CHAPTER 4 BIVARIATE DISTRIBUTION EXTENSION

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1 CHAPTER 4 BIVARIATE DISTRIBUTION EXTENSION 4. Itroductio Numerous bivariate discrete distributios have bee defied ad studied (see Mardia, 97 ad Kocherlakota ad Kocherlakota, 99) based o various methods of costructio. For example, Famoye ad Cosul (995) proposed a bivariate geeralized Poisso distributio (BGPD) by usig the method of trivariate reductio. However, the rage of the correlatio is restricted ad it oly permits positive correlatio betwee the two radom variables, X ad X. The bivariate egative biomial distributio (BNBD) defied by Lee (999) based o copula fuctio is very complicated ad oly be used for over dispersed data. A bivariate Poisso distributio was proposed by Lakshamiaraya et al. (999) with a flexible correlatio that varies over the full rage (, ). The model is formulated as a product of Poisso margials with a multiplicative factor. Famoye () adopted the approach of Lakshamiaraya et al. to costruct a ew bivariate geeralized Poisso distributio (BGPD) which allows a more flexible correlatio structure compared to the existig BGPD. This chapter is a extesio of Chapter where two ew bivariate GIT, distributios are defied ad examied. We shall itroduce the ew bivariate distributios as the alterative bivariate discrete distributios that allow more flexibility i modellig ad less limitatio o the correlatio betwee the two radom variables. 5

2 I sectio, we defie two ew bivariate GIT, distributio (BGITD) based o the covolutio of two bivariate distributios ad the classical trivariate reductio method. The some properties of these two BGITD are preseted i sectio, ad i sectio 4 the characteristic of the distributios are studied. Next, we discuss the estimatio of the parameters of BGITD by maximum likelihood estimatio ad the estimatio based o the probability geeratig fuctio (pgf) i sectio 5. I sectio 6, we illustrate these two methods of estimatio with three real life data together with test of goodess-of-fit. 4. Formulatio of Bivariate GIT, distributios Two formulatios of the bivariate GIT, distributios are cosidered: (a) covolutio of bivariate biomial ad bivariate egative biomial distributios (Type I) ad (b) the method of trivariate reductio (Type II). 4.. Covolutio (Type I) The pgf of GIT, distributio ca be writte as p + pt p pt p ϕ( t) = = + p t p + p p + p p t (4.) If p =, it reduces to the pgf of the biomial distributio ad whe p =, it is the egative biomial pgf. Thus, the GIT, distributio is a covolutio of the biomial p B, p + p ad egative biomial NB (, ) p distributios. I a similar maer, we defie a BGITD (Type I BGITD) as the mixture of bivariate biomial ad bivariate egative biomial distributios. 5

3 Joit probability geeratig fuctio of Type I BGITD The joit pgf is give by q ϕ( t, t) = ( p + pt + pt ) qt qt (4.) where p + p + p = q + q + q = ad, = positive iteger. Joit probability mass fuctio From (4.), we solve for the joit pmf as metioed i (.5) x x u v x x x x! ( u+ v+ )! q q u+ v = u= v= ( x u)!( x v)!( x x u v)! u! v!( )! p p f ( x, x ) q p p p p + + where max( x + x u v),,, = positive iteger ad p + p + p = q + q + q = (4.) 4.. Trivariate Reductio (Type II) The Type II BGITD is formulated by the method of trivariate reductio (Mardia, 97). As explaied i sectio (.4.), we have X = ( Y + Y ) X = ( Y + Y ) where Y, Y ad Y are idepedet GIT, radom variables. Now let ϕ, ϕ ad ϕ be the pgf of Y, Y ad Y respectively, that is p + pt q + qt r + rt ϕ = ; ϕ = ; ϕ =. pt qt rt 5

4 Joit probability geeratig fuctio By defiitio, the joit pgf of ( X, X ) is writte as ϕ( t, t ) p + p ( tt ) q + qt r + rt = p( tt ) qt rt (4.4) where p + p + p = q + q + q = r + r + r = ad,, = positive iteger. Joit probability mass fuctio From (4.4), the joit pmf is derived as + k + x k + x k k k k p p q r F, k; k+ ; mi( x k x k x k, x ) p p x x f ( x, x ) = p q r q r k= q r F, ( x k); ( x k) + ; F, ( x k); ( x k) + ; where p + p + p = q + q + q = r + r + r = ad,, = positive iteger. qq rr (4.5) 4. Properties of the Bivariate Distributios 4.. Properties of Type I BGITD Margial distributios Usig (.) ad (.), the margial pgf of X i, i =, ca easily be foud as q ϕ X ( t) = ( p ) i i + pit q + qi ( t) (4.6) Margial meas ad variaces Usig (4.), we ca fid the expected meas ad variaces of the margial X ad X as follows. For i =, 5

5 E X q = + p i ( i ) q q ( q + q ) Var( X ) = + p ( p ) i i i i i q i Covariace ad correlatio The margial meas ad variaces are used to yield the covariace, Cov( X, X ) ad the correlatio, ρ ( X, X ) of X ad X. q q Cov( X, X ) = p p (4.7) q ρ( X, X ) = q q p p q (( q ( q ) + p ( p ) q )( q ( q ) + p ( p ) q )) (4.8) To further ivestigate the rage of correlatio, we look ito differet parameter sets of Type I BGITD. As illustrated i Table 4., five cases are cosidered. Note that from Table 4., the covariace ad the correlatio of Type I BGITD ca be either positive or egative. The correlatio provides a full rage value which is from - to. Table 4.: Numerical examples case 4 5 p p p q q q

6 Table 4.: Covariace ad correlatio of Type I BGITD case 4 5 = = Cov( X, X ) ρ ( X, X ) =5 > = Cov( X, X ) ρ ( X, X ) = < =5 Cov( X, X ) ρ ( X, X ) From (4.), the first few probabilities are f (,) = p q ( + ) f (,) = f (,) p p q p f (,) = f (,) ( p + p q ) p ( ( p pq + p ( p + pq )) p p f (,) = f (,) ( + ) qq + + p p Coditioal distributio Next, we determie the coditioal distributio o X give X = x by () Applyig the derivative product rule () Successive differetiatio. Firstly, settig ϕ ( t, t) = j( t, t) g( t, t) as a covolutio of two idepedet bivariate radom variables, where j( t, t ) ad g( t, t ) are respectively the pgfs of the biomial ad egative biomial distributios. By defiitio, the pgf of the coditioal distributio of X give X = x is 55

7 ϕ ϕ ( t y) = ϕ (, x ) X (, x ) ( t,) (,) where ϕ ( x, x ) x + x ( t, t ) = ϕ( t, t ) x x t t t = u, t = v Usig the derivative product rule, we have ϕ x x (, x ) k x k ( t, t) = ϕ( t, t) = x j g t t k = k = t = x (4.9) where ( ) j p p t t = = + (4.) i i i ( ) i j = p p + pt ( j), i t = j= g t = q = qt + i i i qt j= (4.) (4.) i g = q q ( + j), i t = (4.) With the substitutio of equatios (4.), (4.), (4.) ad (4.) ito equatio (4.9), the umerator part of equatio (.4) ca be obtaied. A further substitutio of t= ito equatio (4.9) will lead us to solve for the deomiator part of equatio (.4). The, the coditioal distributio of X give X = x ca be determied. Secodly, by successive differetiatio with respect to t ad settig t =, we have ϕ ( t x ) X p + pt q = p + p qt x x x x q p ( + i) + ( j) qt i= p pt j= k x k y k x k y p q + ( j) ( + i) k= k p pt qt j= i= x x x x x q p ( + i) + ( j) q i= p p j= x x x k k x k x p q + ( j) ( + i) k= k p p q j= i= x, x 56

8 ad for x =, we have ϕ ( t x ) X 4.. Properties of Type II BGITD p + pt q = p + p qt Margial distributios Similarly, usig (.) ad (.), the margial pgf of X ad X are give as ϕ ( t) X p + pt q + qt r + r = pt qt r ϕ ( t) X p + pt q + q r + rt = pt q rt Margial meas ad variaces From (4.4), the meas ad variaces of the margial X ad X are obtaied as p q E( X) = + p q p r E( X ) = + p r ( p )( p ) ( q )( q ) Var( X ) = + ( p) ( q) ( p )( p ) ( r )( r ) Var( X ) = + ( p) ( r ) Covariace ad correlatio The covariace ad the correlatio of X ad X are give as ( p )( p ) Cov( X, X ) = ( p) (4.4) ρ( X, X ) = ( p )( p )( q + q )( r + r ) ( ( p )( p)( q + q) + ( q )( q)( p + p) ) ( ( p )( p)( r + r ) + ( r )( r )( p + p) ) (4.5) 57

9 Sice ( p )( p) >, it follows that for this model, the correlatio of X ad X are always positive. The first few probabilities are give by f (,) = p q r ( )( ) (,) (,) f f q = q q ( )( ) (,) (,) f f r = r r ( p + p p) ( q + qq )( r + r r ) f (,) = f (,) + p qr Coditioal distributio The coditioal distributio o X give X = x is determied by usig the derivative product rule as the successive differetiatio of equatio (4.4) is tedious. Settig q ϕ( t, t ) e( t, t ) h( t, t ) + q t = qt (4.6) where e( t, t ) p + p t t = ptt ad h( t, t ) r + r t = rt By defiitio, ϕ (, x ) x ( t, t ) = ϕ( t, t ) x t t = x + qt e( t, t) h( t, t) x qt t t = q = x q + q x t k = e h q t k x k k = t = 58

10 The successive differetiatio of e( t, t ) ad h( t, t ) with respect to t gives us e t = ( p ) = (4.7) i k i i i ( i )! k i k k e = t p p ( p p p) ( j), i t = + (4.8) k = k ( k )! j= h = r (4.9) t = ( ) i k i i ( i )! k i k k h = r r ( r r r ) ( j), i t = + k = k ( k )! j= (4.) (, x ) Similarly, by usig equatios (4.7), (4.8), (4.9) ad (4.), ϕ ( t, t ) ca be solved. The coditioal distributio of X give X = x ca the be easily derived. 4. Characteristics of the Distributios To explore the structure of the distributios graphically, a few figures are preseted by varyig the parameters of the Type I ad Type II BGITD. Note that all the figures are scaled by multiplyig the y-axis with the value. Let the joit pmf be pr( x, x ) with x =,,,..., r ad x =,,,..., s. 4.. Characteristics of Type I BGITD To study the effect of chages of the parameters ad of the Type I BGITD, we set r =8 ad s =8 ad = r + s =6 (as max( x + x) from equatio (4.)) ad = (Figure 4.). I Figure 4., the value of is icreased with fixed whereas i Figure 4. we icrease the value of but fix the value of (same as the value of i Figure 4.). 59

11 It is clear from Figures 4. ad 4. that icreasig the value of shifts the local mode away from the origi, (, ) ad decreases the height of the peak. By comparig Figures 4. ad 4., it is see that the local modes are flatter ad located further away. I coclusio, if the data are located close to the origi ad the local mode is sharp, small positive itegers value of ad are suggested. Larger value of is suggested if the value of the mode is low ad situated slightly away from the origi. Meawhile, large positive itegers for both ad are recommeded oly whe the local mode ad the data are placed quite far away from the origi ad the data form a ice bell-shaped. This observatio will be useful i the empirical modellig of data S S5 S9 S9 S8 S7 S6 S5 S4 S S S A local mode at (,) with value 94. Figure 4.: p =., p =. q =., q =., = 6, = 5 S S9 4 S S S S S S7 S4 S A local mode at (4,4) with value 4. Figure 4.: p =., p =. q =., q =., =4, = 6

12 S S9 5 S6 S S S S S7 S4 S Local modes are (7, 7), (7,8) ad (8,7) with value. Figure 4.: p =., p =. q =., q =., =4, = 4.. Characteristics of Type II BGITD Figures 4.4 to 4. are displayed i order to uderstad the behaviour of Type II BGIT with differet combiatio of the positive itegers, ad. We examie the chages of, ad as give i Table 4.. It is easily see that by varyig, ad, differet characteristics of Type II BGIT models are displayed. I Figure 4.4, where, ad are set at, we achieve a very sharp ad high value of the local mode. However, whe is icreasig ( ad fixed) as i Figure 4.5, the cotours chage to ellipses. Furthermore, the height of its local mode is decreased ad situated further from the origi i compariso with Figure 4.4. I Figure 4.6, is icreased but = = ad it is foud that a flatter mode is formed; the cotour plots show a shift to the x-axis. O the other had, whe = = ad is icreased, Figure 4.7 is see to be a reflectio of Figure 4.6 through the origi, with the heights of the modes uchaged. Figures 4.8 to 4. show the effect of icreasig two parameters ad fixig the remaiig oe. It is observed that i Figure 4.8 the local mode has the same locatio as the oe i Figure 4.5 but the peak of the local mode is much lower ad the elliptic 6

13 cotours are broader. The plot i Figure 4.9 is similar with Figure 4.6 while Figures 4. ad 4.7 are similar (with reflectio). The plots idicate the flexibility of the shape of the Type II BGIT ad small, ad give local modes situated very close to the origi. To apply Type II BGIT i data aalysis the modes of the data will idicate the values of, ad. Table 4.: Chage of the positive itegers, ad Figure S S S S S9 S7 S5 S S S9 S7 S5 S S A local mode at (,) with value 98. Figure 4.4: p =., p =.4, q =., q =.4, r =., r =.4, = = = S S6 S S S8 S5 S S9 S6 S S S7 S4 S A local mode at (9,9) with value 6. Figure 4.5: p =., p =.4, q =., q =.4, r =., r =.4, = 5, = = 6

14 S S7 S4 S S8 S5 S S9 S6 S S S7 S4 S Local modes at (8,) ad (9,) with value. Figure 4.6: p =., p =.4, q =., q =.4, r =., r =.4, =, = 5, = S S7 S4 S S8 S5 S S9 S6 S S S7 S4 S Local modes at (,8) ad (,9) with value. Figure 4.7: p =., p =.4, q =., q =.4, r =., r =.4, =, =, = 5 S 6 S8 S5 4 S S9 8 6 S6 S S S7 4 S46 S S S4 S S A local mode at (9,9) with value 5. Figure 4.8: p =., p =.4, q =., q =.4, r =., r =.4, =, = 5, = 5 6

15 7 S 6 S8 4 S S46 S S6 S S S9 S6 S S S7 S4 S Local modes at (5,9), (6,9) ad (6,) with value 5. Figure 4.9: p =., p =.4, q =., q =.4, r =., r =.4, = 5, = 5, = S 6 4 S8 S5 S S9 S S S S46 S S S7 S4 S S Local modes at (9,5), (9,6) ad (,6) with value 5. Figure 4.: p =., p =.4, q =., q =.4, r =., r =.4, = 5, =, = Parameter Estimatio I this sectio, two methods of parameter estimatio are cosidered. Due to the complicated joit pmf, simulated aealig is proposed for the umerical optimizatio uder these two estimatios. For the Type I BGITD distributio, we shall cosider the case where the parameters ad are fixed ad for the Type II BGITD distributio, the parameters, ad are fixed Maximum Likelihood Estimatio (MLE) For the probability fuctio based estimatio method, we cosider maximum likelihood estimatio where the log-likelihood fuctio 64

16 r s = ij i= j= l L π l f ( i, j) is maximized, where π ij is the observed frequecy i the (i, j) cell for i =,,,, r ; j =,,,, s The Pgf-based Miimum Helliger-type Distace Estimatio (PGFBE) The use of pgf is popular as geerally it has a simpler form tha the probability mass fuctio ad this leads to simpler statistical iferece procedures which shorte computatio times. From Sim ad Og (), the uivariate pgf-based estimator is exteded to bivariate distributio as follows ( ) T = ψ ( t, t ) ϕ( t, t ) dt dt where k k i j ijtt i j ψ ( t, t ) = is the empirical pgf ad ϕ ( t, t) is the pgf of the distributio. 4.5 Numerical examples To illustrate the applicatio i data fittig, three real life data sets are cosidered: a) Number of accidets sustaied by experieced shuters over successive periods of time. (Arbous ad Kerrich, 95) b) Number of patiets i two boxes i a room of the critical care ad emergecy service i the Sa Agusti Hospital (Liares, Spai). (Rodriguez et al., 6). c) Number of times baco ad eggs were purchased o four cosecutive shoppig trips. (Peter ad Bruce, 5) These data are fitted by the bivariate egative biomial distributio (Subrahmaiam, 966), Type I ad Type II BGITD by MLE ad the pgf-based 65

17 miimum Helliger-type distace estimatio. The performace of these parameter estimatio methods will be discussed ad the expected frequecies are displayed i Tables 4.4, 4.6 ad 4.8 respectively. No groupig has bee doe ad the chi-square goodess-of-fit (GOF) tests are carried out to ivestigate how well the observed distributio fits the estimated distributio. The parameters, ad are chose which provide the lowest chi-square value for the fit. For Type I BGITD, the iitial parameter estimate of correspodig to the bivariate egative biomial compoet is first treated as a real umber. The, we choose the positive itegers of parameters ad based o the lowest chi-square value as metioed above. For type II BGITD, Table 4. is used to assist us to choose the suitable positive itegers. I Table 4.4 (Arbous ad Kerrich, 95) X represets the umber of accidets i 5- year period ad X is the umber of accidets i 6- year period From the data, we fid that the sample mea ad variace for X are.9754 ad.969 respectively. Meawhile, for X, the sample mea is x =.75 ad sample variace is ˆ x σ =.655. The sample correlatio is foud to be.585. I Table 4.6, the data were obtaied from a room of Critical Care ad Emergecy Service i Sa Agustí Hospital (Liares, Spai) where the referece populatio is more tha 5, people. Patiets are dealt with i oe of two boxes where they ca stay a maximum of 4 hours without preferece (radom). The hadlig of patiets i shifts worked i ad is cosidered ad the observed frequecies of patiets dealt with i each box are show i Table 4.7. Similarly, we compute the sample mea ad variace for X ad X : x =.66, ˆ x σ =.4768, 66

18 x =.9868 ad ˆ x σ =.47. The sample correlatio is foud to be slightly egative which is -.9. Hece, we fit the data with the Type I BGITD. Table 4.8 exhibits the purchase of baco ad eggs from the Iformatio Resources, Ic., a cosumer pael based i a large U.S. city. A sample of 548 households over four cosecutive store trips was tracked. For each household, a total umber of baco purchases i their four shoppig trips ad the total umber of egg purchases for the same trip were couted. From the data, the sample mea ad variace for X ad X are x =.956, sample correlatio is.. ˆ x ˆ x σ =.44, x =.747 ad σ =.574. I additio, the Example Table 4.4: Number of accidets sustaied by experieced shuters over successive periods of time X Total BNBD* BNBD^ BGITD* BGITD^ BGITD* BGITD^ BNBD* BNBD^ BGITD* BGITD^ BGITD* BGITD^ BNBD* BNBD^ BGITD* BGITD^ BGITD* BGITD^ X 67

19 Table 4.4, cotiued. 9 BNBD* BNBD^ BGITD* BGITD^ BGITD* BGITD^ BNBD* BNBD^ BGITD* BGITD^ BGITD* BGITD^ BNBD* BNBD^ BGITD* BGITD^ BGITD* BGITD^ BNBD* BNBD^ BGITD* BGITD^ BGITD* BGITD^ BNBD* BNBD^ BGITD* BGITD^ BGITD* BGITD^ * = MLE, ^ = PGF based estimator, BGITD = type I BGITD, BGITD = type II BGITD ML estimates: BNBD: ˆp =.964, ˆp =.475, ˆp =.4, ˆv =.876. Type I BGITD: ˆp =.48, ˆp =.75, ˆq =., ˆq =.64, =, =. 68

20 Type II BGITD: ˆp =.9855, ˆp =., ˆq =449, ˆq =.7, ˆr =.555, ˆr =.9, =, =, =. PGF-based estimates: BNBD: ˆp =59, ˆp =., ˆp =.9, ˆv =5.89 Type I BGITD: ˆp =.8, ˆp =.64, ˆq =.6, ˆq =.689, =, =. Type II BGITD: ˆp =.987, ˆp =., ˆq =455, ˆq =.59, ˆr =.549, ˆr =.54, =, =, =. By usig the estimated parameters as show above, we compute the correlatio for the estimated distributios. All of the estimated correlatio values are close to the sample correlatio which is.585. MLE PGFBE Model ρ ( X, X ) Model ρ ( X, X ) BNBD*.8 BNBD^.8 BGITD*.8 BGITD^. BGITD*. BGITD^. Chi-square GOF test A summary of the show i the followig table. χ values, the full model log-likelihood values ad the P-values are Table 4.5: Summary statistics for Table 4.4 Log-likelihood Degree of Model χ value P-value freedom BNBD* BGITD* BGITD* It is to be observed that i Table 4.5, the fit by Type I ad Type II BGITD are sigificatly better tha the BNBD based upo the χ values. The Type II BGITD is preferred over the Type I BGITD as its P-value is higher ad the result is sigificat. 69

21 Example Table 4.6: Number of patiets i two boxes i a room of the critical care ad emergecy service i the Sa Agusti Hospital (Liares, Spai) Box, X Box, X Total BGITD* BGITD^ BGITD* BGITD^ BGITD* BGITD^ BGITD* BGITD^ BGITD* BGITD^ BGITD* BGITD^ BGITD* BGITD^ *= MLE, ^= PGF based estimator, BGITD=type I BGITD, BGITD=type II BGITD ML estimates: Type I BGITD: ˆp =.488, ˆp =.55, ˆq =., ˆq =., =, =. PGF-based estimates: Type I BGITD: ˆp =.58, ˆp =.6, ˆq =., ˆq =., =, =. 7

22 The correlatio for the estimated distributio is compute as follows. Model ρ ( X, X ) BGITD* -. BGITD^ -.5 By comparig the sample correlatio, the estimated correlatio from type I BGITD seems to overestimate the correlatio of the data i Table 4.6. Chi square GOF test A summary of the χ values, log-likelihood values ad the P-values are show below. Table 4.7: Summary statistics for Table 4.6 Log-likelihood Degree of Model χ value P-value freedom BGIT* Table 4.7 shows that the fit of type I BGIT is sigificat where it provides low value of χ ad a high P-value. 7

23 Example : Table 4.8: Number of times baco ad eggs were purchased o four cosecutive shoppig trips 4 Total BNBD* BNBD^ BGITD* BGITD^ BGITD* BGITD^ BNBD* BNBD^ BGITD* BGITD^ BGITD* BGITD^ BNBD* BNBD^ BGITD* BGITD^ BGITD* BGITD^ BNBD* BNBD^ BGITD* BGITD^ BGITD* BGITD^ BNBD* BNBD^ BGITD* BGITD^ BGITD* BGITD^ *= MLE, ^= PGF based estimator, BGITD=type I BGITD, BGITD=type II BGITD ML estimates: BNBD, ˆp =.659, ˆp =.78, ˆp =.44, ˆv =.674. Type I BGITD, ˆp =.64, ˆp =., ˆq =.798, ˆq =.6, =8, =. 7

24 Type II BGITD, ˆp =.9, ˆp =., ˆq =764, ˆq =.4, ˆr =.88, ˆr =., =, =, =. PGF-based estimates: BNBD, ˆp =.699, ˆp =.895, ˆp =., ˆv =.669. Type I BGITD, ˆp =.66, ˆp =., ˆq =.6, ˆq =.4, =8, =. Type II BGITD, ˆp =.898, ˆp =., ˆq =.776, ˆq =., ˆr =.864, ˆr =., =, =, =. The correlatios for the estimated distributios are compute as follows. MLE PGFBE Model ρ ( X, X ) Model ρ ( X, X ) BNBD*. BNBD^.6 BGITD*.4 BGITD^.6 BGITD*.9 BGITD^.4 All of the estimated correlatio values appear to be slightly lower tha the sample correlatio which is.. Chi-square GOF test The summary of the χ, the log-likelihood ad the P- values are show below. Table 4.9: Summary statistics for Table 4.8 Log-likelihood Degree of Model χ value P-value freedom BNBD* BGITD* BGITD* All of the estimated distributios performed well as the P-value, Type I BGITD fits better tha BNBD ad Type II BGITD. χ values are low. Based o the 7

A proposed discrete distribution for the statistical modeling of

It. Statistical Ist.: Proc. 58th World Statistical Cogress, 0, Dubli (Sessio CPS047) p.5059 A proposed discrete distributio for the statistical modelig of Likert data Kidd, Marti Cetre for Statistical