Estimates of the parameters of type-II extreme value distribution from censored samples

Maximum likelihood estimators of a Type-II extreme value distribution are derived from doubly censored samples. The asymptotic variances and covariances of the maximum likelihood estimators are discussed and these are numerically evaluated for different censoring proportions q₁ = 0.0(0. l) (0.9) from below and q₂ = 0.0 (0. l) (0.9- q₁) from above. The asymptotic relative efficiencies of the parameter estimates revealed that lower order statistics are more important for estimating the parameters of Type-II extreme value distribution as compared to higher order statistics.     

Asymptotic joint distribution of sample quantiles and sample mean with applications

Based on a sample from an absolutely continuous distribution F with density f, and with the aid of the Bahadur (Ann. Math. Statist. 37( 1966 ), 577-580) representation of sample quantiles, the asymptotic joint distribution of three statistics, the sample p^th and q^th quantiles (0 < p < q < l) and the sample mean, is obtained. Using the Cramer-Wold device, asymptotic distributions of functions of the three statistics can be derived. In particular, the asymptotic joint distribution of the ratio of sample p^th quantile to sample mean and the ratio of sample q^th quantile to sample mean is presented. Finally, consistent estimators are proposed for the variances and covariances of these limiting distributions.     

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.     

General formulae for expectations, variances and covariances of the mean squares for staggered nested designs

Staggered nested experimental designs are the most popular class of unbalanced nested designs. Using a special notation which covers the particular structure of the staggered nested design, this paper systematically derives the canonical form for the arbitrary m-factors. Under the normality assumption for every random variable, a vector comprising m canonical variables from each experimental unit is normally independently and identically distributed. Every sum of squares used in the analysis of variance (ANOVA) can be expressed as the sum of squares of the corresponding canonical variables. Hence, general formulae for the expectations, variances and covariances of the mean squares are directly obtained from the canonical form. Applying the formulae, the explicit forms of the ANOVA estimators of the variance components and unbiased estimators of the ratios of the variance components are introduced in this paper. The formulae are easily applied to obtain the variances and covariances of any linear combinations of the mean squares, especially the ANOVA estimators of the variance components. These results are eff ectively applied for the standardization of measurement methods.     

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.     

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.     

Moments of the sampled autocovariances and autocorrelations for a Gaussian white-noise process

Cumulants, moments about zero, and central moments are obtained for the mean-corrected serial covariances and serial correlations for series realizations of length n from a white-noise Gaussian process. All first and second moments (and some third, fourth, and higher moments) are given explicitly for the serial covariances; and the corresponding moments for the serial correlations are derived either explicitly or implicitly.     

The effect of tapering on the semiparametric estimators for nonstationary long memory processes

In this paper, we study, by a Monte Carlo simulation, the effect of the order p of “Zhurbenko-Kolmogorov” taper on the asymptotic properties of semiparametric estimators. We show that p  =  [d + 1/2] + 1 gives the smallest variances and mean squared errors. These properties depend also on the truncation parameter m. Moreover, we study the impact of the short-memory components on the bias and variances of these estimators. We finally carry out an empirical application by using four monthly seasonally adjusted logarithm Consumer Price Index series.      

Another Proof of Borel's Strong Law of Large Numbers

Let X be a continuous nonnegative random variable with finite first and second moments and a continuous pdf that is positive on the interior of its support. A nonzero limiting density at the origin and a coefficient of variation (CV) greater than 1 are shown to be sufficient conditions for the distribution truncated below at t > 0 to have a variance greater than the variance of the full distribution. Distributions that satisfy these conditions include those with decreasing hazard rates (e.g., the gamma and Weibull distributions with shape parameters less than 1) and the beta distribution with parameter values p and q for which q > p(p + q + 1). The bound T for which truncation at 0 < t < T increases the variance relative to the full distribution is shown to be greater than the (1 — 1/CV)th percentile of the full distribution.     

Modelling time series of counts with overdispersion

The time series of counts observed in practice often exhibit overdispersion. The INGARCH(p, q) models are able to describe integer-valued processes with overdispersion. Known properties of these models, however, are nearly exclusively restricted to the special case p = q = 1. In this article, we derive a set of equations from which the variance and the autocorrelation function of the general case can be obtained. We investigate the purely autoregressive INGARCH(p, 0) models and show that they are closely related to the standard AR(p) models. For p = 1, we determine the marginal distribution in terms of its cumulants. A real-data example highlights potential fields of application of the INGARCH(p, 0) 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|.     

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.     

Recurrence relations for single and product moments of progressively Type-II censored order statistics from generalized logistic distribution with applications to inference

In this article, we establish several recurrence relations for the single and product moments of progressively Type-II right censored order statistics from a generalized logistic distribution. The use of these relations in a systematic manner allow us to compute all the means, variances, and covariances of progressively Type-II right censored order statistics from the generalized logistic distribution for all sample sizes n, effective sample sizes m, and all progressive censoring schemes (R₁, …, R_m). These moments are then utilized to derive best linear unbiased estimators of the scale and location-scale parameters of the generalized logistic distribution. A comparison of these estimators with the maximum likelihood estimates is then made through Monte Carlo simulations. Finally, the best linear unbiased predictors of censored failure times is discussed briefly.     

Simulating dependent discrete data

This article describes a method for simulating n-dimensional multivariate non-normal data, with emphasis on count-valued data. Dependence is characterized by either Pearson correlations or Spearman correlations. The simulation is accomplished by simulating a vector of correlated standard normal variates. The elements of this vector are then transformed to achieve the target marginal distributions. We prove that the method corresponds to simulating data from a multivariate Gaussian copula. The simulation method does not restrict pairwise dependence beyond the limits imposed by the marginal distributions and can achieve any Pearson or Spearman correlation within those limits. Two examples are included. In the first example, marginal means, variances, Pearson correlations, and Spearman correlations are estimated from the epileptic seizure data set of Diggle et al. [P. Diggle, P. Heagerty, K.Y. Liang, and S. Zeger, Analysis of Longitudinal Data, Oxford University Press, Oxford, 2002]. Data with these means and variances are simulated to first achieve the estimated Pearson correlations and then achieve the estimated Spearman correlations. The second example is of a hypothetical time series of Poisson counts with seasonal mean ranging between 1 and 9 and an autoregressive(1) dependence structure.     

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.     

Interpretation of Canonical Discriminant Functions,Canonical Variates,and Principal Components

Canonical discriminant functions are defined here as linear combinations that separate groups of observations, and canonical variates are defined as linear combinations associated with canonical correlations between two sets of variables. In standardized form, the coefficients in either type of canonical function provide information about the joint contribution of the variables to the canonical function. The standardized coefficients can be converted to correlations between the variables and the canonical function. These correlations generally alter the interpretation of the canonical functions. For canonical discriminant functions, the standardized coefficients are compared with the correlations, with partial t and F tests, and with rotated coefficients. For canonical variates, the discussion includes standardized coefficients, correlations between variables and the function, rotation, and redundancy analysis. Various approaches to interpretation of principal components are compared: the choice between the covariance and correlation matrices, the conversion of coefficients to correlations, the rotation of the coefficients, and the effect of special patterns in the covariance and correlation matrices.     

The left-truncated Weibull distribution: theory and computation

This paper studies the two-parameter, left-truncated Weibull distribution (LTWD) with known, fixed, positive truncation pointT. Important hitherto unknown statistical properties of the LTWD are derived. The asymptotic theory of the maximum likelihood estimates (MLEs) is invoked to develop parameter confidence intervals and regions. Numerical methods are described for computing the MLEs and for evaluating the exact, asymptotic variances and covariances of the MLEs. An illustrative example is given.     

On Reproducing Linear Estimators within the Gauss–Markov Model with Stochastic Constraints

In a Gauss–Markov Model (GMM) with fixed constraints, all the relevant estimators perfectly satisfy these constraints. As soon as they become stochastic, most estimators are allowed to satisfy them only approximately, thereby leaving room for nonvanishing residuals to describe the deviation from the prior information.

Sometimes, however, linear estimators may be preferred that are able to perfectly reproduce the prior information in form of stochastic constraints, including their variances and covariances. As typical example may be considered the case where a geodetic network ought to be densified without changing the higher-order point coordinates that are usually introduced together with their variances and (some) covariances. Traditional estimators are based on the “Helmert” or “S-transformation,” respectively an adaptation of the fixed-constraints Least-Squares estimator.

Here we show that neither approach generates the optimal reproducing estimator, which will be presented in detail and compared with the other reproducing estimators in terms of their MSE-risks.     

Point and Interval Estimation for Bivariate Normal Distribution Based on Progressively Type-II Censored Data

ABSTRACT

The maximum likelihood estimates (MLEs) of parameters of a bivariate normal distribution are derived based on progressively Type-II censored data. The asymptotic variances and covariances of the MLEs are derived from the Fisher information matrix. Using the asymptotic normality of MLEs and the asymptotic variances and covariances derived from the Fisher information matrix, interval estimation of the parameters is discussed and the probability coverages of the 90% and 95% confidence intervals for all the parameters are then evaluated by means of Monte Carlo simulations. To improve the probability coverages of the confidence intervals, especially for the correlation coefficient, sample-based Monte Carlo percentage points are determined and the probability coverages of the 90% and 95% confidence intervals obtained using these percentage points are evaluated and shown to be quite satisfactory. Finally, an illustrative example is presented.