Bayesian inference for the location parameter of a Student-t density Jean-Fran¸cois Angers∗† CRM-2642 February 2000 ∗D´ep. de math´ematiques et de statistique; Universit´ede Montr´eal; C.P. 6128, Succ. ”Centre-ville”; Montr´eal, Qu´ebec, H3C 3J7;[email protected] †This research has been partially funded by NSERC, Canada Abstract Student-t densities play an important role in Bayesian statistics. For example, suppose that an estimator of the mean of a normal population with unknown variance is desired, then the marginal posterior density of the mean is often a Student-t density. In this paper, estimation of the location parameter of a Student-t density is considered when its prior is also a Student-t density. It is shown that the posterior mean and variance can be written as the ratio of two finite sums when the number of the degrees of freedom of both the likelihood function and the prior are odd. When one of them (or both) is even, approximations for the posterior mean and variance are given. The behavior of the posterior mean is also investigated in presence of outlying observations. When robustness is achieved, second order approximations of the estimator and its posterior expected loss are given. Mathematics Subject Classification: 62C10, 62F15, 62F35. Keyworks:Robust estimator, Fourier transform, Convolution of Student-t densities. 1 Introduction Heavy tail priors play an important role in Bayesian statistics. They can be viewed as an alternative to noninformative priors since they lead to estimators which are insensitive to misspecification of the prior parameters. However, they allow the use of prior information when it is available. Because of its heavier tails, the Student-t density is a “robust” alternative to the normal density when large observations are expected. (Here, robustness means that the prior information is ignored when it conflicts with the information contained in the data.) This density is also encountered when the data come from a normal population with unknown variance. In this paper, the problem of estimating the location of a Student-t density, under squared- error loss, is considered. To obtain an estimator which will ignore the prior information when it conflicts with the likelihood information, the prior density proposed in the paper is another Student- t density. However, it has fewer degrees of freedom than the likelihood. Consequently, the prior tails are heavier that those of the likelihood, resulting in an estimator which is insensitive to prior misspecification (cf. O’Hagan, 1979). This problem has been previously studied by Fan and Berger (1990), Angers and Berger (1991), Angers (1992) and Fan and Berger (1992). However, some conditions have to be imposed on the degrees of freedom in order to obtain an analytic expression for the estimator. A statistical motivation of the importance of this problem can be found in Fan and Berger (1990). In Section 2 of this paper, it is assumed that the degrees of freedom of both the prior and the likelihood are odd. Using Angers (1996a), an alternative form (which is sometimes easier to use (cf. Angers, 1996b)) for the estimator is also proposed in this section. In Section 3, it is shown that the effect of large observation of the proposed estimator is limited. In the last section, using Saleh (1994), an approximation is considered for the case where the number of degrees of freedom of the likelihood function is even. 2 Development of the estimator—odd degrees of freedom Let us consider the following model: X | θ ∼ T2k+1(θ, σ), θ ∼ T2κ+1(µ, τ), where σ, µ and τ are known and both k and κ are in N. The notation Tm(η, ν) denotes the Student-t density with m degrees of freedom and location and scale parameters respectively given by η and ν, that is Γ([m + 1]/2) (x − η)2 !−[m+1]/2 f (x | η, ν) = √ 1 + . (1) m ν mπΓ(m/2) mν2 Since the hyperparameters are assumed to be known, we suppose, without loss of generality, that µ = 0 and σ = 1. The general case can be obtained by replacing X by σX + µ and θ by θ + µ in Theorems 2 and 3. In Angers (1996a), the following theorem is proved. 1 Theorem 1. If X | θ ∼ g(x − θ) and if θ | τ ∼ τ −1h(θ/τ), then m(x) = marginal density of X evaluated at x = I0(x), (2) θb(x) = posterior expected mean of θ I (x) = x − i 1 , (3) I0(x) ρ(x) = posterior variance of θ I (x)!2 I (x) = 1 − 2 , (4) I0(x) I0(x) √ −1 (j) (j) where i = −1, Ij(x) = F {hb(τs)gb (s); x}, hb(s) denotes the Fourier transform of h(x), gb (s) the jth derivative of the Fourier transform of g(x) and F −1{fb; x} represents the inverse Fourier transform of fb evaluated at x. Applications of this theorem to several models can be found in Leblanc and Angers (1995) and Angers (1996a, 1996b). In order to compute equations (2),(3) and (4), the Fourier transform of a Student-t density is needed. It is given, along with its first two derivatives, in the following proposition. Since the proof is mostly technical, it is omitted. Proposition 1. If X ∼ Tm(0, σ), then √ ( mσ|s|)m/2 √ fb (s) = K ( mσ|s|), m 2[m−2]/2Γ(m/2) m/2 √ ( mσ|s|)m/2 √ fb0 (s) = −σ sign(s) K ( mσ|s|), m 2[m−2]/2Γ(m/2) [m−2]/2 √ m(m − 1)σ2( mσ|s|)m−2 √ fb00 (s) = mσ2fb(s) − K ( mσ|s|), m 2[m−2]/2Γ(m/2) [m−2]/2 where Km/2(s) denotes the modified Bessel function of the second kind of order m/2. Note that if m = 2k + 1 where k ∈ N, then, using Gradshteyn and Ryzhik (1980, equation (8.468)), we have that √ √ Km/2( mσ|s|) = Kk+1/2( mσ|s|) √ π 1 k (2k − p)! √ = √ X (2σ 2k + 1|s|)p. (5) k+1/2 k (2σ 2k + 1) |s| p=0 p!(k − p)! To obtain the marginal density of x, the posterior mean and variance of θ, we need to compute −1 (j) F {fb2κ+1(τs)fb2k+1(s); x}, for j = 0, 1 and 2. Hence, the following two integrals need to be evaluated: Z ∞ √ k+κ−l+1 Ak,l(x) = cos(|x|s)s Kk−l+1/2( 2k + 1s) 0 √ × Kκ+1/2( 2κ + 1τs)ds, (6) 2 for l = 0 and 1 and Z ∞ √ k+κ+1 Bk(x) = sin(|x|s)s Kk−1/2( 2k + 1s) 0 √ × Kκ+1/2( 2κ + 1τs)ds. (7) Using Angers (1997), we can also show that Theorem 2. (2k + 1)[2k+1]/4(2κ + 1)[2κ+1]/4τ 2κ+1 m (x) = A (x), 2k+1 2k+κπΓ(k + 1/2)Γ(κ + 1/2) k,0 √ Bk(x) θb2k+1(x) = x − 2k + 1 sign(x) , Ak,0(x) !2 2k Ak,1(x) Bk(x) ρ2k+1(x) = (2k + 1) √ − 1 + . 2k + 1 Ak,0(x) Ak,0(x) In order to compute equations (6) and (7), we need the following lemma. This lemma can be proven using Gradshteyn and Ryzhik (1980, equations (3.944.5) and (3.944.6)). Lemma 1. Z ∞ Γ(a + 1) sa cos(xs) e−bs ds = cos([a + 1] tan−1(x/b)), 0 (b2 + x2)[a+1]/2 Z ∞ Γ(a + 1) sa sin(xs) e−bs ds = sin([a + 1] tan−1(x/b)). 0 (b2 + x2)[a+1]/2 Using Lemma 1 and equation (5), the functions Ak,l(x) and Bk(x) can be easily evaluated and they are given in the following theorem. Theorem 3. π 1 A (x) = k,l 2k−l+κ+1 (2k + 1)[2(k−l)+1]/4([2κ + 1]τ 2)[2κ+1]/4 k−l κ X X (2[k − l] − p)! (2κ − q)! p=0 q=0 (k − l − p)! (κ − q)! p + q! 2p+q(2k + 1)p/2([2κ + 1]τ 2)q/2 × √ √ (8) q ([ 2k + 1 + τ 2κ + 1]2 + x2)[p+q+1]/2 " |x| #! × cos [p + q + 1] tan−1 √ √ , 2k + 1 + τ 2κ + 1 π 1 B (x) = k 2k+κ (2k + 1)[2k−1]/4([2κ + 1]τ 2)[2κ+1]/4 k−1 κ X X (2[k − 1] − p)! (2κ − q)! p=0 q=0 (k − 1 − p)! (κ − q)! p + q! 2p+q(2k + 1)p/2([2κ + 1]τ 2)q/2 × √ √ (9) q ([ 2k + 1 + τ 2κ + 1]2 + x2)[p+q+2]/2 " |x| #! × sin [p + q + 2] tan−1 √ √ . 2k + 1 + τ 2κ + 1 3 Using Theorems 2 and 3, the posterior expected mean and the posterior variance can be com- puted using only a ratio of two finite sums. In Section 4, the case where the likelihood function is a Student-t density with an even number of degrees of freedom is considered. In this situation, the posterior quantities cannot be written using finite sums, although they can be expressed as the ratio of two infinite series (cf. Angers, 1997). However, using an approximation for the Student-t density (cf. Saleh94), θb2k and ρ2k(x) can be approximated accurately. Before doing so, we first discuss two limit cases, that is, when |x| is large and when τ → ∞. 3 Special cases The main advantage of using a heavy-tails prior is that the resulting Bayes estimator, under the squared-error loss, is insensitive to the choice of prior when there is a conflict between the prior and the likelihood information.