

Economic˜a, Bucure¸sti, 1994. (Romanian)

[9] G. Guerrieri,Sopra un nuovo metodo concernente la determinazione dei parametri della distribuzione lognormale e delle distribuzioni pearsoniane del III e del V tipo. Annali dell’Istituto di Statistica dell’Universit`a di BariXXXIV(1969/1970), 55–110.

[10] E.J. Gumbel,Statistics of Extremes.Columbia Univ. Press, New York, 1958.

[11] G. Hahn and S.S. Shapiro, Statistical Models in Engineering, 1st Corrected Printing.

Wiley, New York, 1968.

[12] U. Hjorth, A reliability distribution with increasing, decreasing, constant and bath-tub failure rates.Technometrics22(1980),1, 99–112.

[13] N.L. Johnson, S. Kotz and N. Balakrishnan, Distributions in Statistics. Vol. 1, 2nd Edition. Wiley, New York, 1994.

[14] M.S.H. Khan, A generalized exponential distribution. Biometrical J. 29 (1987), 1, 121–127.

[15] R.N. Rodriguez, Systems of frequency curves. In S. Kotz and N.L. Johnson (Eds.), Encyclopedia of Statistical Sciences, Vol.3, pp. 212–225. Wiley, New York, 1983.

[16] M.A. Savageau,A suprasystem of probability distributions.Biometrical J.24(1982),4, 323–330.

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

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[18] V.Gh. Vod˘a,A unified generalization of Burr-Hatke and S-system of distributions.Re- search Report no. 4/2007, “Gh. Mihoc-C. Iacob” Institute of Math. Statistics and Ap- plied Mathematics of the Romanian Academy, Bucharest, 2007.

[19] E.O. Voit, The S-distribution: a tool for approximation and classification of univariat, unimodal probability distributions.Biometrical J.34(1992),6, 855–878.

[20] Sh. Yu and E.O. Voit, A simple, flexible failure model. Biometrical J. 37 (1995), 5, 595–609.

Received 11 October 2007 Romanian Academy

“Gheorghe Mihoc–Caius Iacob”

Institute of Mathematical Statistics and Applied Mathematics Casa Academiei Romˆane Calea 13 Septembrie nr. 13 050711 Bucharest, Romania

von voda@yahoo.com