Statistical Inference Based On Censored Data

Survival analysis is widely used in mechanical research, engineering and many other fields. There are several situations in reliability experiments, life-testing, and survival analysis in which units are lost or removed from the experiments while they are still alive. The loss may occur either out of control or preassigned. For example, the out of control case happened when an unit under study drops out. The other case may occur because of limitation of funds or to save the cost or time etc. In such situations, the censoring schemes take place. In this thesis, we investigate the statistical inference problems based on censored samples from Pareto distribution and Rayleigh distribution.First, we care about the parameter estimation problems of Pareto distri-bution. Different methods of estimation are discussed. They include mid-point approximation estimator, the maximum likelihood estimator and moment esti-mator.1. Mid-point approximation estimator Suppose the exact failure times of the n units on test are known and denoted as y1,y2…,yn,then the log-likelihood functions are obtained as Taking derivatives with respect to a in it and equating them to zero, we obtainThe mid-point estimator is obtained by assuming the Xi failures at the center of the interval (?) and Ri censored items failed at the censoring time ti, therefore, the mid-point estimator (denoted by MPC) αc based on pseudo-complete data is given by (1), where l=1,2,…,n; i=1,2,…,m.Next, suppose that Xi failures occurred at the center (?) of the interval, then the log-likelihood functions are obtained as Taking derivatives with respect to a in (2) and equating them to zero, we obtain the mid-point estimator (denoted by MPP)2. Maximum likelihood estimationBased on progressive type-I interval censored sample, the likelihood functionis Therefore, we can obtain the likelihood equation as Further, by using the EM-algorithm, we can obtain the MLE.3. Method of momentsAfter simple algebraic manipulation, we obtain that, for α＞δ, By equating the sample moment to the corresponding population moment, we have the following equation Iterative procedure can be employed to solve the above equations for a.Second, the estimation of parameters based on a progressively type-I inter-val censored sample from a Rayleigh distribution is studied. We discuss several kinds of different parameter estimation methods of Rayleigh distribution under progressively type-I interval censored sample. They include mid-point approxima-tion estimator, the maximum likelihood estimator, moment estimator, modified moment estimator, Bayes estimator and sampling adjustment moment estimator, maximum likelihood estimator and estimator based on percentile.1. Mid-point approximation methodFirst, suppose the exact failure times of the n units on test are known and denoted as y1,y2,…,yn, then the log-likelihood functions are obtained as Solving the likelihood equation, we obtainBy assuming the Xi failures at the center of the interval (?) and Ri censored items failed at the censoring time ti, therefore, the mid-point estimator (denoted by MPE) αc ased on pseudo-complete data is given by (4), where l=1,2,…,n;i=1,2,…,m.Next, suppose that Xi failures occurred at the center (?) of the interval, then the log-likelihood functions are obtained asBy solving the likelihood equation, we obtain the mid-point estimator (denoted by MPP)2. Maximum likelihood estimationBased on progressive type-I interval censored sample, the log-likelihood func-tions is Taking derivatives with respect to a and equating them to zero, we obtain the likelihood equation as Further, by using the EM-algorithm, we can obtain the MLE.3.Method of momentsAfter simple algebraic manipulation, we obtain that the lth moment of the Rayleigh distribution is By equating the first sample moment to the corresponding population moment, we have the following equation Iterative procedure can be employed to solve the above equations for a.4. Bayes estimationAssumed that the prior of a has the following Gamma G(a0, b0) distribution with the following density: where α＞0. Thus, we can obtain the following joint posterior density of a: where By using importance sampling method, Bayes estimate of a is where Ea denotes the expectation with respect to5. Sampling adjustment moment estimator and maximum likelihood estima-torNote that for the progressively type-I censored sample, the lifetime of the Xi failures in the ith interval(ti-1,ti]are independent and follow a doubly truncated Rayleigh distribution from the left at ti-1and from the right at ti, and the lifetime of Ri censored items in the ith interval (ti-1,ti] are independent and follow a truncated Rayleigh distribution from the left at ti, i=1,2,…,m. Therefore, to obtain the moment and maximum likelihood estimator of a based on complete data, we sample with replacement, as follows:(1). Based on the initial estimate of a (such as the mid-point approximation estimator described in section2.1), Xi censored failure samples can be resampled from a doubly truncated Rayleigh distribution with truncation points ti-1and ti, i=1,2,…,m. Similarly, Ri censored failure samples can be resampled from a left truncated Rayleigh distribution with truncation point ti,i=1,2,…,m. We denote the resampling data as y1, y2,…,yn.(2). The moment and maximum likelihood estimator of a based on the above pseudo complete data isand respectively.6. Estimator based on percentileNote that F(x;α)=1-exp{-αx2},0＜x＜∞. Therefore (?) F(x;α)). Let y1, y2,…,yn be pseudo complete data. Denote y(i) as the ith order statistic. i.e. y(1)≤y(2)≤…≤y(n).If pi denotes some estimate of F(y(i);α), then the estimate of a can be obtained by minimizing with respect to a. We call the corresponding estimators as the percentile estimator (PCE). Here, we mainly consider pi=i/(n+1), which is the expected value of F(y(i)). Therefore, the PCE of α, say αPCE, isLast, Based on the maximum likelihood method, we deal with the problem of parameter estimation and hypothesis testing on the equality of two Rayleigh distribution under Type I censoring samples with missed data.Consider the following two Rayleigh populations whose density functions are of the form where i=1,2, λ>0, λ2>0are unknown parameters.We make n times independent observation from these two populations and stop at time To. Every observation value will be lost with probability1-p. Let (Zi,δi,αi), i=1,2,…, n be the observation values of the first populations, where Zi=min(T0, Xi), Xi denotes the life of the product. Further, let αi=I{Xi≤T0}-I{Xi>T0}.That is, if we obtain the specific failure time, αi=1,otherwise, δi=1. Moreover, if the ith sample is lost, we let δi=0, otherwise, we let δi=0.Similarly, Let (Mj,ηj,βj),j=1,2,…,n be the observation values of the second populations, where Mj=min(T0,Yj), Yj denotes the life of the product. Further, let βj=I{Yj≤T0}-I{Yj>T0}.That is, if we obtain the specific failure time, βj=1, otherwise,βj=-1. Moreover, if the jth sample is lost, we let ηj=0, otherwise, we let ηj=1.Based on the observations (Zi,δi,αi),i=1,2,…,n, we have the following likelihood function: Solve the likelihood equationwe obtain that Similarly, by solving the likelihood equation, we obtain that where andNext, we discuss the limiting properties of λ1and λ2.Theorem1Using the foregoing notation, as n�'∞, we have λ1�'λ1, a.s. Theorem2Using the foregoing notation,as n�'∞,we haveTheorem3Using the foregoing notation,as n�'∞,we have λ2�'λ2,a.s..Theorem4Using the foregoing notation,as n�'∞,we have Further,consider the following hypothesis test H0:λ1-λ2=0←�'H1:λ1-λ2≠0. The above hypothesis test is based on the following results:Theorem5Using the foregoing notation,as n�'∞,we haveParticularly, under H0,aa n�'∞,we have Let△λ=λ1一λ2andIn what follows,we consider the asymptotic confidence interval of△λ For0＜α＜1,let ζα satisfy As n�'∞, we haveTherefore, given the confidence level α, the confidence interval of AA is...