Discriminating among Weibull, log-normal and log-logistic distributions

Transcription

1 Discriminating among Weibull log-normal and log-logistic distributions M. Z. Raqab ab S. Al-Awadhi a Debasis Kundu c a Department of Statistics and OR Kuwait University Safat 36 Kuwait b Department of Mathematics The University of Jordan Amman 942 Jordan c Department of Mathematics Indian Institute of Technology Kanpur Kanpur Pin 286 INDIA Abstract In this paper we consider the problem of the model selection/ discrimination among three different positively skewed lifetime distributions. All these three distributions namely; the Weibull log-normal and log-logistic have been used quite effectively to analyze positively skewed lifetime data. In this paper we have used three different methods to discriminate among these three distributions. We have used the maximized likelihood method to choose the correct model and computed the asymptotic probability of correct selection. We have further obtained the Fisher information matrices of these three different distributions and compare them for complete and censored observations. These measures can be used to discriminate among these three distributions. We have also proposed to use the Kolmogorov-Smirnov distance to choose the correct model. Extensive simulations have been performed to compare the performances of the three different methods. It is observed that each method performs better than the other two for some distributions and for certain range of parameters. Further the loss of information due to censoring are compared for these three distributions. The analysis of a real data set has been performed for illustrative purposes. Keywords: Likelihood ratio method; Fisher information matrix; probability of correct selection; percentiles; model selection; Kolmogorov-Smirnov distance. AMS Subject Classification 2): 62E5; 62F; 62E25. Introduction Among several right skewed distributions the Weibull WE) log-normal LN) and log-logistic LL) distributions have been used quite effectively in analyzing positively skewed lifetime data. These three distributions have several interesting distributional properties and their probability density functions also can take different Corresponding author addresses: mraqab@ju.edu.jo M. Raqab) alawadhidodo@yahoo.com S. Al-Awadhi) kundu@iitk.ac.in D. Kundu).

2 shapes. For example the WE distribution can have a decreasing or an unimodal probability density function PDF) and a decreasing a constant and an increasing hazard function depending on the shape parameter. Similarly the PDF of a LN density function is always unimodal and it has an inverted bathtub shaped hazard function. Moreover the LL distribution has either a reversed J shaped or an unimodal PDF and the hazard function of the LL distribution is either a decreasing or an inverted bathtub shaped. For further details about the distributional behaviors of these distributions one may refer to Johnson et al. 995). Let us consider the following problem. Suppose {x...x n } is a random sample of size n from some unknown lifetime distribution function F ) i.e. F ) = and the preliminary data analysis suggests that it is coming from a positively skewed distribution. Hence any one of the above three distributions can be used to analyze this data set. In this paper we would like to explore among these WE LN and LL distributions which one fits the data best. It can be observed that for certain ranges of the parameters the corresponding PDFs or the cumulative distribution functionscdfs) are very close to each other but can be quite different with respect to other characteristics. Before explaining this with an example let us introduce the following notations. The WE distribution with the shape parameter α > and scale parameter λ > will be denoted by WEαλ). The corresponding PDF and CDF for x > are f WE x;αλ) = αλ α x α e λx)α and F WE x;αλ) = e λx)α respectively. The LN distribution is denoted by LNσ β) with the shape parameter σ > and scale parameter β >. The PDF and CDF of this distribution for x > can be written as f LN x;σβ) = e 2σ2ln x ln β)2 2π σ x and F LN x;σβ) = Φ ln x ln β ) = σ Erfln x ln β 2 σ ) respectivelywhereφ.)isthecdfofastandardnormaldistributionwitherfx) = 2Φ 2 x). The PDF and CDF of the LL distribution denoted by LLξ) with the shape parameter > and scale parameter ξ > for x > are f LL x;ξ) = x respectively. ln x ln ξ) e and F ln x ln ξ) LL x;ξ) = +e ) 2 ln x ln ξ) +e In Figure we have plotted the CDFs of WE4.8.56) LN.27.6) and LL.6.6). It is clear from Figure that all the CDFs are very close to each other. Therefore if the data are coming from any one of these three distributions the other two distributions can easily be used to analyze this data set. Although these three CDFs are quite close to each other the hazard functions of the above three distribution functions see Figure 2 are completely different. Moreover the 2

3 Fx) x Figure : The CDFs of WE4.8.56) LN.27.6) and LL.6.6). 5-th percentile points of WE4.8.56) LN.27.6) and LL.6.6) are and.53 respectively. Clearly the 5-th percentile points of these three distributions are also significantly different. On the other hand if we choose the correct model based on the maximized likelihood ratio the details will be explained later) and the data are obtained from a LL distribution then the probability of correct selection PCS) for different sample sizes are presented in Table. It can be seen that the PCS is as small as only.33 when the sample size is 2. Therefore it is clear that choosing the correct model is a challenging problem particularly when the sample size is small. 6 5 WE4.8.56) 4 hx) 3 2 LN.27.6) LL.6.6) x Figure 2: The hazard functions of WE4.8.56) LN.27.6) and LL.6.6). It is clear that choosing the correct model is an important problem as the effect of mis-classification can be quite severe as we have seen in case of the hazard 3

4 Table : PCS based on Monte Carlo simulations using ratio of maximized likelihood Sample size PCS function or for the percentile points. This issue would be more crucial when the sample sizes are small or even moderate. Therefore the discrimination problem between different distributions received a considerable attention in the last few decades. Cox 962) first addressed the problem of discriminating between the LN and the exponential distributions based on the likelihood function and derived the asymptotic distribution of the likelihood ratio test statistic. Since then extensive work has been done in discriminating among different distributions. Some of the recent work regarding discriminating between different lifetime distributions can be found in Alshunnar et al.2) Pakyari 22) Elsherpieny et al. 23) Sultan and Al-Moisheer 23) Ahmad et al. 26) and the references cited therein. Although extensive work has been done in discriminating between two distributions not much work has been done when more than two distributions are present except the work of Marshall et al. 2) and Dey and Kundu 29). Moreover most of the work till today are based on the likelihood ratio test. The aim of this paper is two fold. First of all we derive the Fisher information matrices of these three distributions and obtain different Fisher information measures both for the complete and censored samples. We also provide the loss of information due to truncation for these three different distributions. It is observed that the Fisher information measure can be used in discriminating purposes. Our second aim of this paper is to compare three different methods namely i) the method based on the Fisher information measures ii) the method based on the likelihood ratio and iii) the method based on the Kolmogorov-Smirnov distance in discriminating among these three different distributions. We perform extensive simulation experiments to compare different methods for different sample sizes and for different parameter values. It is observed that the performance of each method depends on the true underlying distribution and the set of parameter values. Rest of the paper is organized as follows. In Section 2 we derive the Fisher information measures for complete sample for all the three cases and show that how they can be used for discrimination purposes. In Section 3 we provide the discrimination procedure based on likelihood ratio statistics and derive their asymptotic properties. Monte Carlo simulation results are presented in Section 4. In Section 5 we provide the Fisher information measures for censored samples and the loss of information due to truncation. The analysis of a data set is presented in Section 6 and finally we conclude the paper in Section 7. 4

5 2 FI measure for complete sample Let X > be a continuous random variable with PDF and CDF as fx;θ) and Fx;θ) respectively where θ = θ θ 2 ) is a vector parameter. Under the standard regularity conditions see Lehmann99) the FI matrix for the parameter vector θ is θ lnfx;θ) Iθ) = E θ 2 lnfx;θ) [ θ lnfx;θ) θ 2 lnfx;θ)]. In this section we present the FI measures for the WE LN and LL distributions based on a complete data. The FI matrices of WE and LN see for example Alshunnar et al. 2) and Ahmad et al. 26)) can be described as follows: where ) fw f I W αλ) = 2W f 2W f 22W ) and I N σ 2 fn f β) = 2N f 2N f 22N f W = α 2 ψ )+ψ 2 2) ) f 2W = f 2W = λ +ψ)) f 22W = α2 λ 2 and f N = 2 σ 2 f 2N = f 2N = f 22N = β 2 σ 2. Here ψx) = Γ x)/γx) and ψ x) are the psior digamma) and tri-gamma functions respectively with Γ.) being the complete gamma function. The FI matrix for LL distribution based on a complete data in terms of the parameters and ξ is presented in Theorem given below. The proof can be seen easily via differentiation techniques and straight-forward algebra. Theorem : The FI matrix for the LL distribution is ) fl f I L ξ) = 2L f 2L f 22L where f L = 3 2 ) + π2 f 22L = ξ 2 and f 2L = f 2L =. Proof: See in the Appendix. In the arguments similar to the E-optimality of the design of experiment problems we consider the trace of the FI matrix as a measure of the total information in the data about the parameters involved in a specific model. For example the 5

6 trace of the FI matrix of the WE distribution is the sum of the information measure of α when λ is known and λ when α is known. The traces of the FI matrix of the LN and LL are defined similarly. In spite of the fact that the shape and scale parameters are essential tools in many distributional properties these parameters do not characterize the same prominent distributional features of the corresponding distributions. For comparison purposes of distributional characteristics of the three distributions we evaluate the asymptotic variances of the percentile estimators for these distributions. In our case the the p-th < p < ) percentiles of the WE LN and LL distributions are respectively P WE αλ) = ) p λ ln p))/α P LN βσ) = βe σ Φ p) P LL ξ) = ξ. p Therefore Var WE p) Var LN p)var LL p) the asymptotic variances of the logarithm of the p-th percentile estimators of the WE LN and LL distributions respectively can be written as follows: Var WE p) = [ ln P WE α ln P WE λ ] [ f W f 2W f 2W f 22W ] [ ln PWE ] α ln P WE ) λ and Var LN p) = Var LL p) = [ ln PLN σ [ ln PLL ln P LN β ln P LL ξ ] [ f N f 2N f 2N f 22N ] ln P L f L 2L f f 2L f 22L ] [ ] ln PLN σ ln P LN 2) β ln P L ξ. 3) The asymptotic variances for the median or the 95-th percentile estimators of the three distributions can be used for comparison purposes. Using the average asymptotic variance with respect to probability measure W.) proposed by Gupta and Kundu 26) we compare the following measures: and AV WE = Var WE p) dwp) AV LN = Var LN p) dwp) AV LL = Var LL p) dwp) where W.) is a weighted function such that Wp) dp =. For more convenience one may consider the average asymptotic variances of all percentile estimators that is Wp) = < p <. 6

7 To conduct a comparative study of the total information measure between any two distributions we have to compute these measures at their closest values. One way to define the closeness distance) between two distributions is to use the Kullback-Leibler KL) distance see for example White 982 Theorem ) Kundu and Manglick 24)). For notational convenience let θ and θ 2 be the miss-classified parameters of F distribution for given δ and δ 2 of F 2 distribution so that F θ θ 2 ) is closest to F 2 δ δ 2 ) in terms of the Kullback-Leibler distance. Lemma below provides the estimates of the parameters where any two distributions among WE LN and LL are closest to each other. Further the maximum likelihood estimates MLEs) of θ and θ 2 are denoted by ˆθ and ˆθ 2. Lemma [Kundu and Manglick 24)]: i) If the the underlying distribution is WEα λ) then the closest LN distribution in terms of KL distance is LN σ β) where σ = ψ ) α and β = λ eψ) α. 4) ii) If LNσβ) is the valid distribution then the closest WE distribution in terms of KL distance is WE α λ) such that α = σ and λ = β e σ 2. 5) Lemma 2 [Dey and Kundu 29)]: i) Under the assumption that the data are coming from LNσβ) and for n we have ˆ a.s. and ˆξ ξ a.s. where E LN ln f LL X; ξ) ) = maxe LN ln f LL X;ξ)). ξ ii)under the assumption that the data are from LL ξ) we have the following results as n ˆσ σ a.s. and ˆβ β a.s. where E LL ln f LN X; σ β) ) = max E LLln f LN X;σβ)). σβ In fact Dey and Kundu 29) have shown that for σ = β = E LN ln f LL X;ξ) is maximized when =.578 and ξ = while when = ξ = the maximization of E LL ln f LN X;σβ) occurs at σ = 3 and β =. In general if the data are coming from LNσβ) then and ξ can be obtained by maximizing { lnx lnξ E LN ln f LL X;ξ) = E LN ln + lnβ lnξ = ln + 2E N [ln 2ln ) +e } lnx lnξ + β ξ ) e σ X )] where E N.) stands for the expectation under standard normal distribution. They do not have explicit forms and they need to be obtained numerically. 7

8 To obtain σ and β such that LLξ) is closest to LN σ β) we have to maximize E LL ln f LN X;σβ) = ln2π) 2 ln σ E LL ln X) 2σ 2 E LLln X ln β) 2. It is easily seen that E LL lnx) = lnξ. By applying Taylor s series ln z) = j= zj /j it readily follows lnz ln z) = j= π2 = 2 jj +) 2 6. Summing up the most simplified function to be maximized is E LL ln f LN X;σβ) = ln2π) ln σ ln ξ } {lnξ lnβ) 2 + π σ 2 3 It can be easily verified that β = ξ and σ = π/ 3 maximize E LL ln f LN X;σβ). Similarly as in Lemma and Lemma 2 we have the following Lemma related to the WE and LL distributions. Lemma 3: i) Suppose the data come from WEαλ) then as n ˆ a.s. and ˆξ ξ a.s. where E WE ln f LL X; ξ) ) = maxe WE [ln f LL X;ξ)]. ξ ii) If the data come from LLξ) then as n we have ˆα α a.s. and ˆλ λ a.s. where ) E LL ln f WE X; α λ) = max E LL[ln f WE X;αλ)]. αλ Proof: The proof is followed along the lines of Dey and Kundu 29). i) To find and ξ let us define gξ) = E WE ln f LL X;ξ)) = ln ) E WE ln X) ln ξ 2E WE ln + X ξ ) ). The second term in 6) is evaluated to be E WE ln X) = ln λ + ψ) α where C = ψ) = istheEuler sconstant whilethelasttermcanberewritten as E WE ln + X ξ ) ) = ln + 8 y α λξ ) e y dy. 6)

9 Consequently Eq. 6) can take the following simplified form gξ) = ln 2 ln + ) lnλ+ ψ) α y α λξ ) lnξ ) e y dy. 7) For ξitcanbeobtaineddirectlybydifferentiating7)withrespecttoξ andequating theresultingexpressionto. Then ξ canbefoundbysolvingthefollowingequation numerically + ) yα λξ y α λξ ) e y dy = 2. 8) Upon differentiating 7) with respect to using 8) and equating the resulting expression to we compute as a numerical solution to 2 lny α + ) yα λξ y α λξ ) e y dy ψ) α = 9) ii) For given and ξ we maximize E LL [ln f WE X;αλ)] with respect to α and λ. Let hαλ) = E LL [ln f WE X;αλ)] = ln α+αln λ+α )E LL lnx) λ α E LL X α ). By using the facts E LL lnx) = ln ξ and E LL X α ) = ξ α B α+α) α < we have hαλ) = ln α+αlnλ+α )ln ξ λ α ξ α B α+α) where Bab) = wa w) b dw = Γ[a]Γ[b]/Γ[a+b] is beta function. It can be checked that B αα +) = B αα +)[ψ+α) ψ α)]. α Arguments similar to those in i) we find α and λ for which LLξ) is closest to WE α λ) by solving the following normal equations: λ = ξ[b α+α )] /α and ψ+α ) ψ α ) = α. ) 9

10 Table 2: The TI and TV of FI matrices of LN σ β) LL ξ) and WEα) for different α α σ β ξ TIWE TI LN TI LL TV WE TV LN TV LL Here we consider the trace of the information matrix to discriminate between any two distributions. In this case the information content is the sum of the FI measure of the shape parameter assuming the scale parameter to be known and the FI measure of the scale parameter assuming the shape parameter to be known. Another related measure is the the trace of the inverse of the FI information matrix which is also known in the statistical literature as the sum of the asymptotic variances of the MLEs of the shape and scale parameters. To conduct a comparison between the total information TI) and total variance TV) measures between any two distributions it is appropriate to do so at the points where both distributions are closest to each other. The TI and TV for all the three distributions when the data come from WE LN and LL are reported in Tables 2 3 and 4 respectively. It is clearly observed from these tables that the corresponding FI contents are quite different even the distribution functions are closest in terms of the Kullback-Leibler distance. It is obvious that the parametric values are essential in estimating the FI quantities computed in these tables. We will point out in Section 4 how this affects on choosing the correct model. Now we would like to use the Fisher information for discriminating among these three distributions. Let {x...x n } be a random sample from one of these three distribution functions and the observed total information measures for the WELNandLLbedenotedbyTI WE ˆαˆλ)TI LN ˆσ ˆβ)andTI LL ˆ ˆξ)respectively. Consider the following statistics: FD = TI WE ˆαˆλ) TI LN ˆσ ˆβ) FD 2 = TI WE ˆαˆλ) TI LL ˆ ˆξ) FD 3 = TI LN ˆσ ˆβ) TI LL ˆ ˆξ). ) Based on ) we choose WE if FD > and FD 2 > LN if FD < and

11 Table 3: The TI and TV of FI matrices of WE α λ) LL ξ) and LNσ) for different σ σ 2 α λ ξ TILN TI WE TI LL TV LN TV WE TV LL Table 4: The TI and TV of FI matrices of WE α λ) LN σ β) and LL) for different α λ σ β TILL TI WE TI LN TV LL TV WE TV LN

12 FD 3 > LL if FD 2 < and FD 3 < as the preferred distribution. The respective PCSs are defined as follows; PCS FI WE = PFD > FD 2 > data follow WE) PCS FI LN = PFD 2 < FD 3 > data follow LN) PCS FI LL = PFD 2 < FD 3 < data follow LL). 3 Ratio of Maximized Likelihood Method In this section we describe the discrimination procedure based on the ratio of maximized likelihood method. It is assumed that the sample {x...x n } has been obtainedfromoneofthesethreedistributions; namelyweαλ) LNσβ)orLLξ) and the corresponding likelihood functions are L WE αλ) = L LL ξ) = n f WE x i ;αλ) L LN σβ) = i= n f LL x i ;ξ) i= n f LN x i ;σβ) and i= respectively. Let us consider the following test statistics ) ) L WE ˆαˆλ) L WE ˆαˆλ) L LN ˆσ Q = ln L LN ˆσ ˆβ) Q 2 = ln L LL ˆ ˆξ) Q 3 = ln ˆβ) ) L LL ˆ ˆξ) The statistics Q Q 2 and Q 3 can be written as follows: [ Q = n ln 2π ˆα ˆσˆλ)ˆα) ] 2 [ ) Q 2 = n ln ˆα ˆˆλ)ˆα ˆξ) ˆ ]+ˆα ˆ n ) lnx i +2 [ ) ] ˆˆξ) ˆ Q 3 = n ln ˆ 2π ˆσ 2 i= n lnx i +2 i= n ln i= n i= ln + X ) i ˆξ ) ˆ + X i ˆξ ) ˆ Now we choose WE if Q > Q 2 > LN if Q < Q 3 > and LL if Q 2 < Q 3 <. Hence the respective PCS can be defined as follows: PCS LR WE = PQ > Q 2 > data follow WE). PCS LR LN = PQ < Q 3 > data follow LN) PCS LR LL = PQ 2 < Q 3 < data follow LL). Now we have the following results. ). 2

13 Theorem 2: i) Under the assumptions that the data are from WEαλ) Q Q 2 ) is asymptotically bivariate normally distributed with the mean vector E WE Q )E WE Q 2 )) and the dispersion matrix [ ] VarWE Q Σ WE = ) Cov WE Q Q 2 ) Cov WE Q Q 2 ) Var WE Q 2 ) ii) Under the assumptions that the data are from LNσβ) Q Q 3 ) is asymptotically bivariate normally distributed with the mean vector E WE Q )E WE Q 3 )) and the dispersion matrix [ ] VarLN Q Σ LN = ) Cov LN Q Q 3 ) Cov WE Q Q 3 ) Var WE Q 3 ) iii) Under the assumptions that the data are from LLξ) Q 2 Q 3 ) is asymptotically bivariate normally distributed with the mean vector E LL Q 2 )E LL Q 3 )) and the dispersion matrix [ ] VarLL Q Σ LL = 2 ) Cov LL Q 2 Q 3 ) Cov LL Q 2 Q 3 ) Var LL Q 3 ) Proof: The proof can be obtained along the same line as the Theorem of Dey and Kundu 29) the details are avoided. 4 Numerical Comparisons In this section we performed some simulation experiments to compare different methods to discriminate among these three distributions. We have used the method based on Fisher information FI) the ratio of maximized likelihood RML) method and the method based on Kolmogorov-Smirnov MK) distance. The method based ontheksdistanceksd)canbedescribedasfollows. Basedontheobservedsample we compute the KS distances between the i) empirical distribution function EDF) and WEˆαˆλ) ii) EDF and LNˆσ ˆβ) iii) EDF and LLˆ ˆξ). Which ever gives the minimum we choose that particular distribution as the best fitted one. The PCS can be defined along the same manner. It can be proved theoretically that the PCS based on RML does not depend on the parameter values. In all the cases we have taken the scale parameters to be. To compare the performances of the different methods for different sample sizes and for different parameter values we have generated samples from the three different distributions and compute the PCS for each method based on replications. The results are reported in Table 5. Some of the points are quite clear from these experimental results. It is clear that as the sample size increases the PCS increases in each case. It indicates the consistency properties of each method. In case of FI the PCS decreases as the shape parameter increases in all the three cases. 3

14 Although no such pattern exists in case of KSD. It is further observed that the performance of the RML is very poor when the data are obtained from the LL distribution. It seems although we could not prove it theoretically that PCS remains constant in case of KSD when the data are obtained from the LL distribution. If the shape parameter is less than then the method based on FI out performs the RML method. Although nothing can be said when the shape parameter is greater than. 5 Fisher Information Matrix: Censored Sample In engineering economics actuarial studies and medical research censoring is a condition in which an observation is only partially known. With censoring observations result is either lying within an interval or taking other exact values outside the interval. For example suppose a study is performed to study the effect of a treatment on heart stroke. In such a study we consider patients in a clinical trial to study the effect of treatments on stroke occurrence during time period T T 2 ). Those patients who had no strokes during T T 2 ) are censored. Right truncation occurs when the study patients have already stroke experienced after a fixed time say T 2 ). Left truncation occurs when the subjects have been at risk before entering the study. Let X be a right and left censored at fixed times T and T 2 respectively. Therefore we observe the random variable Y as follows: X if T <X<T 2 Y= T ifx < T T 2 if X > T 2. The FI about θ in the observation Y can be written as I C θ;t T 2 ) = I M θ;t T 2 )+I R θ;t 2 )+I L θ;t ) 2) where I M θ;t T 2 ) = I R θ;t 2 ) = I L θ;t ) = [ ] a a 2 a 2 a 22 FT 2 θ) FT θ) [ θ FT 2 θ) θ 2 FT 2 θ) [ θ FT θ) θ 2 FT θ) ] [ ] θ FT 2 θ) θ 2 FT 2 θ) ] [ ] θ FT θ) θ 2 FT θ) FT 2 θ) = FT 2 θ) and a ij = T2 T ) ) lnfx;θ) lnfx;θ) fx;θ)dx θ i θ j 4

15 Table 5: The PCSs under WELN and LL distributions for different sample sizes and for different parameter values. WEα) n = 2 n = 3 n = 5 α RML FI K-S RML FI K-S RML FI K-S LNσ) n = 2 n = 3 n = 5 σ 2 RML FI K-S RML FI K-S RML FI K-S LL ) n = 2 n = 3 n = 5 RML FI K-S RML FI K-S RML FI K-S

16 for ij = 2. Therefore the FI for complete sample or for fixed right censored at time T 2 ) sample or for fixed left censored at time T ) sample with vector of parameters θ can be obtained as I M θ; )I M θ;t 2 )+I R θ;t 2 ) and I M θ;t )+I L θ;t ) respectively by observing the fact that I R θ; ) = and I L θ;) =. In this section first we provide the FI matrices for the WE LN and LL distributions. Theorem 3: Let p = F WE T )p 2 = F WE T 2 ) or p = F LN T )p 2 = F LN T 2 ) or p = F LL T )p 2 = F LL T 2 ). Then the FI matrices of the WE LN and LL distributions for censored sample are obtained respectively to be as follows: i) where ) aw a I MW αλt T 2 ) = 2W I a 2W a RW αλt 2 ) = 22W ) cw c I LW αλt ) = 2W c 2W c 22W a W = ln p2 ) +lnu ulnu) 2 e u du a α 2 22W = α2 ln p ) a 2W = a 2W = λ ln p2 ) ln p ) λ 2 bw b 2W b 2W b 22W ln p2 ) ln p ) ) u) 2 e u du +lnu ulnu) u) e u du b W = p 2) ln 2 p 2 ) ln 2 [ ln p 2 )] α 2 b 22W = α2 p 2 )ln 2 p 2 ) λ 2 b 2W = b 2W = p 2)ln 2 p 2 ) ln[ ln p 2 )] λ c W = p ) 2 ln 2 p ) ln 2 [ ln p )] c α 2 22W = α2 p ) 2 ln 2 p ) p λ 2 p c 2W = c 2W = p ) 2 ln 2 p ) ln[ ln p )] λ p. ii) ) an a I MN αλt T 2 ) = 2N I a 2N a RN αλt 2 ) = 22N bn b 2N b 2N b 22N ) ) cn c I LN αλt ) = 2N c 2N c 22N 6

17 where a N = Φ p 2 ) y 2 ) 2 φy)dy a σ 2 22N = β 2 σ 2 Φ p ) a 2N = a 2L = βσ 2 b N = b 2N = b 2L = c N = Φ p 2 ) Φ p ) y 2 )yφy)dy [ φ Φ p p 2 )σ 2 2 ) )] 2 Φ p 2 ) ) 2 b22l = [ φ Φ p β p 2 )σ 2 2 ) )] 2 Φ p 2 ) ) p σ 2 [ φ Φ p ) )] 2 Φ p ) ) 2 c22l = c 2N = c 2N = βp σ 2 [ φ Φ p ) )] 2 Φ p ) ). Φ p 2 ) Φ p ) y 2 φy)dy β 2 p 2 )σ 2 [ φ Φ p 2 ) )] 2 β 2 p σ 2 [ φ Φ p ) )] 2 iii) ) al a I ML αλt T 2 ) = 2L I a 2L a RL αλt 2 ) = 22L bl b 2L b 2L b 22L ) ) cl c I LL αλt ) = 2L c 2L c 22L a L = 2 p2 / p 2 ) a 2L = a 2L = 2 ξ b L = c L = [ +lnu 2 ulnu +u p / p ) p2 / p 2 ) p 2 )p 2 2 ln 2 p p ) 2 ln 2 p / p ) p 2 p 2 ] 2 +u) du a 2 22L = 2 ξ 2 ] u +u) du 3 [ +lnu 2 ulnu +u 2 b 22L = p2 2 p 2 ) 2 ξ 2 b 2L = b 2L = ) p p 2 c 22L = p p ) 2 2 ξ 2 c 2L = c 2L = p2 / p 2 ) p / p ) p 2 2 p 2 ) ln 2 2 ξ p p ) 2 ln 2 It may be observed that as p and p 2 then a ijw f ijw a ijn f ijn and a ijl f ijl. To compare between the FI and its respective variance of the three distributions we consider three different censoring schemes; Scheme : p = p 2 =.75; Scheme 2: p =.p 2 =.85 and Scheme 3: p =.25p 2 =. These schemes represent right censoring interval censoring and left censoring respectively. The results are displayed in Table 6 under the assumption that the parent distribution is WEα) with α = It is also observed from Table 6 that the FI for the right and left tails for LN are exactly same in this case since p L = p 2R. It can be easily observed from the expressions 7 2 ξ u) 2 +u) 4 du ) p 2 p 2 ) p p.

18 Table 6: Loss of FI of the shape and scale parameters for Weibull distribution Scheme Parameter Scheme Scheme 2 Scheme 3 Shape 46% 78% 38% Scale 25% 78% 9% of the Fisher information matrix that if p L = p 2R where p L = F LN T ) and p 2R = F LN T 2 ) corresponding to left and right censoring cases respectively then both left and right censored data for LN distribution have the same FI about both the parameters. The same is true in case of LL distribution also. It seems it is due to the fact both normal ln LN) and logistic ln LL) distributions are symmetric distributions although we could not prove theoretically. In case of WE distribution the FI gets higher values towards the right tail than the left tail. Now one important question is which of the two parameters has more impact. For this we would like to discuss the loss of information due to truncation in one parameter when the other parameter is known. Suppose the WE distribution is the underlying distribution with fixed truncation points. If the scale shape) parameter is known the loss of information of the shape scale) parameter for WE distribution is Loss WE α) = a W +b W +c W and = f ψ )+ψ 2 2) [ ln p2 ) +lnu ulnu) 2 e u du+ ln p ) p 2 )ln 2 p 2 )ln 2 ln p 2 ))+ p ) 2 ln 2 p )ln 2 ln p )) p Loss WE λ) = a 22W +b 22W +c 22W f 22 [ ] ln p2 ) = u) 2 e u du+ p 2 )ln 2 p 2 )+ p ) 2 ln 2 p ). ln p ) Clearly both losses are free of any parameter and depend only on the truncation parameters. The loss of FI of the shape and scale parameters due to Schemes 2 and 3 are presented in Table 6. It is easily seen that the information due to the interval censoring are similar in both cases while the information due to the right and left censoring are different. It is of interest to see also that the last portion of the data contains higher information of the shape/or scale parameter for the WE distribution. Now for LN distribution if the scale shape) parameter is known the loss of 8 ]

19 Table 7: Loss of FI of the shape and scale parameters for the log-normal distribution Scheme Parameter Scheme Scheme 2 Scheme 3 Shape 3% 8% 5% Scale 6% 72% 3% information of the shape scale) parameter is [ Loss LN σ) = Φ p 2 ) u 2 ) 2 φu) du+ [φφ p 2 ))] 2 Φ p 2 )) Φ p ) p 2 ] [φφ p ))] 2 Φ p )) 2 p and [ ] Φ p 2 ) Loss LN β) = u 2 φu) du+ [φφ p 2 ))] 2 + [φφ p ))] 2. p 2 Φ p ) p For LN distribution the losses of FI of the shape and scale parameters under different schemes are presented in Table 7. It is clear that the maximum information of the shape and scale parameters of LN distribution is occurred in the initial portion of the data. are Proceeding similarly the losses of FI of the shape and scale parameters of LL Loss LL ) = 3+ π2 3 ) [ p 2 p 2 p p ) 2 ln 2 ) p [+lnu 2ulnu p p ] p p 2 +u ]2 + p 2 )p 2 2ln 2 ) + p 2 and Loss LL ξ) = [ p 2 p 2 p p ] u)2 +u) 4 +p2 2 p 2 )+p p ) 2 respectively. The loss of information for the shape and scale parameters for different censoring schemes are presented in Table 8. It is clearly observed that for both the LN and LL the initial portion of the data has more information than the right tail where as for the WE distribution it is the other way. 9

20 Table 8: Loss of FI of the shape and scale parameters for the log-logistic distribution Scheme Parameter Scheme Scheme 2 Scheme 3 Shape 26% 74% 5% Scale 2% 57% 4% 6 Data analysis Here we discuss the analysis of real life data representing the fatigue liferounded to the nearest thousand cycles) for 67 specimens of Alloy T7987 that failed before having accumulated 3 thousand cycles of testing. Their recorded values in hundreds) are First we provide some preliminary data analysis results. The mean median standard deviation and the coefficient of skewness are respectively. The histogram of the above data set is presented in Figure 3 From Figure 3: Histogram of the fatigue data. the preliminary data analysis it is clear that a skewed distribution may be used to analyze this data set. Barreto-Souza et al. 2) fitted the Weibull-geometric extended exponential geometric and WE models to this real data set. Graphically Meeker and Escobar998 p. 49) showed that the LN distribution provides a much better fit than WE distribution. Before progressing further we provide the plot of 2

21 Table 9: The traces and variances of the FI matrices of LN σ β) LL ξ) and W Eα ) for three different censoring schemes α σ β ξ TIWE TI LN TI LL TV WE TV LN TV LL

22 the scaled total time on test TTT) transform. It is well known that the scaled TTT transform plot provides an indication about the shape of the hazard function. For example if the plot is a concave function then it indicates that the hazard function is an increasing function or if the plot is first concave and then convex it indicates that the hazard function is an upside down function etc. We provide the plot of the scaled TTT transform in Figure 4. Although it is not very clear it has an indication that at the beginning it is concave and then it is convex. It indicates that the hazard function is an upside down function Figure 4: Scaled TTT transform plot. Now we fit all the three distributions WE LN and LL to this data set. The MLEs of the models parameters are computed numerically using Newton-Raphson NR) method. The MLEs the Kolmogorov-Smirnov K-S) distances between the fitted and the empirical distribution functions and the corresponding p-values between parentheses) are presented in Table. The empirical and fitted distribution functions are presented in Figure 5. The CDFs of LN and LL are very close to each other and the CDF of WE is quite different than the other two..8 WE LN LL Figure 5: Empirical and fitted distribution functions. 22

23 Table : MLEs K-S statistics and the corresponding p-values of the data set WE model α λ K S ) LN model σ β K S ) LL model ξ K S ) The maximized log-likelihoodmll) values for WE LN and WE distributions are respectively and Therefore based on MLL and K-S statistics it is observed that both LN and LL fit the data equally well although LN provides a slightly better fit than LL. Now let us compute the estimated FI matrices for the WE LN and LL models and they are as follows: ) ) I WE ˆαˆλ) = I LN ˆσ ˆβ) = and I LL ˆ ˆξ) = ) Now if we consider the total asymptotic variances of the three cases they are as follows: TV WE = TV LN =.2267 TV LL =.28. Hence based on all these we conclude that LN is the most preferred among these three distributions for this particular data set. Next let us assess the variance of the p-th percentile estimators of the three distributions for various choices of p. Figure 3 shows the asymptotic variance of the p-th percentile estimators for complete and censored samples. For complete sample it is clear that the WE distribution has higher variance than LN and LL distributions for p <.7. It is also evident from Figure Figure 6 that asymptotic variances of the p-th percentile estimators for censored samples for LN and LL are different while their respective curves tend to be identical for complete sample case. This concludes that both distributions LN and LL can be discriminated easily when 23

24 the censoring data set is available. By taking a censoring observation on [T T 2 ] with p =.5 and p 2 =.9 the FI matrices for WE LN and LL distributions are computed to be I WE ˆαˆλ) = ) I LN ˆβˆθ) = ) and I LL ˆβˆθ) = ) respectively. Based on the FI matrices for complete and censored data sets it is observed the loss of information due to truncation for LN distribution is much more than WE and LL distributions with respect to both parameters while the loss of information of the shape parameter due truncation for WE and LL is much more than that of the scale parameters for the same distributions. Variance.8.6 WE LN LL Variance WE LN LL p p Figure 6: The variances of the pth percentile estimators for WE LN and LL distributions for complete left) and censored right) samples. 7 Conclusions In this article we have considered the problem of discrimination among the WE LN and LL distributions using three different methods. The asymptotic variance of the percentile estimators is also compared for these three distributions. It is 24

25 observed that although the three distributions can be chosen as appropriate fitting models for a specific data set the total information of Fisher information matrix as well as the asymptotic variance of the percentile estimators can be quite different. An extensive simulation experiment has been carried out to compute the PCS by different methods and it is observed that the method based on the Fisher information measure competes with the other existing methods well especially for certain ranges of parameter values. Acknowledgement The authors would like to thank the unknown reviewers for their constructive comments which have helped to improve the manuscript significantly. Appendix Proof of Theorem : Taking the natural logarithm for the PDF of the LL distribution we get lnf LL x;ξ) ln + x ln ξ) ln x ln ξ) 2 ln+eln ). 3) Differentiating both sides of 3) with respect to and ξ respectively we have lnf LL x;ξ) = [+ ln x ln ξ) 2 e ln x ln ξ) ln x ln ξ) +e ln x ln ξ) ] and lnf LL x;ξ) ξ = ln x ln ξ) ξ 2 e +e ln x ln ξ) ). Then f L = [+ 2 ln x ln ξ) ln x ln ξ) 2 ln x ln ξ) e ln x ln ξ) +e ] 2 f LL x;ξ) dx f 22L = 2 ξ 2 [2 ln x ln ξ) e ln x ln ξ) +e ] 2 f LL x;ξ) dx. By simple transformation techniques we may readily obtain f L = [+ln z ) 2 ln z z)] 2 dz and f 2 22L = z z 3 2 ξ 2. 25

26 A more simplified expression of f L can be obtained by using the second derivative approach as follows: f L = E 2 lnfx;ξ) 2 = E ln x ln ξ E[ln x ln ξ ) 2 e ln x ln ξ +e )+ 4 2 E [ ln x ln ξ ln x ln ξ ln x ln ξ e ln x ln ξ +e ) 2 ]. 4) Since the integrand function involved in the second term of the right hand side of 4) is odd function its integral becomes. By using the substitution arguments we con readily obtain the following identities: and E E [ ln x ln ξ [ This leads to ln x ln ξ f L = 2 { ln x ln ξ e ln x ln ξ +e ] ) 2 e ln x ln ξ +e = ln x ln ξ ) 2 ] [ln z) ln z] z) dz = 2 = = [ln z) ln z] 2 z z) dz ] ln z ln z) z z) dz. } ln z ln z) z z) dz. 5) By using the Taylor s series expansion of ln z) = j= zj /j and the identity see Gradshteyn and Ryzhik 994) p.548) we have ln z)z j+ z) dz = = j= k= j= j [ ) ) k k j +k +2) 2 j +2) 2 j +3) 2 ]. The series appeared on the right hand side of the above identity can be computed easily by a straightforward algebra of partial fractions decomposition telescoping series arguments and using the Euler-Riemann zeta function ζ2) = i= /i2 = π 2 /6. This gives ln z)z j+ z) dz = 74 6π

27 and consequently we simplify 5) as follows: f L = ) + π Similarly by differentiating 3) with respect to ξ and respectively and then we have f 2L = f 2L = 2 ξ + 2 [+y +e y e 2y ] dy 2 ξ +e2y and this in turn f 2L = f 2L = by straightforward integration techniques. References [] Ahmad M. A. Raqab M. Z. and Kundu D. 27). Discriminating between the generalized Rayleigh and Weibull distributions: Some comparative studies Communications in Statistics-Simulation & Computation In Press Online Link: [2] Alshunnar F.S. Raqab M.Z. and Kundu D. 2). On the comparison of the Fisher information of the log-normal and Weibull distributions Journal of Applied Statistics [3] Barreto-SouzaW. de Morais A. L. and Cordeiro G.M. 2). The Weibullgeometric distribution J ournal of Statistical Computation and Simulation 85) [4] Cox D. R. 962). Further results on tests of separate families of hypotheses Journal of the Royal Statistical Society Series B Vol. 24 pp [5] Dey A. K. and Kundu D. 29). Discriminating between the log-normal and log-logistic distributions Communications in Statistics-Theory & Methods 392) [6] Elsherpieny E. A. Ibrahim S. A. Radwan N. U. 23). Discrimination between Weibull and log-logistic distributions International Journal of Innovative Research in Science Engineering and Technology 28) [7] Gradshteyn I.S. and Ryzhik I.M. 994) In: A. Jeffrey Ed) Table of Integrals Series and Products 5th edn Academic Press San Diego USA. [8] Gupta R.D. and Kundu D 26). On the comparison of Fisher information of the Weibull and GE distributions Journal of Statistical Planning & Inference [9] Johnson N.L. Kotz S. and Balakrishnan N. 995). Continuous Univariate Distribution Vol. 2nd Ed. New York Wiley. 27

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