SEQUENTIAL ESTIMATION IN A SUBCLASS OF EXPONENTIAL FAMILY UNDER WEIGHTED SQUARED ERROR LOSS *

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1 Iraia Jural f iee & Tehlgy Trasati A Vl.. A Prited i The Islami Republi f Ira 007 hiraz Uiersity QUTIAL TIMATIO I A UBCLA OF POTIAL FAMIL UDR WIGHTD QUARD RROR LO *. MATOLLAHI ** M. JAFARI JOZAI AD. MAHLOOJI Departmet f tatistis Faulty f mis Allameh Tabataba'i Uiersity Tehra I. R. f Ira mail: ematllahi@atu.a.ir Abstrat I a sublass f the sale-parameter expetial family e sider the sequetial pit estimati f a futi f the sale parameter uder the lss futi gie as the sum f the eighted squared errr lss ad a liear st. Fr a fully sequetial samplig sheme sed rder expasis are btaied fr the expeted sample size as ell as fr the regret f the predure. The frmer researhes Gamma ad xpetial distributis a be dedued frm ur geeral results. Keyrds equetial estimati stppig rule regret aalysis expetial family trasfrmed hi-square distributi. ITRODUCTIO The prblem f sequetial estimati refers t ay estimati tehique fr hih the ttal umber f bseratis used is t a degeerate radm ariable. I sme prblems sequetial estimati must be used beause predure that uses a preassiged radm sample size a ahiee the desired bjetie fr example the estimati f the parameter /p i a sequee f Berulli trials. I ther prblems a predure hih uses a preassiged radm sample size may exist but a sequetial estimati predure may be better i sme ays. The prblems f sequetial aalysis ere first studied i the 90s by Barard ad Wald h itrdued the equetial Prbability Rati Test PRT idepedetly Wald ad Wlfitz pred its ptimality ad Haldae ad tei 5 shed h sequetial methds a be used t takle sme usled prblems i pit ad iteral estimati. There is a large bdy f literature this subjet ad it is grig rapidly. Fr a summary f results as ell as a list f referees see Lai. equetial estimati f the sale parameter f xpetial ad Gamma distributis hae bee sidered by tarr ad Wdrfe 7 Wdrfe 8 Gsh ad Mukhpadhyay 9 Isgai ad U 0 Isgai et al. ad U et al.. Uder squared errr lss tarr ad Wdrfe 7 sidered the risk effiiet estimati f the sale parameter i expetial distributi ad studied sme f the first rder prperties f the sequetial predure. Als fr the sequetial estimati f a futi f the expetial parameter U et al. gae the stppig rule ad a suffiiet diti t get a sed rder apprximati t the risk f the sequetial predure. Fr the estimati f the sale parameter the sale iariat squared errr lss is mre apprpriate tha the squared errr lss. I additi there are seeral ases fr hih the estimati f a futi f the sale parameter is desired. it is atural t use δ as a apprpriate relatie squared errr lss Reeied by the editr August 005 ad i fial reised frm April Crrespdig authr

2 90. ematllahi / et al. futi hih is a speial frm f eighted squared errr lss futi. Hee i this paper e sider the lss as the sum f the eighted squared errr lss ad a liear st hih iludes sale iariat squared errr lss. We btai a sequetial pit estimati f a futi f the sale parameter i a sublass f the sale parameter expetial families f distributis hih ilude xpetial Weibull Gamma rmal Ierse Guassia ad sme ther distributis ad determie a stppig rule uder the lss futi. Als a sed rder apprximati t the expeted sample size ad the risk f the sequetial predure as the st per bserati teds t zer are gie. We sh that the results btaied by U et al. i xpetial distributi ad Wdrfe 8 i Gamma distributi uder squared errr lss are speial ases f ur results. I eti the sublass f distributis is itrdued ad the sequetial estimati prblem is speified. I seti e derie asymptti expasis f the expeted sample size ad regret assiated ith the prpsed predure. me speial ases f ur results are gie i seti.. QUTIAL TIMATIO I A UBCLA OF POTIAL FAMIL Let... be a sequee f idepedet ad idetially distributed radm ariables frm a x distributi ith desity g here g is k ad τ is a uk sale parameter. I sme ases τ τ the abe desity redues t Tx fx x e > 0 r here x is a futi f x τ is a k alue ad T is a mplete suffiiet statisti fr ith Gamma - distributi. xamples f distributis f the frm are. xpetial β ith β T x a x. Gamma β ith k ad β T x Γ a. Ierse Gaussia λ ith T x π x λ. rmal 0 σ ith σ T x π β β β 5. Weibull β ith k β ad T x β x. Rayleigh β ith β T x x 7. Geeralized Gamma λ p ith k p ad p p T x x λ Γ p / 8. Geeralized Laplae λ k ith k k ad k a k λ T x. k Γ/ k if... be a radm sample f size frm distributi the the jit desity f... is gie by Tx i / i 0 fx x e > Iraia Jural f iee & Tehlgy Tras. A Vlume umber A prig 007

3 equetial estimati i a sublass f 9 here x Π xi ad i T i Gamma. Csider the estimati f a futi f i the sale parameter say here is a psitie alued ad three times tiuusly differetiable futi f. Gie a sample... f size e at t estimate by ˆ uder the lss futi L ˆ ˆ here > 0 is the k st per uit sample ad is a psitie alued ad t times tiuusly differetiable eight futi. peial ases f the lss are saled iariat squared errr lss ad squared errr lss ˆ L ˆ ˆ L ˆ 5 hsig ad respetiely. te that is the usual estimatr f i.e. ML UMVU hee usig iariae prperty f maximum likelihd estimatrs it is reasable t estimate by ˆ. The risk futi is gie by R R ˆ L ˆ ˆ. We at t fid a apprpriate sample size that ill miimize the risk R. Usig Taylr expasi f abut e btai ˆ Var as ad hee R as. fr suffiietly large R hih leads t the fllig lemma. Lemma -. The risk futi ad fr this alue R i miimized at 0 7 R. 8 ie is uk e a t use the best fixed sample size predure 0. Further there is fixed sample size predure that ill attai the miimum risk R. Thus it is eessary t fid a sequetial 0 prig 007 Iraia Jural f iee & Tehlgy Tras. A Vlume umber A

4 9. ematllahi / et al. samplig rule. We use the fllig stppig rule if m 9 here m is the pilt sample size. if e estimate by ˆ the the risk is gie by R ˆ. 0 I the ext seti e derie sed rder apprximati t the expeted sample size ad the risk f the abe sequetial predure R as 0.. COD ORDR APPROIMATIO I this seti e shall iestigate sed rder asymptti prperties f the sequetial predure. Let Kt t> 0 t t t t t} t ad Z K / K. The frm 9 the stppig rule bemes m Z > if : }. Usig Taylr expasi f abut ad the relatis K t t t t t t K t t t ad K t K t t t t t t t t t t t t t t t t t t t t t t e btai the fllig lemma. T i Lemma -. Let i fr i... ad i the i Z ψ here t K K ψ K K ad is a radm ariable lyig betee ad. Iraia Jural f iee & Tehlgy Tras. A Vlume umber A prig 007

5 equetial estimati i a sublass f 9 Let t t t if : > 0} ad ρ 5 t Fllig assumptis f Aras ad Wdrfe amely A Z m is uifrmly itegrable ε fr sme 0 < ε 0 < here x max x0 m A P ψ < ε} < fr sme 0 < ε < e btai the fllig apprximati t fr all 0 but t uifrmly i. Therem -. If A ad A hld true the here ρ l as 0 l K. 7 K Prf: Obiusly P as ad sie is a radm ariable lyig betee ad therefre P K p K ad hee l. Als K K d W 0 as. ψ K d ψ l. W as. K The rest f the prf is the same as the prf f Therem f U et al. ith replaigσ σ ξ ad h by ψ ad K respetiely is mitted. We shall assess the regret R. By Taylr's therem 8 t prig 007 Iraia Jural f iee & Tehlgy Tras. A Vlume umber A

6 9. ematllahi / et al. here is a radm ariable lyig betee ad. Let ad hse 0 > 0 suh that. We impse the fllig assumpti A : Fr sme a > ad u > sup 0< 0 au u < ad sup <. 0< 0 au Frm A A ad A e hae the fllig therem. Therem -. If A A ad A hld true the as 0 R 7 5 } 9 Prf: Frm 8 e hae R 5 }. 0 Fllig the prf f Therem f U et al. he 0 e btai Iraia Jural f iee & Tehlgy Tras. A Vlume umber A prig 007

7 equetial estimati i a sublass f prig 007 Iraia Jural f iee & Tehlgy Tras. A Vlume umber A 95 7 } l 5 9 } } } } } W } 5 ad } 5.

8 . ematllahi / et al. Iraia Jural f iee & Tehlgy Tras. A Vlume umber A prig ubstitutig it 0 e get 7 R as 0 this mpletes the prf.. PCIAL CA I this seti e sider sme speial ases f the results btaied i seti. These speial ases are. Fr xpetial β -distributi ith β T ad Therem - ad - ith i.e. ith the lss futi 5 bemes the Therem ad f U et al. respetiely. the expeted sample size ad the regret btaied by U et al. are speial ases f ad 9 respetiely. Als he ad the regret bemes R hih is the result btaied by Wdrfe 8.. Fr Gamma - distributi ith k T ad i estimati f uder the lss futi 5 i.e. the regret bemes R hih is the result btaied by Wdrfe 8.. Fr sale iariat squared errr lss e hae ad frm 9 the regret bemes. } } } } 7 R Therefre fr the regret bemes R. te that the lss futi is mre apprpriate fr estimatig the sale parameter tha squared errr lss. The results f seti ad a be exteded t sme ther distributis hih d t eessarily belg t a sale family suh as Paret r Beta distributis. A family f distributis that iludes these distributis as speial ases is the family f trasfrmed hi-square distributis hih is rigially itrdued by Rahma ad Gupta. They sidered the e parameter expetial family x h b x a e x f 7 ad shed that b a has a j Gamma - distributi if ad ly if. j b b 8

9 equetial estimati i a sublass f 97 I ase j is a iteger a b flls a hi-square distributi ith j degrees f freedm. They alled the e parameter expetial family 7 hih satisfies 8 the family f trasfrmed hisquare distributis. Fr example Beta Paret xpetial Lgrmal ad sme ther distributis belg t this family f distributis see Table f Rahma ad Gupta. it is easy t sh that if diti 8 hlds the the e parameter expetial family 7 is i the frm f the sale parameter expetial family ith j T a ad. Hee b ith these substitutis e a exted the results f seti ad t the family f trasfrmed hisquare distributis. RFRC. Barard G. A. 9. my i samplig ith speial referee t egieerig experimetati. Brith. Mi. upply Teh. Rep. QC/R/7 I.. Wald A. 95. equetial test f statistial hyptheses. A. Math. tat Wald A. & Wlfitz J. 98. Optimum harater f the sequetial prbability rati test. A. Math. tat Haldae J. B O a methd f estimatig frequeies. Bimetrika tei C. 95. A t sample test fr a liear hypthesis hse per is idepedet f the ariae. A. Math. tat Lai T. L. 00. equetial Aalysis: me lassial prblems ad e halleges. tatistia iia tarr. & Wdrfe M. 97. Farther remarks sequetial estimati: the expetial ase. A. Math. tat Wdrfe M ed rder apprximati fr sequetial pit ad iteral estimati. A. tatist Gsh M. & Mukhpadhyay equetial estimati f the peretiles f expetial ad rmal distributis. uth Afria tatist. J Isgai. & U C. 99. equetial estimati f a parameter f a expetial distributi. A. Ist. tatist. Math Isgai. Ali M. & U C. 00. equetial estimati f the pers f rmal ad expetial sale parameters. equetial Aalysis U C. Isgai. & Lim D. 00. equetial pit estimati f a futi f the expetial sale parameter. Austria Jural f tatistis Aras G. & Wdrfe M. 99. Asymptti expasis fr the mmets f a radmly stpped aerage. The Aals f tatistis Rahma M.. & Gupta R. P. 99. Family f trasfrmed hi-square distributis. Cmm. tatist. Thery Methds 5-. prig 007 Iraia Jural f iee & Tehlgy Tras. A Vlume umber A