一类删失数据的统计推断
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摘要
本文主要考虑了一类渐进(Progressive)删失数据的统计推断问题.首先,研究了熵损失下渐进Ⅱ型(定数)删失情形威布尔分布尺度参数的估计问题,给出了其熵损失下的最小风险同变(MRE)估计,并证明了这一估计的可容许性；进一步,考虑形状参数的估计方法,给出了两参数都未知时尺度参数的估计,并通过模拟比较了各估计的效果.其次,本文研究了Ⅰ型渐进混合删失情形两参数指数分布参数的精确推断,给出了两参数的极大似然估计(MLE)的精确分布,并在此基础上得到了两参数的精确置信界限,进而得到了p分位数的MLE的精确分布及相应的精确置信界限,并对各置信界限的效果进行了模拟研究.最后,本文讨论了一个误差项为一阶相依的对数正态线性模型,给出了模型中响应变量均值的各种常见估计量的形式及相应的均方误差(MSE)和偏差(Bias),通过最小化渐近均方误差和最小化渐近偏差的方法,得到了两个有效简单易用的估计量,并通过模拟对这些估计量进行了比较.
Censoring data occurs commonly in reliability and survival analysis. Among the different censoring schemes, Type-I and Type-II censoring schemes are the two most popular censoring schemes. In Type-I censoring scheme, the experiment time T is fixed, but the number of failures is random, whereas in Type-II censoring scheme, the experimental time is random but the number of failures m is fixed. Hybrid censoring scheme (Epstein,1954) is a mixture of Type-I and Type-II censoring schemes. In Type-I hybrid censoring scheme, the experimental time is T1*=min{Xm:n,T}; in Type-II hybrid censoring scheme (Childs et al.,2003), the experimental time is T2*max{Xm:n,T}, where T∈(0,∞) and1≤m≤n are prefixed. Xm:n denotes the mth failure time when n units are put on an experiment. One of the common drawbacks of the traditional Type-I,Type-II or hybrid censoring schemes is that they do not have the flexibility of allowing removal of units at points other than the terminal points of the experiment. A natural extension of this point will be a censoring scheme in which units are allowed to be withdrawn from the experiment at different time points, rather than only at the terminal point. Such censoring scheme is referred to progressive censoring scheme (Balakrishnan and Aggarwala,2000).
Progressive Type-II right censoring is a generalization of conventional Type-II censoring. Specifically, consider the situation in which n identical units, with lifetime cumulative distribution function (cdf) F(x) and probability density function (pdf) f(x), are placed on a life-testing experiment. Then, immediately following the first failure, R1surviving units are removed from the test at random; next, immediately following the second observed failure, R2surviving units are removed from the test at random, and so on; finally, at the time of the mth observed failure, all the remaining Rm=n-R1-...-Rm-1-m surviving units are removed from the test. The common drawback of the progressive Type-II censoring, similar to the conventional Type-II censoring is that it may take a lot of time to get to the mth failure time. Kundu and Joarder (2006) proposed a censoring scheme called Type-I progressive hybrid censoring scheme which is a mixture of progressive Type-II and Type-I hybrid censoring schemes. In this case, the experimental time is T1*=min{Xm;m;n,T}, where X1;m;n≤X2;m;n≤...≤Xm:m:n are the failure times resulting from the progressive Type-II censored experiment and often referred as progressive Type-II censored order statistics. In this paper, we make the statistical inference based on progressive Type-II and Type-I progressive hybrid censoring schemes. In the progressive Type-II censoring scheme, we studied the scale parameter estimation of Weibull distribution under entropy loss function; in the Type-I progressive hybrid censoring scheme, we studied the exact inference of the parameters of two-parameter Exponential distribution.
The log-normal distribution as a kind of lifetime distribution has been used quite efficiently in analyzing positively skewed data (Johnson et al.,1995). Log-normality is widely found in studies in many fields such as biology (Koch,1966), medicien (Shen et al.,2006), insurance (Doray,1996), geology (Cressie,2006), hydrology (Gilliom and Helsel,1986), environmentalology (Holland et al.,2000), and so on. Researchers discover that linear models are often fitted to the logarithmic transformed response variables very well, and these are the log-normal linear models. However, in many practical cases, like in the investigations in biology or econometrics, or some others, as the samples are time series or spatial data and they have the influence of continuity of time or space for the explanatory variable, the errors of log-normal linear models are serial correlated and the independent assumption can not be satisfied. At this moment, if people continue to use the original models or assumptions, there will lead to larger bias, even wrong inference. Because first-order correlation is the most common serial correlation in practical investigation, we propose the log-normal linear model whose errors are first-order correlated, and derive the efficient mean estimation of the response variable in original scale. The main results of this paper are as follows:
First of all, we studied the scale parameter estimation of Weibull distribution under entropy loss function based on progressive Type-Ⅱ censoring scheme. Suppose the lifetime of the units Y-Weibull(a, λ), where a is shape parameter, λ is scale parameter. Let Y1:m:n,..., Ym:m:n be m failure times, then Y=(Y1:m:n,...,Ym:m:n) is progressive Type-Ⅱ censored order statistics. When the shape parameter a is fixed, in the progressive Type-Ⅱ censoring scheme, under the entropy loss function the minimum risk equivariant estimator of the scale parameter λ was obtained:
Theorem1Let Z=(Z1,...,Zm)1, where Zi=Yi:m:n/Ym:m:n(i=1,2,...,m); and under the entropy loss function, there exists a finite risk equivariant estimator σ0(Y) of λ. Then, the MRE of λ is σ*(Y)=σ0(Y)[E1(σ0(Y)[Z)]-1, and it is unique almost everywhere.
Let Ta(Y)=Σi=1m Yi:m:n a(1+Ri), and take σ0(Y)=1/(Ta(Y)). Because Ta(Y) and
Z are independent,the exact form of δ*(Y)is
Theorem2δ*(Y) is the unbiased and consislenl eslimalor of λ. We also investigated the admissibility of a class of estimators of the form [cTα(Y)+d]-1(c,d∈R and c, d can not be0at the same time),and two theorems are given:
Theorem3Let c*=1/(m-1), m>1, then (1) when c=0,d>0, the estimators [cTα(Y)+d]1are admissible;(2) when00,the eslimalors [cTα(Y)+d]-1are admissible;(3) when c=c*, d>0, the estimators [cTα(Y)+d]1are admissible;(4) when c=C*, d=0，the eslimalor [cTα(Y)+d]-1are admissible.
Theorem4Under the assumptions of theorem3, As long as one of the following conditions are established, the estimators [cTα(Y)+d]1are not admissible.(1) c<0or d<0;(2)0c*and d>0. When the shape parameter α is unknown,we propose the iterative algorithm and iterative formnula of the maximum likelihood estiimator of α: As the maximum likelihood estimator of α has no explicit forms,we propose the ap-proximate maximum likelihood estimator of α:
where Furthermore,we suggest to use α AML to replace the fixed value of α in order to get the estimator of λ. In this paper,we also compare the performance of each estimator by simulations.
Secondly, we studied the exact inference for the two-parameter exponential distri-bution under Type-I progressive hybrid censoring scheme. Suppose the lifetime of the units X~Exp(μ,θ),where μ is the location parameter, θ is the scale parameter. Let X1:m:n,…,Xm:m:n be m failure times, D the number of failures up to time T. In Type-I progressive hybrid censoring scheme,we obtain the maximum likelihood estimators of μ and θ Furthermore,when m≥2and D>0,the joint moment generating function of ((μ μ)/θ,θ/θ) and the marginal density functions of (μ-μ)/θ and θ/θ are
Theorem5The moment generating function of ((μ-μ)/θ,θ/θ)at (ω1,ω2)is given by
+(1-(ω1)/n)-1(1-(ω2)/m)-(m-1)(1-qn-ω1)+c1(n,m)(1-ω2/m) where, c1(n,j)=multiply from i=1to∞sum from n=1to∞(Rk+1), q=e-(T-μ)/θ,qj=e-(1-(ω2)/σ)((T-μ)/θ),j=1,2...,m. Let
Theorem6The probabilit density function of (μ-μ)/θ is where g (x;a,p)=(ap/T(p))e-axxp-1, x>0; a,p>0is the gamma density function. Let Tj,i=(Rj-i+1)/j ((T-μ)/θ), j=1,2,...,m-1, i=0,1,...,j-1,
Lemma1(Childs et al,2011) Let G (a,p) be a gamma random variable with pdf9(x; a,p). Let U and V be independent random variable such that U~G(a,p) and V~G(β,1). Then, the mgf and pdf of U-V are given by and where pk=(β/(α+β))(α/(α+β))κ.
Theorem7The pdf of θ/θ is given by Based on the above,we obtained the confidence bounds of μ, θ and the pth quantile quantile for the two-paraimeter exponential distribution, and compared the results by simulations.
Finally, we proposed the log-normal linear modelswith first-order correlated er-rors, and obtained the efficient estimation of the mean. Let Ζ=(Ζ1,…,Ζ,n)T be the response vector,xi=(1,xi1,…,xip)T be the covariate vector for observation i. A
log-normal linear model with first-order correlated errors assumes that Y=Δlog(Z)=Xβ+ε,(9)
where x=(x1,...,xn)T,β=(β0,β1...,βp)T, ε=(ε1,...,εn)T and εi:pηi-1+ηi,ηi~i.i.dN(0,σ2),i=1,...,n, p is a constant.
Under this assumption, we can casily know that εi~N(0,σ2(1+p2)), ε~N(0,σ2∑), Y~N(Xβ,σ2∑),
where
For a new set of covariate values x0, the conditional mean is
We mainly consider the estimation of μ(x0). When the coefficient ρis known, We have the ML and GLS estimators of μ(x0) and where, β=(XT∑-1X)-1XT∑-1Y, RSS=(Y-Xβ)T∑-1(Y-Xβ),
The MSE and bias of μML(x0) are For the MSE and bias of μGLS(x0), one can replace n with m.
Because β and σR2are the complete and sufficient statistics of βand σ2, we derived the UMVUE of μ(x0) and
where0F1(α;z) is the Hypergeometric function. The EV estimator of μ(x0) is and In the class of estimators:
we propose two new estimators, one minimizes the MSE approximately and is defined as the other minimizes the bias considerably and is defined as
When ρ is unknown, we use the moment estimation method to propose ρ to be the estimator of ρ. And then, we replace the known ρ above with ρ in order to obtain the estimator of μ(x0).
Censoring data occurs commonly in reliability and survival analysis. Among the different censoring schemes, Type-I and Type-II censoring schemes are the two most popular censoring schemes. In Type-I censoring scheme, the experiment time T is fixed, but the number of failures is random, whereas in Type-II censoring scheme, the experimental time is random but the number of failures m is fixed. Hybrid censoring scheme (Epstein,1954) is a mixture of Type-I and Type-II censoring schemes. In Type-I hybrid censoring scheme, the experimental time is T1*=min{Xm:n,T}; in Type-II hybrid censoring scheme (Childs et al.,2003), the experimental time is T2*max{Xm:n,T}, where T∈(0,∞) and1≤m≤n are prefixed. Xm:n denotes the mth failure time when n units are put on an experiment. One of the common drawbacks of the traditional Type-I,Type-II or hybrid censoring schemes is that they do not have the flexibility of allowing removal of units at points other than the terminal points of the experiment. A natural extension of this point will be a censoring scheme in which units are allowed to be withdrawn from the experiment at different time points, rather than only at the terminal point. Such censoring scheme is referred to progressive censoring scheme (Balakrishnan and Aggarwala,2000).
Progressive Type-II right censoring is a generalization of conventional Type-II censoring. Specifically, consider the situation in which n identical units, with lifetime cumulative distribution function (cdf) F(x) and probability density function (pdf) f(x), are placed on a life-testing experiment. Then, immediately following the first failure, R1surviving units are removed from the test at random; next, immediately following the second observed failure, R2surviving units are removed from the test at random, and so on; finally, at the time of the mth observed failure, all the remaining Rm=n-R1-...-Rm-1-m surviving units are removed from the test. The common drawback of the progressive Type-II censoring, similar to the conventional Type-II censoring is that it may take a lot of time to get to the mth failure time. Kundu and Joarder (2006) proposed a censoring scheme called Type-I progressive hybrid censoring scheme which is a mixture of progressive Type-II and Type-I hybrid censoring schemes. In this case, the experimental time is T1*=min{Xm;m;n,T}, where X1;m;n≤X2;m;n≤...≤Xm:m:n are the failure times resulting from the progressive Type-II censored experiment and often referred as progressive Type-II censored order statistics. In this paper, we make the statistical inference based on progressive Type-II and Type-I progressive hybrid censoring schemes. In the progressive Type-II censoring scheme, we studied the scale parameter estimation of Weibull distribution under entropy loss function; in the Type-I progressive hybrid censoring scheme, we studied the exact inference of the parameters of two-parameter Exponential distribution.
The log-normal distribution as a kind of lifetime distribution has been used quite efficiently in analyzing positively skewed data (Johnson et al.,1995). Log-normality is widely found in studies in many fields such as biology (Koch,1966), medicien (Shen et al.,2006), insurance (Doray,1996), geology (Cressie,2006), hydrology (Gilliom and Helsel,1986), environmentalology (Holland et al.,2000), and so on. Researchers discover that linear models are often fitted to the logarithmic transformed response variables very well, and these are the log-normal linear models. However, in many practical cases, like in the investigations in biology or econometrics, or some others, as the samples are time series or spatial data and they have the influence of continuity of time or space for the explanatory variable, the errors of log-normal linear models are serial correlated and the independent assumption can not be satisfied. At this moment, if people continue to use the original models or assumptions, there will lead to larger bias, even wrong inference. Because first-order correlation is the most common serial correlation in practical investigation, we propose the log-normal linear model whose errors are first-order correlated, and derive the efficient mean estimation of the response variable in original scale. The main results of this paper are as follows:
First of all, we studied the scale parameter estimation of Weibull distribution under entropy loss function based on progressive Type-Ⅱ censoring scheme. Suppose the lifetime of the units Y-Weibull(a, λ), where a is shape parameter, λ is scale parameter. Let Y1:m:n,..., Ym:m:n be m failure times, then Y=(Y1:m:n,...,Ym:m:n) is progressive Type-Ⅱ censored order statistics. When the shape parameter a is fixed, in the progressive Type-Ⅱ censoring scheme, under the entropy loss function the minimum risk equivariant estimator of the scale parameter λ was obtained:
Theorem1Let Z=(Z1,...,Zm)1, where Zi=Yi:m:n/Ym:m:n(i=1,2,...,m); and under the entropy loss function, there exists a finite risk equivariant estimator σ0(Y) of λ. Then, the MRE of λ is σ*(Y)=σ0(Y)[E1(σ0(Y)[Z)]-1, and it is unique almost everywhere.
Let Ta(Y)=Σi=1m Yi:m:n a(1+Ri), and take σ0(Y)=1/(Ta(Y)). Because Ta(Y) and
Z are independent,the exact form of δ*(Y)is
Theorem2δ*(Y) is the unbiased and consislenl eslimalor of λ. We also investigated the admissibility of a class of estimators of the form [cTα(Y)+d]-1(c,d∈R and c, d can not be0at the same time),and two theorems are given:
Theorem3Let c*=1/(m-1), m>1, then (1) when c=0,d>0, the estimators [cTα(Y)+d]1are admissible;(2) when0
Theorem4Under the assumptions of theorem3, As long as one of the following conditions are established, the estimators [cTα(Y)+d]1are not admissible.(1) c<0or d<0;(2)0
where Furthermore,we suggest to use α AML to replace the fixed value of α in order to get the estimator of λ. In this paper,we also compare the performance of each estimator by simulations.
Secondly, we studied the exact inference for the two-parameter exponential distri-bution under Type-I progressive hybrid censoring scheme. Suppose the lifetime of the units X~Exp(μ,θ),where μ is the location parameter, θ is the scale parameter. Let X1:m:n,…,Xm:m:n be m failure times, D the number of failures up to time T. In Type-I progressive hybrid censoring scheme,we obtain the maximum likelihood estimators of μ and θ Furthermore,when m≥2and D>0,the joint moment generating function of ((μ μ)/θ,θ/θ) and the marginal density functions of (μ-μ)/θ and θ/θ are
Theorem5The moment generating function of ((μ-μ)/θ,θ/θ)at (ω1,ω2)is given by
+(1-(ω1)/n)-1(1-(ω2)/m)-(m-1)(1-qn-ω1)+c1(n,m)(1-ω2/m) where, c1(n,j)=multiply from i=1to∞sum from n=1to∞(Rk+1), q=e-(T-μ)/θ,qj=e-(1-(ω2)/σ)((T-μ)/θ),j=1,2...,m. Let
Theorem6The probabilit density function of (μ-μ)/θ is where g (x;a,p)=(ap/T(p))e-axxp-1, x>0; a,p>0is the gamma density function. Let Tj,i=(Rj-i+1)/j ((T-μ)/θ), j=1,2,...,m-1, i=0,1,...,j-1,
Lemma1(Childs et al,2011) Let G (a,p) be a gamma random variable with pdf9(x; a,p). Let U and V be independent random variable such that U~G(a,p) and V~G(β,1). Then, the mgf and pdf of U-V are given by and where pk=(β/(α+β))(α/(α+β))κ.
Theorem7The pdf of θ/θ is given by Based on the above,we obtained the confidence bounds of μ, θ and the pth quantile quantile for the two-paraimeter exponential distribution, and compared the results by simulations.
Finally, we proposed the log-normal linear modelswith first-order correlated er-rors, and obtained the efficient estimation of the mean. Let Ζ=(Ζ1,…,Ζ,n)T be the response vector,xi=(1,xi1,…,xip)T be the covariate vector for observation i. A
log-normal linear model with first-order correlated errors assumes that Y=Δlog(Z)=Xβ+ε,(9)
where x=(x1,...,xn)T,β=(β0,β1...,βp)T, ε=(ε1,...,εn)T and εi:pηi-1+ηi,ηi~i.i.dN(0,σ2),i=1,...,n, p is a constant.
Under this assumption, we can casily know that εi~N(0,σ2(1+p2)), ε~N(0,σ2∑), Y~N(Xβ,σ2∑),
where
For a new set of covariate values x0, the conditional mean is
We mainly consider the estimation of μ(x0). When the coefficient ρis known, We have the ML and GLS estimators of μ(x0) and where, β=(XT∑-1X)-1XT∑-1Y, RSS=(Y-Xβ)T∑-1(Y-Xβ),
The MSE and bias of μML(x0) are For the MSE and bias of μGLS(x0), one can replace n with m.
Because β and σR2are the complete and sufficient statistics of βand σ2, we derived the UMVUE of μ(x0) and
where0F1(α;z) is the Hypergeometric function. The EV estimator of μ(x0) is and In the class of estimators:
we propose two new estimators, one minimizes the MSE approximately and is defined as the other minimizes the bias considerably and is defined as
When ρ is unknown, we use the moment estimation method to propose ρ to be the estimator of ρ. And then, we replace the known ρ above with ρ in order to obtain the estimator of μ(x0).
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