On selecting a prior for the precision parameter of Dirichlet process mixture models

In hierarchical mixture models the Dirichlet process is used to specify latent patterns of heterogeneity, particularly when the distribution of latent parameters is thought to be clustered (multimodal). The parameters of a Dirichlet process include a precision parameter α$\alpha$ and a base probability measure _G0$G_0$. In problems where α$\alpha$ is unknown and must be estimated, inferences about the level of clustering can be sensitive to the choice of prior assumed for α$\alpha$. In this paper an approach is developed for computing a prior for the precision parameter α$\alpha$ that can be used in the presence or absence of prior information about the level of clustering. This approach is illustrated in an analysis of counts of stream fishes. The results of this fully Bayesian analysis are compared with an empirical Bayes analysis of the same data and with a Bayesian analysis based on an alternative commonly used prior.     

Truncated linear estimation of a bounded multivariate normal mean

We consider the problem of estimating the mean θ$\theta$ of an _Np(θ,_Ip)${\mathrm{N}}_{\mathrm{p}}(\theta ,I_p)$ distribution with squared error loss ∥δ−θ∥²$\parallel \delta -\theta {\parallel }^2$ and under the constraint ∥θ∥≤m$\parallel \theta \parallel \le m$, for some constant m>0$m>0$. Using Stein's identity to obtain unbiased estimates of risk, Karlin's sign change arguments, and conditional risk analysis, we compare the risk performance of truncated linear estimators with that of the maximum likelihood estimator _δmle${\delta }_{\mathrm{mle}}$. We obtain for fixed (m,p)$(m,p)$ sufficient conditions for dominance. An asymptotic framework is developed, where we demonstrate that the truncated linear minimax estimator dominates _δmle${\delta }_{\mathrm{mle}}$, and where we obtain simple and accurate measures of relative improvement in risk. Numerical evaluations illustrate the effectiveness of the asymptotic framework for approximating the risks for moderate or large values of p.     

Optimal designs for threshold-determining limiting dilution assays

In common with other non-linear models, the optimal design for a limiting dilution assay (LDA) depends on the value of the unknown parameter, θ$\theta$, in the model. Consequently optimal designs cannot be specified unless some assumptions are made about the possible values of θ$\theta$. If a prior distribution can be specified then a Bayesian approach can be adopted. A proper specification of the Bayesian approach requires the aim of the experiment to be described and quantified through an appropriate utility function. This paper addresses the problem of finding optimal designs for LDAs when the aim is to determine whether θ$\theta$ is above or below a specified threshold, _θ0${\theta }_0$.     

Optimally robust credible sets with a class of priors

We determine a credible set A   that is the “best” with respect to the variation of the prior distribution in a neighborhood Γ$\Gamma$ of the starting prior _π0(θ)${\pi }_0(\theta )$. Among the class of sets with credibility γ$\gamma$ under _π0${\pi }_0$, the “optimally robust” set will be the one which maximizes the minimum probability of including θ$\theta$ as the prior varies over Γ$\Gamma$. This procedure is also Γ-minimax$\Gamma \text{-minimax}$ with respect to the risk function, probability of non-inclusion. We find the optimally robust credible set for three neighborhood classes Γ$\Gamma$, the ε-contamination$\epsilon \text{-contamination}$ class, the density ratio class and the density bounded class. A consequence of this investigation is that the maximum likelihood set is seen to be an optimal credible set from a robustness perspective.     

Distribution of record statistics in a geometrically increasing population

The probability function and binomial moments of the number _Nn$N_n$ of (upper) records up to time (index) n in a geometrically increasing population are obtained in terms of the signless q-Stirling numbers of the first kind, with q   being the inverse of the proportion λ$\lambda$ of the geometric progression. Further, a strong law of large numbers and a central limit theorem for the sequence of random variables _Nn$N_n$, n=1,2,…,$n=1,2,\dots ,$ are deduced. As a corollary the probability function of the time _Tk$T_k$ of the kth record is also expressed in terms of the signless q  -Stirling numbers of the first kind. The mean of _Tk$T_k$ is obtained as a q  -series with terms of alternating sign. Finally, the probability function of the inter-record time _Wk=_Tk-_Tk-1$W_k=T_k-T_{k-1}$ is obtained as a sum of a finite number of terms of q  -numbers. The mean of _Wk$W_k$ is expressed by a q-series. As k   increases to infinity the distribution of _Wk$W_k$ converges to a geometric distribution with failure probability q. Additional properties of the q-Stirling numbers of the first kind, which facilitate the present study, are derived.     

On asymptotic properties of the parameter estimator for a type of SPDE

In this paper, we study a random field _U?(t,x)$U_{?}(t,x)$ governed by some type of stochastic partial differential equations with an unknown parameter θ$\theta$ and a small noise ?$?$. We construct an estimator of θ$\theta$ based on the continuous observation of N   Fourier coefficients of _U?(t,x)$U_{?}(t,x)$, and prove the strong convergence and asymptotic normality of the estimator when the noise ?$?$ tends to zero.     

Confidence intervals in regression utilizing prior information

We consider a linear regression model with regression parameter β=(_β1,…,_βp)$\beta =({\beta }_1,\dots ,{\beta }_p)$ and independent and identically N(0,σ²)$N(0,{\sigma }^2)$ distributed errors. Suppose that the parameter of interest is θ=a^Tβ$\theta =a^T\beta$ where a$a$ is a specified vector. Define the parameter τ=c^Tβ-t$\tau =c^T\beta -t$ where the vector c$c$ and the number t$t$ are specified and a$a$ and c$c$ are linearly independent. Also suppose that we have uncertain prior information that τ=0$\tau =0$. We present a new frequentist 1-α$1-\alpha$ confidence interval for θ$\theta$ that utilizes this prior information. We require this confidence interval to (a) have endpoints that are continuous functions of the data and (b) coincide with the standard 1-α$1-\alpha$ confidence interval when the data strongly contradict this prior information. This interval is optimal in the sense that it has minimum weighted average expected length where the largest weight is given to this expected length when τ=0$\tau =0$. This minimization leads to an interval that has the following desirable properties. This interval has expected length that (a) is relatively small when the prior information about τ$\tau$ is correct and (b) has a maximum value that is not too large. The following problem will be used to illustrate the application of this new confidence interval. Consider a 2×2$2\times 2$ factorial experiment with 20 replicates. Suppose that the parameter of interest θ$\theta$ is a specified simple   effect and that we have uncertain prior information that the two-factor interaction is zero. Our aim is to find a frequentist 0.95 confidence interval for θ$\theta$ that utilizes this prior information.     

Large-sample confidence intervals for risk measures of location–scale families

For a loss distribution belonging to a location–scale family, _Fμ,σ$F_{\mu ,\sigma }$, the risk measures, Value-at-Risk and Expected Shortfall are linear functions of the parameters: μ+τσ$\mu +\tau \sigma$ where τ$\tau$ is the corresponding risk measure of the mean-zero and unit-variance member of the family. For each risk measure, we consider a natural estimator by replacing the unknown parameters μ$\mu$ and σ$\sigma$ by the sample mean and (bias corrected) sample standard deviation, respectively. The large-sample parametric confidence intervals for the risk measures are derived, relying on the asymptotic joint distribution of the sample mean and sample standard deviation. Simulation studies with the Normal, Laplace and Gumbel families illustrate that the derived asymptotic confidence intervals for Value-at-Risk and Expected Shortfall outperform those of Bahadur (1966) and Brazauskas et al. (2008), respectively. The method can also be effectively applied to Log-location-scale families whose supports are positive reals; an illustrative example is given in the area of financial credit risk.     

Comparing several treatments with a control

Consider the partially balanced one-way layout for comparing k   treatments _μi,${\mu }_i,$1?i?k,$1?i?k,$ with a control _μ0${\mu }_0$. We propose a new test which is similar to the test statistics of Marcus [1976. The powers of some tests of the equality of normal means against an ordered alternative. Biometrika 63, 177–183]. By simulation we find that the proposed test has a good power performance when compared with other tests. Moreover, it can produce confidence intervals for _μi-_μ0,1?i?k.${\mu }_i-{\mu }_0,1?i?k.$     

Reduced-rank estimation of the difference between two covariance matrices

We consider m×m$m\times m$ covariance matrices, _Σ1${\Sigma }_1$ and _Σ2${\Sigma }_2$, which satisfy _Σ2-_Σ1=Δ${\Sigma }_2-{\Sigma }_1=\Delta$, where Δ$\Delta$ has a specified rank. Maximum likelihood estimators of _Σ1${\Sigma }_1$ and _Σ2${\Sigma }_2$ are obtained when sample covariance matrices having Wishart distributions are available and rank(Δ)$\text{rank}(\Delta )$ is known. The likelihood ratio statistic for a test about the value of rank(Δ)$\text{rank}(\Delta )$ is also given and some properties of its null distribution are obtained. The methods developed in this paper are illustrated through an example.     

Improving density estimators of discretely observed processes by interpolation

We consider density estimation for a smooth stationary process _Xt$X_t$, t∈R$t\in \mathbf{R}$, based on a discrete sample _Yi=_XΔi$Y_i=X_{\mathrm{\Delta }i}$, i=0,…,n=T/Δ$i=0,\dots ,n=T/\Delta$. By a suitable interpolation scheme of order p  , we augment data to form an approximation _Xp,t$X_{p,t}$, t∈[0,T]$t\in [0,T]$, of the continuous-time process and base our density estimate on the augmented sample path. Our results show that this can improve the rate of convergence (measured in terms of n) of the density estimate. Among other things, this implies that recording n   observations using a small Δ$\Delta$ can be more efficient than recording n independent observations.     

Distribution of concomitants of order statistics and their order statistics

For a random sample of size n$n$ from an absolutely continuous random vector (X,Y)$(X,Y)$, let _Yi:n$Y_{i:n}$ be i$i$th Y$Y$-order statistic and _Y[j:n]$Y_{[j:n]}$ be the Y$Y$-concomitant of _Xj:n$X_{j:n}$. We determine the joint pdf of _Yi:n$Y_{i:n}$ and _Y[j:n]$Y_{[j:n]}$ for all i,j=1$i,j=1$ to n$n$, and establish some symmetry properties of the joint distribution for symmetric populations. We discuss the uses of the joint distribution in the computation of moments and probabilities of various ranks for _Y[j:n]$Y_{[j:n]}$. We also show how our results can be used to determine the expected cost of mismatch in broken bivariate samples and approximate the first two moments of the ratios of linear functions of _Yi:n$Y_{i:n}$ and _Y[j:n]$Y_{[j:n]}$. For the bivariate normal case, we compute the expectations of the product of _Yi:n$Y_{i:n}$ and _Y[i:n]$Y_{[i:n]}$ for n=2$n=2$ to 8 for selected values of the correlation coefficient and illustrate their uses.     

An identity for the Fisher information and Mahalanobis distance

Consider a mixture problem consisting of k classes. Suppose we observe an s-dimensional random vector X   whose distribution is specified by the relations P(X∈A|Y=i)=_Pi(A)$P(X\in A|Y=i)=P_i(A)$, where Y   is an unobserved class identifier defined on {1,…,k}$\{1,\dots ,k\}$, having distribution P(Y=i)=_pi$P(Y=i)=p_i$. Assuming the distributions _Pi$P_i$ having a common covariance matrix, elegant identities are presented that connect the matrix of Fisher information in Y   on the parameters _p1,…,_pk$p_1,\dots ,p_k$, the matrix of linear information in X, and the Mahalanobis distances between the pairs of P  's. Since the parameters are not free, the information matrices are singular and the technique of generalized inverses is used. A matrix extension of the Mahalanobis distance and its invariant forms are introduced that are of interest in their own right. In terms of parameter estimation, the results provide an independent of the parameter upper bound for the loss of accuracy by esimating _p1,…,_pk$p_1,\dots ,p_k$ from a sample of X^′$X^{\prime }$s, as compared with the ideal estimator based on a random sample of Y^′$Y^{\prime }$s.     

The use of an identity in Anderson for multivariate multiple testing

Consider the model where there are I$I$ independent multivariate normal treatment populations with p×1$p\times 1$ mean vectors _μi${\mathit{\mu }}_i$, i=1,…,I$i=1,\dots ,I$, and covariance matrix Σ$\Sigma$. Independently the (I+1)$(I+1)$st population corresponds to a control and it too is multivariate normal with mean vector _μI+1${\mathit{\mu }}_{I+1}$ and covariance matrix Σ$\Sigma$. Now consider the following two multiple testing problems.     

On the extremal behavior of a nonstationary normal random field

Let X={_Xn}n?1$\mathbf{X}=\{X_{\mathbf{n}}{\}}_{\mathbf{n}?\mathbf{1}}$ be a nonstationary random field satisfying a long range weak dependence for each coordinate at a time and a local dependence condition that avoids clustering of exceedances of high values. For these random fields, the probability of no exceedances of high values can be approximated by exp(−τ)$\exp (-\tau )$, where τ$\tau$ is the limiting mean number of exceedances. We present a class of nonstationary normal random fields for which this result can be applied.     

Multivariate orthogonal series estimates for random design regression

In this paper a new multivariate regression estimate is introduced. It is based on ideas derived in the context of wavelet estimates and is constructed by hard thresholding of estimates of coefficients of a series expansion of the regression function. Multivariate functions constructed analogously to the classical Haar wavelets are used for the series expansion. These functions are orthogonal in _L2(_μn)$L_2({\mu }_n)$, where _μn${\mu }_n$ denotes the empirical design measure. The construction can be considered as designing adapted Haar wavelets.     

Testing spatial randomness based on empirical distribution function: A study on lattice data

In this article, we extended the empirical distribution function based test statistic _Ik$I_k$ of Skaug and Tjostheim [1993. Nonparametric test of serial independence based on the empirical distribution function. Biometrika 80, 591–602] in the time series setting to _Dn$D_n$ for spatial lattice data and derived the asymptotic distribution of the proposed test statistic _Dn$D_n$ under the null hypothesis of spatial independence. The size and power of the proposed test statistic under conditional autoregressive model (CAR) were simulated. We applied _Dn$D_n$, Moran's I and Geary's c   to the transformed and well-studied sudden infant death syndrome data from North Carolina and found that _Dn$D_n$ produced a much smaller p$p$-value in testing spatial independence.