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

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1 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 , Kerala, Idia M. Shy Mary Departmet of Statistics, Uiversity of Kerala, Trivadrum , Kerala, Idia 1. INTRODUCTION Let { Xi, i 1} be a sequece of idepedet ad idetically distributed (iid) radom variables havig a absolutely cotiuous cumulative distributio fuctio (cdf) F( x ) ad probability desity fuctio (pdf) f( x ). A observatio. X j is called a upper record if its value exceeds that of all previous observatios. Thus, X j is a upper record if X j > X i for every i< j. A aalogous defiitio deals with lower record values. Record data arise i a wide variety of practical situatios, such as i destructive stress testig, meteorological aalysis, hydrology, seismology, sportig ad athletic evets ad oil ad miig surveys. Iterest i records has icreased steadily over the years sice Chadler (195) first formulated the theory of records. The problem of estimatio of parameters ad predictio of future record values have bee discussed by several authors icludig Balakrisha ad Cha (1998), Sulta et al. (00), Sulta (000), Raqab et al. (007) ad Raqab (00). For more details about record values ad their applicatios, oe may refer to Arold et al. (1998) ad Ahsaullah (1995). Serious difficulties for statistical iferece based o records arise due to the fact that the occurreces of record data are very rare i practical situatios ad the expected waitig time is ifiite for every record after the first. These problems are avoided if we cosider the model of k-record statistics itroduced by Dziubdziela ad Kopociski (1976). For a positive iteger k, the upper k-record times T k ( ) ad the upper k-record values R k ( ) are defied as follows: ad, for 1 T ( k) = k, 0 with probability 1 T ( ) = k mi{ j : j > T 1 k ( k), X j > X + 1:T } T 1( k) 1( k) where : im X deote the i-th order statistic i a sample of size m. The sequece of upper k-records are the defied by

2 506 M. Chacko ad M. Shy Mary R = X +, 0, k 1. k ( ) T k ( ) k 1: T k ( ) I a aalogous way, oe ca defie the lower k-record statistics. Sice the ordiary record values are cotaied i the k-records, by puttig k =1, the results for usual records ca be obtaied as special case. Statistical iferece problems based o k-records have bee cosidered by several authors, see, Maliowska ad Szyal (004), Ahmadi et al. (005), Ahmadi ad Doostparast (008) ad Shy Mary ad Chacko (010). The pdf of R (k ), for 0 is give by (see, Arold et al., 1998) + 1 k k 1 fk ( )( x)= log(1 F( x)) 1 F( x) f( x), < x<! (1) ad the joit pdf of mth ad th k-record values for m< is give by, + 1 k fmk, ( )( x, y)= log(1 F( x)) m!( m 1)! m 1 log(1 F( y)) + log(1 F( x)) m [1 F( y)] 1 F( x) k 1 f( x) f( y), x< y. () I this paper we cosider k-record values arisig from ormal distributio. I sectio, we compute the meas, variaces ad covariaces of k-record values arisig from ormal distributio. I sectio 3, we determie the BLUEs of the mea µ ad stadard deviatio σ of ormal distributio based o k-record values. These BLUEs are the used to predict the future k-record values. I sectio 4 we illustrate the iferece procedures developed i this paper based o k-record values from ormal distributio usig real data.. MOMENTS OF THE K-RECORD VALUES Let R0( k), R1( k),, Rk ( ) be the first (+1) upper k-record values arisig from a sequece of iid stadard ormal radom variables. If we deote Φ () as the cdf ad ϕ() as the pdf of a stadard ormal radom variable, the by usig (1), the pdf of th upper k-record value Rk ( ) + 1 is give by k k 1 fk ( )( x)= log(1 Φ( x)) 1 Φ( x) ϕ( x), < x<, 0.! (3) From (), the joit pdf of mth ad th upper k-record values, Rmk ( ) ad Rk ( ) for m< is give by

3 Estimatio ad predictio based o k-record values from ormal distributio * k fm( k) ( xy) x,, = log 1 Φ m! m 1! Let us deote ( k ( )) ( mk ( ) Rk ( )) C ov R, 0 is give by E R by ( ) ( ) ( y) ( ( )) ( ) ( ( x) ) m m 1 log 1 log 1 Φ + Φ k 1 [1 Φ( y)] ϕ( x) ϕ( y), < x< y<. 1 Φ( x) ar R by k, ( ) α k, V ( k ( )) β, E( Rmk ( ) Rk ( )) 4) by α mk, ( ) ad by β mk, ( ). The, the sigle momets of the th upper k-record value for α ( ) xf * = k ( ) ( ) k x dx, ad the product momets of the mth ad th upper k-record values for * ( ) = mk ( ) ( ) α xyf mk x,, y dy dx, x, m < is give by where fk ( ) () ad fmk, ( ) (,) are give, respectively, i (3) ad (4). We have evaluated umerically the values of α k ( ) ad α mk, ( ) for sample size up to 10. The values of meas, α k ( ) for = 0(1)9 ad k = (1)5 are give i table 1. By usig the values of α k ( ) ad mk, ( ) α we have computed the values of variaces ad covariaces, β mk, ( ) for 0 m 9 ad k = (1)5 ad are give i table. ' REMARK 1. Suppose { k ( ), 0} R is the sequece of lower k-record values arisig from a sequece { X i } of iid stadard ormal variables. The, due to the symmetry of the stadard ormal distributio, oe ca observe that d ( ) ( ) ' d ( ) ( ) ' ( ) ' k= k mk, k ( ) = mk ( ), k ( ) R R ad R R R R (5) ad hece, E ( R ( ) ) ( ) ( ( ) ) ' =, Var R ' = ( ) ad Cov R ' ( ), R ' ( ) = ( ) ( ) α β β k k k k, mk k mk, As a result, the etries i tables 1 ad also yield the meas, variaces ad covariaces of the lower k- record values from the stadard ormal distributio.

4 508 M. Chacko ad M. Shy Mary 3. BEST LINEAR UNBIASED ESTIMATION Let 0( k), 1( k),..., k ( ) R R R be the first ( + 1) upper k-record values from a ormal distributio with mea µ ad variace ad for m< σ. The we have Suppose R k ( ) = ( 0( k ), 1( k ),..., k ( ) ) ( k ) E R = µ + σα, ( ) k ( ) ( k ) V ar R = σ β ( ) k, ( ) ( ) C ov R, R = σ β. mk ( ) k ( ) mk, ( ) T R R R deote the vector of upper k-record values. The where ( ) T k ) 1( k ) ( k ) ( k ) E R ( ) = µ 1 + σα,. α = α 0(, α,..., α ad 1 is a colum vector of (+1) oes. The variacecovariace matrix of R k ( ) is give by ( Rk ) D ( ) = Bσ, where B = (( β i, j( k ), 0 i, j )). Followig the geeralized least-squares approach, the BLUEs of µ ad σ are give, respectively, by (see, Balakrisha ad Cohe, 1991, pp ) ad 1 T 1 T 1 T 1 α B α1 B α B 1α B µ = ( α B α )(1 B 1) ( α B 1) = T T 1 T 1 T 1 ar i i=0 ik ( ) T 1 T 1 T 1 T 1 1 B 1α B 1 B α1 B α = ( α B α )(1 B 1) ( α B 1) T 1 T 1 T 1 = br. i i=0 ik ( ) R R k ( ) k ( ) (6) (7) Furthermore, the variaces ad covariace of the above estimators are give by (see, Balakrisha ad Cohe, 1991, pp )

5 Estimatio ad predictio based o k-record values from ormal distributio 509 T 1 α B α Var( µ )= σ, T 1 T 1 T 1 ( α B α)(1 B 1) ( α B 1) (8) ad T 1 1 B 1 Var( σ )= σ ( α B α)(1 B 1) ( α B 1) T 1 T 1 T 1 (9) T 1 α B 1 Cov( µ, σ )= σ. T 1 T 1 T 1 ( α B α)(1 B 1) ( α B 1) By makig use of the values of meas, variaces ad covariaces preseted i tables 1 ad, we have calculated the coefficiets a i ad b i, i = 0,1,, of BLUEs of µ ad σ for Var( µ ) Var( σ ) = 1(1)9 ; k = (1)5 ad are give i tables 3 ad 4. The values of ad for = 1(1)9 ; k = (1)5 are also computed ad are icorporated i tables 3 ad 4. From tables 3 ad 4, we see that as icreases the variaces of µ ad σ decreases. σ σ (10) 4. PREDICTION OF FUTURE K-RECORD VALUES Predictio of future evets o the basis of the past ad preset kowledge is a fudametal problem of statistics, arisig i may cotexts ad producig varied solutios. Similarly predictio of future records is also a problem of great iterest i may areas such as sports, weather etc. For example, while studyig the record raifalls or sowfalls, havig observed the record values util the preset time, we will be aturally iterested i predictig the amout of raifall or sowfall that is to be expected whe the preset record is broke for the first time i future. Based o the usual records (k=1), the problem of predictig future records has bee studied by several authors. See for example, Ahsaullah (1980), Berred (1998), Sulta et al. (008) ad Raqab ad Balakrisha (008). Ahmadi ad Balakrisha (010) discuss the problem of predictig future order statistics based o observed record values ad similarly, the predictio of future records based o observed order statistics. Ahmadi et al. (01) study the problem of predictig future k-records based o k-record data from a geeral class of distributios uder balaced type loss fuctios. Assume that we have the first (+1) upper k-record values from ormal distributio. The problem of iterest the is to predict the value of the ext upper k-record R + 1(k ), or, more geerally, the value of the mth upper k-record R m(k ) for some m>. For a locatio-scale family with locatio parameter µ ad scale parameter σ, the best liear ubiased predictor (BLUP) of mth record value was cosidered by Balakrisha ad Cha (1998). For k-record values, the BLUP R for some m> ca be writte as of m(k ) R = µ α σ, mk ( ) + mk ( ) (11)

6 510 M. Chacko ad M. Shy Mary where µ ad σ are the BLUEs of µ ad σ based o the first ( + 1) upper k-record values ad α mk ( ) is the mea of mth k-record value from the stadard distributio. 5. ILLUSTRATIVE EXAMPLE I order to illustrate the usefuless of the predictio procedure described i sectio 3, we use the followig data which represet the records of the total aual raifall (i iches) at Oxford, Eglad, for the years ; (see, Arold et al., 1998, p. 173). The upper -records extracted from these data are as follows: 0.77, 5.3, 6.7, 8.07, 30.17, 30.41, 31.77, A simple plot of these eight upper -record values agaist the expected values i table 1 idicate a very strog correlatio (correlatio coefficiet as high as ). Hece, the assumptio that these -record values have come from a ormal N ( µ, σ ) distributio is quite reasoable. Usig the first four upper -record values we predict the 6 th upper -record value. From tables 3 ad 4, we determie the BLUEs of µ ad σ to be ad ( ) ( ) ( ) ( ) µ * = = ( ) ( ) ( ) ( ) σ * = = The by usig (11), the BLUP of 6th value of -record is obtaied as ~ R 5() = 30.60, while the actual value of the 6 th -record is REMARK. Due to the symmetry of the ormal distributio ad hece the relatio i (5), all the iferece procedures based o tables 3 ad 4 ca be suitably chaged to hadle the case whe the lower k- record values are give istead of the upper k-record values. 6. CONCLUSION Record values ad the associated statistics are of iterest ad importat i may real life applicatios. Serious difficulties for statistical iferece based o records arise due to the fact that the occurreces of record data are very rare i practical situatios ad the expected waitig time is ifiite for every record after the first. These problems are avoided if we cosider the model of k- record statistics. I this paper, we have obtaied the BLUEs of the locatio ad scale parameters of ormal distributio based o k-record values. We have also obtaied the BLUP of future k-record value for the ormal distributio. A real data set is used to illustrate the iferetial procedure developed i this paper.

7 Estimatio ad predictio based o k-record values from ormal distributio 511 TABLE 1 Meas of the upper k-record values from N(0,1) k

8 51 M. Chacko ad M. Shy Mary k TABLE Variaces ad covariaces of the upper k-record values from N(0,1) m

9 Estimatio ad predictio based o k-record values from ormal distributio 513 TABLE 3 Var( µ ) Coefficiets for the BLUE of µ ad the values of based o the first (+1) upper k-record values from σ N ( µ, σ ). Var ) σ k a 0 a 1 a a 3 a a 4 5 a 6 a 7 a 8 a ( µ

10 514 M. Chacko ad M. Shy Mary TABLE 4 Var( σ ) Coefficiets for the BLUE of σ ad the values of based o the first (+1) upper k-record values σ from N ( µ, σ ). k b 0 b 1 b b 3 b 4 b 5 b 6 b 7 b 8 Var( σ ) b 9 σ ACKNOWLEDGEMENTS The authors are grateful to the referees for their helpful ad costructive commets.

11 Estimatio ad predictio based o k-record values from ormal distributio 515 REFERENCES J. AHMADI, N. BALAKRISHNAN (010). Predictio of order statistics ad record values from two idepedet sequeces. Statistics, 44, pp J. AHMADI, M. DOOSTPARAST (008). Statistical iferece based o k-records. Mashhad Joural of Mathematical Scieces, 1, o., pp J. AHMADI, M. DOOSTPARAST, A. PARSIAN (005). Estimatio ad predictio i a two parameter expoetial distributio based o k-record values uder LINEX loss fuctio. Commuicatios i Statistics. Theory ad Methods, 34, o. 4, pp J. AHMADI, M. J. JOZANI, E. MARCHAND, A. PARSIAN (01). Predictio of k-records from a geeral class of distributios uder balaced type loss fuctios. Metrika, 70, pp M. AHSANULLAH (1980). Liear predictio of record values for the two parameter expoetial distributio. Aals of the Istitute of Statistical Mathematics, 3, pp M. AHSANULLAH (1995). Record Statistics. Nova Sciece Publishers, New York. B. C. ARNOLD, N. BALAKRISHNAN, H. N. NAGARAJA (1968). (1998). Records, Wiley, New York. N. BALAKRISHNAN, A. C. COHEN (1991). Order Statistics ad Iferece: Estimatio Method. Academic Press, Sa Diego. N. BALAKRISHNAN, P. S. CHAN (1998). O the ormal record values ad associated iferece. Statistics & Probability Letters, 39, pp A. M. BERRED (1998). Predictio of record values. Commuicatios i Statistics. Theory ad Methods, 7, pp K. N. CHANDLER (195). The distributio ad frequecy of record values. Joural of the Royal Statistical Society, Series B, 14, pp W. DZIUBDZIELA, B. KOPOCINSKI (1976). Limitig properties of the k-th record values. Zastosowaia Matematyki, 15, pp I. MALINOWSKA, D. SZYNAL (004). O a family of Bayesia estimators ad predictors for Gumbel model based o the k-th lower record. Applicatioes Mathematicae, 31, o. 1, pp M. Z. RAQAB (00). Ifereces for geeralized expoetial distributio based o record statistics. Joural of Statistical Plaig ad Iferece, 104, o., pp M. Z. RAQAB, N. BALAKRISHNAN (008). Predictio itervals for future records. Statistics & Probability Letters, 15, pp M. Z. RAQAB, J. AHMADI, M. DOOSTPARAST (007). Statistical iferece based o record data from Pareto model. Statistics, 41, pp

12 516 M. Chacko ad M. Shy Mary M. SHY MARY, M. CHACKO (010). Estimatio of parameters of uiform distributio based o k- record values. Calcutta Statistical Associatio Bulleti, 6, pp K. S. SULTAN (000). Momets of record values from uiform distributio ad associated iferece. The Egyptia Statistical Joural, 44, o., pp K. S. SULTAN, G. R. AL-DAYIAN, H. H. MOHAMMAD (008). Estimatio ad predictio from gamma distributio based o record values. Computatioal Statistics & Data Aalysis, 5, pp K. S. SULTAN, M. E. MOSHREF, A. CHILDS (00). Record values from geeralized Power fuctio distributio ad associated iferece. Joural of Applied Statistical Sciece, 11, pp SUMMARY Estimatio ad predictio based o k-record values from ormal distributio I this paper, we itroduce the k-record values arisig from ormal distributio. After computig the meas, variaces ad covariaces of the k-record values, we determie the best liear ubiased estimators for the locatio ad scale parameters of ormal distributio based o k-record values. The best liear ubiased predictor of future k-record values is also determied. Fially, a real data is give to illustrate the iferece procedures developed i this paper. Keywords: k-record values; ormal distributio; best liear ubiased estimatio; best liear ubiased predictio

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