A NEW FORM OF MULTIVARIATE GENERALIZED DOUBLE EXPONENTIAL FAMILY OF DISTRIBUTIONS OF KIND-2

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1 Journal of Rlablty and Statstcal Studs; ISSN (Prnt: , (Onln: Vol. 0, Issu (07: 79-0 A NEW FORM OF MULTIVARIATE GENERALIZED DOUBLE EXPONENTIAL FAMILY OF DISTRIBUTIONS OF KIND- G.S. Davd Sam Jayaumar, A. Solarau and A. Sulthan 3 Jamal Insttut of Managmnt, Jamal Mohamd Collg, Truchraall, Inda Dtt.of Mathmatcs, Jamal Mohamd collg, Truchraall, Inda 3 Jamal Insttut of Managmnt, Truchraall, Inda E Mal: samaya77@gmal.com; Solarama@yahoo.co.n; contact@amsulthan.n Rcvd Octobr 04, 06 Modfd March 8, 07 Acctd Jun 4, 07 Abstract Ths ar rooss a gnralaton of doubl onntal famly of Sarmanov ty contnuous multvarat symmtrc robablty dstrbuton wth rfrnc to a nw form of Sam- Sola s multvarat gnrald doubl onntal famly of dstrbuton of nd- from unvarat cas. Furthr, w hav found t s Margnal, multvarat condtonal dstrbutons, gnratng functons n a closd srs rsson form and also dscussd ts scal cass. Th locaton, scal and sha aramtrs hav layd a sgnfcanc rol and t has dtrmnd th famly of som stng nown and famlar bvarat, mturs of bvarat dstrbutons, gnrald and multvarat dstrbutons as subclass of th roosd multvarat gnrald doubl onntal famly of dstrbutons of nd-. Ky Words: Sarmanov Ty, Sam-Sola s Multvarat Gnrald Eonntal Famly of Dstrbutons, Closd Srs, Sha Paramtr, Mturs of Bvarat Dstrbutons.. Introducton Th gnralaton of gnrald robablty dstrbutons and thr alcaton s vrsatl n all flds scally n fnanc, nsuranc and rs managmnt tc. For th 5 ast dcads, statstcans gav mor concntraton to th gnralaton of onntal dstrbuton and latr thy tndd thr nvstgaton to th doubl onntal dstrbuton n whch th unvarat gnralaton of gnrald doubl onntal dstrbuton dvrtd th ath of dstrbuton thory to nw way. Th authors Ban and Englhard (973 stmatd confdnc ntrval usng mamum llhood stmators and scal aramtr for th doubl onntal dstrbuton. On th othr hand Kanman (977 dvlod th tolranc ntrvals for th doubl onntal dstrbuton. Smlarly, Mr t al., (98 stmatd quantls basd on two ordr statstcs for onntal and doubl onntal dstrbuton. In a dffrnt mannr, Ulrch and Chn (987 dscussd a bvarat doubl onntal dstrbuton and ts gnralaton. Latr, Govndaraulu (00 charactrd doubl onntal dstrbuton usng th ctd valus of th sacng assocatd wth crtan trm ordr statstcs and th rlaton btwn th dffrncs of two roduct momnts of crtan trm ordr statstcs nducd by random samls of arbtrary ss. Basd on

2 80 Journal of Rlablty and Statstcal Studs, Jun 07, Vol. 0( th ast rvws, many authors studd th unvarat, b-varat doubl onntal dstrbuton and thy ald n many aras. In ths ar, th authors mad an attmt to brng out a famly of doubl onntal dstrbuton n a gnrald multvarat form whch lads to lor mor sub-class of Bvarat, mturs of Bvarat, gnrald and multvarat doubl onntal dstrbutons. Th structur of th roosd dstrbuton and ts scal cass ar dscussd n th subsqunt sctons.. Sam-Sola s Multvarat Gnrald Doubl onntal famly of Dstrbutons of Knd- Dfnton.: Lt X,X,X,X ar th random varabls followng contnuous 3 unvarat Gnrald doubl onntal famly of dstrbutons (or Gnrald Doubl Gamma dstrbuton havng th locaton (,scal( λ, and sha ( aramtrs( α, wth man ( and varanc (( / /(/ / α λ ( α / for all (= to.thn th Multvarat Sam-Sola s gnrald doubl onntal famly of dstrbutons of nd- and ts dnsty s dfnd as γ γ / α α (/ λ λ f(,,, = ρ 3 = = ( α / = ( whr,,, γ, γ and ral, λ,, α, > 0, ρ and ρ s th corrlaton co-ffcnt btwn th random varabls. th and Dfnton.: From ( and f = ( /,thn usng mult-dmnsonal Jacoban of transformaton, th Sam-Sola s multvarat Gnrald standard doubl onntal famly of dstrbuton of nd- and ts dnsty s dfnd as 3 / α (/ γ γ λ α λ = ρ = = = ( α / f(,,, ( whr,, γ, γ and ral, λ, α, > 0, Thorm.3: Th cumulatv dstrbuton functon of th Sam-sola s Multvarat Gnrald doubl onntal famly of dstrbutons of nd- can also b rrsntd n trms of ts standardd form and t s gvn as (( α / (( / γ α γ F(,, 3, = ρ (γ / (γ / = = (/ λ ( α / (/ λ ( α / whr, φ( ; λ, α, =,, = = ϕ( ; λ, γ, α, ϕ( ; λ, γ, α, ρ φ( ; λ, α, (3 γ γ and ral,,, 0 ρ λ α >, ρ

3 A nw form of multvarat gnrald doubl onntal 8 Proof: Lt th Multvarat Cumulatv dstrbuton functon s gvn as F(,,, = 3 3 α / (/ u γ γ λ α λ ρu u u du = = = ( α / = F(,,, = α / (/ u γ γ λ α λ ρu u u du = = = ( α / = 3 α/ γ γ (/ λ u α λ ρu u = = ( α / = = F(,,, = 3 (( α γ / ρ (γ/ (γ / = = (/ λ ( α / (/ λ ( α / (( α γ / u du ( ;,,, ( ;,, ϕ λ γ α ϕ λ γ α, ( φ( ; λ, α, ρ = = = φ( ; λ, α, / λ ( α / whr s φ( ; λ, α, = s ds, ( α / 0 / λ (( α γ/ s ( ;,,, = s ds (( α / 0 γ ϕ λ γ α / λ (( α γ / s ( ;,,, = s ds (( α / 0 γ ϕ λ γ α ar th lowr Incomlt Gamma Intgrals. Thorm.4: Th Probablty dnsty functon of th Sam-sola s Multvarat Gnrald condtonal doubl onntal famly of dstrbutons of nd- of X on X, X3, X s γ γ α / α (/ λ λ ρ = = ( α / f(,, = 3 γ γ ρ = = (4 whr,, γ, γ and ral, λ,, α, > 0,, ρ and

4 8 Journal of Rlablty and Statstcal Studs, Jun 07, Vol. 0( Proof: Lt th Multvarat Condtonal dstrbuton of X on X, X3, X s gvn as f(,, = 3 f(,, = 3 f(,,, 3 f(,, 3 γ γ / α α (/ λ λ ρ = = = ( α / d γ γ α / α (/ λ λ ρ = = = ( α / γ γ α / α (/ λ λ ρ = = ( α/ f(,, = 3 γ γ ρ = = whr,,, γ, γ and ral, λ,, α, > 0, ρ Thorm.5: Man and Varanc of th Sam-sola s Multvarat Gnrald condtonal doubl onntal famly of dstrbutons of nd- ar γ ρ (( α γ/ = E( X X, X, X (5 = 3 γ/ γ γ (/ λ ( α/ ρ = = γ ρ (( α γ/ = V( X X, X, X 3 (6 = γ/ γ γ (/ λ ( α / ρ = = Proof: Th Condtonal ctaton and Condtonal varanc of a multvarat dstrbuton ar gvn as = E( X X, X, X f ( /,, d 3 3

5 A nw form of multvarat gnrald doubl onntal 83 E( X X, X, X = 3 γ γ α / α (/ λ λ ρ = = ( α / d γ γ ρ = = γ ρ (( α γ / = E( X X, X, X = 3 γ/ γ γ (/ λ ( α / ρ = = ( V( X X, X, X = E( X X, X, X f(,, d V( X X, X, X = 3 ( E( X X, X, X γ γ / α α (/ λ λ ρ = = ( α / d 3 γ γ ρ = = V( X X, X, X 3 γ ρ (( α γ / = = γ / γ γ (/ λ ( α / ρ = = Thorm.6: If thr ar =q random varabls, such that q random varabls X, X, X, X has a condtonal dndnc on th 3 q varablsx, X, X, X,thn th dnsty functon of th Sam-sola s Multvarat q q q 3 q Gnrald condtonal doubl onntal famly of dstrbutons of nd- s f(,,,,,, = 3 q q q q 3 q γ γ / q q q α α q (/ λ λ ρ ( ( α / = = = q q = q = q γ ρ γ (7

6 84 Journal of Rlablty and Statstcal Studs, Jun 07, Vol. 0( whr,, λ,, α, > 0, ρ, γ, γ and ral, Proof: Lt th multvarat condtonal law for q random varabls X, X, X3, Xq on th varabls Xq, Xq, Xq 3, Xq s gvn as f(,,,,,, = 3 q q q q 3 q f(,,,,,, = 3 q q q q 3 q f(,,,,,,, 3 q q q q 3 q f(,,, q q q 3 q γ γ / q q q q α α (/ λ λ ρ = = = ( α / γ γ q q q q α/ α q (/ λ ρ λ = = = ( α / = d f (,,,,,, = 3 q q q q 3 q γ γ / q q q α α q (/ λ λ ρ ( = = = ( α / γ γ q q ρ = q = q whr,, λ,, α, > 0, ρ, γ, γ and ral, Rmars: Th rsult of (7 can b dducd to th bvarat cas and t s gvn as f(,,,, = q q q 3 q γ γ / α α (/ λ λ ρ ( = ( α / γ γ q q ρ = q = q 3. Constants of Sam-Sola s Multvarat Gnrald Doubl onntal famly of Dstrbutons of nd- Thorm 3.: Th Margnal Co-varanc btwn th gnrald doubl onntal varabls X and X s gvn as

7 A nw form of multvarat gnrald doubl onntal 85 (( α γ / (( α γ / COV ( X,X = ρ ( / / γ γ ( / / λ λ ( α / ( α / (8 Proof: Lt th roduct momnt of th Sam-Sola s Multvarat Gnrald doubl onntal famly of dstrbutons of nd- s gvn as n trms of co-varanc as CO V ( X, X = ( ( f (,, 3, d COVX (, X = γ γ / α α (/ λ λ ( ( ρ d = = = ( α / = (( α γ/ (( α γ / COV( X,X = ρ ( / / γ γ ( / / λ λ ( α / ( α / Rmar: Th rsult can b gnrald to th co-varanc btwn th th and th random varabl ar gvn as (( α γ / (( α γ / COV( X,X = ρ / / γ ( / λ ( / λ γ ( α / ( α / (9 γ / γ / COV( X,X ( / λ ( / λ ( α / ( α / ρ= (( α γ / (( α γ / whr, ρ = Thorm 3.: Th Momnt gnratng functon of th Sam-sola s Multvarat Gnrald doubl onntal famly of dstrbutons of nd- s M, ( t, t t ( ;,,,, = ϕ t λ α = φ ( t ;,, λ, α, φ ( t ;,, λ, α, ( ;,,,, ( ;,,,, ρ = = ϕ t λ α ϕ t λ α Proof: Lt th momnt gnratng functon of a Multvarat dstrbuton s gvn as t M ( t, t, t t = f (,,, d M =,, 3, 3, 3 = ( t, t, t t =,, 3, 3, t γ γ / α α (/ λ = λ ρ d = = = ( α / = (0 (0a

8 86 Journal of Rlablty and Statstcal Studs, Jun 07, Vol. 0( From (0a, t s obsrvd that α / α ( / t λ λ d ( α / α q ( (( / t t q q / ( α / q = ( q!( / λ = α / γ α ( / t λ λ d ( α / ( α / (!( / t q ( t ( α ( γ q / ( γ q / q = q λ = Thus, by Intgraton M, ( t, t t = φ ( t;,, λ, γ, α, φ ( t;,, λ, γ, α, ϕ ( t;,, λ, α, ρ = = = ϕ ( t;,, λ, α, ϕ ( t;,, λ, α, whr φ ( t ;,, λ, γ, α, = t q ( t ( α ( γ q / ( γ q / ( α / q = (q!( / λ t q ( t ( α ( γ q / φ ( t ;,, λ, γ, α, = ( γ q/ ( α / q= (q!(/ λ q ( (( / t t α q ϕ ( t;,, λ, α, = q/ ( α / q= ( q!(/ λ q ( (( / t t α q ϕ ( t ;,, λ, α, = q/ ( α / q= ( q!(/ λ and s th Gamma functon rsctvly. Thorm 3.3: Th Cumulant of Sam-sola s Multvarat Gnrald doubl onntal famly of dstrbutons of nd- s C ( t, t t =, P P φ ( t;,, λ, γ, α, φ ( t ;,, λ, γ, α, ( log ( ϕ ( t;,, λ, α, log ρ = = = ϕ ( t;,, λ, α, ϕ ( t;,, λ, α,

9 A nw form of multvarat gnrald doubl onntal 87 Proof: It s found from C,, ( t, t, t3, t = log ( M,, ( t, t, t3, t 3, 3, Thorm 3.4: Th Charactrstc functon of Sam-sola s Multvarat Gnrald doubl onntal famly of dstrbutons of nd- s φ ( t, t t =, ω ( t;,, λ, γ, α, ω ( t;,, λ, γ, α, ψ ( t;,, λ, α, ρ ψ ( t ;,,,, ( ;,,,, λ α ψ t λ α = = = ( whr t q ( t ( α ( γ q / ω ( t;,, λ, γ, α, = ( γ q/ ( α / q= (q!(/ λ t q ( t ( α ( γ q / ω ( t ;,, λ, γ, α, = ( γ q/ ( α / q= (q!(/ λ q ( (( / t t α q ψ ( t;,, λ, α, = q / ( α / q= ( q!( / λ q ( (( / t t α q ψ ( t ;,, λ, α, = q/ ( α / q= ( q!(/ λ and s th Gamma functon rsctvly. Proof: Lt th charactrstc functon of a multvarat dstrbuton s gvn as φ t = φ,, ( 3, t, t, t3, t (,, 3, = f d = ( t, t, t t =,, 3, 3, t γ γ / α α (/ λ = λ ρ d = = = ( α / = (a From (a, t s obsrvd that α / α ( / t λ λ d ( α / q ( (( / t t α q q / ( α / q = ( q!( / λ α / α ( / t λ λ ( d ( α / ( α / (!( / λ t q ( t ( α ( γ q / ( γ q / q = q = =

10 88 Journal of Rlablty and Statstcal Studs, Jun 07, Vol. 0( Thus, by Intgraton φ ( t, t t =, ω ( t;,, λ, γ, α, ω ( t;,, λ, γ, α, ψ ( t;,, λ, α, ρ = = = ψ ( t;,, λ, α, ψ ( t;,, λ, α, 4. Som Scal cass Rsult 4.: If ρ = 0,whr,=,,, thn thr s no corrlaton btwn th random varabls and th Sam-Sola s multvarat Gnrald doubl onntal dnsty functon s rducd to roduct of un-varat Gnrald doubl onntal dstrbuton. Rsult 4.: From ( and f =, thn th dnsty of Sam-sola s Multvarat Gnrald doubl onntal famly of dstrbutons of nd- rducs to γ γ f (, = ρ α / α / α α ( / λ (/ λ 4 ( α / ( α / whr,,,, γ, γ and ral, λ λ (3,, λ, λ, α, α,, > 0, ρ Ths s calld Sam-Sola s B-varat gnrald doubl onntal famly of dstrbutons of nd-. Rsult 4.3: From (0,f = ( / and = ( /, thn usng twodmnsonal Jacoban of transformaton, th Sam-sola s B-varat Gnrald standard doubl onntal famly of dstrbutons of nd- and ts dnsty s gvn as / / α α γ γ (/ λ (/ λ α α λ λ f(, = ( ρ 4 ( α / ( α / (4 whr,, γ, γ and ral,,, λ, λ, α, α,, > 0, ρ Ths s calld Sam-Sola s B-varat gnrald doubl onntal famly of dstrbutons of nd-.

11 A nw form of multvarat gnrald doubl onntal 89 Rsult 4.4: shown blow som Bvarat dstrbutons as scal cass of Sam-sola s Bvarat gnrald standard doubl onntal famly of dstrbutons of nd- from (4 for dffrnt sttngs of scal and sha aramtrs gvn Standard Lalac dstrbuton, Wth Scal aramtrs (, and Sha aramtrs ( γ, ( 4,, γ, γ and ral, ρ γ,,,,,( γ γ ρ whr Standard normal dstrbuton, Wth Scal aramtrs (, and Sha aramtrs ( γ, ( π γ γ γ,,,,,( ρ whr,, γ, γ and ral, ρ Doubl Gamma dstrbuton, Wth Scal aramtrs ( ν, and Sha aramtrs ν ( γ, γ, α, ( α α γ γ ν ν α α ν ν 4 ( α ( α α,,,( ρ whr,, γ, γ and ral, v, v, α, α > 0, ρ Standard Doubl Raylgh dstrbuton, Wth Scal aramtrs (, and Sha aramtrs ( γ, ( 4 γ γ γ,,,,, ( ρ whr,, γ, γ and ral, ρ Doubl ch-squar dstrbuton, Wth Scal aramtrs (, and Sha aramtrs ( γ, γ, n /, n /,,, ( n/ n/ (/ (/ ( n/ ( n/ 4 ( n / ( n / γ γ ( ρ whr,, γ, γ and ral, n, n > 0, ρ Doubl Mawll-Boltmann dstrbuton, Wth Scal aramtrs ( and Sha aramtrs ( γ, γ γ γ,3,3,,,( ρ, π whr,, γ, γ and ral,, > 0, ρ Doubl Nagaam dstrbuton, Wth Scal aramtrs ( Ω, ( γ, γ, m, m,,, γ γ ( ρ /m Ω /m and Sha aramtrs m m m m ( m / ( m / m m Ω Ω Ω Ω ( m ( m whr,, γ, γ and ral, m, m /, Ω, Ω > 0 ρ

12 90 Journal of Rlablty and Statstcal Studs, Jun 07, Vol. 0( Doubl ch dstrbuton, Wth Scal aramtrs (, and Sha aramtrs γ, γ, α, α,,, ( α / α / ( / ( / α α ( ( α / ( α / γ γ ( ρ whr,, γ, γ and ral, α, α > 0, ρ Doubl Erlang- dstrbuton, Wth Scal aramtrs ( /α, /α and Sha aramtrs ( γ, γ, α, α,,, γ γ ( α ( α α α ( α α α α 4 ( α ( α ( ρ whr,, γ, γ and ral, α, α,, > 0, ρ Doubl-Wbull dstrbuton, Wth Scal aramtrs ( α, α and Sha aramtrs ( γ, γ, α, α, α, α, α α α α γ γ α α ( / ( / α α 4 ( ρ whr,, γ, γ and ral, α, α,, > 0, ρ Rsult 4.5: shown blow som slctd B-varat mtur of dstrbuton as scal cass of Sam-sola s B-varat gnrald standard doubl onntal famly of dstrbutons of nd- from (4 for dffrnt sttngs of scal and sha aramtrs gvn Lalac-normal dstrbuton, Wth Scal aramtrs (, and sha aramtrs ( γ, π γ γ γ,,,,, ( ρ whr,, γ, γ and ral, ρ Lalac-Doubl Gamma dstrbuton, Wth Scal aramtrs (, /ν and sha α aramtrs ( γ, γ,,,,, v α ( v α / ( α γ γ ( ρ whr,, γ, γ and ral, v, α > 0, ρ Lalac-Doubl Raylgh dstrbuton, Wth Scal aramtrs (, and sha aramtrs ( γ, γ,,,,, π γ γ ( ρ

13 A nw form of multvarat gnrald doubl onntal 9 whr,, γ, γ and ral, ρ Lalac-Doubl ch-squar dstrbuton, Wth Scal aramtrs (, and sha aramtrs ( γ, γ,, n /,,,, n/ ( (/ n/ 4 ( n / γ γ ( ρ whr,, γ, γ and ral, n > 0, ρ Lalac-Doubl Mawll Boltmann dstrbuton, Wth Scal aramtrs (, and sha aramtrs ( γ, γ,, 3,,, π γ γ ( ρ whr,, γ, γ and ral, > 0, ρ Lalac-Doubl Nagaam dstrbuton, Wth Scal aramtrs (, Ω / m and sha aramtrs ( γ, γ,, m,,,, m m ( m / m γ γ Ω Ω ( ρ ( m whr,, γ, γ and ral, m /, Ω > 0, ρ Lalac-Doubl ch dstrbuton, Wth Scal aramtrs (, and sha aramtrs ( γ, γ,, α,,, α / (/ α ( α / γ γ ( ρ whr,, γ, γ and ral, α > 0, ρ Lalac- Doubl Erlang- dstrbuton, Wth Scal aramtrs (,/ α and sha aramtrs ( γ, γ,, α,,, α γ γ ( α α ( α ( ρ 4 ( α whr,, γ, γ and ral,, α > 0, ρ Lalac- Doubl-Wbull dstrbuton, Wth Scal aramtrs (, α and sha aramtrs ( γ, γ,, α,, α,

14 9 Journal of Rlablty and Statstcal Studs, Jun 07, Vol. 0( α α γ α(/ α α 4 γ ( ρ whr,, γ, γ and ral,, α > 0, ρ Normal-Doubl Gamma dstrbuton, Wth Scal aramtrs (,/ ν and sha aramtrs ( γ, γ,, α,,, α v α v γ γ ( ρ π( ( α whr,, γ, γ and ral, v, α > 0, ρ Normal-Doubl Raylgh dstrbuton, Wth Scal aramtrs (, and sha aramtrs ( γ, ( π γ γ γ,,,,, ( ρ whr,, γ, γ and ral, ρ Normal-Doubl ch-squar dstrbuton, Wth Scal aramtrs (, and sha aramtrs ( γ, γ,, n /,,, n/ (/ ( n/ ( π( n / γ γ ( ρ whr,, γ, γ and ral, n > 0, ρ Normal-Doubl Mawll dstrbuton, Wth Scal aramtrs aramtrs ( γ, γ γ γ,, 3,,, ( ρ (, and sha π whr,, γ, γ and ral, > 0, ρ Normal Doubl Nagaam dstrbuton, Wth Scal aramtrs (, Ω / m and sha aramtrs ( γ, γ,,m,,, m m ( m / m γ γ Ω Ω ( ρ π( m whr,, γ, γ and ral, m /, Ω > 0, ρ Normal-Doubl ch dstrbuton, Wth Scal aramtrs (, and sha aramtrs ( γ, α/ (/ α ( π( α / γ γ γ,, α,,, ( ρ whr,, γ, γ and ral, α > 0, ρ

15 A nw form of multvarat gnrald doubl onntal 93 Normal- Doubl Erlang- dstrbuton, Wth Scal aramtrs (,/ α and sha aramtrs ( γ, γ,,α,,, α ( α γ γ α α ( ρ π ( α whr,, γ, γ and ral,, α > 0, ρ Normal- Doubl-Wbull dstrbuton, Wth Scal aramtrs (, α and sha aramtrs ( γ, γ,, α,,, α, α α γ γ α(/ α ( ρ π whr,, γ, γ and ral,, α > 0, ρ Doubl Gamma-Doubl Raylgh dstrbuton, Wth Scal aramtrs (/ ν, and sha aramtrs ( γ, γ, α,,,, α ( v α v γ γ ( ρ 4 ( α whr,, γ, γ and ral, v, α> 0, ρ Doubl Gamma-Doubl chsquar dstrbuton, Wth Scal aramtrs (/ ν, and sha aramtrs ( γ, γ, α, n /,,, α n/ v (/ α ( n/ ν γ γ ( ρ 4 ( α ( n / whr,, γ, γ and ral, v, α, n> 0, ρ Doubl Gamma-Doubl Mawll Boltmann dstrbuton, Wth Scal aramtrs ( (/ ν,, and sha aramtrs ( γ, γ,α, 3,,, α v ( v α ( α π γ γ ( ρ whr,, γ, γ and ral, v, α, > 0, ρ Doubl Gamma-Doubl Nagaam dstrbuton, Wth Scal aramtrs (/ ν, Ω / m and sha aramtrs ( γ, γ, α, m,,, m α m v v ( m / α m Ω Ω ( α ( m γ γ ( ρ

16 94 Journal of Rlablty and Statstcal Studs, Jun 07, Vol. 0( whr,, γ, γ and ral, m /, v, α, Ω > 0, ρ Doubl Gamma-Doubl Ch dstrbuton, Wth Scal aramtrs ((/ ν, and sha aramtrs ( γ, γ, α, α,,, α α/ v (/ α α v γ γ ( ρ ( α ( α / whr,, γ, γ and ral, v, α, α > 0, ρ Doubl Gamma-Doubl Erlang- dstrbuton, Wth Scal aramtrs (/ ν,/ α and sha aramtrs ( γ, γ, α, α,,, α α γ γ v ( α 4 ( α ( α ( ρ α α ( v α whr,, γ, γ and ral, v, α, α, > 0, ρ Doubl Gamma-Doubl Wbull dstrbuton, Wth Scal aramtrs (/ ν, and sha aramtrs ( γ, γ, α, α,, α, α α α v γ γ v α α α ( ρ 4 ( α whr,, γ, γ and ral, v, α, α, > 0, ρ α Doubl Raylgh-Doubl ch-squar dstrbuton, Wth Scal aramtrs (, and sha aramtrs ( γ, γ,, n /,,, n/ (/ ( n/ ( 4 ( n / γ γ ( ρ whr,, γ, γ and ral, n > 0, ρ Doubl Raylgh-Doubl Mawll Boltmann dstrbuton, Wth Scal aramtrs (, and sha aramtrs ( γ, γ,, 3,,, π γ γ ( ρ whr,, γ, γ and ral, > 0, ρ Doubl Raylgh-Doubl Nagaam dstrbuton, Wth Scal aramtrs (, Ω / m and sha aramtrs ( γ, γ,, m,,,

17 A nw form of multvarat gnrald doubl onntal 95 m m ( m / m Ω Ω ( m γ γ ( ρ whr,, γ, γ and ral, m /Ω > 0, ρ Doubl Raylgh-Doubl Ch dstrbuton, Wth Scal aramtrs (, and sha aramtrs ( γ, γ,, α,,,, α/ (/ α ( ( α / γ γ ( ρ whr,, γ, γ and ral, α > 0, ρ Doubl Raylgh-Doubl Erlang- dstrbuton, Wth Scal aramtrs (,/ α α and sha aramtrs ( γ, γ,,,,, α ( α γ γ α α ( ρ 4 ( α whr,, γ, γ and ral,, α > 0, ρ Doubl Raylgh-Doubl Wbull dstrbuton, Wth Scal aramtrs (, and sha aramtrs ( γ, γ,, α,, α, α α γ γ α α ( ρ 4 whr,, γ, γ and ral,, α > 0, ρ Doubl ch-squar-doubl Mawll Boltmann dstrbuton, Wth Scal aramtrs (, and sha aramtrs ( γ, γ, n /, 3,,, γ γ ( ρ α / n (/ ( n/ ( n / π whr,, γ, γ and ral, n, > 0, ρ Doubl ch-squar-doubl Nagaam dstrbuton, Wth Scal aramtrs (, Ω / m and sha aramtrs ( γ, γ, n /, m,,, / m n m ( (/ ( m / n/ m Ω Ω ( n / ( m γ γ ( ρ whr,, γ, γ and ral, m /, n, > 0, ρ

18 96 Journal of Rlablty and Statstcal Studs, Jun 07, Vol. 0( Doubl ch-squar-doubl Ch dstrbuton, Wth Scal aramtrs (, and sha aramtrs ( γ, γ, n /, α,,, n/ α/ (/ (/ ( n/ α ( ( n / ( α / γ γ ( ρ whr,, γ, γ and ral, n, α > 0, ρ Doubl ch-squar-doubl Erlang- dstrbuton, Wth Scal aramtrs (,/ α and sha aramtrs ( γ, γ, n /, α,,, n/ α ( (/ ( n/ α γ γ α α ( ρ 4 ( n / ( α whr,, γ, γ and ral, n, α, > 0, ρ Doubl ch-squar-doubl Wbull dstrbuton, Wth Scal aramtrs (, α and sha aramtrs ( γ, γ, n /, α,, α, / n ( / (/ n γ γ ( ρ ( n / π whr,, γ, γ and ral, n, α, > 0, ρ Doubl Mawll Boltmann- Doubl Nagaam dstrbuton, Wth Scal aramtrs (, Ω / m and sha aramtrs ( γ, γ, 3, m,,, m ( / m m m Ω Ω π ( m γ γ ( ρ whr,, γ, γ and ral, m /,, Ω > 0, ρ Doubl Mawll Boltmann- Doubl Ch dstrbuton, Wth Scal aramtrs (, and sha aramtrs ( γ, γ, 3, α,,, α/ (/ α π ( α / γ γ ( ρ whr,, γ, γ and ral,, α > 0, ρ Doubl Mawll Boltmann- Doubl Erlang- dstrbuton, Wth Scal aramtrs (,/ α and sha aramtrs ( γ, γ, 3, α,,,

19 A nw form of multvarat gnrald doubl onntal 97 α α γ γ ( α α π ( α ( ρ whr,, γ, γ and ral,, α, > 0, ρ Doubl Mawll Boltmann- Doubl Wbull dstrbuton, Wth Scal aramtrs (, α and sha aramtrs ( γ, γ, 3, α,, α, α α γ γ α (/ α π ( ρ whr,, γ, γ and ral,, α, > 0, ρ Doubl Nagaam-Doubl ch dstrbuton, Wth Scal aramtrs ( Ω / m, and sha aramtrs ( γ, γ, m, α,,, / m m α ( m / (/ m α γ γ Ω Ω ( ρ ( m ( α / whr,, γ, γ and ral, m /, Ω, α> 0, ρ Doubl Nagaam-Doubl Erlang- dstrbuton, Wth Scal aramtrs ( Ω / m,/ α and sha aramtrs ( γ, γ, m, α,,, m m α α γ γ ( m / ( α m α Ω Ω ( m ( α ( ρ whr,, γ, γ and ral, m /, Ω, α, > 0 ρ Doubl Nagaam-Doubl Wbull dstrbuton, Wth Scal aramtrs ( Ω / m, α and sha aramtrs ( γ, γ, m, α,, α, α m m α γ γ ( m / α(/ m α Ω Ω ( m ( ρ whr,, γ, γ and ral, m /, Ω, α, > 0, ρ Doubl ch- Doubl Erlang- dstrbuton, Wth Scal aramtrs (,/ α and sha aramtrs ( γ, γ, α, α,,, α / α (/ ( α γ γ α α α ( ρ ( α / ( α whr,, γ, γ and ral, α, α, > 0, ρ

20 98 Journal of Rlablty and Statstcal Studs, Jun 07, Vol. 0( Doubl ch- Doubl Wbull dstrbuton, Wth Scal aramtrs (, sha aramtrs ( γ, γ, α, α,, α, α / α α γ γ (/ α(/ α α ( ρ ( α / whr,, γ, γ and ral, α, α, > 0, ρ Doubl Erlang-- Doubl Wbull dstrbuton, Wth Scal aramtrs (/ α, α and sha aramtrs ( γ, γ, α, α,, α, α α α α γ γ ( α α(/ α α 4 ( α ( ρ whr,, γ, γ and ral, α, α, > 0, ρ Rsult 4.6: shown blow som slctd Multvarat dstrbutons as scal cass of Sam-Sola s Multvarat gnrald doubl onntal famly of dstrbuton of nd- from ( for dffrnt sttngs of Locaton, scal and sha aramtrs gvn Gnrald Lalac, dstrbuton wth locaton aramtrs (, scal aramtrs (,,, sha aramtrs ( γ,,,, γ γ ρ ( = = = (/ whr,, α, γ, γ and ral,, > 0, Lalac, dstrbuton wth locaton aramtrs ( scal aramtrs(,, sha aramtrs ( γ,,, γ γ ρ ( = = = whr,,, γ, γ and ral, > 0, Gnrald normal, dstrbuton wth locaton aramtrs ( scal aramtrs (,, sha aramtrs ( γ,,, γ γ (/ (/ ρ ( = = = (/ ρ ρ and

21 A nw form of multvarat gnrald doubl onntal 99 whr, ρ,, γ, γ and ral, λ,, > 0, Normal, dstrbuton wth locaton aramtrs ( scal aramtrs (,, sha aramtrs ( γ,,, γ γ / ρ ( = = π = whr,,, γ, γ and ral, Gnrald Doubl Gamma, dstrbuton wth locaton aramtrs (0, scal aramtrs(, / ν, sha aramtrs ( γ, α,, v v ( whr, α/ γ γ α ρ = = = ( α / γ, γ and ral,, v, α, > 0, Doubl Gamma, dstrbuton wth locaton aramtrs (0, scal aramtrs (, / ν, sha aramtrs ( γ, α,, v α v ( whr, α γ γ ρ = = = ( α, γ, γ and ral, v, α> 0, Doubl Raylgh, dstrbuton wth locaton aramtrs (0, scal aramtrs (,, sha aramtrs ( γ,,, γ γ ρ ( = = = whr,, γ, γ and ral, > 0, Doubl ch-squar, dstrbuton wth locaton aramtrs (0, scal aramtrs (,, sha aramtrs ( γ, n /,, (/ ( whr, n / γ ( n / γ ρ = = = ( n /, γ, γ and ral, n> 0, Doubl Mawll-Boltmann, dstrbuton wth locaton aramtrs (0, scal aramtrs (,, sha aramtrs ( γ, 3,, ρ ρ ρ ρ ρ

22 00 Journal of Rlablty and Statstcal Studs, Jun 07, Vol. 0( γ γ ρ ( = = π = whr,, γ, γ and ral, > 0, Doubl Nagaam, dstrbuton wth locaton aramtrs (0, scal aramtrs (, Ω / m, sha aramtrs ( γ, m,, / m α ( m / γ γ Ω m Ω ρ = = = ( m whr,, γ, γ and ral, m /, Ω> 0, Doubl ch, dstrbuton wth locaton aramtrs (0, Scal aramtrs (,, sha aramtrs ( γ, α,, whr, (/ α/ γ α γ ρ = = = ( α /, γ, γ and ral, α> 0, Doubl Erlang-, dstrbuton wth locaton aramtrs (0, scal aramtrs (,/ α, sha aramtrs ( γ, α,, ( α α ( whr, α γ γ α ρ = = = ( α, γ, γ and ral,, α> 0, Doubl-Wbull, dstrbuton wth locaton aramtrs (0, scal aramtrs (,, sha aramtrs ( γ, α, α, α γ γ α α ρ ( = = = whr,,, α> 0, Doubl rror, dstrbuton wth locaton aramtrs (0, scal aramtrs (,, sha aramtrs ( whr, ρ γ,,, ( γ γ ( / ρ = =, γ, γ and ral, Gnrald Doubl onntal of nd-, dstrbuton wth locaton aramtrs ( scal aramtrs (, λ, sha aramtrs (, α,, / α α (/ λ λ ρ ( = = = ( α / ρ ρ ρ ρ π ρ =

23 A nw form of multvarat gnrald doubl onntal 0 whr,,, γ, γ and ral, λ,, α, > 0, ρ 5. Concluson Th multvarat gnralaton of gnrald doubl onntal famly of dstrbutons n a natural form s mossbl and th authors adotd th Sarmanov systm of multvarat gnralaton of dstrbutons. At frst, th margnal unvarat dstrbutons of th Sam-Sola s Multvarat Gnrald doubl onntal famly of dstrbutons of nd- ar unvarat Gnrald doubl onntal dstrbutons. Scondly, th co-varanc and corrlaton co-ffcnt of any two gnrald doubl onntal varabls wll chang basd on th sha aramtrs of th dstrbuton. Smlarly, basd on th aramtr sttngs of th multvarat gnrald doubl onntal famly of dstrbutons of nd-, th authors furthr drvd th multvarat dstrbutons for th stng unvarat contnuous dstrbutons. Thus th authors rcommnd that th gnralaton of Sarmanov ty famly of symmtrc multvarat dstrbuton ons th way for logcal tnson of th gnralaton of symmtrc famly of all unvarat contnuous robablty dstrbutons n th statstcal ltratur. Rfrncs. Ban, L. J. and Englhardt, M. (973. Intrval stmaton for th twoaramtr doubl onntal dstrbuton, Tchnomtrcs, 5(4, Govndaraulu, Z. (00. Charactraton of doubl onntal dstrbuton usng momnts of ordr statstcs, Communcatons n Statstcs-Thory and Mthods, 30(, Kanman, R. F. (977. Tolranc ntrvals for th doubl onntal dstrbuton, Journal of th Amrcan Statstcal Assocaton, 7(360a, Mr, M. A., Dal, U., and Hassann, K. M. (98. Estmaton of quantls of onntal and doubl onntal dstrbutons basd on two ordr statstcs, Communcatons n Statstcs-Thory and Mthods, 0(9, Ulrch, G., and Chn, C. C. (987. A bvarat doubl onntal dstrbuton and ts gnralaton, In ASA Procdngs on Statstcal Comutng (. 7-9.