Monotone Empirical Bayes Test For Scale Parameter Of Competing Risks Under Random Censorship

Analysis of competing risks data has become the one of the central studies aboutsurvival analysis because of its importance in medicine, engineering and reliability. Con-sidering competing or alternate dying (invalidation), competing risk becomes the naturalgeneralizing of survival analysis. Scale exponential distribution family is very important,especially it includes the exponential distribution as a special case which can be used todescribe many models appearing in survival analysis, reliability theory and so on.EB approach was first introduced to statistical problems by Robbins(1964) and hasgenerated considerable interest among the researchers, and has been applied to numerousreal-life problems. Usually, EB approach assumes that there is a sequence of past dataX1,X2,···,Xn, which comes from past n experiments, is available. Di?ering from themany works, we suppose that the sequence is censored from the right by another sequencewith an unknown distribution function. Kaplan&Meier(1858) has given an approach todeal with this kind of data, called Product Limited (PL) Estimates approach, we'll discussthe Empirical Bayes test for scale parameter of competing risk under random censorshipby combining PL estimates with kernel density estimates. Now we'll introduce it brie?y.Under the model of competing risk: ,we investigate the fol-lowing component decision problem: consider testing H0 :θθ0 against H1 :θ>θ0, inthe scale-exponential family .We adoptthe following weighted linear loss function:L,where r = 0,1, and a > 0 is a constant and D = {d0,d1} is action space, dr meansaccepting Hr.The parameterθis distributed according to a non-degenerate prior G(θ),with support on .Hence the marginal pdf of r.v.X isPutδ(x) = P(acceptingH0|X = x), then the Bayes risk of the testδ(x) is Therefore, the best Bayes test minimizing R(δ(x),G(θ)) would have the formThen the minimum Bayes risk isAssume thatwhere , we easily know . We suppose that the prior G(θ) is non-degenerate, then is strictly increasing. Therefore, there exists a unique point aGjsuch thatThen We analyze the value ofα(x) by giving some examples, then present a reasonablehypothesis:the function has and only has finite known pointsNow we use the EB approach under the random right censor to deal with theproblem that the Bayes testδG(x) given lastly is unavailable to use since the prior G(θ)is unknown.First, we consider namely are the samples come fromthe past n experiments ofIn this paper, we suppose that the sequence is censored fromthe right by with unknown distribution function Wj. It is assumedthat are independent of min and are i.i.d. with the distribution function Hj, A product limit (PL) estimator F?j(x) of thedistribution function Fj(x) can be defined aswhere the ) denote the ordered sample, and being the con-comitant ofNow we use the kernel estimates approach to replace the empirical distributionfunction in by PL estimator Then weobtain an estimator of given by where is a kernel function satisfying some certain conditions.Similarly, we define a kernel estimator for Then we get an estimator of and definethe estimator ofα(x) asIn the following, we assume that the prior G(θ) belongs to the following class of distri-butions F = {G(θ) : 0≤B₁≤A₁≤A₂≤B₂∞}, where B1,B2 are known constants.Then, we propose the EB test as followshence, we have , where En de-notes the expectation with respect to the joint distribution ofSecond, we give several lemmas and some assumptions about the kernel function before analyzing the rate of convergence of the EB test present lastly. Now, we exhibit that the optimal rate of convergence ofδn(x) defined lastly isarbitrarily close to O(n^-1) . It is the key conclusion of the paper.Assumewe havewhere we assume A₁ and A₂ are not the zero points,that is to say, A₁ and A₂ are di?erentfrom a₁ and a_n.Then By Lemmas, we have For 1 <Ï„< 2, note that a1 is a singular point, then bywe know thatÏ„dx is convergent. Furthermore, we found that if continuous and is the continuously di?erentiable, also, then it is easy to see thatTherefore, we draw a conclusion.THEOREM LetδG(x) andδn(x) be define before, respectively. If the followingconditions are satisfied: is continuous; is the s ? th continuously differentiable.Then for , we havewhere s 2 is an arbitrary but fixed integer.If we letÏ„be arbitrarily close to 2 and s be large enough, then the rate of conver-gence can be arbitrarily close to O(n^-1). Now we present a EB test which possesses arate of convergence can be arbitrarily close to O(n^-1) under the condition that the pastsamples are randomly censored from the right.