Bayesian and Robust Bayesian analysis under a general class of balanced loss functions

For estimating an unknown parameter θ, we introduce and motivate the use of balanced loss functions of the form L_{r, w, d0}(q, d)=wr(d₀, d)+ (1-w) r(q, d){L_{\rho, \omega, \delta_0}(\theta, \delta)=\omega \rho(\delta_0, \delta)+ (1-\omega) \rho(\theta, \delta)}, as well as the weighted version q(q) L_{r, w, d0}(q, d){q(\theta) L_{\rho, \omega, \delta_0}(\theta, \delta)}, where ρ(θ, δ) is an arbitrary loss function, δ ₀ is a chosen a priori “target” estimator of q, w ? [0,1){\theta, \omega \in[0,1)}, and q(·) is a positive weight function. we develop Bayesian estimators under L_{r, w, d0}{L_{\rho, \omega, \delta_0}} with ω > 0 by relating such estimators to Bayesian solutions under L_{r, w, d0}{L_{\rho, \omega, \delta_0}} with ω = 0. Illustrations are given for various choices of ρ, such as absolute value, entropy, linex, and squared error type losses. Finally, under various robust Bayesian analysis criteria including posterior regret gamma-minimaxity, conditional gamma-minimaxity, and most stable, we establish explicit connections between optimal actions derived under balanced and unbalanced losses.     

Euler(p,q) Processes and Their Application to Non Stationary Time Series with Time Varying Frequencies

We introduce Euler(p, q) processes as an extension of the Euler(p) processes for purposes of obtaining more parsimonious models for non stationary processes whose periodic behavior changes approximately linearly in time. The discrete Euler(p, q) models are a class of multiplicative stationary (M-stationary) processes and basic properties are derived. The relationship between continuous and discrete mixed Euler processes is shown. Fundamental to the theory and application of Euler(p, q) processes is a dual relationship between discrete Euler(p, q) processes and ARMA processes, which is established. The usefulness of Euler(p, q) processes is examined by comparing spectral estimation with that obtained by existing methods using both simulated and real data.     

Distributions of patterns of two successes separated by a string of <Emphasis Type="Italic">k</Emphasis>-2 failures

Let Z ₁, Z ₂, . . . be a sequence of independent Bernoulli trials with constant success and failure probabilities p = Pr(Z _t  = 1) and q = Pr(Z _t  = 0) = 1 − p, respectively, t = 1, 2, . . . . For any given integer k ≥ 2 we consider the patterns E₁{\mathcal{E}_{1}}: two successes are separated by at most k−2 failures, E₂{\mathcal{E}_{2}}: two successes are separated by exactly k −2 failures, and E₃{\mathcal{E}_{3}} : two successes are separated by at least k − 2 failures. Denote by N_n,k⁽ⁱ⁾{ N_{n,k}^{(i)}} (respectively M_n,k⁽ⁱ⁾{M_{n,k}^{(i)}}) the number of occurrences of the pattern E_i{\mathcal{E}_{i}} , i = 1, 2, 3, in Z ₁, Z ₂, . . . , Z _n when the non-overlapping (respectively overlapping) counting scheme for runs and patterns is employed. Also, let T_r,k⁽ⁱ⁾{T_{r,k}^{(i)}} (resp. W_r,k⁽ⁱ⁾){W_{r,k}^{(i)})} be the waiting time for the r − th occurrence of the pattern E_i{\mathcal{E}_{i}}, i = 1, 2, 3, in Z ₁, Z ₂, . . . according to the non-overlapping (resp. overlapping) counting scheme. In this article we conduct a systematic study of N_n,k⁽ⁱ⁾{N_{n,k}^{(i)}}, M_n,k⁽ⁱ⁾{M_{n,k}^{(i)}}, T_r,k⁽ⁱ⁾{T_{r,k}^{(i)}} and W_r,k⁽ⁱ⁾{W_{r,k}^{(i)}} (i = 1, 2, 3) obtaining exact formulae, explicit or recursive, for their probability generating functions, probability mass functions and moments. An application is given.     

The generalized linear chirp process

The linear chirp process is an important class of time series for which the instantaneous frequency changes linearly in time. Linear chirps have been used extensively to model a variety of physical signals such as radar, sonar, and whale clicks (see 1, 5 and 6). We introduce the stochastic linear chirp model and then define the generalized linear chirp (GLC) process as a special case of the G-stationary process studied by Jiang et al. (2006) to model data with time-varying frequencies. We then define GLC(p,q) processes and show that the relationship between stochastic linear chirp processes and GLC(p,q) processes is analogous to that between harmonic and ARMA models. The new methods are then applied to both simulated and actual data sets.     

Sequential estimation in two-truncation parameter family of distributions under type II censoring

Summary This paper deals with the sequential estimation ofq(ϑ₁, ϑ₂) when the underlying density function is of the formf(x)=q(ϑ₁, ϑ₂)h(x), where ϑ₁ and ϑ₂ are unknown truncation parameters. We study the sequential properties of the stopping rule and the sequential estimator ofq(ϑ₁, ϑ₂). In this study we assume that the sample is type II censored.     

Asymptotic Results for Spatial ARMA Models

Causal quadrantal-type spatial ARMA(p, q) models with independent and identically distributed innovations are considered. In order to select the orders (p, q) of these models and estimate their autoregressive parameters, estimators of the autoregressive coefficients, derived from the extended Yule–Walker equations are defined. Consistency and asymptotic normality are obtained for these estimators. Then, spatial ARMA model identification is considered and simulation study is given.     

Tests for the order of integration against higher order integration

Locally best invariant tests for the null hypothesis of I(p) against the alternative hypothesis of I(q), < q, are developed for models with independent normal errors. The tests are semiparametrically extended for models with autocorrelated errors. The method is illustrated by two real data sets in terms of double unit roots. The proposed tests can be used for determining integration orders of nonstationary time series.     

Geometric ergodicity of nonlinear autoregressive models with changing conditional variances

The authors give easy‐to‐check sufficient conditions for the geometric ergodicity and the finiteness of the moments of a random process x_t = ?(x_t‐1,…, x_t‐p) + ?_tσ(x_t‐1,…, x_t‐q) in which ?: R^p → R, σ R^q → R and (?_t) is a sequence of independent and identically distributed random variables. They deduce strong mixing properties for this class of nonlinear autoregressive models with changing conditional variances which includes, among others, the ARCH(p), the AR(p)‐ARCH(p), and the double‐threshold autoregressive models.     

A new diagnostic tool for VARMA(p,q) models

A new diagnostic method for VARMA(p,q) time series models is introduced. The procedure is based on a statistic that generalizes to a multivariate setting the properties of the usual univariate ARMA(p,q) residual correlations. A multiple version of the cumulative periodogram statistic is also suggested. Simulation studies and one real data application are presented.     

Properties of a class of binary ARMA models

We investigate a class of ARMA-type models for stationary binary time series developed in [M. Kanter, Autoregression for discrete processes mod 2, J. Appl. Probabil. 12 (1975), pp. 371–375, E. McKenzie, Extending the correlation structure of exponential autoregressive-moving-average processes, J. Appl. Prob. 18 (1981), pp. 181–189.], which we shall refer to as BinARMA models. This sparsely parameterized model family is even able to deal with negative autocorrelations, which occur in language modelling, for instance. While the autocorrelation structure of the BinAR(p) models has been studied before in [M. Kanter, Autoregression for discrete processes mod 2, J. Appl. Probabil. 12 (1975), pp. 371–375], we shall present new results on the autocorrelation structure of general BinARMA models. These results simplify in the BinMA(q) case, while the known results concerning BinAR(p) models are included as a special case. A real-data example indicates possible fields of application of these models.     

ON TRANSITION PROBABILITIES OF SKIP-FREE MARKOV CHAINS

Consider an ergodic Markov chain X(t) in continuous time with an infinitesimal matrix Q = (q_ij) defined on a finite state space {0, 1,…, N}. In this note, we prove that if X(t) is skip-free positive (negative, respectively), i.e., q_ij, = 0 for j > i+ 1 (i > j+ 1), then the transition probability p_ij(t) = Pr[X(t)=j | X(0) =i] can be represented as a linear combination of p_0N(t) (p^(m)_(N0)(t)), 0 ≤ m ≤N, where f^(m)(t) denotes the mth derivative of a function f(t) with f⁽⁰⁾(t) =f(t). If X(t) is a birth-death process, then p_ij(t) is represented as a linear combination of p_0N^(m)(t), 0 ≤m≤N - |i-j|.     

Lp (1 < p ⩽ 2) solutions of one-dimensional BSDEs whose generator is weakly monotonic in y and non-Lipschitz in z

ABSTRACT

In this paper, we start with establishing the existence of a minimal (maximal) L^p (1 < p ? 2) solution to a one-dimensional backward stochastic differential equation (BSDE), where the generator g satisfies a p-order weak monotonicity condition together with a general growth condition in y and a linear growth condition in z. Then, we propose and prove a comparison theorem of L^p (1 < p ? 2) solutions to one-dimensional BSDEs with q-order (1 ? q < p) weak monotonicity and uniform continuity generators. As a consequence, an existence and uniqueness result of L^p (1 < p ? 2) solutions is also given for BSDEs whose generator g is q-order (1 ? q < p) weakly monotonic with a general growth in y and uniformly continuous in z.     

D-Optimal Axial Designs for Quadratic and Cubic Additive Mixture Models

The paper discusses D-optimal axial designs for the additive quadratic and cubic mixture models σ_1≤i≤q(β_ix_i + β_iix²_i) and σ_1≤i≤q(β_ix_i + β_iix²_i + β_iiix³_i), where x_i≥ 0, x₁ + . . . + x_q = 1. For the quadratic model, a saturated symmetric axial design is used, in which support points are of the form (x₁, . . . , x_q) = [1 ? (q?1)δ_i, δ_i, . . . , δ_i], where i = 1, 2 and 0 ≤δ₂ <_{δ1 ≤} 1/(q ?1). It is proved that when 3 ≤q≤ 6, the above design is D-optimal if δ₂ = 0 and δ₁ = 1/(q?1), and when q≥ 7 it is D-optimal if δ₂ = 0 and δ₁ = [5q?1 ? (9q²?10q + 1)^1/2]/(4q²). Similar results exist for the cubic model, with support points of the form (x₁, . . . , x_q) = [1 ? (q?1)δ_i, δ_i, . . . , δ_i], where i = 1, 2, 3 and 0 = δ₃ <_δ2 < δ_{1 ≤}1/(q?1). The saturated D-optimal axial design and D-optimal design for the quadratic model are compared in terms of their efficiency and uniformity.     

Orderings for series and parallel systems comprising heterogeneous exponentiated Weibull-geometric components

Let X₁, …, X_n be independent random variables with X_i ～ EWG(α, β, λ_i, p_i), i = 1, …, n, and Y₁, …, Y_n be another set of independent random variables with Y_i ～ EWG(α, β, γ_i, q_i), i = 1, …, n. The results established here are developed in two directions. First, under conditions p₁ = ??? = p_n = q₁ = ??? = q_n = p, and based on the majorization and p-larger orders between the vectors of scale parameters, we establish the usual stochastic and reversed hazard rate orders between the series and parallel systems. Next, for the case λ₁ = ??? = λ_n = γ₁ = ??? = γ_n = λ, we obtain some results concerning the reversed hazard rate and hazard rate orders between series and parallel systems based on the weak submajorization between the vectors of (p₁, …, p_n) and (q₁, …, q_n). The results established here can be used to find various bounds for some important aging characteristics of these systems, and moreover extend some well-known results in the literature.     

On the invertibility of EGARCH(p,q)

Of the two most widely estimated univariate asymmetric conditional volatility models, the exponential GARCH (or EGARCH) specification is said to be able to capture asymmetry, which refers to the different effects on conditional volatility of positive and negative effects of equal magnitude, and leverage, which refers to the negative correlation between the returns shocks and subsequent shocks to volatility. However, the statistical properties of the (quasi-)maximum likelihood estimator (QMLE) of the EGARCH(p, q) parameters are not available under general conditions, but only for special cases under highly restrictive and unverifiable sufficient conditions, such as EGARCH(1,0) or EGARCH(1,1), and possibly only under simulation. A limitation in the development of asymptotic properties of the QMLE for the EGARCH(p, q) model is the lack of an invertibility condition for the returns shocks underlying the model. It is shown in this article that the EGARCH(p, q) model can be derived from a stochastic process, for which sufficient invertibility conditions can be stated simply and explicitly when the parameters respect a simple condition.¹¹Using the notation introduced in part 2, this refers to the cases where α ≥ |γ| or α ≤ ? |γ|. The first inequality is generally assumed in the literature related to the invertibility of EGARCH. This article provides (in the Appendix) an argument for the possible lack of invertibility when these conditions are not met. This will be useful in reinterpreting the existing properties of the QMLE of the EGARCH(p, q) parameters.     

A computational study of a quasi-PORT methodology for VaR based on second-order reduced-bias estimation

In this paper, we deal with the estimation, under a semi-parametric framework, of the Value-at-Risk (VaR) at a level p, the size of the loss occurred with a small probability p. Under such a context, the classical VaR estimators are the Weissman–Hill estimators, based on any intermediate number k of top-order statistics. But these VaR estimators do not enjoy the adequate linear property of quantiles, contrarily to the PORT VaR estimators, which depend on an extra tuning parameter q, with 0≤q<1. We shall here consider ‘quasi-PORT’ reduced-bias VaR estimators, for which such a linear property is obtained approximately. They are based on a partially shifted version of a minimum-variance reduced-bias (MVRB) estimator of the extreme value index (EVI), the primary parameter in Statistics of Extremes. Due to the stability on k of the MVRB EVI and associated VaR estimates, we propose the use of a heuristic stability criterion for the choice of k and q, providing applications of the methodology to simulated data and to log-returns of financial stocks.     

Maximum likelihood estimation for arma models in the presence of ARMA errors

An ARMA(p, q) process observed with an ARMA(c, d) error has an ARMA (p + c, k) representation with k = max(c + q, p + d) whose parameters satisfy some nonlinear constraints. Identification of the model is discussed. We develop Newton-Raphson estimators for the ARMA(p + c, k) process which take advantage of the information contained in the nonlinear restrictions. Explicit expressions for the derivatives of the restrictions are derived.     

Probabilistic Properties of Parametric Dual and Inverse Time Series Models Generated by ARMA Models

For the class of autoregressive-moving average (ARMA) processes, we examine the relationship between the dual and the inverse processes. It is demonstrated that the inverse process generated by a causal and invertible ARMA (p, q) process is a causal and invertible ARMA (q, p) model. Moreover, it is established that this representation is strong if and only if the generating process is Gaussian. More precisely, it is derived that the linear innovation process of the inverse process is an all-pass model. Some examples and applications to time reversibility are given to illustrate the obtained results.     

Limit results for ordered uniform spacings

Let Δ _k:n  = X _k,n  − X _k-1,n (k = 1, 2, . . . , n + 1) be the spacings based on uniform order statistics, provided X _0,n  = 0 and X _n+1,n  = 1. Obtained from uniform spacings, ordered uniform spacings 0 = Δ_0,n  < Δ_1,n  < . . . < Δ _n+1,n , are discussed in the present paper. Distributional and limit results for them are in the focus of our attention.