On a Generalization of Weibull Distribution and Its Applications
|
|
- Clifton Austin
- 6 years ago
- Views:
Transcription
1 International Journal of Statistics and Applications 26, 6(3): DOI:.5923/j.statistics On a Generalization of Weibull Distribution and Its Applications M. Girish Babu Department of Statistics, Government Arts and Science College, Meenchanda, Kozhikode, India Abstract The Weibull and related models have been used in many applications for solving a variety of problems from many disciplines. Here we introduce a new family of distributions, namely Weibull-truncated negative binomial distribution (WTNB) and study some properties of it. Eponential-truncated negative binomial (ETNB) and Marshall-Olkin Weibull (MOW) distribution are special cases of this distribution. We have analyzed a real data set of serum creatinine values and found that this new distribution is a good fit to model the data, compared to Weibull, Gamma, Eponentiated Weibull, ETNB and MOW models. Keywords Eponential truncated negative binomial distribution, Marshall-Olkin family of distributions, Shaon entropy, Weibull distribution. Introduction The Weibull distribution is a very popular distribution for modeling lifetime data and for modeling phenomenon with monotone failure rates. The distribution is named after Waloddi Weibull who was the first to promote the usefulness of this to model the breaking strength of materials (Weibull, 939). A similar model was proposed earlier by Rosin and Rammler (933) in the contet of modeling the variability in the diameter of powder particles being greater than a specific size. The earlier known publication dealing with the Weibull distribution is a work by Fisher and Tippet (928) where this distribution is obtained as the limiting distribution of the smallest etremes in a sample. Gumbel (958) refers to the Weibull distribution as the third asymptotic distribution of the smallest etremes. The Weibull, and related models have been used in many applications, and for solving a variety of problems from many disciplines. Jayakumar and Girish (25) studied some generalizations of Weibull distribution and related time series models. Marshall and Olkin (997) proposed a new method of generating a family of distributions by introducing an additional parameter. Many authors have studied properties of various univariate distributions belonging to the family of Marshall-Olkin distributions, see, Alice and Jose (23, 25), Ghitany et al. (25, 27) and Jayakumar and Thomas (28). A generalization of the Marshall-Olkin distributions was * Corresponding author: giristat@gmail.com (M. Girish Babu) Published online at Copyright 26 Scientific & Academic Publishing. All Rights Reserved introduced by Nadarajah et al. (23) as follows: Let XX, XX 2, be a sequence of independent and identically distributed random variables with survival function FF(). Le NN be a truncated negative binomial random variable with parameters αα (,) and θθ >, that is, αα PP(NN = ) = θθ + αα θθ θθ θθ ( αα), =,2,. Consider UU NN = min(xx, XX 2,, XX NN ). Then PP(UU NN > ) = GG (; αα, θθ) = ααθθ + θθ ααθθ θθ ( αα)ff () = = ααθθ αα θθ FF() + ααff () θθ (.) Similarly, if αα > and NN is a truncated negative binomial random variable with parameters αα and θθ >, then VV NN = ma(xx, XX 2,, XX NN ) also has the survival function given in (.). Here note that in (.), if αα, then GG (; αα, θθ) FF(). If θθ =, then this family reduces to the family of Marshall Olkin distributions. Thus the family of distributions described in (.) is a generalization of the family of Marshall-Olkin distributions. This family can be interpreted as follows. Suppose the failure times of a device are observed. Every time a failure occurs the device is repaired to resume function. Suppose also that the device is deemed no longer useable when a failure occurs that eceeds a certain level of severity. Let XX, XX 2, denote the failure times and NN denote the number of failures. Then UU NN will represent the time to the first failure of the device and VV NN will represent the life time of the device. Thus this family can be used to model both the time to the first failure and the life time. The probability density function of (.) is
2 International Journal of Statistics and Applications 26, 6(3): gg(; αα, θθ) = θθαα θθ ( αα)ff() αα θθ FF()+ααFF() The hazard rate function is given by h(; αα, θθ) = θθ( αα)ff()h FF () FF()+ ααff() FF()+ααFF() θθ θθ + (.2) (.3) where h FF () = ff() is the hazard rate corresponding to F. FF() Nadarajah et al. (23) introduced and studied a new family of distribution as eponential-truncated negative binomial distribution with parameters αα, θθ and λλ by substituting FF() = ee λλλλ, λλ >, > in the survival function (.). That is, ααθθ GG (; αα, θθ, λλ) = αα θθ ee λλλλ + ααee λλλλ θθ (.4) for αα >, θθ >, λλ > and >. The paper is organized as follows. In section 2, we propose a new family of distributions, namely Weibull-truncated negative binomial distribution (WTNB). We study some properties of WTNB, such as the behavior of hazard rate, moments, Shaon and Renyi entropies and distributions of order statistics. Also the maimum likelihood method of estimation is used to obtain the estimates of parameters of WTNB. In section 3, we analyze a real data set of serum creatinine (mg.dl) values and found that WTNB is the most appropriate model for the data. The performance of the WTNB is compared to the well known models such as Weibull, Gamma, eponentiated Weibull, ETNB and Marshall-Olkin Weibull using AIC, BIC, K-S statistic and P-values. 2. Weibull-Truncated Negative Binomial Distribution We propose a new family of distribution named as Weibull-truncated negative binomial (WTNB) distribution with parameters αα >, θθ >, cc > and > by substituting FF() = ee cc, cc >, > in the survival function (.). Then, αα θθ GG (; αα, θθ, cc) = αα θθ ee cc + ααee cc θθ (2.) The probability density function of this distribution is gg(; αα, θθ, cc) = ( αα)θθθθ θθ cc cc ee cc αα θθ ee cc +ααee cc The hazard rate function is given by h(; αα, θθ, cc) = ( αα)θθθθ cc ee cc θθ + (2.2) ee cc +ααee cc [ ee cc +ααee cc θθ ] (2.3) The shape of density and hazard rate functions are shown in the following figures. The cumulative probabilities at different choices of parameters are computed by using MATHCAD software and the results are shown in Table 2.. Some distributions arise as special cases of the WTNB(αα, θθ, cc) Case I : For cc =, ααθθ GG (; αα, θθ) = ( αα θθ ee + ααee ) θθ (2.4) This is the survival function of two parameter Eponential-truncated negative binomial distribution..5.2 g(,.5,.5, ) g(,.5,.5,.5) g(,.5,,.5) g(,.5,.5, 3) Figure 2.. Probability density function of WTNB distribution
3 7 M. Girish Babu: On a Generalization of Weibull Distribution and Its Applications 2.6 h(,.5,, ) h(,.5,.5,.5) h(,.5, 2,.5) h(,.5, 5,.5) Figure 2.2. Hazard rate function of WTNB distribution X Table 2.. Cumulative Probabilities of WTNB distribution θθ =, αα =.5, cc =.5 θθ =.5, αα =.5, cc =.5 θθ =.5, αα =.5, cc =.5 θθ = 2, αα =.5, cc = Case II: For θθ =, GG (, αα, cc) = αα ee cc (2.5) ( αα)ee cc This is the survival function of Marshall-Olkin etended Weibull distribution with parameters αα and cc. Case III: For θθ =, αα = GG (, cc) = ee cc (2.6) This is the survival function of one parameter Weibull distribution. Case IV: For θθ =, αα = 2 GG (, cc) = 2 (2.7) +ee cc This is the survival function of the generalized half logistic distribution. Case V: When αα, WTNB(αα, θθ, cc) reduces to the one parameter Weibull distribution. A random sample from WTNB(αα, θθ, cc) distribution can be simulated as XX = ln ααααθθ +YY αα θθ θθ αα cc for YY~UU(,). (2.8)
4 International Journal of Statistics and Applications 26, 6(3): Moments Suppose that X has the WTNB(αα, θθ, cc) distribution. The n th moment can be written as Taking uu = ee cc, (2.9) reduces to EE(XX ) = ( αα)θθααθθ cc ( αα θθ ) EE(XX ) = ( αα)θθααθθ ( αα θθ ) If αα <, then by the series epansion ( ) mm = mm + kk kk= kk, equation (2.) can be written as mm where uu kk ( log uu) cc Therefore, If αα < αα, then we have EE(XX ) = ( αα)θθααθθ ( αα θθ ) = cc cc cc ( ) (kk+) + +cc ee cc [ ( αα)ee cc ] θθ + ( log (uu)) cc [ ( αα)uu] θθ + + kk θθ ( αα)kk uu kk ( log uu) cc θθ kk= EE(XX ) = ( αα)θθααθθ cc cc cc ( ) ( αα θθ ) EE(XX ) = ( αα)θθ ( αα θθ )αα ( log ( uu)) cc [+ αα uu]θθ + αα (2.9) (2.) θθ + kk ( αα)kk kk= θθ (kk+) +. (2.) Taking uu = vv in equation (2.2) and using equation ( ) of Prudnikov et al. (986), we have 2.2. Shaon Entropy EE(XX ) = = = ( αα)θθ ( αα θθ )αα ( log vv) [ + αα αα ( vv)]θθ+ ( αα)θθ ( αα θθ )αα ( log vv) cc θθ αα θθ kk= cc + kk θθ θθ ( )kk αα αα ( ) kk θθ + kk αα θθ αα kk= = θθ cc cc cc ( ) kk kk+ ( log vv) cc ( vv) kk ( vv) kk (2.2) θθ + kk αα θθ θθ ( )kk αα ( ) jj (kk+ jj) αα kk+ kk jj kk= jj= (2.3) jj!(jj+) cc + Entropy is a measure of variation or uncertainty. The Renyi entropy of a random variable with probability density function gg(. ) is defined as The Renyi entropy of WTNB(αα, θθ, cc) is Letting uu = ee cc, we have II RR (γγ) = II RR (γγ) = The Shaon entropy is given by II RR (γγ) = log γγ ggγγ (), γγ >, γγ. (2.4) γγ log ( αα)θθαα θθ cc cc ee cc ( αα θθ )( ( αα)ee cc ) θθ+ γγ log cc γγ ( αα)θθααθθ αα θθ γγ ( log uu) γγ cc uu γγ ( αα)ee cc γγ (θθ +) (2.5)
5 72 M. Girish Babu: On a Generalization of Weibull Distribution and Its Applications 2.3. Order Statistics ααθθ EE[ log gg(xx)] = log (cc )EE(log XX) + ( αα)θθαα θθ cc EE(XXcc ) + (θθ + ) EElog ( αα)ee XXcc (2.6) Let XX, XX 2,, XX are independent random variables having the WTNB (αα, θθ, cc) distribution. Let XX ii: denote the i th order statistic. The probability density function of XX ii: is! gg ii: (; αα, θθ, cc) = (ii )! ( ii)! gg(; αα, θθ, cc) GGii (; αα, θθ, cc)gg ii (; αα, θθ, cc) = ( ) ii!( αα)θθαα θθ ( + ii) cc. cc ee cc (ii )!( ii)! αα θθ ( αα)ee cc θθ + ii αα θθ ( αα)ee cc θθ Using the binomial series epansion, the probability density function can be written as gg ii: (; αα, θθ, cc) = ii ii ( ) ii! αα θθ( ii) (ii )! ( ii)! ( αα θθ ii ) kk kk= ll= ii ( αα)ee cc θθ. (2.7) ii ( )kk+ll αα θθ(kk+ll+) ll αα θθθθ (kk + ll + ) gg(; αα, θθ(kk + ll + ), cc). (2.8) This shows that XX ii: is a finite miture of WTNB random variables Estimation For a given sample(, 2,, ), the log-likelihood function is given by log LL(αα, θθ, cc) = log ( αα)θθααθθ cc cc + (cc ) log αα θθ ii= ii ii= ii (θθ + ) log ( αα)ee ii cc ii= (2.9) The partial derivatives of the log-likelihood function with respect to the parameters are log LL log LL = +( θθ)ααθθ +θθαα θθ ( αα) αα θθ log LL = log αα cc + (θθ + ) ee ii αα ii= (2.2) ( αα)ee ii cc + log ( αα θθ θθ αα)ee ii ii= (2.2) cc ii cc log ii ee ii cc = + log cc ii= ii ii= cc ii log ii (θθ + )( αα) ii= (2.22) ( αα)ee ii cc Equating these partial derivatives equal to zero and solving these equations numerically, we get the maimum likelihood estimates of αα, θθ and cc. 3. Application In this section, we analyze a real data set and found that WTNB distribution gives the best fit for the data. We consider a real data set of serum creatinine (mg/dl) value of 3 samples reported on February 2 nd, 26 at the Biochemistry Laboratory of Govt. Medical College, Calicut, Kerala. The data are follows: Creatine is a chemical made by the body and is used to supply energy mainly to muscle. Serum creatinine is a chemical waste product of creatine. Creatinine is removed from the body entirely by the kidneys. If kidney function is not normal, the creatinine level will increase. The normal range of creatinine is about.7 to.3 mg/dl. The descriptive measures of the 3 samples are as follows:
6 International Journal of Statistics and Applications 26, 6(3): Initially a histogram of the data is plotted and a normal curve is embedded in it and we observe that normal distribution is not suitable to model the data since the data is highly positively skewed. Table 3.. Descriptive statistics of Creatinine (mg/dl) Mean Median SD Skewness Kurtosis Min Ma Figure 3.. Empirical structure of the serum creatinine data The P-P plot and Q-Q plot of the creatinine data is as follows
7 74 M. Girish Babu: On a Generalization of Weibull Distribution and Its Applications Figure 3.2. P-P plot and Q-Q plot of the serum creatinine data Since the data shows a positively skewed nature, the symmetrical distributions will not be a suitable choice. So we fitted the data using WTNB with three parameters and compared with the well known distributions like Weibull, Gamma, Eponentiated Weibull, Marshall-Olkin Weibull and ETNB. For comparing the goodness of fit of the model we used the information criteria, Akaike Information Criterion (AIC = -2 log L+2k), Bayesian Information Criterion (BIC = -2log L+k log n) and the Kolmogorov-Smirnov statistic, where k is the number of unknown parameters, log L is the log-likelihood function value and n is the sample size. The results are presented in Table 3.2. Table 3.2. Parameter estimates and goodness of fit statistics for various models fitted to the serum creatinine data Weibull Gamma Distribution Parameters AIC BIC K-S P value ff(; λλ, cc) = ccλλ cc cc ee (λλλλ )cc, λλ, cc >. ff(; ββ, λλ) = Eponentiated Weibull λλββ Γ(ββ) ββ ee λλλλ, ββ, λλ >. ff(; λλ, θθ, cc) = ccλλ cc θθ cc ee (λλλλ )cc ee (λλλλ )cc θθ, λλ, θθ, cc >. ETNB ff(; αα, θθ, λλ) = ( αα)θθαα θθ λλee λλλλ ( αα θθ )( ( αα)ee λλλλ ) θθ+, αα, θθ, λλ > Marshall-Olkin Weibull ff(; αα, λλ, cc) = ααααλλcc cc (λλλλ )cc ee [ ( αα)ee (λλλλ )cc ] 2, αα, cc, λλ > WTNB ff(; αα, θθ, cc) = ( αα)θθαα θθ cc cc cc ee ( αα θθ )( ( αα)ee cc ) θθ+, αα, θθ, cc > λλ =., cc =2.55. ββ =5.22, λλ =6.54. λλ =.36, θθ =.66, cc =.95. αα =2.53, θθ =7.7, λλ =3.73. αα=.39, λλ =.9, cc =3.6. αα =.5, θθ =.52, cc = From the above Table 3.2., we can see that the WTNB distribution is the suitable model for the given data. The probability plots for the fitted models are presented in figure 3.3.
8 International Journal of Statistics and Applications 26, 6(3): Weibull distribution Weibuli-truncated negativ Gamma distribution Marshall-Olkin Weibull dis Eponential-truncated nega Eponentiated Weibull dis Figure 3.3. Fitted probability density function for the serum creatinine (mg/dl) data
9 76 M. Girish Babu: On a Generalization of Weibull Distribution and Its Applications We have used the nlm function of R software to compare the empirical distribution and the theoretical distribution of all the si distributions mentioned above. From the results out of the si fitted models, WTNB is closer to the empirical distribution of the serum creatinine data. The following figure shows the closeness of the distribution functions. REFERENCES [] Alice, T. and K.K. Jose (23): Marshall-Olkin Pareto process, Far East Journal of Theoretical Statistics, 9, [2] Alice, T. and Jose, K.K. (25): Marshall-Olkin semi-weibull minification processes, Recent Advances in Statistical Theory and Applications,, 6-7. [3] Fisher, R.A. and Tippett, L.H.C. (928): Limiting forms of the frequency distribution of the largest and smallest member of a sample, Proceedings of the Cambridge Philosophical Society, 24, 8-9. [4] Ghitany, M.E., Al-Hussaini, E.K. and Al-Jaralla, R.A. (25): Marshall-Olkin etended Weibull distribution and its application to censored data, Journal of Applied Statistics, 32, [5] Ghitany, M.E., Al-Awadgi, F.A. and Alkhalfan, I.A. (27): Marshall-Olkin etended Loma distribution and its application to censored data, Communication in Statistical Theory and Methods, 36, [6] Gumbel, E.J. (958): Statistics of Etremes, Columbia University Press, New York. [7] Jayakumar, K. and Girish, B.M. (25): Some generalizations of Weibull distribution and related processes, Journal of Statistical Theory and Applications, 4, Figure 3.4. WTNB distribution function and empirical distribution 4. Conclusions In the present paper, we have studied a new family of distributions, namely Weibull truncated negative binomial distributions. Some properties of this distribution such as hazard rate, moments, Shaon and Renyi entropies, distributions of order statistics are studied. This distribution is found to be the most appropriate model to fit the serum creatinine data compared to the Weibull, Gamma, Eponentiated Weibull, ETNB and MOW models. We hope that the new family will attract wider application in life time modeling. [8] Jayakumar, K. and Thomas, M. (28): On a generalization to Marshall-Olkin scheme and its application to Burr type XII distribution, Statistical Papers, 49, [9] Marshall, A.W. and Olkin, I. (997): A new method for adding parameter to a family of distributions with application to eponential and Weibull families, Biometrika, 84, [] Nadarajah, S., Jayakumar, K. and Ristic, M.M. (23): A new family of lifetime models, Journal of Statistical Computation and Simulation, 83, [] Prudnikov, A.P., Brychkov, Y.A., and Marichev, O.I. (986): Integrals and Series Vol., Gordon and Breach Sciences, Amsterdam, Netherlands. [2] Rosin, P. and Rammler, E. (933): The laws governing the fineness of powdered coal, Journal of the Institute of Fuels, 6, [3] Weibull, W. (939): A statistical theory of the strength of material, Ingeniors Vetenskaps Akademiens Handligar Report No.5, Stockholm.
Further results involving Marshall Olkin log logistic distribution: reliability analysis, estimation of the parameter, and applications
DOI 1.1186/s464-16-27- RESEARCH Open Access Further results involving Marshall Olkin log logistic distribution: reliability analysis, estimation of the parameter, and applications Arwa M. Alshangiti *,
More informationA Marshall-Olkin Gamma Distribution and Process
CHAPTER 3 A Marshall-Olkin Gamma Distribution and Process 3.1 Introduction Gamma distribution is a widely used distribution in many fields such as lifetime data analysis, reliability, hydrology, medicine,
More informationSome Generalizations of Weibull Distribution and Related Processes
Journal of Statistical Theory and Applications, Vol. 4, No. 4 (December 205), 425-434 Some Generalizations of Weibull Distribution and Related Processes K. Jayakumar Department of Statistics, University
More informationTransmuted Exponentiated Gumbel Distribution (TEGD) and its Application to Water Quality Data
Transmuted Exponentiated Gumbel Distribution (TEGD) and its Application to Water Quality Data Deepshikha Deka Department of Statistics North Eastern Hill University, Shillong, India deepshikha.deka11@gmail.com
More informationBivariate Weibull-power series class of distributions
Bivariate Weibull-power series class of distributions Saralees Nadarajah and Rasool Roozegar EM algorithm, Maximum likelihood estimation, Power series distri- Keywords: bution. Abstract We point out that
More informationPROPERTIES AND DATA MODELLING APPLICATIONS OF THE KUMARASWAMY GENERALIZED MARSHALL-OLKIN-G FAMILY OF DISTRIBUTIONS
Journal of Data Science 605-620, DOI: 10.6339/JDS.201807_16(3.0009 PROPERTIES AND DATA MODELLING APPLICATIONS OF THE KUMARASWAMY GENERALIZED MARSHALL-OLKIN-G FAMILY OF DISTRIBUTIONS Subrata Chakraborty
More informationA THREE-PARAMETER WEIGHTED LINDLEY DISTRIBUTION AND ITS APPLICATIONS TO MODEL SURVIVAL TIME
STATISTICS IN TRANSITION new series, June 07 Vol. 8, No., pp. 9 30, DOI: 0.307/stattrans-06-07 A THREE-PARAMETER WEIGHTED LINDLEY DISTRIBUTION AND ITS APPLICATIONS TO MODEL SURVIVAL TIME Rama Shanker,
More informationInternational Journal of Scientific & Engineering Research, Volume 7, Issue 8, August ISSN MM Double Exponential Distribution
International Journal of Scientific & Engineering Research, Volume 7, Issue 8, August-216 72 MM Double Exponential Distribution Zahida Perveen, Mubbasher Munir Abstract: The exponential distribution is
More informationVariations. ECE 6540, Lecture 02 Multivariate Random Variables & Linear Algebra
Variations ECE 6540, Lecture 02 Multivariate Random Variables & Linear Algebra Last Time Probability Density Functions Normal Distribution Expectation / Expectation of a function Independence Uncorrelated
More informationMARSHALL-OLKIN EXTENDED POWER FUNCTION DISTRIBUTION I. E. Okorie 1, A. C. Akpanta 2 and J. Ohakwe 3
MARSHALL-OLKIN EXTENDED POWER FUNCTION DISTRIBUTION I. E. Okorie 1, A. C. Akpanta 2 and J. Ohakwe 3 1 School of Mathematics, University of Manchester, Manchester M13 9PL, UK 2 Department of Statistics,
More informationGumbel Distribution: Generalizations and Applications
CHAPTER 3 Gumbel Distribution: Generalizations and Applications 31 Introduction Extreme Value Theory is widely used by many researchers in applied sciences when faced with modeling extreme values of certain
More informationReliability Analysis Using the Generalized Quadratic Hazard Rate Truncated Poisson Maximum Distribution
International Journal of Statistics and Applications 5, 5(6): -6 DOI:.59/j.statistics.556.6 Reliability Analysis Using the Generalized Quadratic M. A. El-Damcese, Dina A. Ramadan,* Mathematics Department,
More informationMathematical Model for Blood Flow Restriction Exercise Stimulates mtorc1 Signaling in Older Men
International Journal of Scientific Research Publications Volume 7 Issue February 07 356 Mathematical Model for Blood Flow Restriction Exercise Stimulates mtorc Signaling in Older Men Geetha.T * Poongothai.
More informationThe Inverse Weibull Inverse Exponential. Distribution with Application
International Journal of Contemporary Mathematical Sciences Vol. 14, 2019, no. 1, 17-30 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ijcms.2019.913 The Inverse Weibull Inverse Exponential Distribution
More informationECE 6540, Lecture 06 Sufficient Statistics & Complete Statistics Variations
ECE 6540, Lecture 06 Sufficient Statistics & Complete Statistics Variations Last Time Minimum Variance Unbiased Estimators Sufficient Statistics Proving t = T(x) is sufficient Neyman-Fischer Factorization
More informationOn Five Parameter Beta Lomax Distribution
ISSN 1684-840 Journal of Statistics Volume 0, 01. pp. 10-118 On Five Parameter Beta Lomax Distribution Muhammad Rajab 1, Muhammad Aleem, Tahir Nawaz and Muhammad Daniyal 4 Abstract Lomax (1954) developed
More informationTransmuted Pareto distribution
ProbStat Forum, Volume 7, January 214, Pages 1 11 ISSN 974-3235 ProbStat Forum is an e-journal For details please visit wwwprobstatorgin Transmuted Pareto distribution Faton Merovci a, Llukan Puka b a
More informationComparing Different Estimators of Reliability Function for Proposed Probability Distribution
American Journal of Mathematics and Statistics 21, (2): 84-94 DOI: 192/j.ajms.212. Comparing Different Estimators of Reliability Function for Proposed Probability Distribution Dhwyia S. Hassun 1, Nathie
More informationAn Extended Weighted Exponential Distribution
Journal of Modern Applied Statistical Methods Volume 16 Issue 1 Article 17 5-1-017 An Extended Weighted Exponential Distribution Abbas Mahdavi Department of Statistics, Faculty of Mathematical Sciences,
More informationTHE FIVE PARAMETER LINDLEY DISTRIBUTION. Dept. of Statistics and Operations Research, King Saud University Saudi Arabia,
Pak. J. Statist. 2015 Vol. 31(4), 363-384 THE FIVE PARAMETER LINDLEY DISTRIBUTION Abdulhakim Al-Babtain 1, Ahmed M. Eid 2, A-Hadi N. Ahmed 3 and Faton Merovci 4 1 Dept. of Statistics and Operations Research,
More informationHacettepe Journal of Mathematics and Statistics Volume 45 (5) (2016), Abstract
Hacettepe Journal of Mathematics and Statistics Volume 45 (5) (2016), 1605 1620 Comparing of some estimation methods for parameters of the Marshall-Olkin generalized exponential distribution under progressive
More informationInternational Journal of Physical Sciences
Vol. 9(4), pp. 71-78, 28 February, 2014 DOI: 10.5897/IJPS2013.4078 ISSN 1992-1950 Copyright 2014 Author(s) retain the copyright of this article http://www.academicjournals.org/ijps International Journal
More informationResearch Article On a Power Transformation of Half-Logistic Distribution
Probability and Statistics Volume 016, Article ID 08436, 10 pages http://d.doi.org/10.1155/016/08436 Research Article On a Power Transformation of Half-Logistic Distribution S. D. Krishnarani Department
More informationA New Extended Uniform Distribution
International Journal of Statistical Distriutions and Applications 206; 2(3): 3-4 http://wwwsciencepulishinggroupcom/j/ijsda doi: 0648/jijsd20602032 ISS: 2472-3487 (Print); ISS: 2472-309 (Online) A ew
More informationTHE WEIBULL GENERALIZED FLEXIBLE WEIBULL EXTENSION DISTRIBUTION
Journal of Data Science 14(2016), 453-478 THE WEIBULL GENERALIZED FLEXIBLE WEIBULL EXTENSION DISTRIBUTION Abdelfattah Mustafa, Beih S. El-Desouky, Shamsan AL-Garash Department of Mathematics, Faculty of
More informationINVERTED KUMARASWAMY DISTRIBUTION: PROPERTIES AND ESTIMATION
Pak. J. Statist. 2017 Vol. 33(1), 37-61 INVERTED KUMARASWAMY DISTRIBUTION: PROPERTIES AND ESTIMATION A. M. Abd AL-Fattah, A.A. EL-Helbawy G.R. AL-Dayian Statistics Department, Faculty of Commerce, AL-Azhar
More informationThe Marshall-Olkin Flexible Weibull Extension Distribution
The Marshall-Olkin Flexible Weibull Extension Distribution Abdelfattah Mustafa, B. S. El-Desouky and Shamsan AL-Garash arxiv:169.8997v1 math.st] 25 Sep 216 Department of Mathematics, Faculty of Science,
More informationA Quasi Gamma Distribution
08; 3(4): 08-7 ISSN: 456-45 Maths 08; 3(4): 08-7 08 Stats & Maths www.mathsjournal.com Received: 05-03-08 Accepted: 06-04-08 Rama Shanker Department of Statistics, College of Science, Eritrea Institute
More informationThe Kumaraswamy-Burr Type III Distribution: Properties and Estimation
British Journal of Mathematics & Computer Science 14(2): 1-21, 2016, Article no.bjmcs.19958 ISSN: 2231-0851 SCIENCEDOMAIN international www.sciencedomain.org The Kumaraswamy-Burr Type III Distribution:
More informationTHE ODD LINDLEY BURR XII DISTRIBUTION WITH APPLICATIONS
Pak. J. Statist. 2018 Vol. 34(1), 15-32 THE ODD LINDLEY BURR XII DISTRIBUTION WITH APPLICATIONS T.H.M. Abouelmagd 1&3, Saeed Al-mualim 1&2, Ahmed Z. Afify 3 Munir Ahmad 4 and Hazem Al-Mofleh 5 1 Management
More informationMAXIMUM LQ-LIKELIHOOD ESTIMATION FOR THE PARAMETERS OF MARSHALL-OLKIN EXTENDED BURR XII DISTRIBUTION
Available online: December 20, 2017 Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. Volume 68, Number 1, Pages 17 34 (2019) DOI: 10.1501/Commua1_0000000887 ISSN 1303 5991 http://communications.science.ankara.edu.tr/index.php?series=a1
More informationEstimate by the L 2 Norm of a Parameter Poisson Intensity Discontinuous
Research Journal of Mathematics and Statistics 6: -5, 24 ISSN: 242-224, e-issn: 24-755 Maxwell Scientific Organization, 24 Submied: September 8, 23 Accepted: November 23, 23 Published: February 25, 24
More informationPromotion Time Cure Rate Model with Random Effects: an Application to a Multicentre Clinical Trial of Carcinoma
www.srl-journal.org Statistics Research Letters (SRL) Volume Issue, May 013 Promotion Time Cure Rate Model with Random Effects: an Application to a Multicentre Clinical Trial of Carcinoma Diego I. Gallardo
More informationGeneral Strong Polarization
General Strong Polarization Madhu Sudan Harvard University Joint work with Jaroslaw Blasiok (Harvard), Venkatesan Gurswami (CMU), Preetum Nakkiran (Harvard) and Atri Rudra (Buffalo) December 4, 2017 IAS:
More informationThe Compound Family of Generalized Inverse Weibull Power Series Distributions
British Journal of Applied Science & Technology 14(3): 1-18 2016 Article no.bjast.23215 ISSN: 2231-0843 NLM ID: 101664541 SCIENCEDOMAIN international www.sciencedomain.org The Compound Family of Generalized
More informationWeighted Marshall-Olkin Bivariate Exponential Distribution
Weighted Marshall-Olkin Bivariate Exponential Distribution Ahad Jamalizadeh & Debasis Kundu Abstract Recently Gupta and Kundu [9] introduced a new class of weighted exponential distributions, and it can
More informationPlotting data is one method for selecting a probability distribution. The following
Advanced Analytical Models: Over 800 Models and 300 Applications from the Basel II Accord to Wall Street and Beyond By Johnathan Mun Copyright 008 by Johnathan Mun APPENDIX C Understanding and Choosing
More informationStep-Stress Models and Associated Inference
Department of Mathematics & Statistics Indian Institute of Technology Kanpur August 19, 2014 Outline Accelerated Life Test 1 Accelerated Life Test 2 3 4 5 6 7 Outline Accelerated Life Test 1 Accelerated
More informationMath 171 Spring 2017 Final Exam. Problem Worth
Math 171 Spring 2017 Final Exam Problem 1 2 3 4 5 6 7 8 9 10 11 Worth 9 6 6 5 9 8 5 8 8 8 10 12 13 14 15 16 17 18 19 20 21 22 Total 8 5 5 6 6 8 6 6 6 6 6 150 Last Name: First Name: Student ID: Section:
More informationA BIVARIATE MARSHALL AND OLKIN EXPONENTIAL MINIFICATION PROCESS
Faculty of Sciences Mathematics, University of Niš, Serbia Available at: http://wwwpmfniacyu/filomat Filomat 22: (2008), 69 77 A BIVARIATE MARSHALL AND OLKIN EXPONENTIAL MINIFICATION PROCESS Miroslav M
More informationA TWO-STAGE GROUP SAMPLING PLAN BASED ON TRUNCATED LIFE TESTS FOR A EXPONENTIATED FRÉCHET DISTRIBUTION
A TWO-STAGE GROUP SAMPLING PLAN BASED ON TRUNCATED LIFE TESTS FOR A EXPONENTIATED FRÉCHET DISTRIBUTION G. Srinivasa Rao Department of Statistics, The University of Dodoma, Dodoma, Tanzania K. Rosaiah M.
More informationProperties a Special Case of Weighted Distribution.
Applied Mathematics, 017, 8, 808-819 http://www.scirp.org/journal/am ISSN Online: 15-79 ISSN Print: 15-785 Size Biased Lindley Distribution and Its Properties a Special Case of Weighted Distribution Arooj
More informationGeneralized Transmuted -Generalized Rayleigh Its Properties And Application
Journal of Experimental Research December 018, Vol 6 No 4 Email: editorinchief.erjournal@gmail.com editorialsecretary.erjournal@gmail.com Received: 10th April, 018 Accepted for Publication: 0th Nov, 018
More informationThe Geometric Inverse Burr Distribution: Model, Properties and Simulation
IOSR Journal of Mathematics (IOSR-JM) e-issn: 2278-5728, p-issn: 2319-765X. Volume 11, Issue 2 Ver. I (Mar - Apr. 2015), PP 83-93 www.iosrjournals.org The Geometric Inverse Burr Distribution: Model, Properties
More informationOn q-gamma Distributions, Marshall-Olkin q-gamma Distributions and Minification Processes
CHAPTER 4 On q-gamma Distributions, Marshall-Olkin q-gamma Distributions and Minification Processes 4.1 Introduction Several skewed distributions such as logistic, Weibull, gamma and beta distributions
More informationMarshall-Olkin generalized Erlang-truncated exponential distribution: Properties and applications
STATISTICS RESEARCH ARTICLE Marshall-Olkin generalized Erlang-truncated exponential distribution: Properties and applications Idika E. Okorie 1 *, Anthony C. Akpanta 2 and Johnson Ohakwe 3 Received: 8
More informationA New Method for Generating Distributions with an Application to Exponential Distribution
A New Method for Generating Distributions with an Application to Exponential Distribution Abbas Mahdavi & Debasis Kundu Abstract Anewmethodhasbeenproposedtointroduceanextraparametertoafamilyofdistributions
More informationA class of probability distributions for application to non-negative annual maxima
Hydrol. Earth Syst. Sci. Discuss., doi:.94/hess-7-98, 7 A class of probability distributions for application to non-negative annual maxima Earl Bardsley School of Science, University of Waikato, Hamilton
More informationRELIABILITY FOR SOME BIVARIATE EXPONENTIAL DISTRIBUTIONS
RELIABILITY FOR SOME BIVARIATE EXPONENTIAL DISTRIBUTIONS SARALEES NADARAJAH AND SAMUEL KOTZ Received 8 January 5; Revised 7 March 5; Accepted June 5 In the area of stress-strength models, there has been
More informationOn the Comparison of Fisher Information of the Weibull and GE Distributions
On the Comparison of Fisher Information of the Weibull and GE Distributions Rameshwar D. Gupta Debasis Kundu Abstract In this paper we consider the Fisher information matrices of the generalized exponential
More informationThe Gamma Generalized Pareto Distribution with Applications in Survival Analysis
International Journal of Statistics and Probability; Vol. 6, No. 3; May 217 ISSN 1927-732 E-ISSN 1927-74 Published by Canadian Center of Science and Education The Gamma Generalized Pareto Distribution
More informationSolving Fuzzy Nonlinear Equations by a General Iterative Method
2062062062062060 Journal of Uncertain Systems Vol.4, No.3, pp.206-25, 200 Online at: www.jus.org.uk Solving Fuzzy Nonlinear Equations by a General Iterative Method Anjeli Garg, S.R. Singh * Department
More informationAlpha-Skew-Logistic Distribution
IOSR Journal of Mathematics (IOSR-JM) e-issn: 78-578, p-issn: 9-765X Volume 0, Issue Ver VI (Jul-Aug 0), PP 6-6 Alpha-Sew-Logistic Distribution Partha Jyoti Haaria a and Subrata Charaborty b a Department
More informationA world-wide investigation of the probability distribution of daily rainfall
International Precipitation Conference (IPC10) Coimbra, Portugal, 23 25 June 2010 Topic 1 Extreme precipitation events: Physics- and statistics-based descriptions A world-wide investigation of the probability
More informationThe Burr X-Exponential Distribution: Theory and Applications
Proceedings of the World Congress on Engineering 7 Vol I WCE 7, July 5-7, 7, London, U.K. The Burr X- Distribution: Theory Applications Pelumi E. Oguntunde, Member, IAENG, Adebowale O. Adejumo, Enahoro
More informationComparative Distributions of Hazard Modeling Analysis
Comparative s of Hazard Modeling Analysis Rana Abdul Wajid Professor and Director Center for Statistics Lahore School of Economics Lahore E-mail: drrana@lse.edu.pk M. Shuaib Khan Department of Statistics
More informationThe Weibull-Geometric Distribution
Journal of Statistical Computation & Simulation Vol., No., December 28, 1 14 The Weibull-Geometric Distribution Wagner Barreto-Souza a, Alice Lemos de Morais a and Gauss M. Cordeiro b a Departamento de
More informationTopp-Leone Inverse Weibull Distribution: Theory and Application
EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 10, No. 5, 2017, 1005-1022 ISSN 1307-5543 www.ejpam.com Published by New York Business Global Topp-Leone Inverse Weibull Distribution: Theory and Application
More informationTransmuted New Weibull-Pareto Distribution and its Applications
Available at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 932-9466 Vol. 3, Issue (June 28), pp. 3-46 Applications and Applied Mathematics: An International Journal (AAM) Transmuted New Weibull-Pareto Distribution
More informationCHAPTER 5 Wave Properties of Matter and Quantum Mechanics I
CHAPTER 5 Wave Properties of Matter and Quantum Mechanics I 1 5.1 X-Ray Scattering 5.2 De Broglie Waves 5.3 Electron Scattering 5.4 Wave Motion 5.5 Waves or Particles 5.6 Uncertainty Principle Topics 5.7
More informationLecture 3. STAT161/261 Introduction to Pattern Recognition and Machine Learning Spring 2018 Prof. Allie Fletcher
Lecture 3 STAT161/261 Introduction to Pattern Recognition and Machine Learning Spring 2018 Prof. Allie Fletcher Previous lectures What is machine learning? Objectives of machine learning Supervised and
More informationOn Weighted Exponential Distribution and its Length Biased Version
On Weighted Exponential Distribution and its Length Biased Version Suchismita Das 1 and Debasis Kundu 2 Abstract In this paper we consider the weighted exponential distribution proposed by Gupta and Kundu
More informationBivariate Geometric (Maximum) Generalized Exponential Distribution
Bivariate Geometric (Maximum) Generalized Exponential Distribution Debasis Kundu 1 Abstract In this paper we propose a new five parameter bivariate distribution obtained by taking geometric maximum of
More informationWorksheets for GCSE Mathematics. Quadratics. mr-mathematics.com Maths Resources for Teachers. Algebra
Worksheets for GCSE Mathematics Quadratics mr-mathematics.com Maths Resources for Teachers Algebra Quadratics Worksheets Contents Differentiated Independent Learning Worksheets Solving x + bx + c by factorisation
More informationMarshall-Olkin Univariate and Bivariate Logistic Processes
CHAPTER 5 Marshall-Olkin Univariate and Bivariate Logistic Processes 5. Introduction The logistic distribution is the most commonly used probability model for modelling data on population growth, bioassay,
More informationEstimation for Mean and Standard Deviation of Normal Distribution under Type II Censoring
Communications for Statistical Applications and Methods 2014, Vol. 21, No. 6, 529 538 DOI: http://dx.doi.org/10.5351/csam.2014.21.6.529 Print ISSN 2287-7843 / Online ISSN 2383-4757 Estimation for Mean
More informationLecture 11. Kernel Methods
Lecture 11. Kernel Methods COMP90051 Statistical Machine Learning Semester 2, 2017 Lecturer: Andrey Kan Copyright: University of Melbourne This lecture The kernel trick Efficient computation of a dot product
More informationThe Exponentiated Weibull-Power Function Distribution
Journal of Data Science 16(2017), 589-614 The Exponentiated Weibull-Power Function Distribution Amal S. Hassan 1, Salwa M. Assar 2 1 Department of Mathematical Statistics, Cairo University, Egypt. 2 Department
More informationTwo Weighted Distributions Generated by Exponential Distribution
Journal of Mathematical Extension Vol. 9, No. 1, (2015), 1-12 ISSN: 1735-8299 URL: http://www.ijmex.com Two Weighted Distributions Generated by Exponential Distribution A. Mahdavi Vali-e-Asr University
More informationNew goodness-of-fit plots for censored data in the package fitdistrplus
New goodness-of-fit plots for censored data in the package fitdistrplus M.L. Delignette-Muller (LBBE, Lyon) C. Dutang (CEREMADE, Paris) and A. Siberchicot (LBBE, Lyon) July 6th 2018 General presentation
More informationISI Web of Knowledge (Articles )
ISI Web of Knowledge (Articles 1 -- 18) Record 1 of 18 Title: Estimation and prediction from gamma distribution based on record values Author(s): Sultan, KS; Al-Dayian, GR; Mohammad, HH Source: COMPUTATIONAL
More informationMonte Carlo (MC) simulation for system reliability
Monte Carlo (MC) simulation for system reliability By Fares Innal RAMS Group Department of Production and Quality Engineering NTNU 2 2 Lecture plan 1. Motivation: MC simulation is a necessary method for
More informationSome Results on Moment of Order Statistics for the Quadratic Hazard Rate Distribution
J. Stat. Appl. Pro. 5, No. 2, 371-376 (2016) 371 Journal of Statistics Applications & Probability An International Journal http://dx.doi.org/10.18576/jsap/050218 Some Results on Moment of Order Statistics
More information3.2 A2 - Just Like Derivatives but Backwards
3. A - Just Like Derivatives but Backwards The Definite Integral In the previous lesson, you saw that as the number of rectangles got larger and larger, the values of Ln, Mn, and Rn all grew closer and
More informationTerms of Use. Copyright Embark on the Journey
Terms of Use All rights reserved. No part of this packet may be reproduced, stored in a retrieval system, or transmitted in any form by any means - electronic, mechanical, photo-copies, recording, or otherwise
More informationAn Extension of the Freund s Bivariate Distribution to Model Load Sharing Systems
An Extension of the Freund s Bivariate Distribution to Model Load Sharing Systems G Asha Department of Statistics Cochin University of Science and Technology Cochin, Kerala, e-mail:asha@cusat.ac.in Jagathnath
More informationOn Sarhan-Balakrishnan Bivariate Distribution
J. Stat. Appl. Pro. 1, No. 3, 163-17 (212) 163 Journal of Statistics Applications & Probability An International Journal c 212 NSP On Sarhan-Balakrishnan Bivariate Distribution D. Kundu 1, A. Sarhan 2
More informationResearch Article On Stress-Strength Reliability with a Time-Dependent Strength
Quality and Reliability Engineering Volume 03, Article ID 4788, 6 pages http://dx.doi.org/0.55/03/4788 Research Article On Stress-Strength Reliability with a Time-Dependent Strength Serkan Eryilmaz Department
More informationISSN: ISO 9001:2008 Certified International Journal of Engineering and Innovative Technology (IJEIT) Volume 4, Issue 6, December 2014
Bayesian Estimation for the Reliability Function of Pareto Type I Distribution under Generalized Square Error Loss Function Dr. Huda A. Rasheed, Najam A. Aleawy Al-Gazi Abstract The main objective of this
More informationA compound class of Poisson and lifetime distributions
J. Stat. Appl. Pro. 1, No. 1, 45-51 (2012) 2012 NSP Journal of Statistics Applications & Probability --- An International Journal @ 2012 NSP Natural Sciences Publishing Cor. A compound class of Poisson
More informationA GENERALIZED LOGISTIC DISTRIBUTION
A GENERALIZED LOGISTIC DISTRIBUTION SARALEES NADARAJAH AND SAMUEL KOTZ Received 11 October 2004 A generalized logistic distribution is proposed, based on the fact that the difference of two independent
More informationEstimation of the Bivariate Generalized. Lomax Distribution Parameters. Based on Censored Samples
Int. J. Contemp. Math. Sciences, Vol. 9, 2014, no. 6, 257-267 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijcms.2014.4329 Estimation of the Bivariate Generalized Lomax Distribution Parameters
More informationA Step Towards the Cognitive Radar: Target Detection under Nonstationary Clutter
A Step Towards the Cognitive Radar: Target Detection under Nonstationary Clutter Murat Akcakaya Department of Electrical and Computer Engineering University of Pittsburgh Email: akcakaya@pitt.edu Satyabrata
More informationRandom Vectors Part A
Random Vectors Part A Page 1 Outline Random Vectors Measurement of Dependence: Covariance Other Methods of Representing Dependence Set of Joint Distributions Copulas Common Random Vectors Functions of
More informationAnalysis of Survival Data Using Cox Model (Continuous Type)
Australian Journal of Basic and Alied Sciences, 7(0): 60-607, 03 ISSN 99-878 Analysis of Survival Data Using Cox Model (Continuous Type) Khawla Mustafa Sadiq Department of Mathematics, Education College,
More informationThe Odd Exponentiated Half-Logistic-G Family: Properties, Characterizations and Applications
Chilean Journal of Statistics Vol 8, No 2, September 2017, 65-91 The Odd Eponentiated Half-Logistic-G Family: Properties, Characterizations and Applications Ahmed Z Afify 1, Emrah Altun 2,, Morad Alizadeh
More informationIrr. Statistical Methods in Experimental Physics. 2nd Edition. Frederick James. World Scientific. CERN, Switzerland
Frederick James CERN, Switzerland Statistical Methods in Experimental Physics 2nd Edition r i Irr 1- r ri Ibn World Scientific NEW JERSEY LONDON SINGAPORE BEIJING SHANGHAI HONG KONG TAIPEI CHENNAI CONTENTS
More informationOn The Cauchy Problem For Some Parabolic Fractional Partial Differential Equations With Time Delays
Journal of Mathematics and System Science 6 (216) 194-199 doi: 1.17265/2159-5291/216.5.3 D DAVID PUBLISHING On The Cauchy Problem For Some Parabolic Fractional Partial Differential Equations With Time
More informationVariable Selection in Predictive Regressions
Variable Selection in Predictive Regressions Alessandro Stringhi Advanced Financial Econometrics III Winter/Spring 2018 Overview This chapter considers linear models for explaining a scalar variable when
More informationA GENERALIZED CLASS OF EXPONENTIATED MODI ED WEIBULL DISTRIBUTION WITH APPLICATIONS
Journal of Data Science 14(2016), 585-614 A GENERALIZED CLASS OF EXPONENTIATED MODI ED WEIBULL DISTRIBUTION WITH APPLICATIONS Shusen Pu 1 ; Broderick O. Oluyede 2, Yuqi Qiu 3 and Daniel Linder 4 Abstract:
More informationThe Exponentiated Generalized Standardized Half-logistic Distribution
International Journal of Statistics and Probability; Vol. 6, No. 3; May 27 ISSN 927-732 E-ISSN 927-74 Published by Canadian Center of Science and Education The Eponentiated Generalized Standardized Half-logistic
More informationChemical Engineering 412
Chemical Engineering 412 Introductory Nuclear Engineering Lecture 7 Nuclear Decay Behaviors Spiritual Thought Sooner or later, I believe that all of us experience times when the very fabric of our world
More informationNon-Parametric Weighted Tests for Change in Distribution Function
American Journal of Mathematics Statistics 03, 3(3: 57-65 DOI: 0.593/j.ajms.030303.09 Non-Parametric Weighted Tests for Change in Distribution Function Abd-Elnaser S. Abd-Rabou *, Ahmed M. Gad Statistics
More informationExtreme value statistics: from one dimension to many. Lecture 1: one dimension Lecture 2: many dimensions
Extreme value statistics: from one dimension to many Lecture 1: one dimension Lecture 2: many dimensions The challenge for extreme value statistics right now: to go from 1 or 2 dimensions to 50 or more
More informationDistribution Fitting (Censored Data)
Distribution Fitting (Censored Data) Summary... 1 Data Input... 2 Analysis Summary... 3 Analysis Options... 4 Goodness-of-Fit Tests... 6 Frequency Histogram... 8 Comparison of Alternative Distributions...
More informationA NEW WEIBULL-G FAMILY OF DISTRIBUTIONS
A NEW WEIBULL-G FAMILY OF DISTRIBUTIONS M.H.Tahir, M. Zubair, M. Mansoor, Gauss M. Cordeiro, Morad Alizadeh and G. G. Hamedani Abstract Statistical analysis of lifetime data is an important topic in reliability
More informationWeighted Lomax distribution
Kilany SpringerPlus 0165:186 DOI 10.1186/s40064-016-3489- RESEARCH Open Access Weighted Loma distribution N. M. Kilany * *Correspondence: neveenkilany@hotmail.com Faculty of Science, Menoufia University,
More informationGeneral Strong Polarization
General Strong Polarization Madhu Sudan Harvard University Joint work with Jaroslaw Blasiok (Harvard), Venkatesan Gurswami (CMU), Preetum Nakkiran (Harvard) and Atri Rudra (Buffalo) May 1, 018 G.Tech:
More informationWEIBULL BURR X DISTRIBUTION PROPERTIES AND APPLICATION
Pak. J. Statist. 2017 Vol. 335, 315-336 WEIBULL BURR X DISTRIBUTION PROPERTIES AND APPLICATION Noor A. Ibrahim 1,2, Mundher A. Khaleel 1,3, Faton Merovci 4 Adem Kilicman 1 and Mahendran Shitan 1,2 1 Department
More informationKybernetika. Hamzeh Torabi; Narges Montazeri Hedesh The gamma-uniform distribution and its applications. Terms of use:
Kybernetika Hamzeh Torabi; Narges Montazeri Hedesh The gamma-uniform distribution and its applications Kybernetika, Vol. 48 (202), No., 6--30 Persistent URL: http://dml.cz/dmlcz/4206 Terms of use: Institute
More information