Moments of Generalized Order Statistics from a General Class of Distributions

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1 ISSN Jol of Sttt Vole Moet of Geelzed Ode Sttt fo Geel l of Dtto Att Mhd Fz d Hee Ath Ode ttt eod le d eel othe odel of odeed do le e ewed el e of geelzed ode ttt go K 995. I th e exlt exeo fo gle d odt oet of geelzed ode ttt fo fly of dtto he ee oted. Fthe oe dedto d tl e e ded. Keywod Geelzed ode ttt Reod le Sgle oet Podt oet B d Well dtto AMS-Set lfto: 62G3 62E5. Itodto Let 2 e eqee of deedet d detlly dtted do le wth oltely oto dtto fto df Fx d olty dety fto df f x x. Let N 2 2 R M h tht M fo ll {2 }. The 2 e lled geelzed ode ttt go f the ot df ge y F x f x F x f x. o the oe F x x F K 995. Detet of Sttt d Oeto Reeh Algh Ml Uety Algh Id El: heeth@hotl.o

2 Moet of Geelzed Ode Sttt fo Geel l of Dtto 37 Let B e deedet do le hg Bet dtto the t e ee tht Bht et l. 23 ~ F W 2..2 whee W B Kh et l. 28 he oted exlt exeo fo ext oet of geelzed ode ttt fo geel fo of dtto. I th e we he exteded the elt of Kh et l. 28 d he oted ext oet of go fo fly of dtto F x x. Fo oe ddtol elt oe y efe to Keelg 999 K d e 2 e d K 23 e 23 Rq 24 Ath d Il 24 d efeee thee. 2. Moet fo Geelzed Ode Sttt Let the geel l of the dtto e: F x x x 2. whee d e o hoe tht Fx df oe. The fo.2 we he fo 2 ~ B Relto fo Sgle Moet: Theoe 2.: Fo the dtto ge 2. d 2 E whee eeet the tege t of d 2.3

3 38 Fz d Ath 2.4 Poof: Fo 2.2 we he P E E B d hee the elt. Re 2.: At 2.3 we get E 2.5 oted y Kh et l Relto fo Podt Moet: Theoe 2.2: Fo the dtto ge 2. E whee d e the tege t of d eetely. 2.6 Poof: We he fo 2.2 E

4 Moet of Geelzed Ode Sttt fo Geel l of Dtto 39 d hee the elt. Re 2.2: At 2.6 ede to gle oet ge 2.3. Re 2.3: At 2.6 we get E 2.7 oted y Kh et l Illtted Exle Th fly t fo the dtto odeed y Kh et l. 28 t lo lde B d Well dtto. 3. Sgle Moet. B dtto F x x x whee d tege. At d 2. we get E

5 Fz d Ath 4. Well dtto F x x x. Let the we he x L F x e y lto of elt Ath et l. 29! We he! E At oe exeo of the fo. Theefoe lyg L Hotl le d g the elt Rz 996! x 3. we he!! E oted y K 995.

6 Moet of Geelzed Ode Sttt fo Geel l of Dtto Podt oet. B dtto F x x x whee d tege. Hee d Fo 2.6 we he E. Well dtto F x x Hee d ' the E } { ' I ew of the elto Ath et l. 29!

7 Fz d Ath 42 E! } {! Tg the lt d g the elto 3. we get E!!!! 2 Aowledgeet The tho e gtefl to Pofeo A.H. Kh Algh Ml Uety Algh fo h hel d ggeto thoghot the eto of th e. The tho lo owledge wth th the ggeto d oet of the leed efeee whh led to oell oeet of the e. Refeee. Ath H. d Il H. M. 24. Reee elto etwee gle d odt oet of geelzed ode ttt fo geel l of dtto. Meto LII Ath H. Kh R.U. d Aw Z. 29. Ext oet of lowe geelzed ode ttt fo owe fto dtto. Stted fo lto ltt Stttl Aoto Bllet.

8 Moet of Geelzed Ode Sttt fo Geel l of Dtto 3. Bht M. e E. d K U. 23. Dl geelzed ode ttt. Meto LI e E. 23. otto to Geelzed Ode Sttt. Hlttohft Uety of Oldeg Gey. 5. e E. d K U. 23. Mgl dtto of eqetl d geelzed ode ttt. Met K U A oet of geelzed ode ttt. Jol of Sttl Plg d Ifeee K U. d e E. 2. O dtto of geelzed ode ttt. Sttt Keelg odtol dtto of geelzed ode ttt d oe htezto. Met Kh A. H. Aw Z. d Ath H. 28. Ext oet of geelzed d dl geelzed ode ttt fo geel fo of dtto. Jol of Stttl See e.. Rq M. Z. 24. Geelzed exoetl dtto: Moet of ode ttt. Sttt Rz S. M A lge detty ledg to Wlo theoe. The Mthetl Gzette

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