Estimation of the Stress-Strength Reliability for the Dagum Distribution

Transcription

1 156 Journal of Advanced Statistics, Vol 1, No 3, September 2016 Estimation of the Stress-Strength Reliability for the Dagum Distribution Ber Al-Zahrani Samerah Basloom Department of Statistics, King Abdulaziz University, POBox 80203, Jeddah 21589, Saudi Arabia Abstract The paper deals with the problem of estimating the reliability of a single multicomponent stress-strength based on a three-parameter Dagum distribution The maximum likelihood estimators the Bayes estimators of R = P Y < X when X Y are two independent rom variables follow Dagum distribution their asymptotic distributions are obtained Also, the reliability of a multicomponent stress-strength model is estimated using the maximum likelihood Consequently, the asymptotic confidence intervals of the reliability of a single multicomponent stress-strength model are constructed Gibbs Metropolis sampling were used to provide sample-based estimates of the reliability its associated credible intervals Finally, Monte Carlo simulations are carried out for illustrative purposes Keywords: Dagum distribution, maximum likelihood estimator, Bayes estimator, stress-strength models 1 Introduction Dagum distribution was introduced by Dagum 1 for modeling personal income data as an alternative to the Pareto log-normal models This distribution has been extensively used in various fields such as, income wealth data, meterological data, reliability survival analysis The Dagum distribution is also known as the inverse Burr XII distribution, especially in the actuarial literature An important characteristic of Dagum distribution is that its hazard function can be monotonically decreasing, an upside-down bathtub, or bathtub then upside-down bathtub shaped, for details see Domma 2 This behavior of the distribution has led several authors to study the model in different fields In fact, recently, the Dagum distribution has been studied from a reliability point of view used to analyze survival data see Domma et al, 3, Domma Condino 4 Kleiber Kotz 5 Kleiber 6 provided an exhaustive review on the origin of the Dagum model its applications Domma et al 3 estimated the parameters of Dagum distribution with censored samples Shahzad Asghar 7 used TL-moments to estimate the parameter of this distribution Oluyede Ye 8 presented the class of weighted Dagum related distributions Domma Condino 4 proposed the five parameter beta-dagum distribution A continuous rom variable T is said to have a three-parameter Dagum distribution, abbreviated as T Dagumα, β, δ, if its density probability function pdf is given as f t; α, β, δ = βαδt δ αt δ β 1, t > 0, 11 where α > 0 is the scale parameter its two shape parameters β δ are both positive The corresponding distribution function is given by F t; α, β, δ = 1 + αt δ β, t > 0, β, α, δ > 0 12 In this paper we focus our attention on estimating the reliability of a single component stress-strength, R = P Y < X where X Y are two independent rom variables following Dagum distribution with different shape parameter β have the same scale parameter α shape parameter δ Here R is the probability that the strength X of a component during a given period of time exceeds the stress Y We also focus on estimating the reliability of multicomponent stress-strength R s,k, where s k are two intgers s k The problem of estimating R R s,k has attracted the attention of many authors, a

2 Journal of Advanced Statistics, Vol 1, No 3, September good overview on estimating R can be found in the monograph of Kotz et al 9 See also, Pey Uddin 10 Srinivasa Kantam 11 for details on estimation of R s,k It is worth mentioning that the problem here seems to be clearly an application of well known mathematical methods However, there were some challenges in finding the estimatiors The rest of the paper is organized as follows In Section 2, we obtain the maximum likelihood estimatior MLE of R the asymptotic distribution of ˆR The MLE Bayes estimators of R when α is known is considered Also, simulation studies have been presented for illustrative purposes In section 3, the MLEs are obtained employed to get the asymptotic distribution confidence intervals for R s,k Finally, the conclusion comments are provided in Section 4 2 Estimation of R with Different β Common δ α In this section, we investigate the properties of R when the shape parameters β are different, the scale parameter α the shape parameter δ are constants The other cases, ie the parameters α δ are not constants the general case where all parameters are different, can be studied in a similar way to the case presented in this paper 21 Maximum Likelihood Estimator of R Let X Dagumβ 1, α, δ Y Dagumβ 2, α, δ, where X Y are two independent rom variables All three parameters, β, α δ are unknown to us Then it can be shown that R = P Y < X = 0<y<x fx, ydxdy = β 1 β 1 + β 2 21 To compute the MLE of R, suppose that X 1, X 2,, X n is a rom sample from Dagumβ 1, α, δ Y 1, Y 2,, Y m is another rom sample from Dagumβ 2, α, δ Then the log-likelihood function of the observed sample is where ln Lβ 1, β 2, α, δ = n lnβ 1 + n + m lnα + n + m lnδ n m δ + 1 lnx i + lny j L δ = n+m δ β 1 + 1s 1 x, α, δ + m lnβ 2 β 2 + 1s 2 y, α, δ, s 1 x, α, δ = s 2 y, α, δ = n m ln1 + αx δ i, 22 ln1 + αy δ j 23 The MLEs of β, α δ say ˆβ, ˆα ˆδ, respectively, can be obtained as the solutions of the following equations L β 1 = n β 1 s 1 x, α, δ = 0 L β 2 = m β 2 s 2 y, α, δ = 0 L α = n+m α β n x δ i β 1+αx δ m y δ j = 0 1+αy δ i j n lnx i m lny j β n β m y δ lny j j = 0 1+αy δ j αx δ i lnx i 1+αx δ i

3 158 Journal of Advanced Statistics, Vol 1, No 3, September 2016 We obtain n ˆβ 1 = ŝ 1 x, α, δ 25 m ˆβ 2 = ŝ 2 y, α, δ 26 The estimators ˆα ˆδ can be obtained as the solutions of the nonlinear equations given in 24 The plug-in estimation of R, say ˆR, is readily computed as ˆR = 22 Asymptotic Distribution of ˆR ˆβ 1ML ˆβ 1MR + ˆβ 2ML 27 In this section, the asymptotic distribution of ˆθ = ˆβ 1, ˆβ 2, ˆα, ˆδ the asymptotic distribution of ˆR are obtained The Fisher information matrix of θ, denoted by Jθ = EI, θ, is giving below, where I = I i,j i,,2,3,4 is the observed information matrix ie, Iθ = α 2 β 1 α β 2 α δ α the elements of Iθ are as follows I 11 = n + m α 2 + β n α β 1 β1 2 β 2 β 1 δ β 1 α β 2 β 1 β 2 β2 2 δ β 2 α δ β 1 δ β 1 δ δ 2 x δ i αx δ i + β I 11 I 12 I 13 I 14 = I 21 I 22 I 23 I 24 I 31 I 32 I 33 I 34, I 41 I 42 I 43 I 44 m y δ j αy δ j n X δ m i I 12 = I 21 = 1 + αx δ i, I y δ j 13 = I 31 = 1 + αy δ j n x δ i ln x i n x δ 2 I 14 = I 41 = β αx δ i i α ln xi 1 + αx δ i 2 m y δ + β j ln y j m y δ αy δ j j α ln yj 1 + αy δ j 2 I 22 = n β1 2, I 23 = I 32 = 0, I 24 = I 42 = I 33 = m β2 2 I 34 = I 43 = n + m δ 2 + m n αx δ i ln x i 1 + αx δ i αy δ j ln y j 1 + αy δ j β 1 + 1nαx δ lnx2 1 + αx δ β 2 + 1mαy δ lny αy δ + β 2 + 1mα 2 y δ 2 lny αy δ 2 I 44 = n + m n αx δ i lnx i 2 n α 2 β αx δ i α 2 x δ i 2 lnx i αx δ i 2 m αy δ + β j lny j 2 m 1 + αy δ j α 2 y δ j 2 lny j αy δ j 2 + β 1 + 1nα 2 x δ 2 lnx αx δ 2

4 Journal of Advanced Statistics, Vol 1, No 3, September The following integrals can be helpful when finding the elements of the Fisher information matrix Note that ψ is the digamma function, defined as ψx = d/dx lnγ x lnt1 + t β 1 dt = ψβ + γ, β 0 t ln 2 t1 + t β 3 ψ 2 β + γ 2 + π dt = ψβ 6 β + 1β + 2 t 2 lnt1 + t β 3 2ψβ 2γ + 3 dt = ββ 2 + 3β + 2, t lnt1 + t β 2 dt = Ψβ + γ 1, αβ βγ ψβ γβ 1 + 1, ββ + 1β + 2 t 2 ln 2 t1 + t β 3 dt = 6ψ2 β + 12ψβγ + 6γ 2 + π 2 18ψβ 18γ + 6ψ1, β + 6 β3β 2 + 9β + 6 The elements of the Fisher information matrix are obtained by taking the expectations of the observed matrix Doing so will result in the following: J 11 = n + m α 2 nβ 1β α 2 B3, β 1 mβ 2β α 2 B3, β 2 J 12 = nβ 1 α B2, β 1, J 13 = J 31 = mβ 2 α B2, β 2, where B, is the beta function J 14 = J 41 = nβ 1β B2, β 1 lnα + ψβ 1 + γ 1 δα αβ B3, β 1 lnα + 2ψβ 1 2γ + 3 β 1 β β mβ 2β B2, β 2 lnα + ψβ 2 + γ 1 δα B3, β 2 lnα + 2ψβ 2 2γ + 3 β 2 β β αβ Again, the functions B, ψ are the beta function the digamma function J 22 = n β1 2, J 23 = J 32 = 0, J 24 = J 42 = nβ 1 B2, β 1 lnα + ψβ 1 + γ 1 δ β 1 β J 33 = m β2 2, J 34 = J 43 = mβ 2 δ B2, β 2 lnα + ψβ 2 + γ 1 β 1 β the element J 44 is defined as follows J 44 = n + m α 2 + nβ 1β δ 2 B3, β 1 ln 2 α + 2 lnα ψβ 1 + γ 1 β 1 β ψβ ψβ 1 γ + 6γ 2 + π 2 12ψβ 1 12γ + 6ψ1, β 1 6β 1 β ln 2 α 2B3, β ψβ 1 2γ + 3 α β 1 β β ψ2 β ψβ 1 γ + 6γ 2 + π 2 18ψβ 1 18γ + 6ψ1, β β 1 3β β mβ 2β δ 2 B3, β 2 ln 2 α + 2 lnα ψβ 2 + γ 1 β 2 β ψβ ψβ 2 γ + 6γ 2 + π 2 12ψβ 2 12γ + 6ψ1, β 2 6 β 2 β 2 + 1

5 160 Journal of Advanced Statistics, Vol 1, No 3, September ln 2 α 2B3, β ψβ 2 2γ + 3 α β 2 β β ψ2 β ψβ 2 γ + 6γ 2 + π 2 18ψβ 2 18γ + 6ψ1, β β 2 3β β It is worth mentioning that the Dagum family of distributions satisfies all the regularity conditions, for example see Nadarajah Kotz 12 Now, it turns out that we can formulate the limiting joint distribution of estimators Theorem 1 As n m n m p, where p is positive real constant, then m ˆβ1 β 1, n ˆβ 2 β 2, mˆα α, mˆδ δ N 4 0, A 1 β 1, β 2, α, δ, where the covariance matrix A is given as a 11 a 12 a 13 a 14 Aβ 1, β 2, α, δ = a 21 a 22 a 23 a 24 a 31 a 32 a 33 a 34 a 41 a 42 a 43 a 44 Proof We first give the elements of the covariance matrix A a 11 = J 11 lim n,m m = p + 1 α 2 pβ 1β α 2 B3, β 1 β 2β α 2 B3, β 2 a 12 = a 21 = a 13 = a 31 = lim n,m lim n,m J 12 p pβ1 = B2, β 1 n α J 13 m = β 2 α B2, β 2 J 14 a 14 = a 41 = lim n,m m p pβ1 β = B2, β 1 lnα + ψβ 1 + γ 1 δα αβ B3, β 1 lnα + 2ψβ 1 2γ + 3 β 1 β β β 2β B2, β 2 lnα + ψβ 2 + γ 1 pαδ αβ B3, β 2 lnα + 2ψβ 2 2γ + 3 β 2 β β a 22 = J 22 lim n,m n = 1 β1 2 a 34 = a 43 = a 24 = a 42 = lim n,m, a 23 = a 32 = 0 J 34 m p = β 2 δ p J 24 lim n,m n = B 1 δ B2, β 2 lnα + ψβ 2 + γ 1 β 2 β B2, β 1 lnα + ψβ 1 + γ 1 β 1 β a 44 = J 44 lim n,m n = 1 α 2 p + β 1β δ 2 B3, β 1 ln 2 α + 2 lnα ψβ 1 + γ 1 β 1 β 1 + 1

6 Journal of Advanced Statistics, Vol 1, No 3, September ψβ ψβ 1 γ + 6γ 2 + π 2 12ψβ 1 12γ + 6ψ1, β 1 + 6β 1 β ln 2 α 2B3, β ψβ 1 2γ + 3 α β 1 β β ψ2 β ψβ 1 γ + 6γ 2 + π 2 18ψβ 1 18γ + 6ψ1, β β 1 3β β β2 β pδ 2 B3, β 2 ln 2 α + 2 lnα ψβ 2 + γ 1 β 2 β ψ2 β ψβ 2 γ + 6γ 2 + π 2 12ψβ 2 12γ + 6ψ1, β 2 6β 2 β ln 2 2ψβ2 2γ + 3 α 2B3, β α β 2 β β ψ2 β ψβ 2 γ + 6γ 2 + π 2 18ψβ 2 18γ + 6ψ1, β β 2 3β β The proof follows immediately by invoking the asymptotic properties of MLEs the multivariate central limit theorem One of the main results here is Theorem 2, concerning the asymptotic properties of the parameter R Theorem 2 As n, m, n m p, where p is a positive number, then n ˆR R N0, BA, where B A = 1 β U A β 1 + β β 2 a22 a 33 a 44 a 22 a 2 43 a 33 a β 1 a 21 a 33 a 44 a 21 a 43 + a 31 a 42 a 43 a 33 a 41 a 42 β 1 β 2 a21 a 33 a 44 a 21 a a 31 a 2 42a 43 a 33 a 41 a 42 β 1 a11 a 33 a 44 a 11 a 2 43 a 2 31a a 31 a 41 a 43 a 33 a 2 41 U A = a 11 a 22 a 33 a 44 a 11 a 22 a 2 43 a 11 a 33 a 2 24 a 2 12a 33 a 44 + a 2 12a a 12 a 13 a 24 a 34 2a 12 a 33 a 14 a 24 + a 22 a 2 13a a 22 a 13 a 14 a 34 a 22 a 33 a a 2 13a 2 24 Proof Applying the delta method see Casella Roger, 13 using Theorem 1, we can write the asymptotic distribution of ˆR where ˆR = g ˆβ 1, ˆβ 2, ˆα, ˆδ gβ 1, β 2, α, δ = β 1 /β 1 + β 2 as the following: n ˆR R N0, BA, where b A = R β 1 R β 2 R α R δ B A = b t AA 1 b A, = 1 β 1 + β 2 2 β 2 β 1 0 0,

7 162 Journal of Advanced Statistics, Vol 1, No 3, September 2016 A 1 = 1 U A v 11 v 12 v 13 v 14 v 21 v 22 v 23 v 24 v 31 v 32 v 33 v 34 v 41 v 42 v 43 v 44 The elements of the A 1 are obtained as follows v 11 = a 22 a 33 a 44 a 22 a 2 43 a 33 a 2 42 v 12 = v 21 = a 21 a 33 a 44 a 21 a a 31 a 42 a 43 a 33 a 41 a 12 v 13 = v 31 = a 21 a 42 a 43 + a 22 a 31 a 44 a 22 a 41 a 43 a 31 a 2 42 v 14 = v 41 = a 21 a 33 a 42 + a 22 a 31 a 43 a 22 a 33 a 41 v 22 = a 11 a 33 a 44 a 11 a 2 43 a 2 31a a 31 a 41 a 43 a 33 a 2 41 v 23 = v 32 = a 11 a 42 a 43 + a 21 a 31 a 44 a 21 a 41 a 43 a 31 a 41 a 42 v 24 = v 42 = a 11 a 33 a 42 + a 21 a 31 a 43 a 21 a 33 a 41 a 2 31a 42 v 33 = a 11 a 22 a 44 a 11 a 2 42 a 2 21a a 21 a 41 a 42 a 22 a 2 41 v 34 = v 43 = a 11 a 22 a 43 a 2 21a 43 + a 21 a 31 a 42 a 22 a 31 a 41 v 44 = a 11 a 22 a 33 a 2 21a 33 a 22 a 2 31 Therefore, B A = b t AA 1 b A = 1 β U A β 1 + β β 2 a 22 a 33 a 44 a 22 a 2 43 a 33 a β 1 a 21 a 33 a 44 a 21 a a 31 a 42 a 43 a 33 a 41 a 42 β 1 β 2 a 21 a 33 a 44 a 21 a a 31 a 42 a 43 a 33 a 41 a 42 β 1 a 11 a 33 a 44 a 11 a 2 43 a 2 31a a 31 a 41 a 43 a 33 a 2 41 The proof is now completed It should be noted that Theorem 2 can be used to construct the asymptotic confidence intervals for R However, the variance B A is not known it has to be estimated The estimator of B A, say ˆB A, is obtained by replacing β 1, β 2, α δ involved in B A by their corresponding MLEs The 1001 γ% confidence intervals for R are given by ˆBA ˆR Z 1 γ/2, ˆR ˆBA + Z 1 γ/2, 28 n n where Z γ is the γ% percentile of N0, 1 As mentioned in Asgharzadeh et al 14, instead of approximating ˆR R / var ˆR as stard normal variable, it is possible to consider some other normalizing transformation, say gr, of R assume that g ˆR gr / varg ˆR N0, 1, Now we use delta method to approximate the variance of gr, see for example Held Bove 15 It is written as follows var g ˆR = g R 2 var ˆR = g R 2 B A /n

8 Journal of Advanced Statistics, Vol 1, No 3, September As before, the 1001 γ% confidence intervals for gr are given as g ˆR ± Z 1 γ/2 var g ˆR If the function gr is strictly increasing then approximate 1001 γ% confidence intervals for R are derived to be g g 1 ˆR Z 1 γ/2 var ˆR, g 1 g ˆR + Z 1 γ/2 var ˆR Specifically, we follow Asgharzadeh et al 14 consider the following two transformations R 1 Logit transformation: gr = ln 1 R with g 1 R = R1 R 2 Arcsine transformation: gr = sin 1 R with g 1 R = 2 R1 R More details about these transformations one can refer to Lawless 16 or Mukherjee Maiti Bootstrap Confidence Intervals For small sample sizes, confidence intervals carried out based on the asymptotic results are usually expected not to perform well Therefore, we propose to use confidence intervals based on two parametric bootstrap methods These methods are: i the percentile bootstrap method, shortened as Boot-p, based on the idea of Hall 18 ii the bootstrap-t method, shortened as Boot-t, based on the idea of Hall 18 The algorithms for estimating the confidence intervals of R using both methods are summarized below i Boot-p method 1 Use the samples {x 1,, x n } {y 1,, y m } equations 24, to compute the estimates ˆβ 1ML, ˆβ 2ML, ˆα ML ˆδ ML 2 Use the estimates ˆβ 1ML, ˆα ML ˆδ ML to generate a bootstrap sample {x 1,, x n } similarly use ˆβ 2ML, ˆα ML ˆδ ML to generate a bootstrap sample {y 1,, y n } Based on {x 1,, x n } {y 1,, y n } compute the bootstrap estimate of R, say R, using ˆR = ˆβ 1ML ˆβ 1ML + ˆβ 2ML 3 Repeat Step 2, NBOOT times 4 Let g 1 x = P ˆR x be the cdf of ˆR define ˆR Bp x = g1 1 x for a given x Then the approximate 1001 γ% confidence intervals for R are given by γ ˆRBp, 2 ˆR Bp 1 γ 2 ii Boot-t method 1 Use the samples {x 1,, x n } {y 1,, y m } equations 24, to compute ˆβ 1ML, ˆβ 2ML, ˆα ML ˆδ ML 2 Use ˆβ 1ML, ˆα ML ˆδ ML to generate a bootstrap sample {x 1,, x n } similarly use ˆβ 2ML, ˆα ML ˆδ ML to generate a bootstrap sample {y 1,, y n } Based on {x 1,, x n } {y 1,, y n } compute the bootstrap estimate of R, say R, using ˆR = statistic n T = ˆR ˆR var ˆR ˆβ 1ML ˆβ 1ML + ˆβ 2ML the 3 Repeat Step 2, NBOOT times 4 Let g 2 x = P T x be the cdf of T define ˆR Bt x = ˆR + g2 1 x var ˆR n for a given x Then the approximate 1001 γ% confidence intervals for R are given by γ ˆRBt, 2 ˆR Bt 1 γ 2

9 164 Journal of Advanced Statistics, Vol 1, No 3, September Bayes Estimation of R In this subsection, we discuss the estimation of the parameter R using Bayes method assuming that the shape parameters β 1, β 2, δ the scale parameter α are rom variables unknown to us It is assumed that β 1, β 2, δ α have density functions Gammaa 1, b 1, Gammaa 2, b 2, Gammaa 3, b 3 Gammaa 4, b 4 respectively Moreover, it is assumed that β 1, β 2, δ α are independent Based on the above assumptions, we obtain the likelihood function of the observed data as follows Ldata β 1, β 2, α, δ = β n 1 α n+m δ n+m n x δ+1 i s 1 x, α, δ m y δ+1 j s 2 y, α, δ, 29 where s 1 x, α, δ s 2 y, α, δ are defined as before in equations The joint density of data β 1, β 2, α δ can be obtained as Ldata, β 1, β 2, α, δ = Ldata; β 1, β 2, α, δ π 1 β 1 π 2 β 2 π 3 α π 4 δ, 210 where π 1 β 1, π 2 β 2, π 3 α π 4 δ are gamma prior densities for β 1, β 2, α δ respectively Therefore, the joint posterior density of β 1, β 2, α δ given the data is Ldata β 1, β 2, α, δ = 0 0 Ldata, β 1, β 2, α, δ 0 0 Ldata, β 211 1, β 2, α, δdδdαdβ 2 dβ 1 Equation 211 cannot be written in closed form, thus we apply the Gibbs sampling technique to compute the Bayes estimate of R along with the corresponding credible intervals The posterior pdfs of β 1, β 2, α δ can be obtained readily as follows β 1 β 2, δ, α, data Gamman + a 1, b 1 + s 1 x, α, δ, β 2 β 1, δ, α, data Gammam + a 2, b 2 + s 2 y, α, δ, fα β 1, β 2, δ, data α n+m+a3 1 exp α b 3 + n x i + n y j β 1 + 1s 1 x, α, δ β 2 + 1s 2 y, α, δ, fδ β 1, β 2, α, data δ n+m+a4 1 exp δb 4 β 1 + 1s 1 x, α, δ β 2 + 1s 2 y, α, δ Clearly, the above forms of the posterior density do not lead to explicit Bayes estimates of the model parameters For this reason, we prefer to use the Metropolis-Hasting method with normal proposal distribution The algorithm of Gibbs sampling is summarized below: 1 Start with an initial guess β 0 1, β0 2, α0, δ 0 2 Set t = 1 3 Generate β t 1 from Gamma n + a 1, b 1 + s 1 x, α t 1, δ t 1 4 Generate β t 2 from Gamma n + a 2, b 2 + s 2 y, α t 1, δ t 1 5 Use the Metropolis-Hasting method to generate α t from fα t 1 β 1, β 2, δ, data with the Nα t 1, 05 proposal distribution 6 Use the Metropolis-Hasting method to generate δ t from fδ t 1 β 1, β 2, α, data with the Nδ t 1, 05 proposal distribution 7 Compute R t from equation 21 8 Set t = t + 1

10 Journal of Advanced Statistics, Vol 1, No 3, September Repeat Steps 3-8, T times Now the approximate posterior mean variance of R, respectively, are calculated as ÊR data = 1 T T R t, 212 t=1 var ˆ R data = 1 T T t=1 R t ÊR data Using the result in Chen Shao 19, we construct the 1001 γ% highest posterior density HPD credible intervals as R γ T, R 2 1 γ 2 T, where R γ T R 2 1 γ 2 T are the γ 2 T -th smallest integer the 1 γ 2 T -th smallest integer of {R t, t = 1, 2,, T }, respectively It is worthy to point out that the Metropolis algorithm adopts only symmetric proposal distributions Therefore the normal distribution is appropriate It is checked here that the normal proposal distribution with variance σ 2 = 05 is best for the rapid convergence of the Metropolis algorithm 25 Numerical Simulations The comparisons between the the MLEs Bayes estimators of R cannot be done theoretically Thus, we present some simulations to compare the performance of the obtained results We compare the MLEs Bayes estimators in terms of their biases mean squared errors MSE We also compare different confidence intervals, namely; the confidence intervals obtained by using asymptotic distribution of the MLEs, bootstrap confidence intervals the HPD credible intervals in terms of the average confidence lengths The Bayes estimates are computed under the squared error loss function To assess the performance of the methods used in estimation, we use different parameter values, different hyper-parameters different sample sizes Further, we assume two priors to obtain the Bayes estimators HPD credible intervals These priors are typical given as: Prior 1: a j = 00001, b j = 00001, where j = 1, 2, 3, 4, Prior 2: a j = 1, b j = 3, where j = 1, 2, 3, 4 In Table 1, we present the average biases MSEs of the MLEs Bayes estimators based on 1000 replications The results are given under different sample sizes The average confidence/credible lengths the corresponding coverage percentages are given in Table 2 It should be noted that for the two bootstrap methods, we compute the confidence intervals based on 1000 bootstrap iterations The Bayes estimates the corresponding credible intervals are based on T = 1000 samples Also, the confidence level, γ used in finding the confidence intervals or the credible intervals is 095 It is observed that the MLE compares very well with the Bayes estimators in terms of biases MSEs, see Table 1 Also, it is observed that the Bayes estimators based on Prior 2 perform better than those obtained based on Prior 1 Table 2 shows that, the confidence intervals based on the asymptotic distributions of the MLEs work well when n m are getting larger, the Boot-p confidence intervals perform better than the Boot-t confidence intervals, the bootstrap method provide the smallest average lengths, the HPD credible intervals are wider than the other confidence intervals

11 166 Journal of Advanced Statistics, Vol 1, No 3, September 2016 Table 1 Biases MSEs of MLE Bayes estimators of R for different values of parameters BS BS n, m Method MLE prior1 prior2 MLE prior1 prior2 β 1 = 15, β 2 = 2, α = δ = 1 β 1 = 2, β 2 = 15, α = δ = 1 15,15 MSE Bias ,25 MSE Bias ,50 MSE Bias ,15 MSE Bias ,25 MSE Bias ,50 MSE Bias ,15 MSE Bias ,25 MSE Bias ,50 MSE Bias Table 2 Average confidence credible length the coverage percentage MLEs Boot BS n, m Untransformed Logit Arcsin boot-p boot-t prior1 prior2 β 1 = 15, β 2 = 2, α = δ = 1 15,15 Len CP ,25 Len CP ,50 Len CP ,15 Len CP ,25 Len CP ,50 Len CP ,15 Len CP ,25 Len CP ,50 Len CP β 1 = 2, β 2 = 15, α = δ = 1 15,15 Len CP ,25 Len CP ,50 Len CP ,15 Len CP ,25 Len CP ,50 Len CP ,15 Len CP ,25 Len CP ,50 Len CP

12 Journal of Advanced Statistics, Vol 1, No 3, September Multicomponent Stress-Strength Model In this section, we study the multicomponent stress-strength reliability for Dagum distribution when both stress strength variates follow the same population We also obtain the asymptotic confidence intervals for the multicomponent stress-strength reliability using MLE We run a simulation study to compare the estimators Let the rom samples Y, X 1, X 2,, X k be independent, Gy be the continuous distribution function of Y F x be the common continuous distribution function of X 1, X 2,, X k The reliability in a multicomponent stress-strength model developed by Bhattacharyya Johnson 20 is R s,k = P at least s of the x 1, x 2,, x k exceed Y k k = 1 Gy i Gy k i df y 31 i i=s 31 Maximum Likelihood Estimator of R s,k Let X Dagumβ 1, α, δ Y Dagumβ 2, α, δ, where X Y are two independent rom variables with unknown shape parameters β 1 β 2 common scale parameters δ shape parameter α The reliability in multicomponent stress-strength for Dagum distribution using 11 results in: k k R s,k = 1 αy δ β1 i 1 + αt δ β1 k i β 2 αδt δ αt δ β2 1 dt i i=s k k = v 1 t i t k i+v 1 dt i i=s k k = v Bk + v i, i i i=s where t = 1 + αy δ β1, v = β 2 /β 1 B, is the incomplete beta function After some simplification equation 32 is reduced to The MLE of R s,k becomes ˆR s,k = ˆv R s,k = v k i=s k i=s k! k i! k! k i! k k + v j j=i k k + ˆv j j=i , where ˆv = ˆβ 1 ˆβ 2 34 To obtain the asymptotic confidence intervals for R s,k, we proceed as follows: The asymptotic variances AV of the ˆβ 1 ˆβ 2 are given by AV ˆβ 1 = E β 1 1 = β2 1 n, AV ˆβ 2 = E β 2 1 = β2 2 m The asymptotic variance of an estimate of R s,k is given by For details see Rao 21: AV ˆR s,k = AV ˆβ 1 Rs,k β AV ˆβ 2 Rs,k β Thus from equation 35, the asymptotic variance of ˆR s,k can be obtained To avoid the difficulty of derivation of R s,k, we obtain ˆR s,k their derivatives for s, k = 1, 3 2, 4 separately They are

13 168 Journal of Advanced Statistics, Vol 1, No 3, September 2016 given as 3 ˆR 1,3 = ˆv + 3, ˆR2,4 = ˆR 1,3 = 1 3ˆv β 1 β 1 ˆv + 3 2, ˆR 2,4 β 1 = 1 β 1 12ˆv2ˆv + 7 ˆv + 4ˆv ˆv + 4ˆv + 3,, ˆR 1,3 = 1 3 β 2 β 1 ˆv + 3 2, ˆR 2,4 = 1 β 2 β 1 122ˆv + 7 ˆv + 4ˆv Therefore, AV ˆR 1,3 = 9ˆv2 1 ˆv n + 1, m AV ˆR 2,4 = 144ˆv2 2ˆv ˆv + 4ˆv n + 1 m As n m, we have R s,k ˆR s,k /AV ˆR s,k N0, 1 for s, k = 1, 2, The asymptotic 95% confidence interval CI of system reliability R s,k is ˆR s,k 196 AV ˆR s,k Thus, the asymptotic 95% confidence intervals for R 1,3 R 2,4 are, respectively, given by 32 Simulation Study 3ˆv 1 ˆR 1,3 196 ˆv n + 1 m, 12ˆv2ˆv ˆR 2,4 196 ˆv + 4ˆv n + 1 m We generate 3000 rom samples of size 10, 15, 20, 25, 30 each from stress strength populations for β 1, β 2 = 30, 15, 25, 15, 20, 15, 15, 15, 15, 20, 15, 25 15,30 as proposed by Bhattacharyya Johnson 20 The MLEs of β, α δ, say ˆβ, ˆα ˆδ are estimated by solutions of the nonlinear equation These ML estimators of β 1 β 2 are then substituted in v to get the multicomponent reliability for s, k = 1, 3, 2, 4 The average bias average MSE of the reliability estimates over the 3000 replications are given in Table 3 Average confidence length of the simulated 95% confidence intervals of, R s,k are given in Table 3 The true values of reliability in multicomponent stress-strength with the given combinations for s, k = 1, 3 are 0857, 0833, 0800, 0750, 0692, 0643, 0600 for s, k = 2, 4 are 0762, 0725, 0674, 0600, 0519, 0454, 0400 Thus the true value of reliability in multicomponent stress-strength decreases as β 2 increases for a fixed β 1 whereas reliability in multicomponent stress-strength increases as β 1 increases for a fixed β 2 in both cases of s, k Therefore, the true value of reliability decreases as v increases vice versa The average bias average MSE decrease as sample size increases for both situations of s, k Also, it is noted that the bias is negative relatively small in all the combinations of the parameters in both situations of s, k which leads MLE to underestimate the parameters, thus the R s,k This generally proves the consistency property of the MLE of R s,k Whereas, among the parameters the absolute bias MSE decrease as β 1 increases for a fixed β 2 in both cases of s, k The absolute bias MSE increase as β 2 increases for a fixed β 1 in both cases of s, k It is also clear that as the sample size increases the length of CI decreases in all cases

14 Journal of Advanced Statistics, Vol 1, No 3, September Table 3 In each cell the first row represents the average bias, the second row represents the average MSE the third row represents the average confidence length of the simulated 95% confidence intervals of R s,k using MLE β 1, β 2 s, k n, m 30,15 25,15 20,15 15,15 15,20 15,25 15,30 10, , ,3 20, , , , , ,4 20, , , Concluding Remarks In this paper, we have considered the problem of estimation of a single multicomponent stress-strength reliability for a three-parameter Dagum distribution when both stress strength variates follow the same distribution The maximum likelihood Bayes estimators of the stress-strength parameter, R have been derived We have compared the MLEs Bayes estimators in terms of their biases mean squared errors It may be noted, from Table 1 that the maximum likelihood estimates have the smallest mean squared errors as compared with their corresponding Bayes estimates We observed the Bayesian method is sensitive to the choice of the priors Generally the Bayes estimators based on Prior 2 perform better than the Bayes estimators based on Prior 1 For all sample sizes n, m, the bootstrap methods provide the smallest average lengths It is observed that boot-p confidence intervals perform better than those obtained by the boot-t methods Further, the coverage probability is quite close to the given value in all sets of parameters For the multicomponent stress-strength reliability, we have estimated the asymptotic confidence intervals The simulation result in Table 3 indicates that the averages of bias MSE decrease as sample size increases Also, the absolute bias MSE decrease increase as β 1 increases β 2 increases in both situations of s, k Acknowledgments The authors are grateful to the anonymous referees for their valuable comments suggestions which improved the presentation of the paper This study is a part of the Master Thesis of the second named author whose work was supervised by the first named author References 1 C Dagum, A New Model for Personal Income Distribution: Specification Estimation, Economic Applique,

15 170 Journal of Advanced Statistics, Vol 1, No 3, September F Domma, L amento della Hazard function nel modello di Dagum a tre parametri, Quaderni di Statistica, , F Domma S Giordano M M Zenga, Maximum likelihood estimation in Dagum distribution with censored sample, Journal of Applied Statistics, , F Domma F Condino, The beta-dagum distribution: definition properties, Communications in Statistics-Theory Methods, C Kleiber S Kotz, Statistical Size Distributions in Economics Actuarial Sciences John Wiley, Hoboken, New Jersey, C Kleiber, A guide to the Dagum distributions Springer, New York, M N Shahzad Z Asghar, Comparing TL-Moments, L-Moments Conventional Moments of Dagum Distribution by Simulated data, Revista Colombiana de Estadistica B O Oluyede Y Ye, Weighted Dagum related distributions, Afrika Matematika, S Kotz, Y Lumelskii, M Pensky, The Stress-Strength Model its Generalizations-Theory Applications World Scientific, New York, M Pey B Uddin, Estimation of reliability in multicomponent stress-strength model following Burr distribution Proceedings of the First Asian congress on Quality Reliability, pp New Delhi, India R G Srinivasa R R L Kantam, Estimation of reliability in multicomponent stress-strength model: log-logistic distribution, Electronic Journal of Applied Statistical Analysis, S Nadarajah S Kotz, Moments of some J-shaped distributions, Journal of Applied Statistics, G Casella B L Roger, Statistical inference Vol 2 Pacific Grove, CA: Duxbury, A Asgharzadeh, R Valiollahi M Z Raqab, Estimation of the stress-strength reliability for the generalized logistic distribution, Statistical Methodology L Held D S Bove, Applied Statistical Inference: Likelihood Bayes Springer Science & Business Media, J F Lawless, Statistical Models Methods for Lifetime Data, Second ed, John Wiley Sons, New York, S P Mukherjee S S Maiti, Stress-strength reliability in the Weibull case, Frontiers in Reliability P Hall, Theoretical comparison of bootstrap confidence intervals, Annals of Statistics M H Chen, Q M Shao, Monte Carlo estimation of Bayesian credible HPD intervals, Journal of Computational Graphical Statistics G K Bhattacharyya R A Johnson, Estimation of reliability in multicomponent stress-strength model A, C R Rao, Linear Statistical Inference its Applications, Wiley Eastern Limited, India 1973