Improvement On The Best Location Equivariant Prediction Predictors Or Prediction Interval Under The Ordered Parameters

There are situations in which some order restrictionsare assumed about the parameters of the underlyingdistribution. Order restrictions may be provided by eitherprior information about the parameters or the mathematicalstructure of the problem. Such restrictions may enable usto improve on usual estimation procedures. See also the re_ferences in their papers. The most attention in the litera_ture has been given to estimation problems. In this paperwe shall consider prediction problems under the orderedparameters.There are two random variables X and Y whose joint distr_ibution is indexed by an unknown parameterξ .Based on thevalue of X, we want to predict the value of Y.We considersuch a situation that another random variable Z, whose dis_tribution is indexed by an unknown parameter η with η ≥ ξ,is available to the prediction problem. We shall providemethods for improving on the best equivariant predictor orprediction interval by making use of the value of Z.A function called totally positive of order 2 ( TP2 ) playsa fundament role in deriving the main results. A functionK(x,y) is said to be TP2 ifKK (( x x21 ,,yy11))KK((xx12,,yy22))≥0.For all x1 < x2 and y1 < y2. See Karlin (1968) for the compl_ete treatment of TP2 and Barlow and Proschan (1975) for itsapplications to reliability and life testing.Kubokawa and Saleh (1994) applied a very useful methodcalled the integrated expression of risk difference(IERD)method to get an improved estimator.We shall also use theIERD method to derive an improved predictor or an improvedprediction interval.1.Improved predictor of best location equivariantpredictor.Suppose that (X,Y,Z) has the joint densityf ( x? ξ ,y?ξ,z?η) (1.1),where f is known, and ξ and η areunknown location parameters with ξ ≤ η.We shall considerthe problem of improving a predictor δ c =X+c(1.2) byδφ = X +φ( Z?X) (1.3), where c is a constant and φ is afunction. We shall assume that the loss function is of theform L ( y? d) (1.4).Let U = Z?X, V=Y?X.From (1.2),(1.3)and(1.4),therisk function of δ c and δ φ can be written asR (θ ,δc)= Eθ {L (V ?c)}(1.5),R (θ ,δφ)= Eθ {L (V ? φ(U))}(1.6),Where θ ∈ΘL={θ = (ξ ,η)/ξ≤η}.From (2.1) the joint densityof (U,V) is given by g (u ? λ,v)where λ = η?ξ≥0andg (u ,v)= ∫∞? ∞f (t ,t+ v,t+u)dt(1.7). Hence from (1.5) and (1.6)the difference of the risk functions can be expressed asR (θ ,δc)-R (θ ,δφ)=∫ ∫{L ( v?c)?L(v?φ (u))}g (u?λ,v)dudv∞?∞∞? ∞(1.8). Let G (u ,v)=g (ut,v)dt∫0 ?∞(1.9).Then we have thefollowing result.THEOREM 1.1 Assume that(1.10) G (u ,v) is TP2 ,(1.11) φ (u) is non_decreasing anducul i→m ∞φ ()=,(1.12) ∫( ?())(,)≤0∞? ∞L' vφuGuvdv for each u.Then R (θ ,δφ)≤ R (θ ,δc) for any θ ∈ΘL.COROLLARY 1.1 Assume(1.13) G (u ,v) is TP2 ,(1.14) G (u ,x? y) is TP2 in x andy For each u. Then (,)R θδφ0(,)≤ R θδc0 for any θ ∈ΘL.2.Improved prediction interval of best locationequivariant prediction interval.Suppose that (X,Y,Z) has the joint densityf ( x? ξ ,y?ξ,z?η) (2.1),where f is known, and ξ and η areunknown location parameters with ξ ≤ η.We shall considerthe problem of improving an equivariant prediction inrervalδ c =(X+c?d, X+c+d)(2.2) by a intervalδφ = ( X +φ(Z?X)?d,X + φ ( Z?X)+d)(2.3).where the length of δ φ is the same as δ c, c and d areconstants and φ is some function. We want to choose aproper function φ such that Pθ (Y ∈ δφ)≥Pθ (Y ∈ δc)(2.4).That is , We Want to get a improvement predictionintervalof δ c.Let U = Z?X, V=Y?X.From (2.1) the joint densityof (U,V) is given by g (u ? λ,v)where λ = η?ξ≥0andg (u ,v)= ∫∞? ∞f (t ,t+ v,t+u)dt(2.5).The confidence probability of δ φ and δ c can be writtenasPθ (Y ∈ δc)= ∫ ∫+∞?∞+du cc ? ddg(u ?λ ,v)dv (2.6),Pθ (Y ∈δφ)= ∫ ∫+∞?∞+du φφ (( u u ) )? ddg(u?λ ,v)dv (2.7).Pθ (Y ∈ δc)-Pθ (Y ∈δφ)du cdguvdvducdudud∫ ∫ ∫+∞?∞+?+= ( ?φφ (( ))? )(?λ,) (2.8).Let G (u ,v)=∫u g(t,v)dt? ∞ (2.9).Then we have the following result.THEOREM 2.1. Assume that(2.10) G (u ,v) is TP2 ,(2.11 φ (u) is non_decreasing and ucul i→m ∞φ ()=,(2.12) G (u ,φ (u)+ d)≤G (u ,φ (u)?d)for each u.Then Pθ (Y ∈δφ)≥ Pθ (Y ∈ δc) for any θ ∈ΘL.COROLLARY 2 1. Assume(2.13) G (u ,v) is TP2 ,(2.14) G (u ,x+ y) is TP2 in x and y For each u.Then ()Pθ Y∈δφ0≥ ()Pθ Y∈ δc0 for any θ ∈ΘL.