The linear Bayes method was originally proposed by Hartigan (1969), which suggests that in Bayesian statistics one can replace a completely specified prior distribution by an assumption with just a few moments of the distribution. It has been subsequently discussed by Rao (1973) from linear optimization viewpoint. Lamotte (1978) later develops a class of linear estimators, called Bayes linear estimators, by searching, among all linear estimators that have least average total mean squared error. Goldstein (1983) considers the problem of modifying the linear Bayes estimator for the mean of a distribution of unknown form using a sample variance estimate. Heiligers (1993) studies the relationship between linear Bayes estimation and minimax estimation in linear models with partial parameter restrictions. Hoffmann (1996) proposes a well-described subclass of Bayes linear estimators for the unknown parameter vector in the linear regression model with ellipsoidal parameter constraints and obtains a necessary and sufficient condition to ensure that the considered Bayes linear estimators improves the least squared estimator over the whole ellipsoid regardless of the selected generalized risk function. In the framework of empirical Bayes, Samaniego and Vestrup (1999) and Pensky and Ni (2000) respectively construct linear empirical Bayes estimators and establish their superiorities over standard and traditional estimators. In application fields, Busby et al. (2005) proposes the application of Bayes linear methodology to uncertainty evaluation in reservoir forecasting. Zhang and Wei (2005) also drive the unique Bayes linear unbiased estimator of estimable functions for the singular linear model. Wei and Zhang (2007) employs linear Bayes procedure to define Bayes linear minimum risk estimation in a linear model and discusses its superiorities. Recently, Zhang et al. (2011, 2012) extend the research on a linear Bayes estimator to the partitioned linear model and multivariate linear models, respectively.

In this paper, along the same line as in Wang and Singh (2014), we use the normal linear model as an example to demonstrate how to apply a linear Bayes method to simultaneously estimate all the parameters involved in the model and elaborate advantages and potential disadvantages.

Let W be a known p-dimension subspace of Rⁿ. Suppose that we observe the random vector

This model is called the normal linear model as defined by Arnold (1980) and adopted by many other authors as well. Let X be a basis matrix for W. Then X is an n × p matrix of rank p, and there exists a unique β ∈ R^p such that μ = Xβ. Hence, an equivalent version of the normal linear model can be presented, and we observe the random vector

Define β̂ = (X′X)⁻¹X′Y and σ̂² = (||Y||² – ||Xβ̂ ||²)/(n – p) and note that the fact that (β̂,σ̂²) is a complete sufficient statistic for the above linear model. Hence, the classical estimators for the parameters β and σ² are β̂ and σ̂², which are the uniformly minimum variance unbiased estimator (UMVUE) in the sense of minimizing mean squared error.

From the Bayesian viewpoint, note that in most cases past experience about the parameters β and σ² are often available. Let f₀(β,σ²) be the joint prior of β and σ² and the loss function be

where D is a positive definite matrix and θ̂ denotes the estimate of the vector θ = (β′, σ²)′. Then, by virtute of the Bayes theorem, the usual Bayes estimators (UBE) for β and σ², say β̂_UB and σ^UB2, can be calculated by

where g(β,σ²|y) denotes the conditional joint posterior density of β and σ² given Y. However, it is difficult to handle complicated or non-standard integrations. Normally, in these cases approximate Bayes estimators are suggested such as Lindley’s approximation and Tierney and Kadane’s approximation, see Lindley (1980) and Tierney and Kadane (1986) for details. Simulation-based methods such as the Gibbs sampling procedure and Metropolis method also have emerged in the past twenty years, see Martinez and Martinez (2007). Traditional Bayes estimators (UBE) are somewhat complicated and inconvenient to use in these situations.

In the following, enlightened by Rao (1973), we employ the linear Bayes method to propose a linear Bayes estimator (LBE) for the parameters β and σ² simultaneously as well as investigate superiorities. We also extend our discussions on the application of linear Bayes method to some other useful distributions.

The survey is organized as follows: In Section 2 we define the LBE for the parameter vector θ = (β′, σ²)′ and establish it superiorities over the classic UMVUE and MLE. Numerical comparisons between the LBE and the usual Bayes estimator (UBE) are presented in Section 3. Extended discussions and remarks are made in Section 4.

Throughout this paper, for two nonnegative definite matrices A₁ and A₂ of the same size, we say A₁ ≥ A₂ if and only if A₁ – A₂ is a nonnegative definite matrix.

2.1. The proposed LBE

Denote θ = (β′, σ²)′. In what follows we assume that the prior G(θ) belongs to the distribution family:

Put T = (β̂′, σ̂²)′ and define the linear Bayes estimator (LBE) of θ, say θ̂_LB, be of the form θ̃ = BT + b satisfying

where align and b are (p+1)×(p+1) and (p+1)×1 undetermined matrices respectively, E_(Y,θ) denotes the expectation with respect to the joint distribution of Y and θ and the loss function is given by (1.3).

Thus, we have the following conclusion.

Note that

where we use θ̂_U to denote the UMVUE of θ = (β′, σ²)′.

Theorem 2

Let θ̂_LBand θ̂_Ube given by (2.2) and (2.7) respectively. If n ≥ p+1, then θ̂_LBis superior to θ̂_Uin terms of MSEM criterion, i.e. MSEM(θ̂_LB) ≤ MSEM(θ̂_U).

Theorem 3

Let θ̂_LBand θ̂_MLbe given by (2.2) and (2.12) respectively. If n ≥ p + 1, then θ̂_LBis superior to θ̂_MLin terms of MSEM criterion, i.e. MSEM(θ̂_LB) ≤ MSEM(θ̂_ML).

To illustrate Theorem 2 and Theorem 3 we investigate the case of two-dimensional normal linear model, i.e.

where we assume that (β₀, β₁)′ ~ N((1, 2)′, Cov(β₀, β₁)) with Cov(β₀, β₁) having three alternative values and σ² ~ U(a, b) with three different pairs a and b. We also assume that (β₀, β₁) and σ² are uncorrelated, i.e. Cov(β₀, β₁, σ²) = diag(Cov(β₀, β₁), Var(σ²)), and x = (−4, −3, −2, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16).

Define the percentages of improvement of θ̂_LB over θ̂_U and θ̂_ML, respectively, by

For different sample size n (= 5, 10, 20), the corresponding computation results for IP(θ̂_U) and IP(θ̂_ML) under the three different priors are presented in Table 1, where tr(Cov(β₀, β₁, σ²)) is used as an index of the variation of the prior information.

As stated in Theorems 2 and 3, since the above priors belong to the family (2.1), both MSEM(θ̂_U)– MSEM(θ̂_LB) and MSEM(θ̂_ML) – MSEM(θ̂_LB) are always nonnegative definite. From Table 1, firstly we can see that when the sample size n is fixed, as expected, both IP(θ̂_U) and IP(θ̂_ML) decrease as the variation of the prior increases (i.e, tr(Cov(β₀, β₁, σ²)) tends to be larger); secondly, for the same prior information, it is only natural that as the sample size n grows, which means that the sample information gets more, both IP(θ̂_U) and IP(θ̂_ML) decrease; finally, it seems that IP(θ̂_U) is larger than IP(θ̂_ML), the reason may be due to MSEM(θ̂_U) ≥ MSEM(θ̂_ML) for our case.

For the model (1.2), note that under the loss L(θ̂, θ) and the prior G(θ), the usual Bayes estimator (UBE) of θ, say θ̂_UB, would be equal to E(θ|Y). In this Section, for given priors G(θ), we present some numerical comparisons between the LBE θ̂_LB and the UBE θ̂_UB, the latter is calculated by employing an MCMC sampling method.

Suppose p = 2 and let us consider the normal linear model

Denote β = (γ₀, γ₁)′. We assume that σ² follows an inverse-Gamma distribution with density

and given σ² the conditional distribution of β is N₂(β̃₀, σ²∑₀).

Note that the posterior density of (β,σ²) given Y is

where X′=(11⋯1x1x2⋯xn) denotes the design matrix.

However, it is almost impossible to calculate θ^UB=(β^UB′,σ^UB2)′ analytically, so we have to obtain it numerically.

Note that the posterior conditional densities of β given σ² and σ² given β are respectively proportional to

where β˜=(Σ0-1+X′X)-1(Σ0-1β˜0+X′Xβ^), Σ˜=σ2(Σ0-1+X′X)-1 and c1=2λ0+(n-2)σ^2+(β-β˜0)′Σ0-1(β-β˜0)+(β-β^)′X′X(β-β^).

The Gibbs sampler was originally developed by Geman and Geman (1984) as applied to image processing and the analysis of Gibbs distributions on a lattice. It is brought into mainstream statistics through the articles of Gelfand and Smith (1990) and Gelfand et al. (1990). The Gibbs sampler can also be shown to be a special case of the Metropolis-Hastings algorithm, see Gilks et al. (1996) and Robert and Casella (1999). In describing the Gibbs sampler, we follow the treatment in Casella and George (1992).

• Step 1. Choose the initial values of β and σ² and denote the values of β and σ² at the j^th step by β_j and σj2, respectively.

• Step 2. Generate β_j+1 and σj+12 from f1(β∣σj2,y) and f₂(σ²|β_j, y), respectively.

• Step 3. Repeat Step 2 for N times.

• Step 4. Calculate the Bayes estimator of l(β,σ²) by 1/(N-m0)∑j=m0+1Nl(βj,σj2), where l(β,σ²) denotes any a function of β and σ² and m₀ is the burn-in period.

Note that under the above priors, it is readily seen that Eθ=(Eβ′,Eσ2)′=(β˜0′,t/[λ0-2])′ and Cov(θ) = diag(t/(λ₀ – 2) · ∑₀, t²/{(λ₀ – 2)²(λ₀ – 3)}) since Cov(β) = ECov(β|σ²) = ∑₀Eσ² and Corr(β,σ²) = 0.

In the following table we first calculate the values of θ̂_LB and the corresponding numerical results of θ̂_UB for different prior parameters and then present ‖θ^LB-θ^UB‖=‖β^LB-β^UB‖2+(σ^LB2-σ^UB2)2, which is defined as an index of degree of approximation between θ̂_LB and θ̂_UB.

The above numerical comparisons indicate two trends, one is that for the same prior, ||θ̂_LB – θ̂_UB|| tends to be smaller as sample size gets larger, the other is that given sample size, ||θ̂_LB –θ̂_UB|| increases as the prior variance becomes larger. In the process of simulation, we find that the value of ||θ̂_LB–θ̂_UB|| is affected by the value of (σ^LB2-σUB2)2; however, the value of ||β̂_LB – β̂_UB||² is always small, which means the LBE β̂_LB is rather close to the UBE β̂_UB and there could be a certain difference between the LBE σ^LB2 and the UBEσ^UB2 for our cases.

Remark 5

Two cases are considered for the two-parameter exponential family. In case (I) we assume that the parameters μ and λ have independent prior distributions, where μ follows an exponential distribution and λ has an inverted Gamma prior. In case (II), we suppose that, given λ, the conditional prior of μ is an inverted Gamma density and λ follows an inverted Gamma prior. For the above two cases, numerical simulations show that ||θ̂_LB –θ̂_UB||s are small, which means that as a linear approximation of θ̂_UB, θ̂_LB works better.

Remark 6

In the case of the uniform distribution U(θ₁, θ₂), numerical computations show that θ̂_LB works very good for both independent prior and non-independent prior. For example, for the single parameter uniform distribution U(0, θ₂), we assume the prior π(θ₂) has finite second-order moment and mimic the above discussions, then the LBE for the parameter θ₂ is θ̂_2,LB = a₀X_(n) + b₀ with

Specifically, let π(θ2)=t2t1θ2-t1-1 exp(-t2/θ2)/Γ(t1) and together with f (x_(n)|θ₂) and the squared loss, we know that the UBE θ̂_2,UB is

where we utilize the relationship between the inverse Gamma and the χ² distribution (Mao and Tang, 2012). Say, let n = 5, x_(n) = 2 and t₁ = 3 and t₂ = 8, simple computations show that a₀ = 1.1351, b₀ = 0.2163 and P(χ²(14) ≤ 8) = 0.1107 and P(χ²(16) ≤ 8) = 0.0511. Hence, we have θ̂_2,LB = 2.4865 and θ̂_2,UB = 2.4758, which show that the LBE is very close to the UBE.

Remark 7

For the two-parameter inverse Gaussian distribution IG(α₁, α₂), similarly, numerical studies show that LBE is adequate.

Remark 8

In above simulation, it should be noted that the problem of deciding when to stop the chain is an important issue and is the topic of current research in MCMC methods. If the resulting sequence has not converged to the target distribution, then the estimators and inferences we get from it are suspect. Let γ represent the characteristic of the target distribution (mean, moments, quantiles, etc.) in which we are interested. An obvious method to monitor convergence to target distribution is to run multiple sequences of the chain and plot γ versus the iteration number.

This paper uses the normal linear model Y ~ N(Xβ,σ²I_n) as an example to investigate the application of linear Bayes method, where we employ the linear Bayes method to simultaneously estimate regression parameter β and the variance parameter σ² as well as propose a linear Bayes estimator for the parameter vector θ = (β′, σ²)′. The proposed linear Bayes estimator is shown superior to the classical estimators UMVUE and MLE, respectively, in terms of the mean squared error matrix criterion. Numerical simulations are presented to verify the validity of the linear Bayes estimator. The procedure used in this paper includes normal distribution as its special case and can be extended easily to other useful distributions (such as log-normal, inverse Gaussian distribution and two-parameter exponential family), which are frequently used parametric lifetime models in survival analysis and reliability theory. We also discuss and remark the applications of linear Bayes method to the two-parameter exponential family, uniform distribution and the inverse Gaussian distribution. Compared with the usual Bayes estimator, we find that

• The proposed linear Bayes estimator is simple and easy to calculate as well as a good approximation in many situations; the linear Bayes method works especially well for the case of uniform distribution.

• We can always define a linear Bayes estimator if there exists sufficient statistic for the parametric model; subsequently, the conclusions of Theorems 2 and 3 always hold.

However, an advantage of the usual Bayes estimator over the linear Bayes estimator is that the former allows for noninformative (improper) priors. Of note is that the linear Bayes estimator may be an inadequate approximation in some situations even for the cases of proper priors. Hence there is still scope for the linear Bayes method to be improved. For instance, for some cases a quadratic Bayes estimator would be a better alternative. However, for the case of normal linear model, one can consider to add more other statistics into the definition of T, for example, we can replace T = (β̂′,σ̂²)′ by T₁ = (β̂′, ||β̂||², σ̂²)′ or T₂ = (β̂′, σ̂², σ̂⁴)′ to redefine a new linear Bayes estimator. We note that the loss function often plays an important role in Bayesian analysis; consequently, some interesting loss functions such as the balanced loss and the linex loss can be integrated with the linear Bayes method in future studies.