Maximum likelihood estimation from record-breaking data for the generalized Pareto distribution

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1 METRON - Iteratioal Joural of Statistics 004, vol. LXII,. 3, pp NAGI S. ABD-EL-HAKIM KHALAF S. SULTAN Maximum likelihood estimatio from record-breakig data for the geeralized Pareto distributio Summary -I this paper, we obtai the MLEs of parameters for the geeralized Pareto distributio GPD) based o record-breakig data record values). The, we discuss the properties of these estimates. Next, we compare the MLEs of the locatio scale parameters with the BLUEs give by Sulta Moshref 000). I additio, we use the MLEs to costruct cofidece itervals for the locatio scale parameters of GPD. Key Words - Upper record values; Maximum likelihood estimates; Biased ubiased estimates; Best liear ubiased estimates; Iterval estimatio; Miimum variace boud relative efficiecy.. Itroductio A rom variable X is said to have the GPD if its probability desity fuctio pdf) is of the followig form see Picks 975): f x)= { + )} x θ +/), x θ, for >0, θ<x <θ / for <0, e x θ)/, x θ, for = 0, 0, otherwise,.) while the stard form of the GPD is give from.) by substitutig = θ = 0. Some related distributios are listed below see also Johso, Kotz Balakrisha 994). Received December 003 revised December 004.

2 378 NAGI S. ABD-EL-HAKIM KHALAF S. SULTAN. For >0, GP distributio is kow as Pareto type II or Lomax distributio.. For =, GP distributio coicides with the uiform distributio o θ,θ + ). 3. As 0, GP distributio leads to a two-parameter expoetial distributio. The geeralized Pareto distributio was itroduced by Picks 975). Some of its applicatios iclude its uses i the aalysis of extreme evets, i the modelig of large isurace claims, to describe the aual maximum flood at river gaugig statio. Hoskig Wallis 987) studied the parameter quatile estimatio for the two-parameter geeralized Pareto distributio, Smith 987) has discussed the maximum likelihood estimatio for the GPD uder simple rom samplig. For some iterestig graphical represetatio of the geeralized Pareto desities see Reiss 989). Record values arise aturally i may real life applicatios ivolvig data relatig to weather, sports, ecoomics life testig studies. May authors have studied record values associated statistics; for example, see Chler 95), Ahsaullah 980, 988, 990, 993, 995), Arold, Balakrisha Nagaraja 99, 998). Ahsaullah 980, 990), Balakrisha Cha 993), Balakrisha, Ahsaullah Cha 995) have discussed some iferetial methods for expoetial, Gumbel, Weibull logistic distributios, respectively. Maximum likelihood estimates of parameters for some useful distributios, icludig oe two parameter expoetial, oe two parameter uiform, ormal, logistic Gumbel distributios are discussed i Arold, Balakrisha Nagaraja 998). Balakrisha Ahsaullah 994) have established some recurrece relatios satisfied by the sigle double momets of upper record values from the stard form of the GPD. I this paper, we derive the MLEs of parameters of GPD give i.) based o record values, the we discuss the efficiecy of these estimates. Also, we compare our results by the BLUEs of the locatio scale parameters obtaied by Sulta Moshref 000). Fially, we use the MLEs to costruct cofidece itervals for the locatio scale parameters of GPD.. MLEs Let X U), X U),...X U) be the first upper record-brakig values from the GPD give i.), for coveiece let us deote X Ui) by X i, i =,,...,. The the pdf of the -th upper record value is give by f x) = Ɣ) log{ Fx)} f x),.) where f.) is give by.) F.) is the correspodig cdf.

3 MLE from record-breakig data for the geeralized Pareto distributio 379 The likelihood fuctio i this case may be writte as Lθ,, )= + ) x θ / ) xi θ +, 0, i= e x θ)/, = 0..) From.), we discuss the followig cases:. Whe = 0 Two-parameter expoetial distributio): Arold, Balakrisha Nagaraja 998) have obtaied the MLEs of θ to be ˆθ = x ˆ = x x )/. They also have discussed the ubiasedess variaces. For the sake of completeess comparisos, we preset their results as give below: E ˆθ) = θ +, Var ˆθ) =, MSE ˆθ) =,.3) E ˆ) = ), Var ˆ) = ), MSE ˆ) =..4) I this case, we propose the ubiased estimate of to be = x x,.5) hece Var ) = MSE ) =..6) The miimum variace boud for the estimate of MVB) is give by the relative efficiecy of with respect to MVB )) is give by. For θ we propose the followig MLEs θ = x ˆ = + )x x,.7) θ = x = x x..8)

4 380 NAGI S. ABD-EL-HAKIM KHALAF S. SULTAN From.7).8), we have E θ )=θ +, Var θ )= + ) MSE θ )= +),.9) E θ ) = θ Var θ ) = MSE θ ) =..0) From the above discussio, we ote that ˆ represets a biased estimate for while represets a ubiased estimate for but MSE ˆ) < MSE ), while θ is biased estimate for θ θ is ubiased estimate for θ but MSE θ )< MSE θ ). Also, we ca see that ˆ are cosistet. Remark. whe = 0, the estimators i.3).4) are either asymptotically cetered or cosistet, while the estimators i.9).0) are ot cosistet.. Whe 0: maximizig the logarithm of the likelihood fuctio i.) with respect to θ,, respectively, gives i= e ˆ + x x i x i ˆθ ˆθ = x,.) ˆ ˆ = e ˆ x ˆθ),.) = e ˆ ˆe..3) ˆ I order to discuss the efficiecy of the MLEs of θ,wecosider the followig cases: a) are kow: from.), it is easy to show that E ˆθ) = θ +, <,.4) with variace give by Var ˆθ) =, < /,.5) ) ) MSE ˆθ) =, < /..6) ) )

5 MLE from record-breakig data for the geeralized Pareto distributio 38 From.4), we may propose the ubiased estimate of θ as θ = x,.7) with the same variace give i.5). Notice that, the results give i.3) ca be easily obtaied from.4),.5).6) by lettig 0. b) θ are kow: if θ are kow, the from.), we have E ˆ) = ), e.8) Var ˆ) = ) ), e ).9) MSE ˆ) = ) ) e + e..0) e ) I this case, we propose the ubiased estimate of to be = x θ) )..) It ca be show that Var ) = ) ) ) )..) c) is kow: if θ is ukow is kow, the from.), we have E ˆ)= ) ),.3) e Var ˆ)= ) ) ) ) +) + ) ) e ),.4) MSE ˆ)= ) ) ) ) +) + ) ) + ) ) e + e )..5)

6 38 NAGI S. ABD-EL-HAKIM KHALAF S. SULTAN From.3),.4).5), we have lim E ) ˆ =,.6) 0 lim Var ) ˆ =,.7) 0 lim MSE ) ˆ = 0,.8) which are the same as the results give i.4). I this case, we cosider the ubiased estimates of θ to be x x ) =,.9) ) ) θ = + ) x ) ) ) ) x..30) Hece Var ) = MSE ) = ) ) ) ) +) + ) ),, ) ) Var θ)= ) + ) ) / ) ) ) ) )..3).3) d) θ are kow is ukow: solvig the equatio.3) gives the MLE of.

7 MLE from record-breakig data for the geeralized Pareto distributio 383 I the followig two theorems, we discuss the miimum variace boud MVB) of the MLEs of both : Theorem. For positive, the lower boud of the variace of ˆ is give by MVB ˆ) = 3,.33) ) + ) 3 MVB ˆ) = + ) + ), as 0, 0, as..34) Proof. See Appedix B. Theorem. For > /, the lower boud of the variace of ˆ is give by MVB ˆ) =,.35) + ), as 0, MVB ˆ) =, as,>0, 0, as, 0..36) Proof. See Appedix A. 3. Simulatio comparisos I order to show the efficiecy of our results, we calculate the variaces of the MLEs of the locatio scale parameters of GPD compare them with those of the BLUEs θ obtaied by Sulta Moshref 000). Table gives the variaces of the BLUEs MLEs for = 3, 4, 5, 6 7.

8 384 NAGI S. ABD-EL-HAKIM KHALAF S. SULTAN Table : Variaces of the BLUEs MLEs whe θ = 0 = Locatio Parameter Scale Parameter Varθ ) Var θ) Var ) Var ) From Table, we ca see that the variaces of the BLUEs MLEs decrease as icreases, icrease whe icreases. I coclusio, we ca say that the variaces of BLUEs obtaied by Sulta Moshref 000) the ubiased MLEs preseted i this paper are very close, but the MLEs are simpler to evaluate tha the BLUEs. Also, as we ca see from Table, if, = 0. =, the Var ˆθ) = Var ˆ) = that is because whe <0wehave lim Var ˆθ) = lim Var ˆ) =. 4. Iterval estimatio I this sectio, we costruct cofidece itervals for the locatio scale parameters of GPD give i.). 4.. A cofidece iterval for θ whe are kow Cofidece iterval for θ whe are kow may be costructed through the statistic T = θ µ θ, 4.) θ where µ θ θ represet the mea the stard deviatio of the ubiased estimate of θ give i.7).

9 MLE from record-breakig data for the geeralized Pareto distributio 385 It is easy to show that the distributio of T is the GPD with locatio parameter, scale parameter ) shape parameter. A α)00% cofidece iterval for θ i this case is obtaied to be x α/), x ) α/), 4.) where x is the first upper record. 4.. A cofidece iterval for whe θ are kow Cofidece iterval for whe θ are kow may be costructed usig the statistic τ = µ, 4.3) where µ θ θ represet the mea the stard deviatio of the ubiased estimate of give i.). It is easy to show that the distributio of τ is the th record value of the GPD give i.) with locatio parameter θ, scale parameter shape parameter, where θ = ) ) ) =. 4.4) ) ) The α)00% cofidece iterval for i this case is obtaied to be x θ) ) +, ) ) τ α/ ) 4.5) x θ) ) + ) ) τ α/ where x is the th upper record the percetage poit τ α is the solutio of the oliear equatio αɣ) = Ɣ, log + ) τ θ ), 4.6) where θ are give by 3.4) Ɣ, a) is the icomplete gamma fuctio defied by Ɣ, a) = a 0 x exp x)dx.

10 386 NAGI S. ABD-EL-HAKIM KHALAF S. SULTAN 4.3. Cofidece itervals for θ whe is kow I this case, the α)00% cofidece iterval for θ are give, respectively, by x α/), x ) α/), 4.7) x θ) ) +, ) ) τ α/ x θ) ) 4.8) ) + ) ) τ α/ where τ α is the solutio of the equatio 4.6) θ are give, respectively, by.9).30). Appedices A. Proof of Theorem. The pdf of the i-th record value from.) ca be writte as f i x)= log + y) i + y) +/), x >θ for >0, Ɣi) θ<x <θ / for <0, A.) where y = x θ)/. From A.), it is easy to prove that E log + Y i ) = i, A.) ) E =, >, A.3) + Y i + ) i ) E =, > /. A.4) + Y i + ) i From the likelihood equatio give i.), we may write E ) log L = E i= + Y i ) + + Y ). A.5)

11 MLE from record-breakig data for the geeralized Pareto distributio 387 By usig A.3) A.4) i A.5), we get hece E ) log L MVB ) = + ) =, A.6), A.7) + ) lim MVB ) = 0, A.8) which gives the boud i case of two parameters expoetial distributio. Also, from A.6), we have lim MVB ) = {, > 0, 0, < 0. A.9) B. Proof of Theorem. From the likelihood equatio give i.), we may write ) log L E = E i= + Y i Y Y i = X i θ)/. ) + 3 ) 3 log + Y ) + Y, ) B.) By usig A.), A.3) A.4) i B.), we get ) log L E = i= ) i + + ) ) = 3 ) + ) i ) + ) ) ) + ) ), B.)

12 388 NAGI S. ABD-EL-HAKIM KHALAF S. SULTAN hece for positive, we get MVB) = ) + ), B.3) lim MVB) = ) + ). Also, from B.3), we have lim MVB) = 0. B.4) B.5) Ackowledgmets The authors would like to thak the referees for their helpful commets, which improved the presetatio of the paper. The secod author would like to thak the Research Ceter, College of Sciece, Kig Saud Uiversity for fudig the project Stat/4/7). REFERENCES Ahsaullah, M. 980) Liear predictio of record values for the two parameter expoetial distributio, A. Ist. Statist. Math.,, Ahsaullah, M. 988) Itroductio to Record Values, Gi Press, Needham Heights, Massachusetts. Ahsaullah, M. 990) Estimatio of the parameters of the Gumbel distributio based o the m record values, Comput. Statist. Quart., 3,3 39. Ahsaullah, M. 993) O the record values from uvariate distributios, Natioal Istitute of Stards Techology Joural of Research Special Publicatios, 866, 6. Ahsaullah, M. 995) Record values, IThe Expoetial Distributio: Theory, Methods Applicatios, N. Balakrisha AP. Basu eds), Gordo Breach Publishers Newark, New Jersey, pp Arold, B. C., Balakrisha, N., Nagaraja, H. N. 99) AFirst Course i Order Statistics, Joh Wiley Sos, New York. Arold, B. C., Balakrisha, N., Nagaraja, H. N. 998) Records, Joh Wiley Sos, New York. Balakrisha, N. Ahsaullah, M. 994) Recurrece relatios for sigle product momets of record values from geeralized Pareto distributio, Commu. Statist. Theor. - Meth., 3, Balakrisha, N., Ahsaullah, M., Cha, P. S. 995) O the logistic record values associated iferece, Appl. Statist. Sci.,,33 48.

13 MLE from record-breakig data for the geeralized Pareto distributio 389 Balakrisha, N. Cha, P. S. 993) Record values from Rayleigh Weibull distributios associated iferece, Natioal Istitute of Stards Techology Joural of Research Special Publicatios, 866, 4 5. Chler, K. N. 95) The distributio frequecy of record values, J Roy. Statist. Soc. B., 4, 0 8. Hoskig, J. R. M. Wallis, J. R. 987) Parameter quatile estimatio for the geeralized Pareto distributio, Techometrics, 9, Johso, N. L., Kotz, S., Balakrisha, N. 994) Cotiuous Uivariate Distributios, Vol., Secod editio, Joh Wiley & Sos, New York. Picks, J. 975) Statistical iferece usig extreme order statistics, A. Statist., 3,9 3. Reiss, R. D. 989) Approximate Distributios of Order Statistics: With Applicatios to Noparametric Statistics, Spriger-Verlag Berli. Smith, R. L. 987) Estimatig tail of probability distributios, A. Statist., 5, Sulta, K. S. Moshref, M. E. 000) Record values from geeralized Pareto distributio associated iferece, Metrika, 5, NAGI S. ABD-EL-HAKIM Departmet of Mathematics El-Miia Uiversity El-Miia Egypt) KHALAF S. SULTAN Departmet of Statistics Operatios Research College of Sciece Kig Saud Uiversity P.O.Box 455 Riyadh 45 Saudi Arabia) ksulta@ksu.du.sa

ESTIMATION AND PREDICTION BASED ON K-RECORD VALUES FROM NORMAL DISTRIBUTION

STATISTICA, ao LXXIII,. 4, 013 ESTIMATION AND PREDICTION BASED ON K-RECORD VALUES FROM NORMAL DISTRIBUTION Maoj Chacko Departmet of Statistics, Uiversity of Kerala, Trivadrum- 695581, Kerala, Idia M. Shy