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A note on bias reduction of maximum likelihood estimates for the scalar skew t distribution

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Abstract

The skew t distribution is a flexible parametric family to fit data, because it includes parameters that let us regulate skewness and kurtosis. A problem with this distribution is that, for moderate sample sizes, the maximum likelihood estimator of the shape parameter is infinite with positive probability. In order to try to solve this problem, Sartori (2006) has proposed using a modified score function as an estimating equation for the shape parameter. In this note we prove that the resulting modified maximum likelihood estimator is always finite, considering the degrees of freedom as known and greater than or equal to 2.

Introduction

The univariate skew normal distribution was introduced by Azzalini (1985), as a family with the appealing property of strictly including the normal law, as well as a wide variety of skewed densities. A random variable Z is said to be skew normal with shape parameter λ, written ZSN(λ), if it has density function f(z;λ)=2ϕ(z)Φ(λz),zR,where ϕ and Φ denote the density function and the distribution function of a standard normal law, respectively. Later on, Azzalini and Dalla Valle (1996) defined the multivariate skew normal distribution as a multivariate extension of the skew normal distribution. The skew t distribution is a natural generalization of the skew normal distribution, which allows us to model the tails of the distribution as well as its skewness. The specific form which we will consider is the one studied by Azzalini and Capitanio (2003). Equivalent forms are studied in Branco and Dey, 2001, Branco and Dey, 2002 and Gupta (2003).

Despite of their nice properties, the skew normal distribution and the skew t distribution both present some problems with the estimation of the shape parameter. Here we concentrate on the following one: for moderate sample sizes, the maximum likelihood estimator is infinite with positive probability. This undesirable situation occurs when all observations are positive or negative. The probability that all observations be positive or negative decreases as n increases, but for moderate n this probability is nonnegligible. As noted by Sartori (2006), given n and λ, the sample size and the shape parameter, respectively, the probability of having λ^= is the same for the skew normal likelihood and the skew t likelihood, irrespectively of the degrees of freedom of the skew t, that will be considered fixed and known along this paper. In order to have an idea of such probability, Table 1 displays these probabilities for several sample sizes and several values of λ. Observe that for large values of λ the probability of λ^= is very high.

To solve this problem for the scalar case, Sartori (2006) has proposed using a modified score function as an estimating equation for the shape parameter. The resulting modified maximum likelihood estimator for the shape parameter of the skew normal distribution has been proved to be always finite. The method has also been numerically applied to the skew t-distribution, assuming the degrees of freedom fixed. In all simulated samples, a finite solution of the modified estimating equation was found. So, it seems that there is some empirical evidence that the method also works for the skew t. Nevertheless, to our knowledge (see also Section 3 in Azzalini and Genton, 2008), no theoretical proof of the finiteness of the resulting shape estimator has been provided. The aim of this paper is to prove it.

The paper is organized as follows. In Section 2 we give the density of the skew t distribution and derive an expression of the modified score function for the shape parameter. In Section 3 we prove the finiteness of the resulting shape estimator when the degrees of freedom are greater than or equal to 2. Some open related questions are listed in Section 4. Appendix A gives a sketch of some algebraic calculations and Appendix B contains a result of Soms (1976) that is used to prove the finiteness in Section 3. No empirical results are included because the paper by Sartori (2006) is very complete in this respect.

Section snippets

The skew t distribution and the modified score function

A continuous random variable Y is said to have a skew t distribution, written YST(μ,σ,λ,ν), if it has density function f(y;μ,σ,λ,ν)=21σt(z;ν)T(λu;ν+1),yR,where z=yμσ,u=zν+1z2+ν1/2.t(x;ν) and T(x;ν) are the density function and the distribution function of a t distribution with ν degrees of freedom, respectively, t(x;ν)=1πΓ{(ν+1)/2}Γ(ν/2)1+x2ν(ν+1)/2,xR.Along this paper we will consider that μ=0, σ=1 and that the degrees of freedom ν are fixed. Let y1,y2,,yn be a random sample from YST(0,1

Finiteness of the solution

Suppose that all observations are positive (the case with all observations negative is symmetric). Then the likelihood equation S(λ)=0, has no solution because the score function is always positive. Moreover, S(λ)0+, as λ+. From now on, all limits are taken when λ+. It is easy to see that S(λ)=O(λ(ν+2)). Next we prove that M(λ)=Ω(λ1), ν2, where f=Ω(g){f=O(g) and g=O(f)}. This implies that S(λ)0 because M(λ) is negative for positive values of λ. Finally, since S(0)=S(0)>0, there

Further research

Here we have proven the finiteness of every solution of the modified score function S(λ)=0, for the scalar case assuming the degrees of freedom known and greater than or equal to 2. The condition ν2 is needed to bound some integrals in our proof. Thus, two questions remain unsolved: the study of the scalar case for 0<ν<2 and the study of the multivariate case (for both the skew normal distribution and the skew t distribution). The uniqueness of the solution in each case is also an open

Acknowledgments

The authors thank the anonymous referees for their careful reading, constructive comments and suggestions which helped to improve the presentation. The research in this paper has been partially supported by Grant MTM2008-00018 (Ministerio de Ciencia e Innovación, Spain).

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