Ion Vladimirescu, Radu Tunaru

Transcription

1 Commet.Math.Uiv.Caroliae 44, 2003) Estimatio fuctios ad uiformly most powerful tests for iverse Gaussia distributio Io Vladimirescu, Radu Tuaru Abstract. The aim of this article is to develop estimatio fuctios by cofidece regios for the iverse Gaussia distributio with two parameters ad to costruct tests for hypotheses testig cocerig the parameter λ whe the mea parameter µ is kow. The tests costructed are uiformly most powerful tests ad for testig the poit ull hypothesis it is also ubiased. Keywords: iverse Gaussia distributio, estimatio fuctios, uiformly most powerful test, ubiased test Classificatio: 62F03, 62F25. Itroductio The iverse Gaussia distributio was first derived by Schrodiger 95) i coectio to the first hittig time i Browia motio. I statistics it was derived by Wald 947) for sequetial testig, by Hadwiger 940) ad Tweedie 957). Some moographs dedicated to this subject are Chhikara ad Folks 989), Seshadri 994) ad Seshadri 999). A bivariate iverse Gaussia distributio is ivestigated i Essam ad Nagi 98). Although i the literature there are several goodess-of-fit tests, see Edgema et al. 988), O'Reilly ad Rueda 992), Pavur et al. 992), Mergel 999) ad Heze ad Klar 200), ad some other empirical distributio fuctio tests suchaskolmogorov-smirov test, the Cramer-vo Mises test, the Aderso- Darlig test ad the Watso test have bee ivestigated i Gues et al. 997), there are o uiformly most powerful tests developed for testig i the iverse Gaussia cotext. The iverse Gaussia distributio has may applicatios i actuarial statistics for example Ter Berg 980, 984) ad it has bee also used lately i mathematical fiace due to its useful properties such as closure uder covolutio ad flexibility i modelig positively-skewed ad leptokurtic sets of data. The mai aim of this paper is to propose estimatio fuctios by cofidece regios ad some uiformly most powerful tests for the λ parameter whe the mea parameter µ is kow. Various poit, uidirectioal ad bidirectioal tests are cosidered for testig hypotheses.

2 54 I. Vladimirescu, R. Tuaru All probability desity fuctios i this paper are cosidered with respect to the Lebesgue measure o the relevat metric space, most ofte this is R the set of real umbers. For ay radom variable g we deote by F g the cumulative distributio fuctio of g ad by ρ g the probability desity fuctio of g. The iverse Gaussia distributio G λ,µ has the followig probability desity fuctio ) λ 2 ρx : λ, µ) = exp [ λ x µ) 2 ] 2πx 3 2µ 2 x 0, ) x). Uder this parameterizatio, the iverse Gaussia distributio has mea µ ad variace µ 3 /λ. Its shape is modeled by the value of λ/µ. The cumulat geeratig fuctio is the iverse of that of the ormal or Gaussia distributio ad this is the reaso for the ame of this distributio, the iverse Gaussia. The ext results are useful to prove the mai results of this paper. Theorem. Let Ω, F,P) be a probability space ad f :Ω R be a radom variable havig the distributio G λ,µ. The a) cf G cλ,cµ for ay c>0; b) λ f µ) 2 µ 2 f χ 2 ), where χ 2 ) is the chi-square distributio with oe degree of freedom. The first poit was proved i Tweedie 957) while the secod ca be foud i Shuster 968). For ay positive iteger the cumulative distributio fuctio of the chi-square distributio χ 2 ) is deoted by F ). Usig the characteristic fuctio it ca be easily show that if f,...,f are radom variables idepedet ad idetically distributed with distributio G λ,µ the ) f i G λ,µ. i= Cosider the statistical model give by 2) The mappigs 0, ), B 0, ), G λ,µ λ, µ > 0}) ). pr i :0, ) 0, ) defied by pr i x ) ) = x i for ay x ) = x,...,x ) 0, ) ad ay i =,..., are idepedet, idetically distributed with distributio G λ,µ.

3 Estimatio fuctios ad uiformly most powerful tests for iverse Gaussia distributio 55 Theorem above implies that 3) pr i G λ,µ. i= Applyig the the secod poit of Theorem we get that 4) λ ) 2 i= µ pr i i= χ 2 ). pr i Next we eed to defie the fuctios π ; ) :0, ) 0, ) 0, ) by where μx = i= x i. I other words, 5) π ; λ) =λ for ay λ>0. π x ) ; λ) = λ ) 2 μx µ μx, 2. Mai results for estimatio fuctios ) 2 i= µ pr i i= pr i I this sectio we are preparig the way to the mai results providig cofidece regios ad uiformly most powerful tests. Lemma. Let be a positive iteger ad µ be a positive real umber. The 6) π ; λ) > 0, G λ,µ a.e. for ay λ>0. Proof: Obviously π x ) ; λ) 0 for ay x ) 0, ) ad λ>0. I additio G λ,µ x) 0, ) π x ) ; λ) =0}) = G λ,µ π ; λ)) ) 0}) = χ 2 )0}) =0. Similarly with the costructio above, if is a positive iteger ad λ is a positive real umber we ca defie ~π ; ) :0, ) 0, ) 0, ) by

4 56 I. Vladimirescu, R. Tuaru where μx = i= x i. Oce agai ~π x ) ; µ) = λ ) 2 μx µ μx 7) ~π ; µ) =λ ) 2 i= µ pr i i= χ 2 ) pr i for ay µ>0. ) Theorem 2. Let 0, ), B 0, ), G λ,µ λ, µ > 0}) be a statistical model. a) If µ>0 ad α 0, ) are kow ad 0 <u<vsuch that χ 2 )[u, v]) = α the the mappig δ :0, ) 2 0, ) defied as 8) δ x ) )=λ >0 u π x ) ; λ) v} is a estimatio fuctio by cofidece regios at the level of sigificace α for the parameter λ. b) If λ>0ad α 0, ) are kow ad 0 <u<v are real umbers such that χ 2 )[u, v]) = α the the mappig δ ~ :0, ) 2 0, ) defied as 9) ~ δ x ) )=µ >0 u ~π x ) ; µ) v} is a estimatio fuctio by cofidece regios at the level of sigificace α for the parameter µ. Proof: a) From 5) it follows that π ; λ)isb 0, ), B 0, ) )-measurable while 4) implies that G λ,µ π ; λ) = χ 2 ) for ay λ>0. Thus π ; ) is a pivotal fuctio for the parameter λ. Takig ito accout that χ 2 [u, v])= α we coclude that δ ) is a estimatio fuctio by cofidece regios at level of sigificace α for the parameter λ. b) The proof is similar with that for a) replacig π ; λ) by~π ; µ). Combiig the above theorem with Lemma we get that [ ] δ x ) μx )= µ μx ) 2 u, μx µ μx ) 2 v, G λ,µ a.e.

5 Estimatio fuctios ad uiformly most powerful tests for iverse Gaussia distributio 57 Let h ;β be the quatile of order β for the χ 2 ) distributio ad α,α 2 0, ) such that α + α 2 = α. The μδ x ) )= [ ] μx µ μx ) 2 h μx ;α, µ μx ) 2 h ; α 2, G λ,µ a.e. provides a estimatio method by cofidece regios at the level of cofidece α for the parameter λ. Moreover, the above theorem may be used to coclude that the mappig δ μ : 0, ) 2 0, ) defied as [ ) )] μδ vμx uμx =, μx [ + μx λ μx ), + μx uμx λ λ )] vμx λ provides a estimatio method by cofidece regios at level of sigificace α for the parameter µ. 3. Prelimiary results for testig hypotheses Lemma 2. Let Ω, F,P) be a probability space, λ a positive real umber ad f :Ω Ra radom variable such that λf χ 2 ). The, the probability desity fuctio of the radom variable f is 0) ρx; λ) = ) λ 2 x 2 exp λx 2π 2 ) 0, )x). Proof: F f x) =P f <x)=p λf < λx) 0, ) x) ) λ 2 = x 2 exp λx 2π 2 ) 0, ) x). Cosider the probability measure ν λ o the set of real umbers R havig the probability desity ρ ; λ). For ay positive parameter λ it is obvious the that suppν λ )=[0, ) ad, if T λ : R Ris a fuctio defied as T λ x) =λx, the ν λ Tλ = χ 2 ). Hece, if the radom variable h has the distributio ν λ the the radom variable λh has the distributio χ 2 ).

6 58 I. Vladimirescu, R. Tuaru Lemma 3. Let ν λ be a probability distributio havig the probability desity ) ρx; λ) = The statistical model ratio. Proof: Cosider 0 <λ <λ 2. The ρx; λ 2 ) ρx; λ ) = ) λ 2 x 2 exp λx 2π 2 ). ) 0, ), B 0, ), ν λ λ>0} has a mootoe likelihood λ2 λ ) /2 exp λ 2 λ x) =h 2 λ,λ 2 T x)), where T : 0, ) R, Tx) = x is a B 0, ), B R )-measurable fuctio ad the fuctio h λ,λ 2 : R Rdefied by h λ,λ 2 x) = λ2 λ ) /2 exp λ 2 λ x) 2 is icreasig. 4. Mai results for testig hypotheses Theorem 3. Cosider the statistical model 2) ) ) 0, ), B 0, ), G λ,µ λ>0} with µ>0 ad kow. Let > 0 be a fixed value of parameter λ ad α a level of sigificace. a) For testig the ull hypothesis H 0 : λ 0, ] versus the alterative H : λ, ), the pure test ϕ = C with 3) C = x ) 0, ) μx µ μx ) 2 < h } ;α is uiformly most powerful. b) For testig the ull hypothesis H 0 : λ [, ) versus the alterative H : λ 0, ), the pure test ϕ = C with 4) C = x ) 0, ) μx µ μx ) 2 > h } ; α

7 Estimatio fuctios ad uiformly most powerful tests for iverse Gaussia distributio 59 is uiformly most powerful. c) Let 0 <λ <λ 2 be some kow values. For testig the hypothesis H 0 : λ 0,λ ] [λ 2, ) versus H : λ λ,λ 2 ) at the level of sigificace α, a uiformly most powerful test is the pure test ϕ = C where } 5) C = x ) 0, ) μx µ μx ) 2 [c,c 2 ] ad c < 0 ad c 2 are determied from the coditios F c λ ) F c 2 λ )=α, F c λ 2 ) F c 2 λ 2 )=α. d) Let 0 <λ <λ 2 be some kow values. For testig the hypothesis H 0 : λ [λ,λ 2 ] versus H : λ 0,λ ) λ 2, ) at the level of sigificace α, a uiformly most powerful test is the pure test ϕ = C where C = x ) 0, ) ) } 2 μx µ μx <c 6) x ) 0, ) ) } 2 μx µ μx >c 2 ad c <c 2 < 0 are determied from the coditios F c λ ) F c 2 λ )= α, F c λ 2 ) F c 2 λ 2 )= α. e) Let λ>0 be some kow value. For testig the hypothesis H 0 : λ = versus H : λ> at the level of sigificace α, a ubiased, uiformly most powerful test is the pure test ϕ = C where C = x ) 0, ) ) } 2 μx µ μx <c 7) x ) 0, ) ) } 2 μx µ μx >c 2 ad c <c 2 < 0 are determied from the coditios F c ) F c 2 )= α, λ F c λ) F c 2 λ)) λ=λ0 =0.

8 60 I. Vladimirescu, R. Tuaru Proof: a) Cosider the probability desity fuctio ν λ give i ) ad the statistical model ) 8) 0, ), B 0, ), ν λ λ>0}. If v :0, ) 0, ) is a applicatio defied by v x ) )= ) 2 μx µ μx the the above statistical model ca be rewritte as ) 0, ), B 0, ), G λ,µ v λ>0}. Usig Lemma 3 this statistical model has a mootoe likelihood ratio with respect to the statistic T x) = x. Applyig Lehma's theorem see Lehma, 959, for further details), we get that the pure test ϕ 0 = C0 with C 0 = x >0 πx) >c} where c<0 is determied from the coditio ν λ0 C 0 ) = α, is uiformly most powerful at the level of sigificace α for testig H 0 : λ 0, ] agaist the alterative H : λ, ). Observe that α = ν λ0 C 0 )=ν λ0 x >0 x>c}) =ν λ0 x >0 x< c }) = χ 2 )x >0 x< c }) = F c ). Now, c = h ;α or c = h :α ad therefore C 0 = x >0 x< h ;α }. At the same time α = ν λ0 C 0 )=G λ0,µ v ) x >0 x< h ) ;α } = G,µ x ) 0, ) v x ) ) < h ) ;α }. Thus, the uiformly most powerful critical regio at the level of sigificace α, for testig the ull hypothesis H 0 : λ 0, ]versus H : λ, ) is give by 3).

9 Estimatio fuctios ad uiformly most powerful tests for iverse Gaussia distributio 6 b) Startig with the statistical model 8) the pure test ϕ 0 = C0 where C 0 = x >0 T x) <c} ad c beig determied from the coditio ν λ0 C 0 )=α is uiformly most powerful at the level of sigificace α for testig H 0 : λ [, ) versus H : λ 0, ). First remark that α = ν λ0 x >0 x<c}) =ν λ0 x >0 x> c }) = χ 2 )x >0 x> c }) = F c ). This meas that c = h ; α or c = h ; α. Thus, C 0 = x>0 x> h ; α From this poit the proof cotiues similarly as i a). }. c) The statistical model 8) is of expoetial type sice 9) ρx; λ) = cλ)dx) exp Qλ)T x)), ) where cλ) = λ /2; 2π dx) = x 2 ; T x) = x; Qλ) = λ 2 for ay λ>0, x>0. It is obvious that d ad T are measurable ad that Q is icreasig, so usig a theorem from Lehma, 959, p. 28, gives a uiformly most powerful test at the level of sigificace α for testig H 0 : λ 0,λ ] [λ 2, ) versus H : λ [λ,λ 2 ]. The test is ϕ 0 = C0 where C 0 = x >0 c <Tx) <c 2 } with c,c 2 beig calculated from the coditios 20) ν λ C 0 )=α, ν λ2 C 0 )=α. Proceedig as i the proof of poits a), b) we get that c < 0 ad that the equatios 20) are equivalet to I additio F c λ ) F c λ )=α, F c λ 2 ) F c 2 λ )=α. α = ν λ C 0 )=G λ,µ v )x >0 c 2 <x< c }) = G λ,µ x) 0, ) c 2 <v x)) < c }) = G λ,µ x) 0, ) c < v x)) <c 2 })

10 62 I. Vladimirescu, R. Tuaru ad aalogously α = G λ 2,µ x) 0, ) c < v x)) <c 2 }). For the statistical model 0, ), B 0, ), G λ,µ λ>0}) ) the result ow follows. d) The proof is very similar to the above for c) observig that Q is a cotiuous ad icreasig fuctio ad applyig a well-kow theorem from Lehma, 959. e) Startig agai from the theorem from Lehma, 959, we ca say that the pure test ϕ 0 = C0, where C 0 = x >0 T x) <c } x>0 T x) >c 2 } ad c,c 2 are calculated from the coditios 2) 22) ν λ0 C 0 )=α λ ν λc 0 )) λ=λ0 =0 is uiformly most powerful ad ubiased at the level of sigificace α for testig the ull hypothesis H 0 : λ = versus H : λ>0. The first equatio from 2) is equivalet to Take ito accout that F c ) F c 2 )= α. ν λ C 0 )=ν λ x >0 x> c } x>0 x< c 2 }) = ν λ x >0 λx > λc })+ν λ x >0 λx < λc 2 }) = χ 2 )x >0 x> λc })+χ 2 )x >0 x< λc 2 }) = F λc 2 )+ F λc 2 ). Thus, the secod equatio from 2) is equivalet to λ F c 2 λ) F c λ)) λ=λ0 =0. Similarly to the proof detailed for a) above, we get the stated result.

11 Estimatio fuctios ad uiformly most powerful tests for iverse Gaussia distributio Coclusio The iverse Gaussia distributio has bee used for may decades i actuarial statistics ad it makes its way through mathematical fiace. This distributio is a flexible positive-support probabilistic model with two parameters. Although i the literature there are several goodess-of-fit tests ad some other empirical distributio fuctio tests such as Kolmogorov-Smirov test, the Cramer-vo Mises test, the Aderso-Darlig test ad the Watso test, there are o uiformly most powerful tests developed for testig i the iverse Gaussia cotext. The theorems proved i this paper fill a gap i the literature about uiformly most powerful tests. The theoretical results proved here may be used for model selectio, so makig a useful lik to the practical world of actuary ad fiace. I additio, i the first part of the paper, estimatio fuctios through cofidece regios are costructed for the parameters of the iverse Gaussia distributio. Refereces [] Ter Berg P., 980, Two pragmatic approaches to logliear claim cost aalysis, Asti Bulleti, [2] Ter Berg P., 994, Deductibles ad the iverse Gaussia distributio, Asti Bulleti 24, o. 2, [3] Chhikara R.S., Folks J.L., 974, Estimatio of the iverse Gaussia distributio fuctio, J. Amer. Statist. Assoc. 69, 345, [4] Chhikara R.S., Folks J.L., 989, The Iverse Gaussia Distributio: Theory, Methodology ad Applicatios, New York, Marcel Dekker. [5] Edgema R., Scott R., Pavur R., 988, A modified Kolmogorov-Smirov test for the iverse Gaussia distributio with ukow parameters, Commu. Statist.- Simula. 7, [6] Essam K. Al Hussaii, Nagi S. Abd-El-Hakim, 98, Bivariate iverse Gaussia, Aals of Istitute of Statistics ad Mathematics 33, Part A, [7] Gues H., Dietz D.C., Auclair P., Moore A., 997, Modified goodess-of-fit tests for the iverse Gaussia distributio, Computatioal Statistics ad Data Aalysis 24, [8] Hadwiger H., 940, Naturliche Ausscheidefuktioe fur Gesamtheite ud die Losug der Ereuerugsgleichug, Mitteiluge der Vereiigug schweizerischer Versicherugsmathematiker 40, [9] Heze N., Klar B., 200, Goodess-of-Fit Tests for the iverse Gaussia distributio based o the empirical Laplace trasform, A. Statist., forthcomig. [0] Lehma E.L., 959, Testig Statistical Hypotheses, New York, Wiley. [] Mergel V., 999, Test of goodess of fit for the iverse-gaussia distributio, Math. Commu. 4, [2] Pavur R., Edgema R., Scott R., 992, Quadratic statistic for the goodess-of-fit test for the iverse Gaussia distributio, IEEE Tras. Reliab. 4, [3] O Reilly F., Rueda R., 992, Goodess-of-fit for the iverse Gaussia distributio, Caad. J. Statist. 20, [4] Rao C.R., 973, Liear Statistical Iferece ad Its Applicatios, New York, Wiley. [5] Schrodiger E., 95, Zur Theorie der Fall-ud Steigversuche a Teilche mit Browscher Bewegug, Physikalische Zeitschrift 6,

12 64 I. Vladimirescu, R. Tuaru [6] Seshadri V., 994, Iverse Gaussia Distributios, Oxford, Oxford Uiversity Press. [7] Seshadri V., 999, The Iverse Gaussia Distributio Statistical Theory ad Applicatios, Lodo, Spriger. [8] Shuster J., 968, O the iverse Gaussia distributio, J. Amer. Statist. Assoc. 63, [9] Tweedie M.C.K., 957, Statistical properties of iverse Gaussia distributios I, II, Aals of Mathematical Statistics 28, [20] Wald, A., 947, Sequetial Aalysis, Wiley, New York. I. Vladimirescu: Uiversity of Craiova, Faculty of Mathematics ad Iformatics, Str. A. I. Cuza 3, 00 Craiova, Romaia R. Tuaru, address for correspodece: Busiess School, Middlesex Uiversity, The Burroughs, Lodo NW4 4BT, Eglad r.tuaru@mdx.ac.uk Received Jauary 29, 2002)