On Likelihood Ratio and Stochastic Order. for Skew-symmetric Distributions. with a Common Kernel

Transcription

1 In. J. Conemp. Mah. Sciences Vol. 8 3 no HIKARI Ld hp://d.doi.org/.988/icms.3.38 On Likelihood Raio and Sochasic Order or Skew-symmeric Disribuions wih a Common Kernel Werner Hürlimann RSlobal Swizerland Seeeldsrasse 69 CH-88 ürich Swizerland whurlimann@bluewin.ch Copyrigh 3 Werner Hürlimann. This is an open access aricle disribued under he Creaive Commons Aribuion License which permis unresriced use disribuion and reproducion in any medium provided he original work is properly cied. Absrac Characerizaions o he likelihood raio order and he usual sochasic order or he comparison o skew-symmeric disribuions wih a common kernel are derived. Two likelihood raio perurbaion invarian properies are also obained. Mahemaics Subec Classiicaion: Primary 6E5 Secondary 6E Keywords: skew-symmeric disribuion likelihood raio order sochasic order dangerousness order increasing conve order perurbaion invariance. Inroducion Azzalini and Capianio [5] deine he densiy uncion o he skew-symmeric disribuion SSD wih coninuous symmeric abou densiy kernel absoluely coninuous skewing disribuion wih an even densiy ' g and odd skewing weigh w by w }. The linear special case w λ wih skewing parameer λ has been

2 958 Werner Hürlimann inroduced by Azzalini [3] [4] and includes he skew normal by O Hagan and Leonard [6]. The SSD depends on he skewing disribuion and he skewing weigh only hrough he perurbaion uncion w } such ha and he relecive propery + holds. Conversely any uncion ha saisies hese condiions ensure ha is a densiy which represens he SSD ormulaion adoped by Wang e al. 4. In ac any probabiliy densiy uncion admis a uniquely deined SSD represenaion as shown irs by Wang e al. 4 Proposiion 3. Azzalini and Regoli [6] reine his resul o he represenaion o a densiy wih arbirary suppor in heir Proposiion. Skew-symmeric disribuions have been sudied by Nadaraah and Koz [4] [5] or he normal and Cauchy kernels and by Ali e al. [] [] or he gamma and Laplace kernels. Some urher analysis o skewsymmeric disribuions wih a common kernel has been underaken by Umbach [9] and Hürlimann [] []. So ar only a limied amoun o resuls esablish ormal properies o he SSD given qualiaive properies o he kernel skewing disribuion and skewing weigh or equivalenly given he kernel and perurbaion uncion. Even ransormaion invariance see Secion is he mos prominen such propery. We derive characerizaion resuls or SSD s wih a common kernel namely or he likelihood raio order Proposiion. and he usual sochasic order Theorem 4.. Two likelihood raio perurbaion invarian properies are also obained Corollary.. The encounered mahemaical ools include new varians o he Karlin-Noviko [] once-crossing condiion and he Karlin- Noviko-Soyan-Taylor crossing condiions used in a previous characerizaion o he sop-loss order or increasing conve order beween arbirary random variables.. Characerizaion o likelihood raio order Unless oherwise saed denoe hroughou real random variables on some probabiliy space wih suppors S S. We assume ha have skew-symmeric disribuions SSD in he sense o Azzalini and Capianio [5] and Wang e al. [] as uniied in Azzalini and Regoli [6]. Our analysis is resriced o SSD s wih a common kernel. This means ha here eis a symmeric abou densiy uncion called kernel and relecive perurbaion uncions saisying he condiions where he noaion is used hroughou such ha he probabiliy densiy uncions pd o are given by

3 On likelihood raio and sochasic order 959 S y y y y S.. In case a perurbaion uncion is isel a probabiliy disribuion he ransorm is equal o is survival uncion. Equivalenly o. here eis a coninuous skewing disribuion wih an even densiy ' g and odd skewing weighs w w such ha y w y w } S y} y S.. ollowing Azzalini and Regoli [5] Equaion 9 his equivalence is underpinned by he sandard choice made hroughou o a uniorm U random variable wih disribuion + w + [ R and.3 where A denoes he indicaor uncion o he se A. Wih his properies o perurbaion uncions direcly ranslae o properies o skewing weighs and vice versa. We begin wih wo simple resuls on he likelihood raio order which is deined as ollows e.g. Shaked and Shanhikumar [8] equaion.c.. Deiniion.. Le be wo coninuous random variables wih densiies. One says ha is smaller han in likelihood raio order wrien i and only i one has y y y. lr Proposiion. Likelihood raio order characerizaion. Suppose ha are SSD wih a common kernel absoluely coninuous perurbaion uncions wih derivaives ' ' g g y y. The ollowing condiions are equivalen: LR precedes in likelihood raio order lr LR The raio o he perurbaion uncions / is monoone increasing in over he union o he suppors S S LR3 The reverse hazard raes o he perurbaion uncions saisy he inequaliies r g / r g / S S

4 96 Werner Hürlimann Proo. By Deiniion. he likelihood raio order or SSD s wih a common kernel is equivalen wih he condiion / y / y ha is wih LR. On he oher hand in case / is monoone increasing so is is logarihmic. The equivalence o LR and LR3 ollows rom he relaion d d ln / } g / g / r r. As a consequence we derive rom he preceding resul wo perurbaion invarian sochasic order characerizaions or he SSD class. Under perurbaion invariance we mean general saemens abou he random variables and associaed o he kernel densiies and he SSD densiies ha remain valid over a large class o perurbaion uncions. or eample he well-known even ransormaion invariance saes ha T d T equaliy in disribuion or all even real uncions T whaever he perurbaion uncion is e.g. Azzalini and Regoli [6] Proposiion. Corollary. Likelihood raio order invariance. Le be he pd o he SSD random variable and le be he random variable associaed o he kernel densiy. Then he ollowing likelihood raio order invarian characerizaions hold: LRI is an increasing perurbaion on S lr + LRI ma lr is a decreasing on S [ Proo. To show LRI i suices o observe ha he perurbaion uncion generaes he kernel densiy o and invoke he characerizaion LR o Proposiion.. Similarly he choice + + [ generaes he random variable and LRI ollows rom Proposiion.. Remark.. The suicien condiion in LRI is essenially equivalen wih Theorem 8 in Nekoukhou and Alamasaz [5]. Though hese auhors adop he ormulaion. he resul holds or he ormulaion. in virue o he relaionship w in.3. Though raher rivial he necessary condiion in LRI is new.

5 On likelihood raio and sochasic order A Karlin-Noviko once-crossing condiion or sochasic order Le denoe he survival uncions associaed o he random variables. One says ha is smaller han in sochasic order wrien s i and only i. Azzalini and Regoli [6] Proposiion 4 show ha s holds provided is smaller han on he inerval [ wrien S ha is i [. In he presen Secion we show ha here is a more sringen propery namely a once-crossing condiion which also implies he sochasic order. Deiniion 3.. Le be a SSD random variable wih kernel densiy and perurbaion uncion. The weighed mean o wih respec o on [ is deined and denoed by d. 3. Deiniion 3.. Le be wo perurbaion uncions. Then is said o be less dangerous han on he inerval [ wih respec o he kernel densiy wrien D i and only i i such ha he ollowing once-crossing condiion holds ha is < ii There eiss The ne resul is compleely similar o he once-crossing condiion by Karlin- Noviko [] also called Lemma o Ohlin [7] which implies an increasing conve order relaionship ic or equivalenly sop-loss order sl beween wo arbirary random variables e.g. Hürlimann [7] Lemma.. Lemm 3.. SSD Karlin-Noviko condiion Le be SSD wih a common kernel and perurbaion uncions such ha D. Then one has. s Proo. irs o all here and hroughou Secion 3 and 4 i suices o resric he aenion o he inerval [. Indeed rom he characerizaion o SSD s wih a common kernel Azzalini and Regoli [6] Proposiion 3 one knows ha

6 96 Werner Hürlimann Δ Δ Hence Δ on [ implies Δ on and vice versa. Le be he crossing poin beween and. or one has } Δ d. I < one sees ha Δ. } } } } } } d d d d d d A converse o Lemma 3. also holds. Suppose ha s and assume ha he perurbaion uncions cross only once on he righ say a. Then one has necessarily D. Indeed he condiion Δ d } can only be saisied i. Since is he only crossing poin one mus have < hence condiion ii o Deiniion 3. holds. urhermore one has necessarily } Δ d which is condiion i o Deiniion 3.. Togeher his shows ha D. 4. Characerizaion o sochasic order The line o hough o Secion 3 can be eended o he class o perurbaion uncions wih iniely many crossing poins. or his he noion o dangerousness beween perurbaion uncions is eended o a noion o increasing conve

7 On likelihood raio and sochasic order 963 order or sop-loss order. A irs he noion o sop-loss ransorm is eended o a noion o weighed sop-loss ransorm. Deiniion 4.. Le p be a weigh uncion deined on he inerval [ p any perurbaion uncion he uncional π p d [. or called weighed sop-loss ransorm o. The quaniy π p is called weighed mean o. In case p and is a disribuion on [ is he uncional π d is he well-known sop-loss ransorm o and π p is is mean. or a SSD random variable wih kernel and perurbaion uncion and he choice p he weighed sop-loss ransorm o is simply wrien d [ π. In his siuaion π is he weighed mean o considered in Deiniion 3.. In he ollowing le S be he class o all perurbaion uncions wih inie weighed mean wih respec o a given kernel such ha any wo o hem cross iniely many imes on [ inie crossing condiion. Deiniion 4. Sop-loss order beween perurbaion uncions. Le S be wo perurbaion uncions wih inie weighed means wih respec o he kernel densiy and le < <... < n be he ordered crossing poins. Then is said o precede in sop-loss order on he inerval [ wih respec o he kernel densiy wrien sl i and only i one o he ollowing condiions is ulilled: Case : The irs sign change o he dierence occurs rom o + here is an even number o crossing poins n m and one has he inequaliies π π π + π +... m beween he weighed sop-loss ransorms o. Case : The irs sign change o he dierence occurs rom + o here is an odd number o crossing poins n m + and one has he inequaliies π π π π.... In case + + m n he condiion o Deiniion 3. holds.

8 964 Werner Hürlimann The ne characerizaion o sochasic order among SSD random variables is compleely similar o he Karlin-Noviko-Soyan-Taylor crossing condiions or sop-loss order beween wo arbirary random variables see Hürlimann [7] Theorem. where he inequaliies in Case and Case mus be replaced by he correc ones.7 and. saed in he given proo. Theorem 4. Sochasic order characerizaion. Le be SSD wih a common kernel and perurbaion uncions S. Then one has s i and only i sl. Proo. Two cases mus be disinguished. Case : The irs sign change o he dierence occurs rom o + I he las sign change occurs rom + o. Oherwise one has s d > d or n hence n m is even. Consider he SSD random variables m + and... m wih common kernel and perurbaion uncions. or... m he once-crossing condiion beween + and is ulilled wih crossing poin. Rearranging inegrals one obains he ollowing weighed mean ormulas or wih inde or simpliied noaion d d + d π + π. Now by Lemma 3. and is converse one has inequaliies + are ulilled ha is + s i and only i he π π π + π +... m and π π m m

9 On likelihood raio and sochasic order 965 which are equivalen o he inequaliies in Case o Deiniion 4.. Noing he obvious sochasic relaion s one obains he ordered sequence o SSD random variables s m s... s s m + which is valid under he inequaliies o Case in Deiniion 4.. Invoking he ransiive propery o he sochasic order implies he saed equivalence in he presen siuaion. Case : The irs sign change o he dierence occurs rom + o I s hen rom Case he las sign change occurs rom + o hence n m + is odd. Similarly o Case consider SSD random variables m + and... m wih common kernel and perurbaion uncions <. or... m he once-crossing condiion beween + and is ulilled wih crossing poin +. Using he weighed mean ormulas or ha is π + π and Lemma 3. he condiions or + s ha is + are hereore + π π π π π + π... m and π π. m m This is equivalen o he inequaliies in Case o Deiniion 4.. One obains he ordered sequence o SSD random variables s m s... s s m + which is valid under he inequaliies o Case in Deiniion 4.. The saed equivalence or Case is shown and he proo is complee. Le us conclude wih some remarks. Similar ordered sequences o random variables as in he proo characerize he conve order sop-loss order by equal means as shown in Müller [3] see also Shaked and Shanhikumar [8] Theorem 3.A.45 and he Complemens p.75. The saed iniely many crossing

10 966 Werner Hürlimann condiions are more eplici and compuable. A recen illusraion wih more han one crossing poin is ound in Hürlimann [9] Table 4.. The class o perurbaion uncions S is big enough or mos pracical applicaions. or heoreical purposes and he sake o compleeness le us menion ha Theorem 4. eends o he general case o perurbaion uncions wih ininiely many crossing poins. The proo is compleely similar o he wo-sage limiing procedure used in Hürlimann [7] [8]. One reduces he ininiely many crossing poins siuaion o he inie case by invoking he dominaed convergence heorem and he coninuiy propery o he skew-symmeric survival uncional d. Reerences [] M. Ali M. Pal and J. Woo Skewed releced disribuions generaed by releced gamma kernel Pakisan J. Sais [] M. Ali M. Pal and J. Woo Skewed releced disribuions generaed by he Laplace kernel Ausrian J. Sais [3] A. Azzalini A class o disribuions which includes he normal ones Scand. J. Sais [4] A. Azzalini urher resuls on a class o disribuions which includes he normal ones Saisica Bologna [5] A. Azzalini and A. Capianio Disribuions generaed by perurbaion o symmery wih emphasis on a mulivariae skew disribuion J. Royal Sais. Soc. Ser. B Eended version a URL: hp://ariv.org/abs/9.34 [6] A. Azzalini and. Regoli Some properies o skew-symmeric disribuions Ann. Ins. Sa. Mah [7] W. Hürlimann On sop-loss order and he disorion principle ASTIN Bullein [8] W. Hürlimann A noe on generalized disorion risk measures inance Research Leers [9] W. Hürlimann Mied versus composie Pareo ype disribuions: goodnesso-i and ordering o risk comparisons Orienal Journal o Saisical Mehods Theory and Applicaions -7.

11 On likelihood raio and sochasic order 967 [] W. Hürlimann On he CVaR measure o skew-symmeric amilies Advances and Applicaions in Saisics [] W. Hürlimann Tail approimaion o he skew-normal by he skew-normal- Laplace: applicaion o Owen s T uncion and he bivariae normal disribuion J. Sais. and Economeric Mehods 3 -. [] S. Karlin and A. Noviko eneralized conve inequaliies Paciic Journal o Mahemaics [3] A. Müller Ordering o risks: a comparaive sudy via sop-loss ransorms Insurance: Mahemaics and Economics [4] S. Nadaraah and S. Koz Skewed disribuions generaed by he normal kernel Sais. Probab. Le [5] S. Nadaraah and S. Koz Skewed disribuions generaed by he Cauchy kernel Braz. J. Probab. Sa [5] V. Nekoukhou and M.H. Alamasaz A amily o skew-symmeric-laplace disribuions Saisical Papers [6] A. O'Hagan and T. Leonard Bayes esimaion subec o uncerainy abou parameer consrains Biomerika [7] J. Ohlin On a class o measures o dispersion wih applicaion o opimal reinsurance ASTIN Bullein [8] M. Shaked and J.. Shanhikumar Sochasic Orders Springer-Verlag New ork 7. [9] D. Umbach The eec o he skewing disribuion on skew-symmeric amilies Soochow J. Mah [] J. Wang J. Boyer and M. enon A skew-symmeric represenaion o mulivariae disribuions Saisica Sinica Received: Ocober 7 3