The Kumaraswamy Inverse Weibull Poisson Distribution With Applications

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1 Indiana University of Pennsylvania Knowledge IUP Theses and Dissertations (All) The Kumaraswamy Inverse Weibull Poisson Distribution With Applications Walter T. Bera Indiana University of Pennsylvania Follow this and additional works at: Recommended Citation Bera, Walter T., "The Kumaraswamy Inverse Weibull Poisson Distribution With Applications" (2015). Theses and Dissertations (All) This Thesis is brought to you for free and open access by Knowledge IUP. It has been accepted for inclusion in Theses and Dissertations (All) by an authorized administrator of Knowledge IUP. For more information, please contact cclouser@iup.edu, sara.parme@iup.edu.

2 THE KUMARASWAMY INVERSE WEIBULL POISSON DISTRIBUTION WITH APPLICATIONS A Thesis Submitted to the School of Graduate Studies and Research in Partial Fulfillment of the Requirements for the Degree Master of Science Walter T. Bera Indiana University of Pennsylvania August 2015

3 Indiana University of Pennsylvania School of Graduate Studies and Research Department of Mathematics We hereby approve the thesis of Walter T. Bera Candidate for the degree of Master of Science Mavis Pararai, Ph.D. Associate Professor of Mathematics Christoph E. Maier, Ph.D. Associate Professor of Mathematics Russell Stocker, Ph.D. Assistant Professor of Mathematics ACCEPTED Randy L. Martin, Ph.D. Dean School of Graduate Studies and Research ii

4 Title: The Kumaraswamy Inverse Weibull Poisson Distribution with Applications Author: Walter T. Bera Thesis Chair: Dr. Mavis Pararai Thesis Committee Members: Dr. Russell Stocker Dr. Christoph E. Maier The aim of this thesis is to propose a new distribution called the Kumaraswamy inverse Weibull Poisson (KIWP) Distribution. The distribution properties including hazard functions, reverse hazard functions, survival functions, quantile functions, moments, distributions of order statistics, mean deviations, Lorenz and Bonferroni curves and Fischer information are presented. The maximum likelihood method is used to estimate the model parameters of this new distribution. The special cases of the KIWP distribution including the inverse Weibull Poisson (IWP), Kumaraswamy Frechet Poisson (KFP), Kumaraswamy inverse exponentiated Poisson (KIEP) and Kumaraswamy inverse Rayleigh Poisson (KIRP) distributions are presented. A Monte Carlo simulation study is presented to exhibit the performance and accuracy of the maximum likelihood estimates of the KIWP model parameters. Real data examples are used to show the usefulness of the proposed model. iii

5 ACKNOWLEDGMENTS I owe my deepest gratitude to my thesis adviser, Dr. Mavis Pararai for her support and guidance during the completion of my thesis. I would also like to thank my committee members Dr. Russell Stocker and Dr. Christoph Maier for their investment of time in providing guidance and direction. I am thankful to Dr. John Chrispell for assisting me with the formatting of my document and also to my colleague Gayan Warahena Liyanage who taught me a lot. Finally, I am grateful to my family for their unwavering support and encouragement. iv

6 TABLE OF CONTENTS Chapter Page 1 INTRODUCTION Introduction Thesis Outline THE KUMARASWAMY INVERSE WEIBULL POISSON DISTRIBUTION General Class of Distribution Expansion of the Density Function Monotonicity Properties Survival, Hazard and Reverse Hazard Functions Moments, Moment Generating Function and Conditional Moments Quantile Function Moments Moment Generating Function Conditional Moments Order Statistics Sub-Models of the KIWP distribution Mean and Median Deviations Bonferroni and Lorenz curves MEASURES OF UNCERTAINTY Rényi Entropy Shannon Entropy Reliability Maximum Likelihood Estimation Fisher Information Matrix Asymptotic Confidence Intervals Concluding Remarks MONTE CARLO SIMULATION STUDY Monte Carlo Simulation Study Concluding Remarks APPLICATIONS TO LIFETIME DATA Cancer Patients Data Set Guinea Pigs Data Set Yarn Specimen Data Set Glass Fibers Data Set Concluding Remarks v

7 Chapter Page 6 CONCLUSIONS AND SUGGESTIONS FOR FURTHER STUDY Suggestions for Further Study REFERENCES APPENDICES R Algorithms Code for Simulation Study vi

8 LIST OF TABLES Table Page 1 Moments of the KIWP Distribution for α = 1.5, β = 6.5 and θ = Moments of the KIWP Distribution for a = 1.0 and b = Monte Carlo Simulation Results: Average Bias, RMSE and AW Remission Times (in months) of Bladder Cancer Patients Estimates Of Models For Cancer Patients Data Survival Times (in days) of Guinea Pigs Infected with Virulent Tubercle Bacilli 62 7 Estimates Of Models For Guinea Pigs Data Number of Cycles of Failure for Yarn Specimens Estimates Of Models For Yarn Specimens Data Strengths of Glass Fibers measured by the National Physical laboratory Estimates Of Models For Glass Fibers Data vii

9 LIST OF FIGURES Figure Page 1 Plot of the cdf for different parameters Plot of KIWP density for different parameters Plot of the hazard function for different parameter values Fitted densities of cancer patients Data Probability plots of cancer patients data Fitted densities of guinea pigs data Probability plots of guinea pigs data Fitted densities of yarn specimens data Probability plots of yarn specimen data Fitted densities of glass fibers data Probability plots of glass fibers data viii

10 CHAPTER 1 INTRODUCTION Introduction Statistical lifetime distributions are employed broadly in data modeling. They are greatly utilized in areas such as reliability engineering, survival analysis, social sciences and a host of other applications. Of particular interest are applications of lifetime distributions in reliability engineering, including the measurement of survival time of electrical components. In medicine, an interesting application of lifetime distributions is in the calculation of survival times of patients post surgery. Khan et al. (2008) defined reliability as the probability that a system or process performs its prescribed duty without failure for a given time given it is operated correctly in a specified environment. In social sciences, an interesting application is modelling the lifetime of marriages, Almalki and Nadarajah (2014). One of the most important distributions used in modeling lifetime data is the Weibull distribution whose cumulative distribution function (cdf) and probability density function (pdf) are respectively given by F (x) = 1 exp( αx β ) and f(x) = αβx β 1 exp( αx β ), where x > 0, α > 0, and β > 0. Notice that α is a scale parameter and β is a shape parameter. Liu (1997) explained the use of the Weibull distribution as being an appropriate model of failure in instances where an item consists of numerous components and each component has an identical failure time distribution and the item fails when the weakest part fails. If X is a random variable following a Weibull Distribution with parameters α, β > 0, under the 1

11 transformation 1/X, the inverse Weibull distribution proposed by Keller and Kamath (1982) is obtained. The cdf and pdf of the inverse Weibull distribution are respectively given by F (x; α, β) = exp( αx β ), x > 0, α > 0, β > 0, (1.1) f(x; α, β) = αβx β 1 exp( αx β ). x > 0, α > 0, β > 0 (1.2) The hazard function of the inverse Weibull distribution is given by: h(x, α, β) = βα β x β 1 exp[ (αx) β ] 1 exp[ (αx) β ] The inverse Weibull distribution has proved to be very useful in the modeling of lifetime data. It has been compounded with other continuous distributions to produce new lifetime distributions. Pararai et al. (2014) proposed the gamma inverse Weibull distribution by compounding the inverse Weibull distribution density function with the gamma generator proposed by Ristić and Balakrishnan (2012). The gamma inverse Weibull distribution is a 3 parameter distribution with parameters λ, β and σ. Some of the submodels of the gamma inverse Weibull (GIW) include the inverse Weibull distribution, the gamma Frechet (GF) distribution and the inverse Rayleigh (IR) distribution. The hazard rate function assumes unimodal and upside down bathtub shapes. The cdf and pdf of the GIW distribution are respectively given by F GIW (x) = 1 γ( log(exp[ (αx) β ]), δ) Γ(δ) then, g GIW (x) = βx 1 Γ(δ) [ λx β ] δ exp [ λx β ]. 2

12 where λ > 0, β > 0, and δ > 0. The GIW distribution was applied to a data set from Bjerkedal et al. (1960) on the survival times in days of guinea pigs injected with different doses of tubercle bacilli. The GIW distribution was also applied to a data set from Lawless (1982) on the number of millions of revolutions before failure of each of 23 ball bearings in a life testing experiment. The GIW was found to be superior to all the models it was compared against. The Kumaraswamy inverse Weibull (KIW) distribution was proposed and studied by Shahbaz et al. (2012). Its cdf and pdf are respectively given by F KIW (x) = 1 [ 1 exp( aαx β ) ] b and f KIW (x) = abαβx β 1 exp( αx β ) [ exp( αx β ) ] a 1 [ 1 {exp( αx β )} a] b 1. The KIW distribution was applied to survival time data and salary data by Shahbaz et al. (2012). Cordeiro and de Castro (2011) proposed a generalized class of distribution called the Kum-G distribution which has a cdf of the following form F K G (x) = 1 [1 G(x) a ] b and the corresponding pdf of the generalized class of distribution is as follows f Kum G (x) = abg(x)g(x) a 1 [1 G(x) a ] b 1. Shahbaz et al. (2012) substituted in the cumulative distribution function of the inverse Weibull distribution into the probability density function of the Kum-G distribution to 3

13 obtain the Kumaraswamy inverse Weibull distribution. The beta inverse Weibull distribution was proposed and studied by Khan (2010). Hanook et al. (2013) further investigated the properties of the beta inverse Weibull distribution which is given by: G (x) = B F (x)(a, b) B(a, b) where B(a, b) = Γ(a + b) Γ(a)Γ(b) and F(x) is the baseline cdf. The cdf of the BIW distribution is given by F BIW = GIW 0 t a 1 (1 t)dt. Another interesting modification of the inverse Weibull distribution is the length biased inverse Weibull distribution proposed by Kersey and Oluyede (2013). Consider the weight function ω(x) = x and the inverse Weibull distribution with the pdf given in 1.2. The general form of the length biased inverse Weibull (LBIW) pdf is given by: g ω (x, α, β) = f(x, α, β) µ F where x 0, α > 0, β > 1 and µ F is the mean. The corresponding cdf for the LBIW distribution is given by: G LBIW = βα β+1 Γ(1 1/β) x 0 t β exp( (αx) β )dt. The hazard function of the LBIW is given by: h(x, α, β) = x β exp( (αx) β ) x t β exp( (αt) β ). 4

14 Khan et al. (2008) studied the inverse Weibull distribution and ascertained its flexibility in modelling reliability data. They further discovered that the inverse Weibull (IW) distribution approaches different distributions when its shape parameter changes. New families of distributions have been formulated by compounding the Poisson distribution with other continuous distributions to provide more flexibility for lifetime data. The zero truncated version of the Poisson distribution is defined as the zero-truncated binomial distribution for the Poisson distribution. Jones (2009) advocated for the Kumaraswamy distribution on the grounds that firstly, it had closed form solutions of the distribution and quantile functions; secondly, the simplicity of the formulas for the moments and thirdly, the fact that the distribution had a simple normalizing constant. Jones further contrasted the Kumaraswamy distribution with the beta distribution. The motivation of this study arises from the advantages observed in the Kumaraswamy inverse Weibull distribution in modelling lifetime data as well as the flexibility in the hazard function which has numerous shapes such as the bathtub shape, increasing and decreasing shape. We propose and study a new class of distribution which inherits these important and desirable properties and contains several submodels with a large number of shapes. Thesis Outline The outline of this thesis is as follows: In chapter two, we present some basic information related to generalized distributions and Kumaraswamy generalized distributions. Properties of the (KIWP) distribution are derived in Chapter 3, including expansion of density, hazard function, monotonicity property, shapes, moments, reliability, mean deviations, Bonferroni and Lorenz curves. Measures of uncertainty such as Renyi entropy and Shannon entropy as well as Fisher information are presented. The mle maximum likelihood is used to estimate the parameters of the (KIWP) and related distributions. Finally, real data examples are discussed to illustrate the applicability of this class of models. 5

15 CHAPTER 2 THE KUMARASWAMY INVERSE WEIBULL POISSON DISTRIBUTION General Class of Distribution The probability distribution function (pdf) and corresponding cumulative distribution function (cdf) of the two-parameter inverse Weibull distribution are respectively given by f(x; α, β) = αβx β 1 exp( αx β ), x > 0, α > 0, β > 0 (2.1) and F (x; α, β) = exp( αx β ), x > 0, α > 0, β > 0. (2.2) Suppose that the random variable X has the inverse Weibull (IW) distribution where its pdf and cdf are given in equations 2.1 and 2.2. Given N, let X 1,..., X N be independent and identically distributed random variables from the IW distribution. Let N be distributed according to the zero truncated Poisson distribution with pdf P (N = n) = θ n e θ, n = 1, 2,..., θ > 0. n!(1 e θ ) Let X=max(Y 1,..., Y N ), then the cdf of X N = n is given by G X N=n (x) = exp( nαx β ) x > 0, α > 0, β > 0, 6

16 which is the inverse Weibull distribution. The inverse Weibull-Poisson distribution denoted by IWP(α, β, θ) is defined by the marginal cdf of X, that is, G IW P (x; α, β, θ) = 1 exp[θe αx β ] (2.3) 1 e θ for x > 0, β > 0, θ > 0. The graph of the distribution function for various values of the parameters α, β, θ, a and b is given in Figure 1. Figure 1. Plot of the cdf for different parameters. The IWP density function is given by g IW P (x; α, β, θ) = θαβx β 1 e αx β exp[θe αx β ] e θ 1 (2.4) 7

17 for x > 0, β > 0, θ > 0, where α is the scale parameter and β is the shape parameter.the graph of the density function for various values of the parameters α, β, θ, a and b is given in Figure 2. Figure 2. Plot of KIWP density for different parameters. Expansion of the Density Function The expansion of the pdf of KIWP distribution is presented in this section. For b > 0 a real non-integer, we use the series representation (1 z) a 1 = ( ) a 1 ( 1) j z j ; a > 0, z < 1 (2.5) j j=0 8

18 We thus have: f KIW P (x) = abg(x)[g(x)] a 1 [1 G(x) a ] b 1 ( ) b 1 = ( 1) j abg IW P (x)[g IW P (x)] aj+a 1. j j=0 f KIW P (x) = ( ) b 1 ( 1) j abαβθx β 1 e αx β exp[θe αx β ] j=0 j [ 1 exp(θe αx β ) 1 e θ ] aj+a 1 e θ 1 (2.6) f KIW P (x) = ( ) b 1 ( 1) aj+a+j 1 abαβθx β 1 e αx β exp[θe αx β ] j j=0 [ 1 exp(θe αx β ) ] aj+a 1 (e θ 1) aj+a = j=0 k=0 exp ( )( ) b 1 aj + a 1 ( 1) aj+a+j+k 1 abαβθx β 1 e αx β j ] [θ(k + 1)e αx β. k (e θ 1) aj+a Notice that e t t m = m!. m=0 9

19 Applying the above identity to the last part of 2.6 yields: f KIW P (x) = j=0 k=0 m=0 ( )( b 1 aj + a 1 j k ) ( 1) aj+a+j+k 1 abθ m+1 (k + 1) m (e θ 1) aj+a m! αβx β 1 e (m+1)αx β ( )( ) b 1 aj + a 1 ( 1) aj+a+j+k 1 abθ m+1 (k + 1) m = j k (e θ 1) aj+a (m + 1)! j=0 k=0 m=0 αβ(m + 1)x β 1 e (m+1)αx β = ω j,k,m (a, b, θ)g(x; α[m + 1], β), (2.7) j,k,m=0 where ( )( ) b 1 aj + a 1 ( 1) aj+a+j+k 1 abθ m+1 (k + 1) m ω j,k,m (a, b, θ) = j k (e θ 1) aj+a (m + 1)! (2.8) are the weights and g(x; α(m + 1), β) is the inverse Weibull pdf with scale parameter α(m+1) and shape parameter β. Consequently, the KIWP density can be written as a linear combination of inverse Weibull density functions. The mathematical properties of the KIWP follow directly from those of the IW distribution. Monotonicity Properties Let The monotonicity properties of the KIWP distribution are discussed in this section. V (x) = ( ) 1 exp[θe ( αx) β ] 1 e θ (2.9) From equation (2.9) we can rewrite the KIWP pdf as f KIW P (x; α, β, θ, a, b) = abαβθx β 1 e αx β [1 V (x)(1 e θ )] e θ 1 V (x) a 1 (1 V (x) a ) b 1 10

20 for x > 0, α > 0, β > 0, θ > 0, a > 0, b > 0. It follows that log f KIW P (x) = log(abαβθ) log x β 1 αx β + log[1 V (x) + V (x)e θ ], log(e θ 1) + (a 1) log V (x) + (b 1) log(1 V (x) a ) and d log f KIW P (x) dx = β + 1 x + V (x)(e θ 1) 1 V (x) + V (x)e + (a 1)V (x) θ V (x) a(b 1) V (x)a 1 V (x) 1 V (x) a. ( ) 1 Substituting V (x) = dv (x)/dx = αβθx β 1 e αx β exp[θe αx β ] into 1 e θ equation 2.10, we have d log f KIW P (x) dx = β + 1 x + βαx β 1 + V (x)(e θ 1) 1 V (x) + V (x)e θ + (a 1)V (x) V (x), a(b 1) V (x)v (x) a 1 (x) 1 V (x) a. Taking the 2nd partial derivative gives: d 2 log f KIW P (x) dx = β + 1 x 2 αβ(β + 1)x β 2, + V (x)(e θ 1)(1 V (x) + V (x)e θ ) V (x) (e θ 1)(V (x)e θ 1) (1 V (x) + V (x)e θ ) 2, +(a 1) V (x)v (x) (V (x)) 2 V (x) 2, a(b 1) (a 1)V (x)a 2 (V (x)) 2 (1 V (x) a ) av (x) a 1 (V (x)) 2 V (x) a 1 (1 V (x) a ) 2 11

21 Thus: d 2 log f KIW P (x) dx = β + 1 x 2 αβ(β + 1)x β 2 + q(x) where q(x) = V (x)(e θ 1)(1 V (x) + V (x)e θ ) V (x)(e θ 1)(V (x) e θ 1) (1 V (x) + V (x)e θ ) 2, +(a 1) V (x)v (x) (V (x)) 2 V (x) 2, a(b 1) (a 1)V (x)a 2 (V (x)) 2 (1 V (x) a ) av (x) a 1 (V (x) ) 2 V (x) a 1 (1 V (x) a ) 2. Since α > 0, β > 0, θ > 0, a > 0 and b > 0, we have V (x) = dv (x) dx = ( ) 1 exp[θe ( αx) β ] αβθ(αx) β 1 exp[θe αx β ], 1 e θ 1 e θ for all x > 0. If x 0, then V (x) = ( ) 1 exp[θe ( αx) β ] 1. 1 e θ If x, then V (x) = ( ) 1 exp[θe ( αx) β ] 0. 1 e θ Since lim x + exp[θe ( αx) β ] = 1 then lim x + (1 exp[θe ( αx) β ]) = 0. Thus V (x) is monotonically increasing from 0 to 1. Note that, since 0 < V (x) < 1,0 < V a 1 (x) < 1, for all a > 1, 0 < 1 V a 1 (x) < 1, for all a > 0 and V (x) > 0, we have 0 < [1 V a (x)] b 1 < 1. for all a > 1, and b > 1. If α > 0, f KIW P (x; α, β, θ, a, b) could attain a maximum, a minimum 12

22 or a point of inflection according to whether d 2 log f KIW P (x) dx 2 < 0, d 2 log f KIW P (x) dx 2 > 0 or d 2 log f KIW P (x) dx 2 = 0. Survival, Hazard and Reverse Hazard Functions The hazard and reverse hazard functions for the KIWP distribution will be presented in this section. The survival function of the KIWP distribution is given by: S(x) = F (x) = 1 F KIW P (x) = [ 1 ( ) 1 exp[θe (αx) β a ] b ]. 1 e θ The hazard and reverse hazard functions of the KIWP are given respectively by f KIW P (x; α, β, θ, a, b) h(x) = 1 F KIW P (x; α, β, θ, a, b) [ ] = abθαβx β 1 e (αx) β exp[θe (αx) β ] 1 exp[θe (αx) β a 1 ] e θ 1 1 e θ [ ( ) 1 exp[θe (αx) β a ] 1 ] 1 1 e θ ( ) a 1 abθαβx β 1 e (αx) β exp[θe (αx) β ] exp[θe (αx) β ] 1 = ( a ( ) a, e θ 1) exp[θe (αx) β ] 1 and τ(x) = f KIW P (x; β, θ, a, b) F KIW P (x; β, θ, a, b) ( = abθαβx β 1 e (αx) β exp[θe (αx) β ] 1 exp[θe (αx) β ] e θ 1 1 e θ [ ( ) 1 exp[θe (αx) β a ] b 1 ] 1 1 e θ { [ ( ) 1 exp[θe (αx) β a ] b } 1 ] 1 1, 1 e θ 13 ) a 1

23 for x > 0, α, β > 0, θ > 0, a > 0 and b > 0. The graph of the hazard function for various values of the parameters α, β, θ, a and b is given in Figure 3. Figure 3. Plot of the hazard function for different parameter values. The graph of the hazard function for different values of the parameters exhibits various shapes such as monotonically increasing, bathtub shape, increasing-decreasing, monotone decreasing and upside down bathtub shapes. This is an attractive feature that renders the KIWP distribution suitable for monotonic and non-monotonic hazard behaviors which are more likely to be encountered in real life situations. 14

24 Moments, Moment Generating Function and Conditional Moments In this section, we present the moments, moment generating function and conditional moments of the KIWP distribution. Moments are necessary and important in any statistical analysis, especially in applications. They can be used to study the most important features and characteristics of a distribution such as tendency, dispersion, skewness and kurtosis. Quantile Function The quantile function of the KIWP distribution is obtained by solving the equation F (x) = u, where 0 < u < 1. We therefore have 1 [ 1 ( ) 1 exp[θe αx β a ] b ] = u. 1 e θ Hence isolating the term exp(θe αx β ) yields: 1 exp(θe αx β ) 1 e θ = [ 1 (1 u) 1/b ] 1/a. Thus, [ ] 1/a exp(θe αx β ) = 1 (1 e θ ) 1 (1 u) 1/b, Taking natural logarithms both sides and dividing by θ yields: e αx β = 1 ) 1/a } {1 θ log (1 e θ ) (1 (1 u) 1/b 15

25 Thus, [ { ) 1/a }] 1 αx β = log θ log 1 (1 e θ ) (1 (1 u) 1/b Dividing both sides by α x β = 1 [ { ) 1/a }] 1 α log θ log 1 (1 e θ ) (1 (1 u) 1/b. The quantile function of the KIWP distribution is obtained by solving for x in the cdf to obtain x = ( [ { ( ) 1/a }]) 1/β 1 1 α log θ log 1 (1 e θ ) 1 (1 u) 1/b. Moments In this section, moments and related measures including coefficients of variation, skewness and kurtosis for the KIWP distribution are presented. A table of values for mean, standard deviation, coefficient of variation (CV), coefficient of skewness (CS) and coefficient of kurtosis (CK) is also presented. The r th moment of a random variable X following the KIWP distribution, denoted by E(X r ) = µ r is given by E(X r ) = = = 0 j,k,m=0 x r f KIW P (x; α, β, θ, a, b)dx ω j,k,m (a, b, θ)αβ(m + 1) ω j,k,m (a, b, θ) j,k,m=0 0 0 x r β 1 e (m+1)αx β dx x r e (m+1)αx β αβ(m + 1)x β 1 dx. 16

26 By letting u = (m + 1)αx β, we have du = αβ(m + 1)x β 1 dx and x = u 1/β [α(m + 1)] 1/β. Thus 0 x r e (m+1)αx β αβ(m + 1)x β 1 dx = [α(m + 1)] r β u (1 r β ) 1 e u du 0( = [α(m + 1)] r/β Γ 1 r ), β where β > r. Hence, E(X r ) is given by µ r = j,k,m=0 ( ω j,k,m (a, b, θ)[α(m + 1)] r β Γ 1 r ), (2.10) β where β > r and ω j,k,m (a, b, θ) is defined in equation 2.8.The mean of the KIWP distribution is µ = µ 1 = j,k,m=0 ( ω j,k,m (a, b, θ)[α(m + 1)] 1/β Γ 1 1 ), β for β > 1. The variance, CV, CS, and CK are given by σ 2 = µ 2 µ 2, CV = σ µ = µ 2 µ 2 µ = µ 2 µ 2 1, CS = E [(X µ)3 ] [E(X µ) 2 ] 3/2 = µ 3 3µµ 2 + 2µ 3 (µ 2 µ 2 ) 3/2, and CK = E [(X µ)4 ] [E(X µ) 2 ] 2 = µ 4 4µµ 3 + 6µ 2 µ 2 3µ 4 (µ 2 µ 2 ) 2, respectively. Table 1 lists the first six moments of the KIWP distribution for selected values of the parameters obtained by fixing α = 1.5, β = 6.5 and θ = 1.5. Table 2 lists the first six 17

27 moments of the KIWP distribution for selected values of the parameters obtained by fixing a = 1.0 and b = 1.0. These values can be determined numerically using R and MATLAB. Table 1. Moments of the KIWP Distribution for α = 1.5, β = 6.5 and θ = 1.5. µ s a = 1.0, b = 1.5 a = 1.0, b = 3.5 a = 1.0, b = 1.0 a = 2.5, b = 1.0 µ µ µ µ µ µ SD CV CS CK Table 2. Moments of the KIWP Distribution for a = 1.0 and b = 1.0. µ s α = 0.8, β = 6.5 α = 1.3, β = 7.5 α = 2.0, β = 8.5 α = 1.5, β = 7.0 µ µ µ µ µ µ SD CV CS CK

28 Moment Generating Function The moment generating function of the KIWP distribution can be obtained from the r th moment which is given as follows: E(e tx ) = = i=0 t i i! E(Xi ) i=0 j,k,m=0 ( ω j,k,m (a, b, θ)[α(m + 1)] i/β Γ 1 i ), (2.11) β β > i where ω j,k,m (a, b, θ) is defined by equation 2.8 Conditional Moments For income and lifetime distributions, it is of interest to obtain the conditional moments and mean residual life function. The r th conditional moment for KIWP distribution is given by E(X r X > t) = = 1 F KIW P (t) 1 F KIW P (t) t abθm+1 (k + 1) m (m + 1)! x r f KIW P (x)dx j=0 k=0 m=0 t ( )( ) b 1 aj + a 1 ( 1) aj+a+j+k 1 j k (e θ 1) aj+a αβ(m + 1)x r β 1 e (m+1)αx β dx. (2.12) Considering the integral part, we thus have = t t αβ(m + 1)x r β 1 e (m+1)αx β dx x r e (m+1)αx β αβ(m + 1)x β 1 dx. 19

29 By letting u = (m + 1)αx β, we have du = αβ(m + 1)x β 1 dx and x = u 1/β [α(m + 1)] 1/β. When x = t, u = α(m + 1)t β and when x =, u = 0. We therefore have = t 0 x r e (m+1)αx β αβ(m + 1)x β 1 dx (m+1)αt β [α(m + 1)] r β u r β e u du = [α(m + 1)] r β 0 ( = [α(m + 1)] r β γ (m+1)αt β u r β e u du 1 r β, α(m + 1)t β ), after applying the lower incomplete gamma function γ(s, a) = a 0 t s 1 e t dt. The r th conditional moment for the KIWP distribution is given by E(X r X > t) = [ 1 ( ) 1 exp[θe αt β a ] b ] 1 e θ j,k,m=0 [α(m + 1)] r β γ ( 1 r β, α(m + 1)t β ), ω j,k,m (a, b, θ) where ω j,k,m (a, b, θ) is defined by 2.8. The mean residual life function is E(X r X > t) t. Order Statistics Order statistics are used to find the characteristics such as minimum and maximum time to failure of electronic components for example lightbulbs in reliability theory. This may be achieved by testing n lightbulbs and simultaneously putting them on test and collecting the time to failure as the successive failures occur. The observations may then be ordered 20

30 as X 1, X 2,..., X n whereby X 1 denotes the minimum time to failure and X n denotes the maximum time to failure. The trials are independent and identically distributed. The pdf of the k th order statistic from the KIWP distribution is f k:n (x) = n!f KIW P (x) (k 1)!(n k)! [F KIW P (x)] k 1 [1 F KIW P (x)] n k. Using the identity (1 z) a 1 = ( ) a 1 ( 1) p z p, p p=0 we have = = n!f KIW P (x) ( ) n k ( 1) p [F KIW P (x)] p+k 1 (k 1)!(n k)! p p=0 n!f KIW P (x) ( ) n k ( 1) p (k 1)!(n k)! p p=0 { [ ( ) 1 exp[θe αx β a ] b } p+k 1 ] e θ n!f KIW P (x) ( )( ) n k p + k 1 ( 1) p+q (k 1)!(n k)! p q p=0 q=0 [ ( ) 1 exp[θe αx β a ] bq ] 1. 1 e θ 21

31 Substituting the KIWP pdf into the above equation gives n! ( )( ) n k p + k 1 = ( 1) p+q (k 1)!(n k)! p q p=0 q=0 [ abθαβx β 1 e αx β exp[θe αx β ] 1 exp[θe αx β ] e θ 1 1 e θ [ ( ) 1 exp[θe αx β a ] bq+b 1 ] 1 1 e θ ] a 1 n! ( )( ) n k p + k 1 = ( 1) p+q+s (k 1)!(n k)! p q p=0 q=0 s=0 ( ) bq + b 1 abθαβx β 1 e αx β exp[θe αx β ] s e θ 1 [ ] 1 exp[θe αx β as+a 1 ] 1 e θ n! = ( 1) p+q+s+a+as 1 (k 1)!(n k)! p=0 q=0 s=0 ( )( )( ) n k p + k 1 bq + b 1 p q s [ ] abθαβx β 1 e αx β exp[θe αx β as+a 1 ] 1 exp(θe αx β ) (e θ 1) as+a n! = ( 1) p+q+s+a+as+i 1 (k 1)!(n k)! p=0 q=0 s=0 i=0 ( )( )( )( ) n k p + k 1 bq + b 1 as + a 1 p q s i abθαβx β 1 e αx β exp[θ(i + 1)e αx β ]. (e θ 1) as+a Using the expansion e x = t=0 x t t!, 22

32 e αx β exp[θ(i + 1)e αx β ] = = [θ(i + 1)] j e α(j+1)x β j=0 j! [θ(i + 1)] j (j + 1)e α(j+1)x β. j!(j + 1) j=0 Thus, f k:n (x) = = n! (k 1)!(n k)! p=0 q=0 s=0 i=0 j=0 ( )( )( )( ) n k p + k 1 bq + b 1 as + a 1 p q s i abθ j+1 (i + 1) j (j + 1)!(e θ 1) αβ(j + as+a 1)x β 1 e α(j+1)x β Q(p, q, s, i, j)f IW (x; α(j + 1), β), p,q,s,i,j=0 ( 1) p+q+s+a+as+i 1 where Q(p, q, s, i, j) = n! ( 1) p+q+s+a+as+i 1 abθ j+1 (i + 1) j (k 1)!(n k)! (j + 1)!(e θ 1) ( )( )( )( as+a ) n k p + k 1 bq + b 1 as + a 1. p q s i Thus, the pdf of the k th order statistic from the KIWP distribution is clearly a linear combination of inverse Weibull pdfs with parameters α(j + 1), β > 0. The r th moment of the distribution of the k th order statistic is given by E(X r k:n) = = p,q,s,i,j=0 p,q,s,i,j=0 Q(p, q, s, i, j) 0 x r f IW (x; α(j + 1), β)dx ( Q(p, q, s, i, j)[α(j + 1)] r β Γ 1 r ), β where β > r. 23

33 Sub-Models of the KIWP distribution In this section, some sub-models of the KIWP distribution for selected values of the parameters α, β, θ, a and b are presented. (1) a = b = 1 When a = b = 1, we obtain the inverse Weibull Poisson (IWP) distribution whose cumulaltive distribution function (cdf) and probability density function (pdf) are respectively given by F (x, α, β) = exp( αx β ), x > 0, α > 0, β > 0, f(x, α, β) = αβx β 1 e α x β. (2) b = 1 When b = 1, we obtain the exponentiated inverse Weibull Poisson (EIWP) distribution which belongs to the resilience parameter family and whose cdf is given by [ ] 1 exp(θe αx β a ) F (x; α, β, θ, a) =, 1 e θ with corresponding pdf [ ] f(x; α, β, θ, a) = abθαβx β 1 e αx β exp(θe αx β ) 1 exp(θe αx β a 1 ), e θ 1 1 e θ for x > 0, α > 0, β > 0, θ, a > 0. (3) a = 1 When a = 1, we obtain the new lifetime distribution belonging to the frailty parameter family with cdf 24

34 [ 1 1 ( )] 1 exp[θe αx β b ], 1 e θ and pdf [ ( f(x) = bθαβx β 1 e αx β exp[θe αx β ] 1 exp[θe αx β ] 1 e θ 1 1 e θ )] b 1 for x > 0, α, β > 0, θ > 0, a > 0, b > 0. (4) β = 2 When β = 2, we obtain the Kumaraswamy inverse Rayleigh Poisson (KIRP) distribution whose cdf is F (x; α, θ, a, b) = 1 [ 1 ( ) 1 exp[θe αx 2 a ] b ], 1 e θ with corresponding pdf [ f(x) = 2abθαx 3 e αx 2 exp[θe αx 2 ] 1 exp[θe αx 2 ] e θ 1 1 e θ [ ( ) 1 exp[θe αx 2 a ] b 1 ] 1, 1 e θ ] a 1 for x > 0, α > 0, θ, a > 0, b > 0. (5) β = 1 When β = 1, we obtain the Kumaraswamy inverse exponential Poisson (KIEP) distribution whose cdf is 25

35 with corresponding pdf [ F (x; α, θ, a, b) = 1 1 ( ) 1 exp[θe αx 1 a ] b ] 1 e θ [ f(x) = abθαx 2 e αx 1 exp[θe αx 1 ] 1 exp[θe αx 1 ] e θ 1 1 e θ [ ( ) 1 exp[θe αx 1 a ] b 1 ] 1, 1 e θ ] a 1 for x > 0, α > 0, θ, a > 0, b > 0. (6) α = 1 When α = 1, we obtain the Kumaraswamy Frechet Poisson (KFP) distribution whose cdf is 1 [ 1 ( ) 1 exp[θe x β a ] b ] 1 e θ with corresponding pdf [ f KF P (x) = abθβx β 1 e x β exp[θe x β ] 1 exp[θe x β ] e θ 1 1 e θ [ ( ) 1 exp[θe x β a ] b 1 ] 1, 1 e θ ] a 1 for x > 0, β > 0, θ > 0, a > 0, b > 0. (7) α = b = 1 When α = b = 1, we get the exponentiated Frechet Poisson (EFP) distribution whose cdf is F EF P (x; a, β, θ) = ( ) a 1 exp[θe x β ]. 1 e θ 26

36 The corresponding pdf is aθβx β 1 e x β exp[θe x ] e θ 1 [ 1 exp[θe x β ] 1 e θ ] a 1 (2.13) for x > 0, a > 0, β, θ > 0. (8) α = a = b = 1 When α = a = b = 1, we get the Frechet Poisson (FP) distribution whose cdf is F F P (x; a, β, θ) = 1 exp[θe x β ]. 1 e θ The corresponding pdf is f F P (x) = θαβx β 1 e x β exp[θe αx ] e θ 1 for x > 0, θ > 0. (9) β = b = 1 When β = b = 1, we get the exponentiated inverse exponential Poisson (EIEP) distribution whose cdf is F EIEP (x; a, β, θ) = ( ) a 1 exp[θe x 1 ]. 1 e θ 27

37 The corresponding pdf is [ f EIEP (x) = aθαx 2 e αx 1 exp[θe αx ] 1 exp[θe αx 1 ] e θ 1 1 e θ ] a 1 for x > 0, a > 0, β > 0, θ > 0. (10) β = a = b = 1 When β = a = b = 1, we get the inverse exponential Poisson (IEP) distribution whose cdf is F EIRP (x; a, β, θ) = 1 exp[θe x 1 ]. 1 e θ The corresponding pdf is f EIRP (x) = θαx 2 e αx 1 exp[θe αx ] e θ 1. for x > 0, θ > 0. (11) β = 2, b = 1 When β = 2, b = 1, we get the exponentiated inverse Rayleigh Poisson (EIRP) distribution whose cdf is F EIRP (x; a, β, θ) = ( ) a 1 exp[θe x 2 ]. 1 e θ 28

38 The corresponding pdf is [ f EIRP (x) = 2aθαx 3 e αx 2 exp[θe αx ] 1 exp[θe αx 2 ] e θ 1 1 e θ ] a 1 for x > 0, a > 0, α > 0, θ > 0. (12) β = 2, a = b = 1 When β = 2, a = b = 1, we get the inverse Rayleigh Poisson (IRP) distribution whose cdf is F IRP (x; α, θ) = 1 exp[θe αx 2 ]. 1 e θ The corresponding pdf is f IRP (x) = 2θαx 3 e αx 2 exp[θe αx ] e θ 1 for x > 0, a > 0, α > 0, θ > 0. (13) a = b = 1 and θ 0 +, we get the (IW) inverse Weibull distribution. (14) b = 1 and θ 0 +, we get the (EIW) exponentiated inverse Weibull distribution. (15) β = 2 and θ 0 +, we get the (KIR) Kum-inverse Rayleigh distribution. (16) β = 1 and θ 0 +, we get the (KIE) Kum-inverse exponential distribution. (17) α = 1 and θ 0 +, we get the (KF) Kum-Frechet distribution. 29

39 (18) α = b = 1 and θ 0 +, we get the (EF) exponentiated-frechet distribution. (19) β = b = 1 and θ 0 +, we get the (EIE) exponentiated-inverse exponential distribution. (20) β = a = b = 1 and θ 0 +, we get the (IE) inverse exponential distribution. (21) β = 2, b = 1 and θ 0 +, we get the (EIR) exponentiated inverse Rayleigh distribution. (22) β = 2, a = b = 1 and θ 0 +, we get the (IR) inverse Rayleigh distribution. Mean and Median Deviations The amount of dispersion in a population can be measured by the totality of deviations from the mean and the median. If X has the KIWP distribution, we can derive the mean deviation about the mean µ = E(X) and the mean deviation about the median M = Median(X) = F 1 (1/2) from δ 1 = = µ 0 µ 0 = µf (µ) x µ f KIW P (x)dx f(x)dx = 2µF (µ) 2µ + 2 = 2µF (µ) 2µ µ 0 xf(x)dx + xf(x)dx + 2 µ j,k,m=0 µ xf(x)dx µ xf(x)dx µ xf(x)dx µ [1 F (µ)] µ f(x)dx ω j,k,m (a, b, θ)[α(m + 1)] 1 β γ ( 1 1 β, α(m + 1)µ β ), (2.14) 30

40 and δ 2 = = 0 M = M 0 M x M f KIW P (x)dx (M x)f(x)dx + 0 = MF (M) f(x)dx 0 M 0 M (x M)f(x)dx xf(x)dx + xf(x)dx + 2 = 2MF (M) M µ + 2 M M M xf(x)dx = 2M 1 2 M µ + 2 xf(x)dx = µ + 2 = µ + 2 by using equation M j,k,m=0 xf(x)dx M xf(x)dx M M xf(x)dx M [1 F (M)] f(x)dx ω j,k,m (a, b, θ)[α(m + 1)] 1 β γ ( 1 1 β, α(m + 1)M β ), Bonferroni and Lorenz curves Lorenz curves are income inequality measures that are also useful and applicable to other areas including reliability, demography, medicine and insurance. Lorenz and Bonferroni curves are given by L(F (x)) = x tf(t)dt 0, and B(F (x)) = E(X) L(F (x)), F (x) or L(p) = 1 µ q 0 tf(t)dt, and B(p) = 1 q tf(t)dt, pµ 0 31

41 respectively, where q = F 1 (p). Now using equation 2.14, we can re-write the Lorenz and Bonferroni curves as and q B(p) = 1 tf(t)dt pµ 0 = 1 [ ] xf(x)dx xf(x)dx pµ 0 q [ = 1 µ ω j,k,m (a, b, θ)[α(m + 1)] 1 β γ (1 1 ) ] pµ β, α(m + 1)q β. j,k,m=0 q L(p) = 1 tf(t)dt µ 0 = 1 [ ] xf(x)dx xf(x)dx µ 0 q [ = 1 µ ω j,k,m (a, b, θ)[α(m + 1)] 1 β γ (1 1 ) ] µ β, α(m + 1)q β, j,k,m=0 where ω j,k,m (a, b, θ) is defined in equation

42 CHAPTER 3 MEASURES OF UNCERTAINTY Rényi Entropy In this section we present some measures of uncertainty. The concept of entropy plays a vital role in information theory. The entropy of a random variable is defined in terms of its probability distribution and can be shown to be a good measure of randomness or uncertainty. Rényi entropy is an extension of Shannon entropy. Rényi entropy is defined to be I R (v) = 1 ( ) 1 v log [f KIW P (x; a, b, α, β, θ)] v dx, v 1, v > 0. 0 Rényi entropy tends to Shannon entropy as v 1. Notice that [f(x)] v = [ ] v [ abαβθ 1 exp[θe x βv v αx β ] e αvx β exp[θve αx β ] e θ 1 1 e θ [ ( ) 1 exp[θe αx β a ] bv v ] 1. 1 e θ ] av v We thus have = = = [ 1 ( ) 1 exp[θe αx β a ] bv v [ ] 1 exp[θe αx β ] 1 e θ 1 e θ ( [ ] bv v 1 exp[θe )( 1) i αx β ai+av v ] i 1 e θ i=0 ( ) [ bv v ( 1) i+ai+av v 1 exp(θe αx β ) i (e θ 1) ai+av v i=0 ( )( ) bv v ai + av v ( 1) i+j+ai+av v i=0 j=0 i j ] av v ] ai+av v (e θ exp(θje αx β). 1) ai+av v 33

43 Thus [ ] v abαβθ ( )( ) bv v ai + av v ( 1) [f(x)] v i+j+ai+av v = (e θ 1) a i j (e θ 1) ai i=0 j=0 ) x βv v exp (θ(j + v)e αx β e αvx β. Using series expansions, we obtain: [f(x)] v = [ ] v abαβθ ( )( ) bv v ai + av v ( 1) i+j+ai+av v θ k (j + v) k (e θ 1) a i j k!(e θ 1) ai x βv v e α(v+k)x β. i=0 j=0 k=0 Let u = α(v + k)x β du = αβ(v + k)x β 1 dx, and x = [α(v + k)] 1 β u 1 β. We have 0 x βv v e α(v+k)x β dx = = = 1 αβ(v + k) 0 [α(v + k)]1 v+ αβ(v + k) 1 v vβ [α(v + k)] β β x βv+β v+1 e α(v+k)x β αβ(v + k)x β 1 dx 1 v β 0 u v 1 v 1+ β ( vβ + v 1 Γ β du ). Therefore the Rényi entropy for the KIWP distribution is given by I R (v) = {[ ] v 1 abαβθ 1 v log (e θ 1) a i=0 ( 1)i+j+ai+av v θ k (j + v) k k!(e θ 1) ( )} ai vβ + v 1 Γ. β 34 j=0 k=0 ( )( ) bv v ai + av v 1 v vβ [α(v + k)] β β i j

44 Shannon Entropy Shannon entropy is a crucial concept in information theory. It measures the uncertainty associated with a random variable or the expected value of information encoded in a message in information theory. Shannon entropy is defined as H [f KIW P ] = E [ log(f KIW P (X; α, β, θ, a, b))]. Thus we have [ ] e θ 1 H [f KIW P (X)] = log + (β + 1)E [log(x)] + αe [ x β] θe abαβθ [ { }] 1 exp(θe αx β ) (a 1)E log 1 e θ [ {[ ( ) a ]}] 1 exp[θe αx β ] (b 1)E log 1. 1 e θ [e αx β] Note that log(x + 1) = ( 1) q+1 x q, 1 < x 1. q q=1 E [ log { }] 1 exp(θe αx β ) 1 e θ = log(1 e θ ) = log(1 e θ ) E q=1 q=1 s=0 [ ] exp(θqe αx β ) q ] (qθ) s E [e αsx β qs! = log(1 e θ ) ( 1) u θ s q s 1 (αs) u E[X βu ]. s!u! q=1 s=0 u=0 35

45 E [ log {[ 1 ( ) a ]}] 1 exp[θe αx β ] 1 e θ = = b=1 b=1 1 b E ( ) ab 1 exp(θe αx β ) 1 e θ { [ ] } ab ( 1) ab E 1 exp(θe αx β ). b(e θ 1) ab Using the identity (1 z) a 1 = ( ) a 1 ( 1) i, i i=0 we have b=1 ( 1) ab [1 exp(θe αx β )] ab b(e θ 1) ab = = ( ) ab ( 1) ab+c [ ] c b(e θ 1) E exp(θe αx β ) ab c=0 ( 1) ab+c+g (θc) d (αd) g E [ X βg]. bd!g!(e θ 1) ab b=1 b=1 c=0 d=0 g=0 Also, E [ e αx β] = ( 1) l α l E [ X βl]. l! l=0 36

46 E [log(x)] = E [log(1 + (x 1))] ( 1) p+1 (x 1) p = p p=1 p ( ) p ( 1) p+1 ( 1) p w E [X p ] = w p p=1 w=0 p ( ) p ( 1) 2p+1 w E [X p ] = w p p=1 w=0 p ( ) p ( 1) w E [X p ] =. w p p=1 w=0 Using results from the r th moment in 2.10, the Shannon entropy of the KIWP distribution is given by H [f(x)] = log [ ] e θ 1 abαβθ (β + 1) [α(m + 1)] p β Γ ( 1 p β p j,k,m=0 p=1 w=0 ) + j,k,m=0 ( ) p ( 1) w ω j,k,m (a, b, θ) w p ω j,k,m (a, b, θ) m + 1 θ j,k,m=0 l=0 { + (a 1) log(1 e θ ) + } Γ(1 + u) [α(m + 1)] u { + (b 1) Γ(1 + g) [α(m + 1)] g ( 1) l α l ω j,k,m (a, b, θ) l![α(m + 1)] l Γ(1 + l) j,k,m=0 b=1 c=0 d=0 g=0 }. j,k,m=0 q=1 s=0 u=0 ω j,k,m (a, b, θ)( 1) u θ s q s 1 (αs) u s!u! ω j,k,m (a, b, θ)( 1) ab+c+g (θc) d (αd) g bd!g!(e θ 1) ab Note that Γ(a + 1) = a! 37

47 Reliability Reliability may be defined as the probability that a piece of equipment operated under specific conditions performs to a specified standard for a particular period of time. It is a number between 0 and 1. It focuses on maximizing the inherent repeatability or consistency in an experiment. To maintain reliability internally, a researcher will use as many repeat sample groups as possible. It is a crucial part of any research design. If any test is not reliable, then its validity is questionable thus making the entire experiment a waste of time. Reliability R when X and Y have independent KIW P (α 1, β 1, θ 1, a 1, b 1 ) and KIW P (α 2, β 2, θ 2, a 2, b 2 ) is given by: R = P (X > Y ) = = 0 0 f x (x, α 1, β 1, θ 1, a 1, b 1 )F Y (x, α 2, β 2, θ 2, a 2, b 2 )dx a 1 b 1 θ 1 α 1, β 1 x (β+1) e α 1x β 1 exp(θ 1 e α 1x β 1 ) exp(θ 1e α 1x β1 1) [ 1 { e θ 1 1 a 1 1 [ 1 1 exp(θ 2e α 2x β2 ) e θ exp(θ 1e α 1x β1 1) e θ 1 1 a 2 }] b2. a 1 ] (b1 1) Using the binomial expansion (1 z) b = i=0 ( b i) ( 1) i (z) i, [ 1 exp(θ 1e α 1x β1 1) e θ 1 1 a 1 ] (b1 1) = i=0 ( 1) i ( b1 1 i ) exp(θ1 e α 1x β 1 1) 1 e θ 1 a 1 i. Thus [ 1 { 1 1 exp(θ 2e α 2x β2 ) e θ 2 1 a 2 }] b2 = j=0 { ( ) b2 ( 1) j j 1 1 exp(θ 2e α 2x β2 ) 1 e θ 2 a 2 } j. 38

48 { 1 Therefore, [ 1 { ( 1 exp(θ 2 e α 2x β 2 a 2 )} j ) = 1 e θ exp(θ 2e α 2x β2 ) e θ 2 1 k=0 a 2 }] b2 = ( ( ) j )( 1) k 1 exp(θ 2 e α 2x β a2 2 ) k 1 e θ 2 l=0 l k k=0 j=0 ( j k )( b2 j )( a2 k (1 e θ 2 ) a 2k exp(θ 2 e α 2x β 2 ) l. l ) exp(( 1) j+k+1 We further expand the expression by applying series expansions ( ) exp(θ 1 e α 1x β a1 i 1 1) 1 e θ 1 = ( 1) a 1i+m = ( 1) a 1i+m m=0 m=0 ( ) a1 { } m exp(θ 1 me α 1x β 1 ) (1 e θ m 2) a 1i ( ) a1 i exp(θ 1 me α 1x β 1 )(1 e θ m 1) a1i. Expanding further by binomial expansions yields: ( 1 1 exp(θ 1e α 1x β1 e θ 1 1 ) a1 1 = (e θ 1 1) 1 a 1 ( ) a1 1 ( 1) n exp(θ 1 ne α 1x β 1 ). n n=0 Thus we have R = a 1 b 1 θ 1 α 1 β 1 n m l k j n=0 m=0 l=0 k=0 j=0 i=0 ( )( (e θ 1 1) 1 a 1 (1 e θ 2) a 2k a 1 i a1 1 b1 1 n i 0 x (β+1) ( 1) a 1 1+i+n+ai+m+j+k+l )( a1 i m )( j k )( b2 exp(θ 1 ne α 1x β 1 ) exp(θ 1 me α 1x β 1 ) exp(θ 2 le α 2x β 2 )dx. j )( ) a2 k l Notice that e t t m = m!. m=0 39

49 Thus, applying the above identity to exp(θ 1 ze α 2x β 2 ) yields exp(θ 1 ze α 2x β 2 ) = θ1z p p e α 1px β 1. p! p=0 Likewise, we can apply the same identity to the expressions for exp(θ 1 e α 1x β 1 ), exp(θ 1 me α 1x β 1 ) and exp(θ 2 le α 1x β 2 ). Thus, Thus, let 0 f x (x, α 1, β 1, θ 1, a 1, b 1 )F Y (x, α 2, β 2, θ 2, a 2, b 2 )dx be denoted 0 f x (.)F Y (.)dx 0 f x (.)F Y (.) = a 1 b 1 θ 1 α 1 β 1 i,j,k,l,m,n=0 p,q,r=0 ( 1) a 1(1+j)+i+j+k+m+n+l 1 ( )( )( )( ) a1 1 b1 1 b2 l i j l m ( ) am (e θ 2 1) am n (e θ 1 1) 1 a 1 a 1 j θ v 2 0 θ p+q+r 1 z p m r n v p!q!r!v! x (β+1) e α 1(1+p+q+r)x β 1 e α 2vx β 2 dx. By invoking the identity e t = t m m=0, we can express exp( α m! 2vx β 2 ) as exp( α 2 vx β 2 ) = ( α 2 ) t v t x tβ 2 t=0 t! = ( 1) t (α 2 ) t v t x tβ 2. t! t=0 R = a 1 b 1 θ 1 α 1 β 1 ( b1 1 0 j )( b2 l i,j,k,l,m,n=0 p,q,r=0 )( l m )( am n θ p+q+r x β 1 tβ 2 1 e α(1+p+q+r)x β 1 dx. 1 z p m r n v p!q!r!v!t! ) (e θ 2 1) am (e θ 1 1) 1 a 1 a 1 j θ2 v ( 1) a 1(1+j)+i+j+k+m+n+l 1 ( ) a1 1 i 40

50 For the integration of the expression 0 x β 1 tβ 2 1 e α 1(1+p+q+r)x β 1 dx, we will let u = α 1 (1 + p + q + r)x β 1, and so du = α 1 β 1 (1 + p + q + r)x β 1 1 dx. Therefore dx = 1 α 1 β 1 (1 + p + q + r) xβ 1+1 du = (1 + p + q + r) 1 1 β β 1 α u ( tβ 2 β 1 +1) 1 e u du. Therefore, we arrive at the expression below for the reliability: R = a 1 b 1 θ 1 α 1 β 1 ( )( a1 1 b1 1 i j ( ) 1 β α 1 tβ2 1 Γ + 1. β 1 i,j,k,l,m,n=0 p,q,r,t=0 )( b2 l )( l m θ v 2θ p+q+r 1 )( am n α t v t z p m r l v ( 1) a 1(1+j)+i+j+k+m+n+l 1 v!q!p!r!t! ) (e θ 2 1) am (e θ 1 1) 1 a 1 a 1 j (1 + p + q + r) 1 β 1 41

51 Maximum Likelihood Estimation In this section, the maximum likelihood method is used in estimating the parameters a, b, α, β, θ. The pdf of the KIWP distribution can be rewritten as [ ] f(x) = abθαβx β 1 e αx β exp(θe αx β ) exp(θe αx β a 1 ) 1 e θ 1 e θ 1 [ ( ) exp(θe αx β a ] b 1 ) 1 1 e θ 1 [ ] [ ] a 1 abθαβ = x β 1 e αx β exp(θe αx β ) exp(θe αx β ) 1 (e θ 1) ab [ ( ) a ] b 1 (e θ 1) a exp[θe αx β ] 1 [ ] [ ] abθαβ = x β 1 e αx β exp(θe αx β )V a 1 (x) (e θ 1) a V a (x), (e θ 1) ab where V (x) = exp(θe αx β ) 1. Let x 1,, x n be a random sample from the KIWP distribution. The log-likelihood function is given by L = n log(a) + n log(b) + n log(α) + n log(β) + n log(θ) nab log(e θ 1) n n n n (β + 1) log(x i ) α + θ e αx β i + (a 1) log[v (x i )] + (b 1) i=1 n i=1 i=1 The associated score function is given by x β i i=1 { } log (e θ 1) a V a (x i ). (3.1) i=1 U(Θ) = ( L a, L b, L α, L β, L ). (3.2) θ 42

52 The components of the score function are the partial derivatives with respect to each of the parameters. The maximum likelihood estimates of Θ can be obtained by solving the non-linear system of equations U n (Θ) = 0. Since the equations 3.1 and 3.2 are not in closed form, the solutions can be found by use of a numerical method such as the Newton Rhapson method. The elements of the score vector are given by; L a = n a nb log(eθ 1) + + (b 1) i=1 n log[v (x i )] i=1 n { } (e θ 1) a log(e θ 1) V a (x) log[v (x i )], (e θ 1) a [V (x i )] a L b = n b na log(eθ 1) + n i=1 { } log (e θ 1) a V a (x i ), L α = n n α i=1 (b 1) x β i n i=1 θ n i=1 x β i a[v (x i )] a 1 V (x i ) α (e θ 1) a [V (x i )] a, exp( αx β i ) + (a 1) n i=1 V (x i ) α V (x i ) L β = n n β log(x i ) + α i=1 + (a 1) n i=1 n i=1 x β i log(x i ) + αθ n i=1 x β i V (x i ) n β V (x i ) (b 1) a[v (x i )] a 1 V (x i ) β (e θ 1) a [V (x i )] a i=1 log(x i ) exp( αx β i ) and 43

53 L θ = n n θ nabeθ e θ 1 + exp( αx β i ) + (a 1) + a(b 1) n i=1 i=1 n i=1 e θ (e θ 1) a 1 [V (x i )] a 1 V (x i ) θ (e θ 1) a [V (x i )] a. V (x i ) θ V (x i ) The mixed partial derivatives are given by: 2 L = n n a 2 a + 2 i=1 V (x i ) V (x i ), θ 2 L a b = n log(e θ 1) + + i=1 n log[v (x i )] i=1 n { } (e θ 1) a log(e θ 1) V a (x) log[v (x i )], (e θ 1) a [V (x i )] a 2 L n ][ {[(e a α = (b 1) θ 1) a [V (x i )] a a[v (x i )] a 1 log[v (x i )] V (x i) α i=1 + [V (x i )] a 1 V (x ] [ ] i) (b 1) (e θ 1) a log(e θ 1) V a (x) log[v (x i )] α [ a[v (x i )] a 1 V (x i) α ]}[ (e θ 1) a [V (x i )]] 2 + n i=1 V (x i ) α V (x i ), 2 L n ][ {[(e a β = (b 1) θ 1) a [V (x i )] a a[v (x i )] a 1 log[v (x i )] V (x i) β i=1 + [V (x i )] a 1 V (x ] [ ] i) (b 1) (e θ 1) a log(e θ 1) V a (x) log[v (x i )] β [ a[v (x i )] a 1 V (x i) β ]}[ (e θ 1) a [V (x i )]] n i=1 V (x i ) β V (x i ),

54 2 L a θ = V (x i ) θ V (x i ) eθ nb + (b 1) n e θ 1 i=1 [(eθ 1) a V a (x i )] A [(e θ 1) a [V a (x i )] 2 (eθ 1) log(e θ 1) V a (x i ) log[v (x i )] a(e θ 1) a 1 e θ av a 1 (x i ) V (x i) θ, [(e θ 1) a [V a (x i )] 2 where A is denoted by A = a(e θ 1) log(e θ 1) + (e θ 1) a + (e θ e θ 1) e θ 1 av a 1 (x i ) V (x i) log[v (x i )]. θ V (x i ) V a β (x i ) V (x i ) 2 L = n b 2 b 2, 2 L n b α = a[v (x i )] a 1 V (x i ) α (e θ 1) a [V (x i )], a i=1 2 L n b β = a[v (x i )] a 1 V (x i ) β (e θ 1) a [V (x i )], a i=1 2 L b θ = naeθ e θ 1 + a n i=1 (e θ 1) a 1 e θ [V (x i )] a 1 V (x i ) θ (e θ 1) a [V (x i )] a, 45

55 2 L = n α 2 α 2 θ n a(b 1) i=1 n i=1 x 2β i exp( αx β i ) + (a 1) n V (x i ) 2 V (x i ) α 2 ( ) 2 V (x i ) α [V (x i=1 i )] 2 ){ ( V {[((e θ 1) a [V (x i )] a (a 1)[V (x i )] a 2 (xi ) α } ( ) 2 ] + [V (x i )] a 1 2 V (x i ) V + a 2 [V (x α 2 i )] 2a 2 (xi ) α [ ] 2 } (e θ 1) a [V (x i )] a ), ) 2 2 L = n α 2 α 2 θ n a(b 1) i=1 n i=1 x 2β i exp( αx β i ) + (a 1) n V (x i ) 2 V (x i ) α 2 ( ) 2 V (x i ) α [V (x i=1 i )] 2 ){ ( V {[[V (x i )] a 1 ((e θ 1) a [V (x i )] a (a 1)[V (x i )] 1 (xi ) α } ( ) 2 ] + 2 V (x i ) V + a 2 [V (x α 2 i )] 1 (xi ) α [ ] 2 } (e θ 1) a [V (x i )] a ), ) 2 2 L n α β = [log(x i )] i=1 n i=1 x β i + (a 1) n i=1 2 V (x i ) V (x α β i) V (x i) α [V (x i )] 2 ] V (x i ) β n (a 1) V a 2 (x) V (x i) V (x i ) + V a 1 (x) 2 V (x i ) β α α β [(e θ 1) a [V (x i=1 i )]] n V a 1 (x) V (x i ) [a(e θ 1) a 1 e θ av a 1 (x α i ) V (x i) ] θ a(b 1), [(e θ 1) a [V (x i )]] 2 i=1 46

56 2 L α θ = n i=1 x β i exp( αx β i ) + (a 1) n V (x i ) 2 V (x i ) i=1 V (x i) α θ α V (x i ) θ [V (x i )] 2, n n α x β i [log(x i )] 2 + αθ i=1 i=1 ( ) n V (x i ) 2 V (x i ) L β + a 1) 2 β, [V (x)] 2 2 L = n β 2 β 2 i=1 x β i [log(x i )] 2 exp( αx β i )(α 1) 2 L β θ = α n i=1 x β i exp( αx β i ) log(x i ) + (a 1) n i=1 V (x i ) 2 V (x i ) β θ [V (x)] 2 V V β θ n (a 1)V a 2 (x i ) V V θ a(b 1) + [V (x β i)] a 1 2 V β θ [(eθ 1) a V (x i ) a ] [(e θ 1) a [V (x i=1 i )] a ] [ ] a(e θ 1) a 1 e θ av a 1 (x i ) V ) av a 1 (x θ i ) V β [(e θ 1) a [V (x i )] a ] and 2 L = n θ 2 α + nab e θ n 2 (e θ 1) + (a 1) 2 i=1 2 V (x i ) θ 2 V (x i ) V V θ θ [V (x)] 2 + a(a 1)(eθ 1) a 2 e 2θ + a(e θ 1)e θ a(a 1)V a 2 (x i ) ( ) V 2 θ av a 1 (x i ) 2 V θ 2. [(e θ 1) a [V (x i )] a ] 2 47

57 For the one variable partials, we obtain V (x i ) a V (x i ) b V (x i ) α V (x i ) θ V (x i ) β = 0, = 0, = exp[θe αx β ]θx β exp[ αx β ], = e αx β exp[θe αx β ], = θαβx β 1 e αx β exp[θe αx β ] Fisher Information Matrix In this section, we present a measure for the amount of information. This information can be used for interval estimation and hypothesis testing for the model parameters a, b, α, β, θ. Let X be a random variable with the KIWP density f KIW P (.; Θ), where Θ = (θ 1, θ 2, θ 3, θ 4, θ 5 ) T = (a, b, α, β, θ) T. The Fisher information matrix (FIM) is the 5 5 symmetric matrix denoted as follows: 2 L a 2 2 L b a 2 L α a 2 L β a 2 L θ a 2 L a b 2 L b 2 2 L α b 2 L β b 2 L θ b 2 L a α 2 L b α 2 L α 2 2 L βα 2 L θ α 2 L a β 2 L b β 2 L α β 2 L β 2 2 L θ β 2 L a θ 2 L b θ 2 L α θ 2 L β θ 2 L θ 2 48

58 The elements [ ] 2 log(f KIW P (x, Θ)) I ij (Θ) = E θ θ i θ j of the Fisher information matrix (FIM) can be obtained by considering the second order partial derivatives of the log likelihood function in equation 3.1. The elements can be obtained numerically by utilizing MATLAB software. The total FIM I n (Θ) can be approximated by: [ J n ( ˆΘ) 2 log L θ i j When dealing with actual real world data, the matrix above is obtained after convergence of the Newton Rhapson method in MATLAB. θ=ˆθ ]. 5 5 Asymptotic Confidence Intervals In this section, we present the asymptotic confidence intervals for the parameters of the KIWP distribution. The expectations in the Fisher s information Matrix (FIM) can be obtained numerically. Let ˆΘ = (â, ˆb, ˆα, ˆβ, ˆθ) be the maximum likelihood estimate of Θ = (a, b, α, β, θ). Under the usual regularity conditions and that the parameters are in the interior of the parameter space, but not on the boundary, we thus have n( ˆΘ Θ) d N 5 (0, I 1 (Θ)) where I(Θ) is the expected Fisher information matrix. The asymptotic behavior is still valid if I(Θ) is replaced by the observed information matrix evaluated at ˆΘ, that is J( ˆΘ). The multivariate normal distribution with mean vector (0) = (0, 0, 0, 0, 0) T and covariance matrix I 1 (Θ) can be used to construct confidence intervals for the model 49