The Uniform Truncated Negative Binomial Distribution and its Properties

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1 Joural of Mathematics ad Statistics Origial Research Paper The Uiform Trucated Negative Biomial Distributio ad its Properties Bahady I. Kamel, Shaba E. Abo Youssef ad Mohamed G. Sief Departmet of Mathematics, Faculty of Sciece, Al-Azhar Uiversity, Nasr City, Cairo, Egypt Departmet of Mathematics, Faculty of Sciece, Fayoum Uiversity, Fayoum, Egypt Article history Received: Revised: 9--6 Accepted: 9--6 Correspodig Author: Mohamed G. Sief Departmet of Mathematics, Faculty of Sciece, Fayoum Uiversity, Fayoum, Egypt Abstract: A ew three parameters cotiuous distributio which is amed the Uiform Trucated Negative Biomial (UTNB distributio is itroduced. It has the uiform ad the Marshall-Olki exteded uiform distributios as sub-models. Some reliability ad statistical properties of the ew distributio are derived, icludig the shape behavior of the desity ad hazard rate fuctios, the mea residual life, momets ad momet geeratig fuctio, quatiles ad related measures. The limitig distributios of the sample extremes, stochastic orderigs, etropies ad stress-stregth reliability are derived. Maximum likelihood estimatio is performed. Applicatio to cesored data illustrate the potetiality of the proposed distributio is provided. Keywords: Uiform Distributio, Trucated Negative Biomial, Stress- Stregth, Hazard Rate, Mea Residual Life, Maximum Likelihood Itroductio The basic distributios (expoetial, Weibull, gamma, log ormal, Lomax ad others used widely i reliability ad survival aalysis, are described by Cox ad Oakes (984, have a limited rage of behavior ad caot represet all the situatios foud i applicatios. For example, the expoetial has a restricted hazard fuctio, beig costat. The Weibull is ofte described as flexible, but its hazard fuctio is i fact restricted to beig mootoically icreasig or mootoically decreasig, or costat. The limitatios of the stadard distributios arouse the iterest i developig exteded or geeralized distributios by addig further parameters. Marshall ad Olki (997 itroduced a ew techique by addig a parameter to a existig distributio F to geerate a family of distributios. The exteded family takig the form: ( G x αf( x ( F( x, α + α > F x ( I the literature, may ew parametric extesios of various uivariate distributios were proposed usig this method such as: Marshall-Olki exteded Lomax, Ghitay et al. (7, Marshall-Olki exteded uiform (MOEU, Jose ad Krisha (, Marshall-Olki Pareto, Ghitay (6, Marshall- Olki logistic, Alice ad Jose (5a, Marshall-Olki semi-weibull, Alice ad Jose (5b, Marshall-Olki q- Weibull, Jose et al. ( ad Marshall-Olki Weibull, Ghitay et al. (5. Moreover, Jayakumar ad Mathew (8 itroduced the expoetiated Marshall-Olki scheme which has Marshall-Olki family as a special case give by: (, α, γ G x αf( x ( α F( x γ ( where, γ, α >,x R. Also, a ew family of distributios was proposed by Nadarajah et al. (3 by addig a extra parameter to the Marshall-Olki family of distributios whose survival fuctio is give by: α G( x ( F( x + αf( x α α, >, x R, α (3 The family (3 is a geeralizatio of the Marshall- Olki family of distributios (. Namely, if, the the itroduced family reduces to the Marshall-Olki family of distributios. Moreover, Nadarajah et al. (3 itroduced three-parameter geeralizatio of the expoetial distributio as a particular case of the ew 6 Bahady I. Kamel, Shaba E. Abo Youssef ad Mohamed G. Sief. This ope access article is distributed uder a Creative Commos Attributio (CC-BY 3. licese.

2 Bahady I. Kamel et al. / Joural of Mathematics ad Statistics 6, (4: 9.3 DOI:.3844/jmssp family. Also, Jose ad Sivadas (5 used the family (3 to itroduce the egative biomial Marshall-Olki Rayleigh distributio. The uiform distributio is regarded as the simplest probability model ad is related to all distributios by the fact that the cumulative distributio fuctio, take as a radom variable, follows uiform distributio over (,. This result is basic to the iverse method of the radom variable geeratio. Some researchers utilized uiform distributio i coectio with sample quasirages. The distributio is also applied to determie power fuctios of tests of radomess. It is also applied i a power compariso of tests of o-radom clusterig. There are also umerous applicatios i oparametric iferece, such as Kolmogorov-Smirov test for goodess of fit. The uses of uiform distributio to represet the distributio of roud off errors ad i coectio with the probability itegral trasformatios, are also well-kow. We proposed a ew versio of the family (3 by selectig the survival fuctio of the uiform (, i x Equatio 3, where F( x,< x<, so we get: α x x α ( + α G x (4 where, <x<, α, >. We refer to the distributio with the survival fuctio (4, as the uiform trucated egative biomial distributio with parameters,α, ad will be deoted by UTNB (,α,. I this article, we itroduce a ew distributio which is amed the Uiform Trucated Negative Biomial (UTNB distributio. Some statistical characteristics of the proposed distributio are obtaied ad the maximum likelihood estimatio of its parameters is discussed. The article is outlied as followig: I sectio, the probability desity fuctio (pdf ad its behavior are studied. Sectio 3 discusses the hazard rate fuctio ad obtais the mea residual life fuctio (mrlf. The momet geeratig fuctio is expressed as a ifiite sum ad a explicit form for the rth momet is obtaied i sectio 4. I sectio 5, we give a expressio for the quatile fuctio ad also some related measures. Sectio 6 provides the limitig distributios of the sample extreme. The stochastic orderigs are derived i sectio 7. I sectio 8, the stress-stregth reliability R P(Y<X of a system is discussed whe both the stregth X of the system ad the stress Y, which act o it, are idepedet, oidetical UTNB distributed radom variables. Réyi ad Shao etropies are determied i sectio 9. Sectio discusses the maximum likelihood estimatio of the model s parameters. I sectio, the proposed model has bee fitted to radomly cesored data to motivate its usefuless. Fially, the cocludig remarks are addressed i sectio. The Shapes of the Desity Fuctio The pdf of the UTNB distributio with is : ( α α α g( x α + x α ( where, α, >, <x<. Remark ( + (5 The UTNB distributio havig the uiform ad MOEU distributios as sub-models as: If α, the the UTNB (,α, U (, If, we get the MOEU distributio The followig propositio gives simple sufficiet coditio for the shape behavior of the pdf (5 Propositio The pdf g(x of the UTNB (α,, is icreasig (decreasig if α>(α<, idepedet o ad. Proof Takig the first derivative of log(g the we have: ( ( log ( + α, < x < α + ( α x ( g x If α (, the (log(g'< for all x (, ad hece, g(x is decreasig with g( (α-/(α -α ad g(. If α> the (log(g'> for all x (,,>, it follows that g(x is icreasig. Figure shows the behavior of the pdf for selected values of the parameters α,,. The Hazard rate ad Mea Residual Life Fuctios The hazard rate fuctio (hrf of the UTNB distributio is give by: ( h x ( α ( α (( α x α α x + where, α, >,<x<. (6 9

3 Bahady I. Kamel et al. / Joural of Mathematics ad Statistics 6, (4: 9.3 DOI:.3844/jmssp The followig propositio gives simple coditio uder which the hrf (6 is icreasig or bathtub. Propositio The hrf h(x of the UTNB (α,, is icreasig (bathtub if α (+- (<. Proof By takig the first derivative of h(x we get: ( h x ( α φ ( α x α + α α + ( ( x ( x ( α x where, φ( x ( + α +, < x<. If φ( ( +α -, the h'(x ad hece, h(x is icreasig for all <x<. If φ( <, thus the fuctio φ(x has oe root say, / ( α ( + x o ad it is egative for x<x o ad α positive for x>x o. This implies that the fuctio h'(x is egative for x<x o ad positive for x>x o. Thus, we coclude that h(x is bathtub shaped. Figure shows the graph of the hrf for the UTNB distributio for selected values of the parameters α,,. Fig.. Some possible shapes of the UTNB pdf 9

4 Bahady I. Kamel et al. / Joural of Mathematics ad Statistics 6, (4: 9.3 DOI:.3844/jmssp Fig.. The hrf of the UTNB distributio for selected values of the parameters α,, A importat ageig property for the UTNB distributio is the mea residual life fuctio (mrlf, which was itroduced by Watso ad Wells (96 to aalyse bur-i problems. It has bee studied by reliabilists, statisticias, survival aalysts ad others. It is defied simply as the expected value of the remaiig lifetime beyod a age t. For X UTNB(α,,, the residual life radom variable at age t, is deoted by: µ X X t X > t The mrlf is defied formally as: ( t E( X tx > t ( t t α G( x dx α x dx G t t ( t + α α+ t α + α+ t ( α( α α+ t where, α, >, t. Momet Geeratig Fuctio ad Momets I this sectio, we preset a ifiite sum for the momet geeratig fuctio (mgf of the UTNB distributio. Also, we derive a explicit form of the rth momet of the UTNB distributio. I the sequel, α we use the substitutio u α + x. The mgf is defied as: ( α tx tx M t,,, E( e e g( x dx α t ( α (, α,, t u ( α e u du α ( α ( + tx( α α α e α x dx + ( α α e ( α M t, α α e α t ( α t α! ( α ( The rth momet for the UTNB radom variable X is give by: r r E( X x g( x dx ( α ( α Γ( ( α ( + r( α α α x α x dx + ( α r α u u r α ( α r α α Γ( Bα ( r, r+ Γ ( r+ Γ( r where, fuctio. z a b Bz( a, b ( t t dt r du is the icomplete beta Quatiles, Kurtosis, Skewess The qth quatile of the UTNB distributio is give by: 93

5 Bahady I. Kamel et al. / Joural of Mathematics ad Statistics 6, (4: 9.3 DOI:.3844/jmssp ( α (7 α ( ( α q( q Q q G q where, q ad G ( is the iverse fuctio of the distributio fuctio. As special cases, we have: The qth quatile of the uiform distributio (α is give as x q q The qth quatile of the MOEU distributio ( is αq give by xq ( α q+ I particular, the media of UTNB distributio is give by: / / ( α ( α Xmed + α Also, we ca get the media of the uiform ad that of the MOEU as special cases respectively as Xmed α ad Xmed α +. A ew quatile measures for the dispersio of a distributio aroud the values µ ± σ will be preseted. The Bowley skewess, Keepig (96 is give by: 3 Q Q + Q SK 3 Q Q 4 4 Also, the Moors kurtosis, Moors (988 is give by: Q + Q Q + Q K 3 Q Q 4 4 Figure 3 displays the Bowley skewess ad Moors kurtosis for selected values of the parameters which shows how the skewess ad kurtosis are idepedet o ad deped oly o α ad. Limitig Distributio of Sample Extremes Let X, X,, X be a radom sample of size from the UTNB distributio, the the sample miima ad the sample maxima which are a log stad area i applicatio of probability ad statistics, are respectively: ( ( X mi X, X,..., X, X max X, X,..., X : : Fig. 3. Some possible shapes of the Bowley skewess ad Moors kurtosis for the UTNB distributio 94

6 Bahady I. Kamel et al. / Joural of Mathematics ad Statistics 6, (4: 9.3 DOI:.3844/jmssp Usig the asymptotic result of X : ad X :, Arold et al. (99, as followig: For the miimal order statistic X :, we have: * * { : } c limp X a + bt exp t, t >, c> (8 * (of the Weibull type where a F ( ad b F (/ F ( if ad oly if: * F ( is fiite For all t>, c>, we have: lim + ε ( ( + εt ( ( + ε F F F F t For the maximal order statistic X :, we have: { } k limp X: a + bt exp t t >, k > (of the Weibull type where a F ( ad: if ad oly if: b F ( F F ( is fiite. For all t>, k>, we have: lim + ε ( ( εt ( ( ε F F F F t k c (9 I the followig propositio we derived the limitig distributios of X : ad X : from the UTNB distributio. Propositio 3 Let X : ad X : be the smallest ad the largest order statistic from the UTNB distributio, the we have: * : t lim P{ X bt} e, t> where: : where: * α b α α + lim { } t P X a + bt e, t > / Proof a, b α α + We have G ( is fiite ad: + + ε ε + / ε αε α G( εt + lim tlim t + G( ε ( α tε α * Usig Equatio 8 ad 7 we have a ad: * α b α α + We have G ( is fiite ad: + + ε ε / + ( α ε + G( εt lim t lim t + G( ε ( α tε + Thus, we obtai that k, also usig (7 we get a / ad b α α, therefore statemet + ( flows from (9. Remark For, i.e., for the MOEU distributio, the ormig costats b *, b are give respectively by: α α(, α + α( + (see, e.g., Jose ad Krisha (. If α, i.e., the uiform distributio, the ormig * costats will be b, b. * * X a b ad Let the limitig distributio of ( : / (X : -a /b are, respectively, deoted as G * (t ad G(t. The, it is well-kow from Arold et al. (99 that for ay fiite i>, the limitig distributios of the radom variables (X i: -a /b ad (X -i+: -a /b are, respectively, give by: i * * * { i: + } ( G ( t limp X a bt j i { i+ : + } ( lim P X a bt G t j * j { log[ G ( t]} { log G( t} j! j j! 95

7 Bahady I. Kamel et al. / Joural of Mathematics ad Statistics 6, (4: 9.3 DOI:.3844/jmssp From propositio (3 we coclude that, for ay fiite i>, the limitig distributio of the ith ad the (-i + th order statistic from the UTNB distributio, respectively, are give by: i j * tt limp{ Xi: bt } e P( Z < i, t > j! j i j tt lim P{ X i+ : a + bt }, e P( Z < i t> j! j where, b *, a ad b are give i propositio (3 ad Z is discrete radom variable follows Poisso distributio with mea t. Stochastic Orderigs A very importat tool for judgig the comparative behavior is the stochastic orderig. If X ad Y are two cotiuous radom variables, the we ca say that X is smaller tha Y i the: Stochastic orderig (X st Y if FX( t FY( t, for all t Hazard rate orderig (X hr Y if h x (t h Y (t, for all t Mea residual life orderig (X mrl Y if m X (t m Y (t, for all t fx( t Likelihood ratio orderig (X lr Y if f ( t decreasig, for all t Accordig to the well-kow result of Shaked ad Shathikumar (994 for establishig stochastic orderig of distributio we have: X Y X Y X lr hr mrl X Y st Y Y ( I the followig propositio, we show that the UTNB distributio is ordered with respect to the strogest "likelihood ratio" orderig. Propositio 4 Let X UTNB(α,, ad Y UTNB(α,,, the (X lr Y ad hece (X hr Y, (X mrl Y ad (X st Y if ad oly if oe of the followig cases is satisfied: If (α α,, ad (α>, < or (α<, > If (α α, ad (α>,, or (α<,, If (, ad (α <α Proof It is sufficiet to show that (X lr Y if gx( t g ( t decreasig, for all t i this cases. The we get the required results. Y Stress-Stregth Reliability If a system subject to a stress, the reliability of the system is defied as the probability that it accomplishes its required fuctio adequately without failure. For stress-stregth models both the stregth of the system, X ad the stress, Y, imposed o it by its operatig eviromets are cosidered to be radom variables. The reliability, R, of the system ca be stated as, R P(Y<X. Now, Let X ad Y are two idepedet but ot idetically distributed UTNB radom variables represet the stregth of a system ad the stress actig o it respectively. Suppose that X UTNB(α,, ad Y UTNB(α,,. That is the survival fuctios of X ad Y are respectively give by: ad: α α G( x α x + α α α G( y α y + α The the reliability of the system will be: R PY ( < X ( P X > Y Y y. g ( y dy G( y. g ( y dy α ( α + + ( α ( α ( + ( α y α + + ( α α dy ( α ( α ( α y α + Etropies The etropy of a radom variable X is a measure of ucertaity or radomess of dyamical system modeled by this radom variable. We derive two popular etropy measures that are Réyi etropy, Sog ( ad Shao s etropy, Shao (95 of the UTNB radom variable. The Réyi etropy is defied as: R γ ( g ( x dx γ ( I ( γ ( log where, γ> ad γ. If X UTNB(α,, the we have: 96

8 Bahady I. Kamel et al. / Joural of Mathematics ad Statistics 6, (4: 9.3 DOI:.3844/jmssp ( γ( γ α α + α g ( x dx α + x dx ( α γ γ γ ( α α α α x ( α + Puttig γ γ γ γ( + dx α w α + x, the we have: ( γ γ γ γ α α γ( + g ( x dx w dw γ γ ( α α γ γ γ γ( + ( α α ( α γ γ ( α + ( γ( Therefore, Réyi etropy is give by: γ γ I R ( ( log ( γ γ γ γ( + ( α α α γ γ ( ( ( + α γ ( Now we cosider the Shao etropy which is defied as E[-log(g(X]. It is the special case of the Réyi etropy for γ. For the UTNB radom variable we have: ( α α ( g( X ( E log log (( E ( α + log α + log α + X First, we derive the expectatio: α log( α E log α + X α Therefore, we get: ( ( g( X ( α ( α ( α α E log log log log( + ( + α Maximum Likelihood Estimatio (3 (4 Suppose that X, X,, X be the failure time of idividual i ad let Y, Y,, Y be the correspodig cesorig time. Assumig that X i s ad Y i s are idepedet. If X i s UTNB ad Y i are assumed to have a o-iformative distributio. Oe observes the pairs (T i, i where T i mi(x i,y i ad I I(X i Y i is the cesorig idicator. Give the data (t,δ, (t,δ,, (t,δ the the likelihood fuctio is give by: δi α i i i L (,, [ g( t] G( t, α,, δi δi ( + ( α α α α + t i ( α δi i α α α ti + α ad the log-likelihood fuctio is the give by: ( α δilog + log( α α l log( α δi( + log α + ti i α + ( δi log α + ti Takig the first partial derivative of the loglikelihood fuctio with respect to α,, we have: ad: ti ( + δi α δ i + α α α α α + t i l α i ti α ( δi α + t i α α + ti l ( log α δi α + δilog α t + i α α log α + ti α α + ti α ( δi α + ti i αti ti α ( δi α t i δ + i α l α + t i i αti ti ( + δi α α t + i Maximum likelihood estimates (mles ˆ ˆ ˆ,, α are obtaied umerically as the solutio of the o-liear equatios: 97

9 Bahady I. Kamel et al. / Joural of Mathematics ad Statistics 6, (4: 9.3 DOI:.3844/jmssp l l l,, α Applicatio to Cesored Data I this sectio, we fit the proposed UTNB model to radomly cesored data represetig the serum reversal times (i years of a radom sample of 48 childre cotamiated with HIV from vertical trasmissio at the uiversity hospital of Ribeirão Preto school of medicie from 986 to, Silva (4: : Cesored data The data set cotais 84 right cesored observatios which costitute 56.7% of the sample size (heavy cesorig. We have fitted the uiform ad the MOEU distributios to the data set usig MLE ad the compared it with the UTNB distributio. I additio, for model selectio, we use the Akiake Iformatio Criterio (AIC, the Bayesia Iformatio Criterio (BIC, the Cosistet Akiake Iformatio Criterio (CAIC ad the Haa- Qui Iformatio Criterio (HQIC defied as: ( ˆ α ˆ ˆ k ( ˆ α ˆ ˆ k ( ( ˆ ˆ ˆ k l α k ( ˆ ˆ ˆ α AIC l,, +, BIC l,, log, CAIC,, +, HQIC l,, + klog(log( where, k is the umber of parameters i the model ad is the sample size. The model with lowest AIC or (BIC, CAIC ad HQIC value is the oe that better fits the data. Table show that the fitted UTNB distributio should be selected based o either the AIC or (BIC, CAIC ad HQIC procedure for the give data. To test the ull hypothesis H : The data follows the uiform distributio versus H a : The data follows the UTNB distributio, we used the Likelihood Ratio Test statistic (LRT which has a asymptotic chi-square distributio with degrees of freedom. H is rejected at a sigificace level of.5% if LRT>. The χ (,.5% uiform distributio is rejected i favor of the UTNB distributio for this data set (LRT 77.53> χ (, This result shows that UTNB is a great improvemet over the uiform distributio for this cesored data set. Let be the total umber of breakig stress of carbo fibers whose survival times, ucesored data, are available. Retable the survival times i order of icreasig magitude such that t t t. The Kapla ad Meier (958, estimator (KME, also kow as the product limit estimator, gave a oparametric estimator of the survival fuctio of the uderlyig as follows: δ( i S( t, t > i+ tt : i t Table. A compariso betwee the Log-likelihood, AIC, BIC, CAIC ad HQIC for the fitted UTNB, MOEU ad uiform distributios Models Parameters MLE Log-likelihood AIC BIC CAIC HQIC UTNB ˆα ˆ. ˆ.99 MOEU ˆα ˆ.96 Uiform ˆ

10 Bahady I. Kamel et al. / Joural of Mathematics ad Statistics 6, (4: 9.3 DOI:.3844/jmssp (a (b (c Fig. 4. P-P plot of KME versus fitted UTNB, MOEU ad uiform survival fuctios (a fitted uiform (b fitted MOEU (c fitted UTNB Fig. 5. The estimated hrf of the UTNB distributio based o the give data 99

11 Bahady I. Kamel et al. / Joural of Mathematics ad Statistics 6, (4: 9.3 DOI:.3844/jmssp Figure 4 shows respectively the P-P plot of the KME of the fitted UTNB, MOEU ad uiform distributios for the give data. The figures idicate the goodess-of-fit of the fitted UTNB as compared with the fitted uiform ad MOEU survival fuctios. ˆ ˆ α + ˆ.73 accordig to Also, whereas ( propositio ( the estimated hrf is icreasig as show i Fig. 5. Coclusio For the first time, we propose a three-parameter cotiuous distributio which geeralizes the uiform ad the Marshall-Olki exteded uiform distributios. We refer to the ew model as the Uiform Trucated Negative Biomial (UTNB distributio. Provide several statistical properties for the proposed model icludig reliability measures: The desity, the hazard rate, mea residual life, momet geeratig fuctio, momets, quatiles, kurtosis, skewess, limitig distributio of sample extremes, stochastic orderigs, stress-stregth reliability ad etropies. Estimatio via maximum likelihood is straightforward. Cesored data applicatio of the UTNB distributio shows that it could provide a better fit tha the uiform ad the MOEU distributios. We hope that the proposed exteded model may attract wider applicatios i survival aalysis. Ackowledgemet The authors are grateful to the referees for their commets ad suggestios, which certaily improved the presetatio of the cotet of the paper. Author s Cotributios Bahady I. Kamel: Offerig ideas, ecouragemet, proof read text ad equatios. Shaba E. Abo Youssef: Revise, improve ad editig thefial drafts of the paper. Mohamed G. Sief: Aalyzed ad iterpreted the result, prepared the mauscript ad resposible for the mauscript correctio, proof readig ad paper submissio. Ethics This article is origial ad cotais upublished material. The correspodig author cofirms that all of the other authors have read ad approved the mauscript ad o ethical issues ivolved. Refereces Alice, T. ad K. Jose, 5a. Marshall-Olki logistic processes. STARS It. J., 6: -. Alice, T. ad K. Jose, 5b. Marshall-Olki semi- Weibull miificatio processes. Recet Adv. Stat. Theory Applied. Arold, B.C., N. Balakrisha ad H.N. Nagaraja, 99. A First Course i Order Statistics. st Ed., SIAM, Philadelphia, ISBN-: , pp: 79. Cox, D.R. ad D. Oakes, 984. Aalysis of Survival Data. CRC Press, ISBN-: 4449X, pp: 8. Ghitay, M., E. Al-Hussaii ad R. Al-Jarallah, 5. Marshall-Olki exteded Weibull distributio ad its applicatio to cesored data. J. Applied Stat., 3: DOI:.8/ Ghitay, M., 6. Marshall-Olki exteded Pareto distributio ad its applicatio.proceedigs of the 3th Iteratioal Coferece of the Forum for Iterdiscipliary Mathematics o Iterdiscipliary Mathematical ad Statistical Techiques, Sept. -4, New Uiversity of Lisbo-Tomar Polytechic Istitute, Portugal. Ghitay, M.E., F. Al-Awadhi ad L. Alkhalfa, 7. Marshall-Olki exteded Lomax distributio ad its applicatio to cesored data. Commu. Stat. Theory Meth., 36: DOI:.8/ Jayakumar, K. ad T. Mathew, 8. O a geeralizatio to Marshall-Olki scheme ad its applicatio to Burr type XII distributio. Stat. Papers, 49: DOI:.7/s Jose, K., S.R. Naik ad M.M. Ristić,. Marshall- Olki q-weibull distributio ad max-mi processes. Stat. Papers, 5: DOI:.7/s Jose, K. ad E. Krisha,. Marshall-Olki exteded uiform distributio. ProbStat Forum, 4: Jose, K. ad R. Sivadas, 5. Negative biomial Marshall-Olki Rayleigh distributio ad its applicatios. Ecoomic Quality Cotrol, 3: DOI:.55/eqc-5-9 Kapla, E.L. ad P. Meier, 958. Noparametric estimatio from icomplete observatios. Am. Stat. Assoc., 53: DOI:.8/ Keepig, E.S., 96. Itroductio to Statistical Iferece. st Ed., Courier Corporatio, New York, ISBN-: , pp: 45. Marshall, A. ad I. Olki, 997. A ew method for addig a parameter to a family of distributios with applicatio to the expoetial ad Weibull families. Biometrika, 84: DOI:.93/biomet/ Moors, J.J.A., 988. A quatile alterative for kurtosis. Statisticia, 37: 5-3. DOI:.37/ Nadarajah, S., K. Jayakumar ad M.M. Ristić, 3. A ew family of lifetime models. J. Stat. Comput. Simulat., 83: DOI:.8/

12 Bahady I. Kamel et al. / Joural of Mathematics ad Statistics 6, (4: 9.3 DOI:.3844/jmssp Shaked, M. ad J.G. Shathikumar, 994. Stochastic Orders ad their Applicatios. Academic Press, New York, ISBN-: 6386, pp: 545. Shao, C.E., 95. Predictio ad etropy of prited eglish. Bell Syst. Tech. J., 3: DOI:./j tb366.x Silva, A.N., 4. Estudo evolutivo das criaças expostas ao HIV e otificadas pelo cleo de vigilâcia epidemiol gica do HCFMRP-USP [dissertatio]. Uiversidade de Sمo Paulo. Sog, K.S.,. Réyi iformatio, loglikelihood ad a itrisic distributio measure. J. Stat. Pla. Iferece, 93: DOI:.6/S (69-5 Watso, G.S. ad W.T. Wells, 96. O the possibility of improvig the mea useful life of items by elimiatig those with short lives. Techometrics, 3: DOI:.8/

ANOTHER WEIGHTED WEIBULL DISTRIBUTION FROM AZZALINI S FAMILY

ANOTHER WEIGHTED WEIBULL DISTRIBUTION FROM AZZALINI S FAMILY Sulema Nasiru, MSc. Departmet of Statistics, Faculty of Mathematical Scieces, Uiversity for Developmet Studies, Navrogo, Upper East Regio, Ghaa,