Weighted Inverse Weibull Distribution: Statistical Properties and Applications

Transcription

1 Theoretical Mathematics & Applications, vol.4, no.2, 214, 1-3 ISSN: print), online) Scienpress Ltd, 214 Weighted Inverse Weibull Distribution: Statistical Properties and Applications Valeriia Sherina 1 and Broderick O. Oluyede 2 Abstract In this paper, the weighted inverse Weibull WIW) class of distributions is proposed and studied. This class of distributions contains several models such as: length-biased, hazard and reverse hazard proportional inverse Weibull, proportional inverse Weibull, inverse Weibull, inverse exponential, inverse Rayleigh, and Fréchet distributions as special cases. Properties of these distributions including the behavior of the hazard function, moments, coefficients of variation, skewness, and kurtosis, Rényi entropy and Fisher information are presented. Estimates of the model parameters via method of maximum likelihood ML) are presented. Extensive simulation study is conducted and numerical examples are given. Mathematics Subject Classification: 62E99; 6E5 Keywords: Weighted distribution; Inverse Weibull distribution; Maximum likelihood estimation 1 Georgia Southern University. valery.sherina@gmail.com 2 Department of Mathematics and Statistics at Georgia Southern University. boluyede@georgiasouthern.edu Article Info: Received : March 24, 214. Revised : May 17, 214. Published online : June 3, 214.

2 2 Weighted Inverse Weibull Distribution 1 Introduction The inverse Weibull IW) distribution can be readily applied to modeling processes in reliability, ecology, medicine, branching processes and biological studies. The properties and applications of IW distribution in several areas are given in the literature Keller [8], Calabria and Pulcini [2], [3], [4], Johnson [7], Khan et al. [9]). A random variable X has an IW distribution if the corresponding probability density function pdf) is given by fx; α, ) = α x 1 exp[ αx) ], x, α >, >. If = 1, the IW pdf becomes inverse exponential pdf, and when = 2, the IW pdf is referred to as the inverse Raleigh pdf. The IW cumulative distribution function cdf) is given by F x; α, ) = exp[ αx) ], α >, >, where α and are the scale and shape parameters, respectively. If α = 1, we have the Fréchet distribution function. Motivated by various applications of weighted distributions, Oluyede [12], Patil and Rao [13]) to biased samples in several areas including reliability, exponential tilting weighting) in finance and actuarial sciences, we construct and present the statistical properties of this new class of distributions called weighted inverse Weibull WIW) distribution and apply it to real lifetime data in order to demonstrate its usefulness. 1.1 Weighted Distribution Weighted distribution can be used to deal with model specification and data interpretation problems. Patil and Rao [13] used weighted distributions as stochastic models in the study of harvesting and predation. Let Y be a nonnegative random variable with its natural pdf fy; θ), where θ is a parameter in the parameter space Θ, then the pdf of the weighted random variable Y w is given by: f w wy, )fy; θ) y; θ, ) =, ω where the weight function wy, ) is a positive function, that may depend on the parameter, and < ω = EwY, )) < is a normalizing constant. A

3 Valeriia Sherina and Broderick O. Oluyede 3 general class of weight functions wy) is defined as follows: wy) = y k e ly F i y)f j y). Setting k = ; k = j = i = ; l = i = j = ; k = l = ; i i 1; j = n i; k = l = i = and k = l = j = in this weight function, one at a time, implies probability weighted moments, moment-generating functions, moments, order statistics, proportional hazards and proportional reversed hazards, respectively, where F y) = P Y y) and F y) = 1 F y). If wy) = y, then Y = Y w is called the size-biased version of Y. The rest of the paper is organized as follows. In section 2, probability weighted moments and the weighted inverse Weibull WIW) are developed. Some statistical properties, including mode, hazard and reverse hazard functions of the WIW distribution are presented. Glaser s Lemma [6] is applied to the WIW distribution to determine the behavior of the hazard function. Distribution of functions of WIW random variables are presented in section 3. In section 4, moments, entropy measures and Fisher information are given. Section 5 contains estimation of the parameters of the WIW distribution via the methods maximum likelihood. Simulation study is presented in section 6. Numerical examples are given in section 7, followed by concluding remarks. 2 Weighted Proportional Inverse Weibull Distribution In this section, probability weighted moments PWMs) of the proportional inverse Weibull distribution and WIW distribution are presented. The mode, hazard and reverse hazard functions are given. The proportional inverse Weibull IW) distribution has a cdf given by Gx; α,, γ) = [F x)] γ = exp[ γαx) ], for α >, >, γ >, and x. Let α γ = θ, then PIW distribution reduces to the IW distribution with cdf and pdf Gx; θ, ) = exp[ θx ],

4 4 Weighted Inverse Weibull Distribution and gx; θ, ) = θx 1 exp[ θx ], respectively, for θ >, >, and x. 2.1 Probability Weighted Moments where The PWMs of the IW distribution are given by E[X k G l X)G m X)] = E[X k G l+j X)] = j= 1) j Γm + 1) Γm + 1 j)γj + 1) E[Xk G l+j X)], x k e θx j+l) θx 1 e θx dx. Now, we make the substitution u = θx l + j + 1), so that E[X k G l+j X)] = θ k l + j + 1) k 1 = θ k l + j + 1) k 1 Γ Therefore, PWMs of the IW distribution are E[X k G l X)G m X)] = Remark: Special cases θ k l + j + 1) k 1 Γ u k e u du ). 1 k 1) j Γm + 1) Γm + 1 j)γj + 1) j= 1 k ), for > k. 1. When l = m =, the k th noncentral moments of the IW distribution is 2. When l = k =, we have E[G m X)] = E[X k ] = γ k α Γ 1 k ), for > k. k j= 1) j Γm + 1) Γm + 1 j)γj + 1) j + 1) 1,

5 Valeriia Sherina and Broderick O. Oluyede 5 3. When l =, we have E[X k G m X)] = for > k. j= 4. When m =, we obtain 1) j Γm + 1) Γm + 1 j)γj + 1) θ k k j + 1) 1 Γ 1 k ), E[X k G l X)] = θ k k l + 1) 1 Γ 1 k ), > k. 5. When k = m =, we have E[G l X)] = l + 1) When l i 1, m n i, we have E[X k G i 1 X)G n i X)] = 1) j Γn i + 1) Γn i + 1 j)γj + 1) j= 1 k θ k j + i) k 1 Γ ), > k. 7. When k =, we obtain E[G l X)G m X)] = j= 1) j Γm + 1) Γm + 1 j)γj + 1) l + j + 1) When k = l = 1, we get E[XGX)G m X)] = 1) j Γm + 1) Γm + 1 j)γj + 1) j= 1 1 θ k j + 2) 1 1 Γ ), for > Weighted Inverse Weibull Distribution In this section, we present the WIW distribution and some of its properties. We are particularly interested in the WIW distribution obtained via the weight

6 6 Weighted Inverse Weibull Distribution function wx) = xg IW x). In general, the WIW pdf is given by g W IW x) = xk G l x)g m x)gx) E[X k G l X)G m X)] ) m x k 1 θ 1 e θx e θl+1)x = 1) j Γm + 1) γ k k l + j + 1) 1 ), j= Γ 1 k Γm + 1 j)γj + 1) for > k. When m =, k = l = 1 the corresponding WIW pdf and cdf are given by and g W IW x) = xgx)gx) E[XGX)] = 1 2θ)1 x e 2θx Γ 1 1 ), 11) G W IW x; θ, ) = = x 2θ) 1 1 y e 2θy Γ 1 1 ) dy ) Γ 1 1, 2θx ), for > 1, 12) Γ 1 1 respectively, where Γs, x) = x ts 1 e t dt is the upper incomplete gamma function. Graphs of the WIW pdfs for selected values of the parameters θ and show different shapes of the curves depending on the values of the parameters [Figure 1]. If the distribution of the a random variable is given by equation 12), we write X W IW θ, ) W IW θ,, 1, 1) Mode of the WIW Distribution Consider the WIW pdf given in equation 11). Note that lng W IW x)) = 1 1 ) ln2θ) + ln) lnx) 2θx ln Γ 1 1 )). 13) Differentiating equation 13) with respect to x, we obtain ln g W IW x; θ, ) x = x + 2θx. 14) x

7 Valeriia Sherina and Broderick O. Oluyede 7 Figure 1: Pdfs of the WIW Distribution Now, set equation 14) equal and solve for x, to get Note, that x = 2θ) 1. 2 ln g W IW x; θ, ) x 2 = x )θx <. x 2 When < x < 2θ) 1, ln gx;θ,) >, so g x W IW x; θ, ) is increasing, and when x > 2θ) 1, gw IW x; θ, ) is decreasing, therefore g W IW x; θ, ) achieves a maximum when x = 2θ) 1, so that x is the mode of WIW distribution Hazard and Reverse Hazard Functions The hazard function of the WIW distribution is given by λ GW IW x; θ, ) = 2θ) 1 1 x e 2θx Γ 1 1 ) Γ 1 1 ),, 2θx for θ >, > 1, and x. The behavior of the hazard function of the WIW distribution is established via Glaser s Lemma [6]. Note that η GW IW x; θ, ) = g x; θ, ) W IW g W IW x; θ, ) = x 1 2θx 1, and η G W IW x) = x 2 + 2θ + 1)x 2.

8 8 Weighted Inverse Weibull Distribution Now, η G x) = implies x W IW = 2γ + 1)) 1, for θ >, > 1. Note that, when < x < x, η G x) >, η W IW G x W IW ) = and when x > x, η G x) <. Consequently, WIW hazard function has an upside down bathtub shape. The graphs of the hazard function show upside down bathtub shape W IW for the selected values of the parameters [Figure 2]. The reverse hazard func- Figure 2: Graphs of Hazard Functions of the WIW Distribution tion is given by for θ >, > 1, and x. τ GW IW x; θ, ) = k 2θ)1 x e 2θx Γ 1 1 ),, 2θx 3 Distribution of Functions of Random Variables In this section, distributions of functions of the IW and WIW random variables are presented. Consider the IW pdf: gx; θ, ) = θx 1 e θx, x >, θ >, >. 1. Let Y = θx. Then the pdf of Y is g 1 y) = θx 1 e θx θ) 1 x +1 = e θx = e y,

9 Valeriia Sherina and Broderick O. Oluyede 9 for y >, that is, if X IW θ, ), then Y = θx EXP 1), unit exponential distribution. 2. Let X W IW θ, ), that is 1 g W IW x; θ, ) = 2θ)1 x e ) 2θx, Γ 1 1 and Y = θx, then the resulting pdf of Y is given by 1 e 2y g 2 y; ) = 21 ), > 1, and y >. Γ 1 1 Note, that if l = k =, then g 1 y) = g 2 y), y >. 3. Now, let Z = 2θX, where X W IW θ, ). The pdf of the random variable Z is given by gz; θ, ) = 2θz) 2θ) e z ) = z1 1 e z ), Γ 1 1 Γ 1 1 for z >, > 1. Thus, if X W IW θ, ), then Z GAM1 1, 1). 4 Moments, Entropy and Fisher Information In this section, we present the moments and related functions as well as entropies and Fisher information for the WIW distribution. 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. 4.1 Moments and Moment Generating Function The c th non-central moment of the WIW distribution is given by EX c ) = 2θ) 1 1 xc x e 2θx Γ 1 1 ) dx.

10 1 Weighted Inverse Weibull Distribution Making the substitution u = 2θx, du = 2θx 1 dx, so that x = we obtain ) 1 2θ, u EX c ) = = 2θ) c u c+1 exp[ u] Γ 1 1 ) dx 2θ) c Γ 1 c + 1 ) Γ 1 1 ), 2) where > c + 1. Let δ c = Γ 1 c + 1 ), then the mean and variance are µ X = 2θ) 1 δ 1 δ, and σx 2 = 2θ) 2 δ δ 2 δ1) 2, respectively. The coefficient of δ 2 δ δ 2 δ1 2 variation CV) is CV =. The coefficient of skewness CS) is given δ 1 by CS = 2δ3 1 3δ δ 1 δ 2 + δδ 2 3, and the coefficient of kurtosis CK) is CK = [δ δ 2 δ1] δδ 3 4 4δδ 2 1 δ 3 + 6δ δ1δ 2 2 3δ1 4. [δ δ 2 δ1] 2 2 The graphs of CV, CS and CK versus are given in Figures 3 and 4. We can see decreasing coefficients CV, CS, CK) for increasing values of. Figure 3: Graphs of CV and CS for WIW Distribution The moment generating function MGF) of the WIW distribution is M X t) =

11 Valeriia Sherina and Broderick O. Oluyede 11 j= Figure 4: Graphs of CK for WIW Distribution t j j! E [Xj ], where E [X j ] = 2θ) j Γ 1 j + 1 ) Γ 1 1 ), > j + 1. Table 1 gives the mode, mean, standard deviation STD), CV, CS and CK for some values of the parameters θ,, k and l, k, l = or 1). We can see from Table 1, as increases, the values of CV, CS and CK decrease. We can also see that as θ increases, the values for mode and mean increase, for the selected values of the model parameters. 4.2 Shannon Entropy Shannon entropy [15] for WIW distribution is given by Hg W IW ) = log 2θ) 1 1 x e 2θx Γ 1 1 ) 2θ) 1 1 x e 2θx Γ 1 1 ) dx = [A + B + C],

12 12 Weighted Inverse Weibull Distribution Table 1: Mode, Mean, STD, Coefficients of Variation, Skewness and Kurtosis θ k l Mode Mean STD CV CS CK where A, B and C are obtained below: ) 2θ) 1 1 2θ) 1 1 x e 2θx A = log = log ) + δ δ 1 1 ) log2θ) logδ ), dx B = = 2θ)2 2θ)) 1 1 x e 2θx δ 1 δ e 2θx x 2 dx. 2θx ) dx Let u = 2θx, then du = 2θx 1 dx, x = ) 1 2θ, and we obtain u

13 Valeriia Sherina and Broderick O. Oluyede 13 B = Γ 2 δ 1. Also, C = 2θ) 1 1 x e 2θx δ ) logx) dx, = 1 [log2θ) logu)] e u u 1 du, δ where u = 2θx. Using the fact that Γ n) t) = log n x)x t 1 exp x) dx, the integral becomes C = δ δ log2θ). δ δ Consequently, Shannon entropy for WIW distribution is given by Γ 2 1 ) ) δ δ log + δ log2θ) δ Hg W IW ) =. δ 4.3 Rényi Entropy Rényi entropy [14] generalizes Shannon entropy. Renyi entropy of order t, where t > and t 1 is given by Note that H R g) = 1 1 t log [ ] g t x) dx. [ ] g t x) dx = 2θ) 1 1 t x e 2θx dx. W IW Let u = 2 θ t x, then the integral becomes g t x) dx = t 1 2θ) 1 t t t+1 W IW δ t Γ Rényi entropy for WIW distribution reduces to δ t 1 ). H R g W IW ) = log) + 1 t + 1 log2θ) + 1 t) logt) + 1 ) t 1 1 t log Γ t 1 t logδ ).

14 14 Weighted Inverse Weibull Distribution 4.4 Fisher Information Let Θ = θ, ). Fisher information matrix FIM) for WIW distribution is given by: [ IΘ) = Iθ, ) = I θθ I θ I θ I ], where the entries of the Iθ, ) are given in Appendix B. 5 Estimation of Parameters In this section, we obtain estimates of the parameters for the WIW distribution. Method of maximum likelihood ML) estimation is presented. Asymptotic confidence intervals and likelihood ratio test are also given. 5.1 Maximum Likelihood Estimation Let x 1, x 2,, x n be a random sample from a WIW distribution and Θ = θ, ). The log-likelihood function is ln L = lθ) = n ln) + n n ) ln2θ) n ln Γ 1 1 ) n n lnx i ) 2θ x i. i=1 The normal equations are ln L θ ln L = i=1 1 1ˆ ) ṋ θ 2 = ṋ + ṋ 2 ln2ˆθ) +2ˆθ n i=1 n i=1 n i=1 ) x ˆ i lnx i ) x ˆ i =, lnx i ) ṋ Ψ 1 1ˆ ) 2 =. 24) Therefore, if is known, we can obtain an estimate for ˆθ from 24) ˆθ = n ˆ 1) 2 ˆ, n ˆ i=1 x i

15 Valeriia Sherina and Broderick O. Oluyede 15 and if θ is known, we can find an estimate for using Newton s method. When all parameters are unknown, numerical methods must be used to obtain the MLE ˆθ, and ˆ of the parameters θ, and, respectively, since the system of equations does not admit any closed form solutions. 5.2 Asymptotic Confidence Intervals and Likelihood Ratio Test The multivariate normal distribution can be used to obtain confidence intervals and confidence regions for the parameters of the WIW distribution. The approximate 11 δ)% two-sided confidence intervals for the parameters θ, and are given by ˆθ ± z δ/2 [ V arˆθ) ] 1/2 and ˆ ± zδ/2 [ V ar ˆ) ] 1/2. respectively, where V ar ) are the diagonal [ elements of ] I 1 ˆΘ) or J 1 ˆΘ), corresponding to each parameter, JΘ) = 2 lθ) Θ= is the observed θ i θ ˆΘ j information matrix and z δ/2 is the upper δ 2 th percentile of the standard normal distribution. The likelihood ratio LR) statistic for testing θ = 1 is given by w = 2[lnLˆθ, ˆ, 1, 1)) lnl1,, 1, 1))], where ˆθ, and ˆ are the unrestricted estimates, and is restricted estimate. The LR test reject the null hypothesis if w > χ 2 ɛ, where χ 2 ɛ denote the upper 1ɛ% point of a χ 2 distribution with 1 degrees of freedom. 2x2 6 Simulation Study Various simulation were conducted for different sample sizes n=5, 1, 2, 3, 5, 1) to study the performance of IW and WIW distributions. We simulated 1 samples for Model 1 with the true values of the parameters θ =.5, = 3, k = 1, and l = 1, Model 2 with the true values of the parameters θ = 1.2, = 4.6, k = 1, and l = 1, and Model 3 with the true

16 16 Weighted Inverse Weibull Distribution values of the parameters θ = 1, = 2, k = 1, and l = 1. From the results of the simulations presented in Tables 2, 4 and 6, we can see that average bias for the parameters are very small, it is negative for the parameter l in the Model 1, the average bias for the parameters and k in the Model 2 is negative also. The root mean squared errors RMSEs) decreases as the sample size n increases. Also, as the sample size gets larger the mean estimates of the parameters gets closer to the true parameter values. When k = or l =, the simulation results are presented in Tables 3, 5 and 7. 7 Applications In this section, we present examples to illustrate the flexibility of the WIW distribution and its sub-models for data modeling. The first data set from Bjerkedal [1] represents the survival time, in days, of guinea pigs injected with different doses of tubercle bacilli. It is known that guinea pigs have high susceptibility of human tuberculosis. The data set consists of 72 observations. For the second example, the data is a subset of the breast feeding study from the National Longitudinal Survey of Youth, the complete data set is available in Klein and Moeschberger [1]. The data set considered here consists of the times to weaning 927 children of white-race mothers who choose to breast feed their children. The duration of the breast feeding was measured in weeks. Estimates of the parameters of WIW distribution standard error in parentheses), Akaike Information Criterion AIC), Consistent Akaike Information Criterion AICC), Bayesian Information Criterion BIC) are given in Table 8 for the first data set and in Table 9 for the second data set. Plots of the fitted densities and histogram of the data are given in Figures 5 and 6. Probability plots Chambers et al. [5]) are also presented in Figures 5 and 6. For the probability plot, we plotted for example, G W IW x j) ) = Γ1 1ˆ, 2ˆθx ˆ j) ) Γ1 1ˆ ) against j.375 n +.25, j = 1, 2,, n, where x j) are the ordered values of the observed data. We also computed a measure of closeness of each plot to the

17 Valeriia Sherina and Broderick O. Oluyede 17 Table 2: Simulation results for Model 1: Mean estimates, Average Bias and RMSEs IW WIW Sample size, n Parameter Mean Average Bias RMSE Mean Average Bias RMSE 5 θ k l θ k l θ k l θ k l θ k l θ k l diagonal line. This measure of closeness is given by the sum of squares n [ )] 2 j.375 SS = G W IW x j) ). n +.25 j=1

18 18 Weighted Inverse Weibull Distribution Table 3: Simulation results for Model 1: Mean estimates, Average Bias and RMSEs of the parameters θ, with k = and l = WIW, l= WIW, k= Sample size, n Parameter Mean Average Bias RMSE Mean Average Bias RMSE 5 θ k l θ k l θ k l θ k l θ k l θ k l For the first dataset, the LR test statistic of the hypothesis H : W IW 1,, 1, 1)

19 Valeriia Sherina and Broderick O. Oluyede 19 Table 4: Simulation results for Model 2: Mean estimates, Average Bias and RMSEs IW WIW Sample size, n Parameter Mean Average Bias RMSE Mean Average Bias RMSE 5 θ k l θ k l θ k l θ k l θ k l θ k l against H a : W IW θ,, 1, 1), is w = = The p-value is 2.86x1 34 <.1. Therefore, we reject H in favor of H a : W IW θ,, 1, 1).

20 2 Weighted Inverse Weibull Distribution Table 5: Simulation results for Model 2: Mean estimates, Average Bias and RMSEs of the parameters θ, with k = and l = WIW, l= WIW, k= Sample size, n Parameter Mean Average Bias RMSE Mean Average Bias RMSE 5 θ k l θ k l θ k l θ k l θ k l θ k l We can also test the hypothesis H : W IW θ, 1.5, 1, 1) against H a : W IW θ,, 1, 1), is w = = 35.. The p-value is 3.29x1 9 <.1. Therefore, we

21 Valeriia Sherina and Broderick O. Oluyede 21 Table 6: Simulation results for Model 3: Mean estimates, Average Bias and RMSEs IW WIW Sample size, n Parameter Mean Average Bias RMSE Mean Average Bias RMSE 5 θ k l θ k l θ k l θ k l θ k l θ k l reject H in favor of H a : W IW θ,, 1, 1). The values of the statistics AIC, AICC and BIC shows that sub-models W IW θ,,, 1) and W IW θ,,, )

22 22 Weighted Inverse Weibull Distribution Table 7: Simulation results for Model 3: Mean estimates, Average Bias and RMSEs of the parameters θ, with k = and l = WIW, l= WIW, k= Sample size, n Parameter Mean Average Bias RMSE Mean Average Bias RMSE 5 θ k l θ k l θ k l θ k l θ k l θ k l are equally better fits for this data. probability plot is smallest for the W IW θ,,, 1) model. Also, the value of SS given in the

23 Valeriia Sherina and Broderick O. Oluyede 23 Table 8: Estimates of models for Bjerkedal data set Estimates Statistics Model θ k l 2 log L AIC AICC BIC SS WIWθ,,,) WIWθ,,1,1) WIWθ,1.5,1,1) WIW1,,1,1) WIWθ,,1,) WIWθ,,,1) Figure 5: Fitted density and probability plots for guinea pigs survival time For the second dataset, the LR test statistic of the hypothesis H : W IW 1,, 1, 1)

24 24 Weighted Inverse Weibull Distribution Figure 6: Fitted density and probability plots for breast feeding data against H a : W IW θ,, 1, 1), is w = = The p-value is <.1. Therefore, we reject H in favor of H a : W IW θ,, 1, 1). According to the values of the statistics AIC, AICC and BIC the models W IW θ,,, 1) and W IW θ,,, ) are good fits for the second dataset, however the value of SS corresponding to the W IW θ,,, 1) model is the smallest. 8 Conclusion Statistical properties of the weighted generalized inverse Weibull distribution and its sub-models including the pdf, cdf, moment, hazard function, reverse hazard function, coefficient of variation, skewness and kurtosis, Fisher information, Shannon entropy, Renyi entropy and s-entropy are given. Estimation of the parameters of the models are also presented. Extensive simulation studies is carried out to examine the bias and mean square error of the maximum likelihood estimators of the parameters. Applications of the WIW distribution to real data shows a good fit. The proposed class of distributions contains a large number of distributions with potential applications to several areas of probability and statistics, finance, economics, and medicine. Acknowledgements. The authors are grateful to the referees for some useful comments on an earlier version of this manuscript which led to this improved version.

25 Valeriia Sherina and Broderick O. Oluyede 25 Table 9: Estimates of models for the breast feeding data set Estimates Statistics Model θ k l 2 log L AIC AICC BIC SS WIWθ,,,) ).2136) WIWθ,,1,1) ).28) WIWθ,1.5,1,1) ) WIW1,,1,1) ) WIWθ,,1,) ).28) WIWθ,,,1) ).2136) References [1] Bjerkedal, T., Acquisition of Resistance in Guinea Pigs Infected with Different Doses of Virulent Tubercle Bacilli, American Journal of Hygiene, 72, 196), [2] Calabria, R., Pulcini, G., Confidence Limits for Reliability and Tolerance Limits in the Inverse Weibull Distribution, Engineering and System Safety, 24, 1989), [3] Calabria, R., Pulcini, G., On the Maximum Likelihood and Least Squares Estimation in Inverse Weibull Distribution, Statistica Applicata, 2, 199), [4] Calabria, R., Pulcini, G. Bayes 2-Sample Prediction for the Inverse Weibull Distribution, Communications in Statistics-Theory Meth., 236),

26 26 Weighted Inverse Weibull Distribution 1994), [5] Chambers, J., Cleveland, W., Kleiner, B., and Tukey, P., Graphical Methods of Data Analysis, Chapman and Hall, London, [6] Glaser, R.E., Bathtub and Related Failure Rate Characterizations, Journal of American Statistical Association, 75, 198), [7] Johnson, N.L., Kotz, S., Balakrishnan, N., Continuous Univariate Distributions-1, Second Edition, John Wiley and Sons, Inc., Hoboken, NJ, [8] Keller, A.Z., Giblin, M.T., Farnworth, N.R., Reliability Analysis of Commercial Vehicle Engines, Reliability Engineering, 1, 1985), [9] Khan, M.S., Pasha, G.R., Pasha, A.H., Theoretical Analysis of Inverse Weibull Distribution, WSEAS Transactions on Mathematics, 72), 28), [1] Klein, J.P., Moeschberger, M.L., Survival Analysis: Techniques for Censoring and Truncated Data, 2nd ed., Springer-Verlag New York, Inc., New York, 23. [11] Lehmann, E.L., Theory of Point Estimation, Springer-Verlag New York, Inc., New York, [12] Oluyede, B. O., On Inequalities and Selection of Experiments for Length Biased Distributions, Probability in the Engineering and Informational Sciences, 132), 1999), [13] Patil, G. P., Rao, C. R., Weighted Distributions and Size-biased Sampling with Applications to Wildlife Populations and Human Families, Biometrics, 342), 1978), [14] Renyi, A., On Measures of Information and Entropy, Proceedings of the Fourth Berkeley Symposium on Mathematics, Statistics and Probability, 1, 1961), [15] Shannon, E., A Mathematical Theory of Communication, The Bell System Technical Journal, 271), 1948),

27 Valeriia Sherina and Broderick O. Oluyede 27 Appendix A. Some Basic Utility Notions Here, some basic utility notions and definitions are presented. The n th - order derivative formula of gamma function is given by: Γ n) s) = z s 1 log z) n exp z) dz. This derivative is used in this paper. The lower incomplete gamma function and the upper incomplete gamma function are γs, x) = x t s 1 e t dt and Γs, x) = x t s 1 e t dt, respectively. Also, 1 z) b 1 = 1) j Γb) j= Γb j)γj + 1) zj, for real non-integer b > and z < 1. Appendix B. Fisher information The FI for the WIW distribution, that X contains about the parameters Θ = θ, ) is obtained below. We have the following partial derivatives of ln [gx; θ, )] with respect to the parameters: ln g W IW x; θ, ) θ = 1 1 θ 2x, 27) ln g W IW x; θ, ) = ln2θ) + 2θx lnx) 2 ) + 1 Ψ 1 1 lnx) 2, 28) We differentiate 27) with respect to θ and, we obtain: 2 ln g W IW x; θ, ) θ 2 = 1 1 θ 2, 2 ln g W IW x; θ, ) θ = 1 θ 2 + 2x lnx),

28 28 Weighted Inverse Weibull Distribution Differentiating 28) with respect to, we get: 2 ln g W IW x; θ, ) = 2 ln2θ) 2θx ln 2 x) 2 3 4θx lnx) 1 ) 2 Ψ 1, Ψ1 1 ). 4 3 Now, we compute the following expectations: E [ X ], E [ X ln X) ], E [ X ln 2 X) ] in order to obtain FIM Iθ, ). E [ X ] = = x g W IW x)dx x 2 [2θ] 1 1 e 2θx dx ). Γ 1 1 Let u = 2θx, du = 2θx 1 dx and x = ) 1 2θ. The integral becomes: u E [ X ] = = ) 1 2θ 1 2θ) 1 e u du ) u Γ 1 1 ) 2θ) 1 Γ 2 1 ). Γ 1 1 E [ X ln X) ] = = x lnx)g W IW x)dx x 2 2θ) 1 k lnx) e 2θx dx ). Γ 1 1

29 Valeriia Sherina and Broderick O. Oluyede 29 Making the same substitution u = 2θx, we obtain: E [ X ln X) ] = = ) 1 2θ 1 2θ) 1 e u du ) u Γ 1 1 2θ ) ) 1 ln du u ) ) 2θ) 1 ln 2θ) 1 Γ 2 1 ) Γ 1 1 ) 2θ) 1 Γ ). Γ 1 1 E [ X ln 2 X) ] = = x ln 2 x)g W IW x)dx x 2 [2θ] 1 1 ln 2 x) e 2θx dx ). Γ 1 1 Making the same substitution one more time we get: E [ X ln 2 X) ] = = ) 1 2θ 1 2θ) 1 e u du ) u Γ 1 1 2θ ) ) 1 ln 2 du u 2θ) 1 ) Γ lnu) ln 2θ) 1 [ ) u e u ln 2θ) 2 1 ) + 1 ] 2 ln2 u) du

30 3 Weighted Inverse Weibull Distribution = ) ) 2θ) 1 ln 2θ) 2 1 Γ 2 1 ) Γ 1 1 ) ) 2θ) 1 2 ln 2θ) 1 Γ 2 1 ) Γ 1 1 ) 2θ) 1 Γ 2 2) 1 + ). 2 Γ 1 1 Now, the entries of FIM are given below: ) ) I = ln2θ) Ψ 1, 1 1 2Ψ ) 3 ) ln 2 2θ)Γ ln2θ)γ ) ) 2 Γ 1 1 Γ 1 1 ) ) ) Γ 2 2) 1 ln2θ)γ 2 1 2Γ ) + ) + ), 2 Γ 1 1 Γ 1 1 Γ 1 1 I θθ = 1 1 θ 2, ) ) I θ = 1 ln2θ)γ 2 1 Γ θ ) ). θγ 1 1 θγ 1 1