MOMENTS OF NONIDENTICAL ORDER STATISTICS FROM BURR XII DISTRIBUTION WITH GAMMA AND NORMAL OUTLIERS

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1 Journa of Matheat and Statt, 9 (): 5-6, 3 ISSN Sene Puaton do:.3844/p Puhed Onne 9 () 3 ( MOMENTS OF NONIDENTIAL ORDER STATISTIS FROM BURR XII DISTRIBUTION WITH GAMMA AND NORMAL OUTLIERS Jaoo, A.A. and Z.A. A-Saary Departent of Statt, Gr oege of Sene-Kng, Adu-Azz Unverty, Jeddah, Saud Araa Reeved --9, Reved 3--9; Aepted 3-4- ABSTRAT There are oe dtruton wth no pe oed for for dtruton funton uh a the Nora and Gaa dtruton. Th w e the proe f we want to fnd oent of nondenta order tatt n the preene of Gaa and Nora outer oervaton. We ued the dea of approxatng Nora and Gaa dtruton wth Burr type XII dtruton. We get nge oent for order tatt fro ape of ndependent nondentay dtruted Burr XII rando varae that ontan p-outer fro Nora or Gaa dtruton. Approxatng thee dtruton wth Burr XII dtruton and then we opared the reut y prevou ethod. Keyword: Approxaton, Nondenta Order Statt, Burr XII Dtruton, Nora Dtruton, PERMINANTS, Gaa Dtruton. INTRODUTION In th hapter we ued the dea of approxatng Nora and Gaa dtruton wth Burr XII to get There are no pe oed for ext for the nge oent for order tatt fro ape of nora dtruton funton and the gaa dtruton ndependent nondentay dtruted Burr XII rando funton o that approxaton to G (x) ut e ued to varae that ontan p-outer fro nora or Gaa fnd oent of the rth denta order tatt. An dtruton. approxatng thee dtruton wth Burr approah of otanng oe approxaton to the nora XII dtruton and then we opared the reut y thoe dtruton wa preented y Burr (94). Burr (967; ung the Baraat and Adeader (4). 973), Burr reauated the vaue of hape paraeter Barnett and Lew (994) have defned an outer n for Burr XII dtruton. Thee vaue gve a oer a et of data to e an oervaton or uet of approxaton to the nora dtruton. Another oervaton whh appear to e nontent wth the approah aed on approxatng gaa dtruton reander of the et of data. They ao dere evera wth the Burr fay dtruton ha een preented y ode for outer; ee (Mohref and Sutan, 7). Tadaaa and Raerg (975) and Wheeer (975), Denty funton and ont denty funton of order who approxate Gaa wth two paraeter wth Burr tatt arng fro a ape of a nge outer have wth two paraeter. Tadaaa (977) ued the een gven y Shu (978) and Hartey and Davd (978). generazed four paraeter B-dtruton to One ay ao refer to Vaughan and Venae (97) for approxate gaa dtruton and put a tae for ore genera expreon of dtruton of order eeted ntegra vaue of hape paraeter and other tatt ung peranent expreon. Arnod and paraeter whh Burr XII dtruton funton Baarhnan (989) have otaned the denty funton approxate to the exat Gaa dtruton funton. of X when the ape of ze n ontan undentfed orrepondng Author: Jaoo, A.A., Departent of Statt, Gr oege of Sene- Kng Adu-Azz Unverty, Jeddah, Saud Araa Sene Puaton 5

2 Jaoo, A.A. and Z.A. A-Saary /Journa of Matheat and Statt 9 (): 5-6, 3 nge outer. They ao otaned the ont denty funton of X and X :n, r< n. Baarhnan and Baauraanan (995) ha derved oe reurrene reaton atfed y the nge and produt oent of order tatt fro the rght trunated exponenta dtruton. Ao he ha dedued the reurrene reaton for the utpe outer ode (wth ppage of oervaton), ee ao Baarhnan (994). hd et a. () have derved oe reurrene reaton for the nge and produt oent of order tatt fro n ndependent and non-dentay dtruted Loax and the rght-trunated Loax rando varae. We aue X, X,,X p are ndependent wth proaty denty funton f (x) whe X n -p+,,x n are ndependent were are fro oe odfed veron of f (x) whh a g(x) n whh the hape paraeter have een hfted n vaue. Fnay, oe pea ae are dedued. The proaty denty funton of the rth order tatt X, under the utpe outer ode an e wrtten a, ee hd (996) Equaton : f [ ] p (x) = n ( np, r) = ax (, rp) f (x){f(x)} {G(x)} Sene Puaton r { F(x)} { G(x)} + np p r+ + n ( np,r) = ax (, rp) g(x){f(x)} {G(x)} r np p r+ { F(x)} { G(x)}, r n, p =,,, n, < x< Where Equaton : =! (r - -)!(n - p - -)!(p - r + +) (n - p)!p! () () (n - p)!p! =! (r - -)!(n - p - )!(p - r + ) Settng p = n () we otan the orrepondng pdf n ae of the nge outer gven y Shu (978). In th tudy, we onder the ae when the varae X, X,, X n-p are ndependent oervaton fro Burr XII wth four paraeter dtruton wth denty Equaton 3: 5 ρ xa ρ f(x) = ( + ) xa, a x, ρ >, >, > (3) and X n-p+,, X n are fro the ae dtruton wth denty Equaton 4: τ x a g(x) = ( + ) τ x a, a x, τ>, >,> (4) The orrepondng uuatve dtruton funton F (x) and G(x) are gven a Equaton 5 and 6: ρ x a F(x ) = ( + ), x a, ρ>, >, > -τ (5) x -a G( x )=- (+ ), (6) a x,τ>, >, > The reaton etween f (x) and F (x) gven y Equaton 7: ρ x a f ( x ) = [F( x )] x a f ( x ) (7) Sary, the reaton etween g(x) and G(x) Equaton 8: τ x a g (x ) = [G (x )] x a g (x ) (8) In the foowng eton, we ue (3) and (4) to derve the nge and produt oent of order tatt fro Burr XII dtruton under the utpe outer ode. Th tuaton nown a a utpe outer ode wth

3 Jaoo, A.A. and Z.A. A-Saary /Journa of Matheat and Statt 9 (): 5-6, 3 ppage of p oervaton; Barnett and Lewe (994). Th pef utpe outer ode wa ntrodued y Launer and B (979)... Snge Moent We derve the th oent of the r th order tatt under utpe outer ode (wth a ppage of p oervaton). Let () [p]; ( r n) denote the th nge oent of order tatt n the preene of p-outer oervaton fro BurrXII dtruton The foowng theore gve an expt for of () [p]. Theore For r n, p =,,, n and =,, the nge ( ) oent gven y Equaton 9: [p] n ( n p, r ) ( ) r : n [ p ] = ρ = ax (, r p ) r r + ( ) = = a β ( φ +, + ) = + τ n ( n p, r ) = ax (, r p ) r r + = ( ) = a β ( φ +, + ) = where, =, Equaton : φ = ρ(n p + ) + τ(p r ) +, (n p)!p! =! (r )!(n p )!(p r + + ) (n p)!p! =! (r )!(n p )!(p r + ) Proof Startng fro (), we have Equaton : ( ) r :n [ p ] = x f [ p ]( x ) dx = n ( np,r ) = ax (, rp) np p r+ + { F( x )} { G ( x )} dx + n ( np, r) = ax (, r p ) np p r+ { F( x )} { G ( x )} x f ( x ){F( x )} {G ( x )} r x g ( x ){F( x )} {G ( x )} r dx () (9) () Ung the reaton Equaton : [ F( x )] = ρ x a x a f (x )( + ) τ x a [G ( x )] g (x ) = x a ( + ) ( ) We get Equaton 3 and 4: n (n p, r) = ax (, r p) a [p] = x f (x ){F(x )} {G (x )} r { F(x )} np ρ x a x a f ( x ) ( + ) p r + + { G ( x )} dx n ( np, r ) = ax (, r p ) a + x τ x a [G ( x )] x a (+ ) {F( x )} {G (x )} r np p r + { F(x )} { G ( x )} dx ( ) ρ [p] = n ( np, r ) = ax (, r p) x a + a x (x a ) ( ) {F( x )} {G (x )} r np p r + + { F( x )} { G ( x )} dx τ + n ( np,r ) = ax (, r p ) x a + a x (x a ) ( ) r np {F( x )} {G (x )} p r + + { G (x )} dx Now y wrtng: { F( x )} () (3) (4) Sene Puaton 53

4 Jaoo, A.A. and Z.A. A-Saary /Journa of Matheat and Statt 9 (): 5-6, 3 F (x) = - (-F (x)) and G (x) = -(-G (x)) n (4) and expand we get Equaton 5: ( ) ρ [p] = n ( np, r ) = ax (, r p) x a + a x (x a ) ( ) r r ( ) + = = { F(x )} n p + p r { G ( x )} dx n ( n p, r ) τ + = ax (, rp ) a n p + p r { G ( x )} dx x a x (x a ) ( + ) r r ( ) + = { F(x )} a = We now that Equaton 6-8: = ρ r r ( ) + = τ x a x a F(x ) = ( + ) G ( x ) = ( + ) n ( n p, r ) ( ) ρ [p] = = ax (, rp) x ( x a ) [ ρ(n p + ) + τ x a (p r+ + + ) + ] ( + ) dx τ + n ( np, r) = ax (, r p ) x (x a ) a = r r ( ) + = [ ρ(n p + ) + τ x a (p r+ + + ) + ] ( + ) dx Le: (5) (6) x a x a y = + = y y x a y y = x a = y y y x = + a y y dx = dy y y at x = a y=, at x = y= n ( np,r ) ( ) [p] = ax (, r p) = ρ r r ( ) = = y ( ) a + y [ ρ(n p + ) y + τ ( p r ) + ] + τ n ( np,r ) = ax (, r p ) r = = y Let: ( p r ) + ] dy r ( ) y ( ) a + y [ ρ(n p + ) + τ y ( ) + a y dy = y = a = y n ( np, r) ( ) [p] = ρ = ax (, r p) + + (7) Sene Puaton 54

5 Jaoo, A.A. and Z.A. A-Saary /Journa of Matheat and Statt 9 (): 5-6, 3 r r ( ) = [ ρ(n p + ) + a ( y) = τ ( p r ) + ] y dy n ( np,r ) + τ = ax (, r p ) r r ( ) = = a [ ρ(n p + ) + y ( ) y = τ ( p r ) + ] Sene Puaton dy + + Let Φ = ρ(n-p-+)+τ(p-r+++)+ Equaton 9: Φ ( ) y y dy = β Φ, + n ( np, r) ( ) [p] = ρ = ax (, rp) r r + ( ) = = a β( φ +, + ) = n ( np, r) r +τ = ax (, r p) = = r + ( ) a β( φ +, + ) Rear (8) (9) If we put a =, = n (9) and onder = we get oent of order tatt fro Burr XII dtruton wth two paraeter n the preene of outer oervaton a Equaton : 55 n ( np, r ) ( ) r :n [p] = ρ = ax (, r p) r r = = + ( ) β( φ, + ) +τ n ( np,r ) r = ax (, r p ) = = r β φ + + ( ) (, ) 4 Spea ae () We dedue oe pea ae fro the nge oent gven n (9) and () a foow: Settng p =, we get the nge oent of order tatt when x, x,,x n have Burr dtruton a Equaton and : r r ( ) ρ n! [] = (r )!(n r )! = a ( ) = () β( φ +, + ) Φ = ρ(n r + + ) + r r ( ) ρn! [] = (r )!(n r )! = β φ + + Φ = ρ(n r + + ) + ( ) (, ) () If we put p = n, we have the ae reaton aove ut wth paraeter τ If we put p =, we have the reaton for nge outer Exape () Let X, X ~Barr XII wth =.39533, ρ = 3, a =.3445, = and X 3 ~Gaa(β =, α = ). Fnd :3. Souton A nge outer n th ape that X 3 ~Gaa(β =, α = ).

6 Jaoo, A.A. and Z.A. A-Saary /Journa of Matheat and Statt 9 (): 5-6, 3 Tae. Reut of Tadaaa (977) for the paraeter ρ, a,,,, σ when α nteger and β = α a σ * * Nora E Fg. Burr dtruton wth ( = , τ = ) Fg. Nora dtruton wth ( =.64477, σ =.699) Sene Puaton 56

7 Jaoo, A.A. and Z.A. A-Saary /Journa of Matheat and Statt 9 (): 5-6, 3 But we now that Tadaaa (977) approxate Gaa dtruton β =, α = wth Burr XII wth four paraeter a =.39533, ρ = , a =.3445, = Tae and Fg. and. So when X 3 ~Gaa (β =, α = ) X 3 ~ Burr IIX wth paraeter ( =.39533, ρ = , a =.3445, = ). Now ung Equaton 9: n (, ) ( ) :3 [] = 3 ( ) r r ( ) = =.3445 = = ax (, ) + β( φ +, + ) n (,) ( ) = ax (, ) r r = = = ( ) β( φ +, + ) ( ) Φ = 3( 3 + ) ( ) + = 3( ) + ( )(++)+ ( ) :3 ( ) [] = r r ( ) = = n (, ) = ax (, ) = β( φ +, + ) ( ) r r ( ) = = n (,) = ax (, ) = β( φ +, + ) ( ) + :3 [] 3 ( ) ( ) = = = =.3445 ( β φ + =, + ) ( ) ( ) = = = = β( φ +, + ) ( ).3445 :3 [] = 3 ( ) = β ( 6 + ( ) , + ) β + = ( ) ( 6 ( ) +, + ) Fro (): = = = = ( ).3445 :3 [] = 6 ( ) = β ( 6 + ( ) , + ) β + = ( ) ( 6 ( ) +, + ) If = :.3445 :3[] = 6 ( ) = β ( 6 + ( ) , + ) ( ) Sene Puaton 57

8 Jaoo, A.A. and Z.A. A-Saary /Journa of Matheat and Statt 9 (): 5-6, 3 Tae. The expeted vaue and the varane n the preene of utpe outer fro Burr XII wth four paraeter when =, n = 5, =.39533, ρ = 3, a =.3445, = , τ = p r r:5 [p] Varane p r r:5 [p] Varane p r r:5 [p] Varane β ( 6 + ( ) = +, + ) = More reut an e een n Tae for n = 5, r =,,,5 and p =,,. Tae gven eow dpay the vaue of the nge oent of order tatt n (.9) when = ; n = 5; τ = ; α =.3445; = and = Exape () Let X,X ~Barr XII wth = , ρ = 5 and X 3 ~Nora (.64477,.699). Fnd :3 Souton A nge outer n th ape that X 3 ~Nora (.64477,.699). But we now that Burr (94) approxate the nora dtruton wth Burr XII wth two paraeter a = and ρ = fro whh the ore aurate vaue of, σ, α 3 and α 4 an e otaned a =.64477, σ =.699, α 3 =., α 4 =.. So when X 3 ~ Nora ( =.64477, σ =.699) X 3 ~ Burr ( = ), τ = Now ung Equaton : n (, ) r-- ( ) :3 = ax (, -) = = [] = r-- + (-) β Φ- -, + n (,) r (-) = ax (, ) = = β Φ- -, r-- ( ) ( )( ) ( ) ( )( ) Φ = = ( ) :3 = = = = [] (-) β Φ- -, = = = (-) β Φ- -, + ( ) :3 [] = 5 () β 5() ()+ - -, ()β 5() ()+ - -, Fro (): = = = = ( ) [] = β :3 5() () +, β 5() () +, If = : ( ) :3 ( ) [] = β , =.553 Sene Puaton 58

9 Jaoo, A.A. and Z.A. A-Saary /Journa of Matheat and Statt 9 (): 5-6, 3 Tae 3. The expeted vaue and the varane n the preene of utpe outer fro Burr XII wth two paraeter when =, n = 5, = and ρ = 3, τ = p r r:3 [p] Varane p r r:3 [p] Varane p r r:3 [p] Varane More reut an e een n Tae 3 for n = 5, r =,,,5 and p =,,... Moent of Order Statt fro Burr XII wth Four Paraeter Ung Theore of Baraat and Adeader (4) Let X, X,, X n e ndependent nondentay dtruted r.v,.. The th oent of a order ( ) tatt, for r n and =,, gven y (Baraat and Adeader, 4) Equaton 3: = n ( ) (n r+ ) ( ) I ( ) (3) nr = n r+ I ( ) = Where Equaton 4: < < < n x G ( x )dx, =,,,n t t= (4) G (x) = -F (x), wth (, I,, n ) a perutaton of (,,, n ) for whh < < n. We onder the ae when the varae X, X,, X n-p are ndependent oervaton fro Burr XII wth four paraeter dtruton wth denty Equaton 5: ρ x a ρ f (x ) = ( + ) x a, a x, ρ >, >, > (5) The orrepondng uuatve dtruton funton F (x) gven a: x a ρ F(x ) = ( + ), x a, ρ>, >, > Ung (4) n (): I ( ) = < < < n x a t = ρ t x ( + ) dx a x a Suttutng y = : x = y+ a dx = dy at x = a y y =, at x = y = The aove Equaton redue to: I ( ) = = < < < n t t = ( y+ a ) (+ y ) dy < < < n a ( y) ( y ) + = ρ t = t < < < n dy ( y) ρ a ρ t (+ y ) dy = t = Upon ung: α ( y) (+ y ) dy = β ( α, ) where, β (a, ) the reguar eta funton: Sene Puaton 59

10 Jaoo, A.A. and Z.A. A-Saary /Journa of Matheat and Statt 9 (): 5-6, 3 Tae 4. The expeted vaue and the varane n the preene of utpe outer ung Baraat and Adeader (4) and ethod fro Burr XII wth four paraeter when =, n=5, =.39533, ρ = 3, a =.3445, = , τ = p r r:3 [p] Varane p r r:3 [p] Varane p r r:3 [p] Varane I ( ) = < < < n a β ( ρ, ) t = t = n ( ) (n r + ) = ( ) nr = n r + < < < n a (, ) t β ρ = t = If = : n (n r + ) = ( ) nr = n r+ Exape t < < < n t = β ( ρ, ) Let X,X ~Barr XII wth =.39533, ρ = 3, a =.3445, = and X 3 ~ Gaa (β =, α = ). Fnd :3. Souton X 3 ~ Gaa (β =, α = ) X 3 ~Burr IIX wth paraeter ( =.39533, ρ = , a =.3445, = ): :3 3 = ( ) = 3 < < < 3 3 β ρ, t t= = < < < β ρ, t t= = (, ) β ρ + ρ + ρ = β ( , ) =.5563 More reut an e een n Tae 4 for r =,,,5 and p =,,.. ONLUSION We an fnd oent of order tatt fro ndependent and nondentay dtruted rando varae for any dtruton wth no pe oed fro ung approxatng dea wth Burr XII or any other dtruton. 3. REFRENES Arnod, B.. and N. Baarhnan, 989. Reaton, Bound and Approxaton for Order Statt. t Edn., Sprnger-Verga, New Yor, ISBN-: , pp: 73. Baarhnan, N. and K. Baauraanan, 995. Order tatt fro non-denta power funton rando varae. oun. Stat. Theor. Meth., 4: DOI:.8/ Baarhnan, N., 994. Order tatt fro nondenta exponenta rando varae and oe appaton. oput. Stat. Data Ana., 8: DOI:.6/ (94)97-4 Baraat, H. and Y. Adeader, 4. oputng the oent of order tatt fro nondenta rando varae. Stat. Meth. App., 3: 5-6. DOI:.7/ Sene Puaton 6

11 Jaoo, A.A. and Z.A. A-Saary /Journa of Matheat and Statt 9 (): 5-6, 3 Barnett, V. and T. Lew, 994. Outer n Statta Data. 3rd Edn., Wey and Son, hheter, ISBN- : , pp: 584. Burr, I.W., 94. uuatve frequeny funton. Anna Math. Stat., 3: 5-3. Burr, I.W., 967. A uefu approxaton to the nora dtruton funton, wth appaton to uaton. Tehnoetr, 9: DOI:.8/ Burr, I.W., 973. Paraeter for a genera yte of dtruton to ath a grd of á3 and á4. oun. Stat, : -. hd, A, 996. Advane n tatta nferene and outer reated ue. Ph.D. The, MMater Unverty. hd, A., N. Baarhnan and M. Mohref,. Order tatt fro non-denta rght-trunated Loax rando varae wth appaton. Stat. Paper, 4: Hartey, H.O. and H.A. Davd, 978. ontruton to Survey Sapng and Apped Statt. t Edn., Aade Pre, San Frano, ISBN-: 4758, pp: 38. Launer,.A. and T.D. B, 979. Infuene of Seeted Envronenta Fator on the Atvty of a Propetve Fh Toxant, -(Dgeranyano)- Ethano, n Laoratory Tet. t Edn., Departent of the Interor, Wahngton, pp: 4. Mohref, M.E. and K.S. Sutan, 7. Moent of order tatt fro Rayegh dtruton n the preene of outer oervaton. Departent of Matheat, A-Azhar Unverty, Shu, V.S., 978. Rout Etaton of a Loaton Paraeter n the Preene of Outer. t Edn., Iowa, A, pp: 37. Tadaaa, P.R. and J.S. Raerg, 975. An approxate ethod for generatng gaa and other varae. J. Stat. oput. Su., 3: Tadaaa, P.R., 977. An approxaton to the oent and the perente of gaa order tatt. Sanhya: IJS, 39: Vaughan, R.J. and W.N. Venae, 97. Peranent expreon for order tatt dente. J. Roya Stat. So., 34: Wheeer, D.J., 975. An approxaton for uaton of gaa dtruton. J. Stat. oput. Su., 3: 5-3. DOI:.8/ Sene Puaton 6