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The problem of estimating the change point in a sequence of independent observations is considered. Hinkley (1970) demonstrated that the maximum likelihood estimate of the change point is associated with a two-sided random walk in which the ascending and descending epochs and heights are the key elements for its evaluation. The aim here is to expand the information generated from the random walks and from fluctuation theory and applied to the change point formulation. This permits us to obtain computable expressions for the asymptotic distribution of the change point with respect to convolutions and Laplace transforms of the likelihood ratios. Further, if moment expressions of the likelihood ratios are known, explicit representations of the asymptotic distribution of the change point become accessible up to the second order with respect to the moments. In addition, the rate of convergence between the finite and infinite distribution of the change point distribution is established and it is shown to be of polynomial order. [ABSTRACT FROM AUTHOR]

The article offers a method for estimating the volatility covariance matrix of vectors of financial time series data using a change point approach. The proposed method supersedes general varying-coefficient parametric models, such as GARCH, whose coefficients may vary with time, by a change point model. In this study, an adaptive pointwise selection of homogeneous segments with a given right-end point by a local change point analysis is introduced. Sufficient conditions are obtained under which the maximum likelihood process is adaptive against the covariance estimate to yield an optimal rate of convergence with respect to the change size. This rate is preserved while allowing the jump size to diminish. Under these circumstances, argmax results of a two-sided negative Brownian motion or a two-sided negative drift random walk under vanishing and non-vanishing jump size regimes, respectively, provide inference for the change point parameter. Theoretical results are supported by the Monte–Carlo simulation study. A bivariate data on daily log returns of two US stock market indices as well as tri-variate data on daily log returns of three banks are analyzed by constructing confidence interval estimates for multiple change points that have been identified previously for each of the two data sets. [ABSTRACT FROM AUTHOR]

The aim of the article is to analyze inhomogeneous time series data caused by the presence of an unknown change point. We assume that the time series data are from a gamma distribution with an unknown time point of change in the scale and/or shape parameters. A complete change point methodology is proposed including change detection methodology based on the likelihood ratio statistic as well as estimation of the unknown change point by the method of maximum likelihood estimation (mle). Furthermore, we provide asymptotic distribution of the change point mle when a change occurs in the scale parameter of the gamma distribution. Extensive simulations have been conducted to show excellent agreement between the distribution of the change point under finite sample sizes and its asymptotic counterparts. The simulations are conducted under a change in the scale parameter as well as a change in both scale and shape parameters. A comparison analysis between known parameters and estimated parameters indicates that the error committed is negligible. Two examples, one from the financial market and another from climatology, are analyzed to adequately illustrate the proposed inferential methodology. [ABSTRACT FROM AUTHOR]

The purpose of this study is to characterize multivariate generalized hyperbolic (MGH) distributions and their conditionals by considering the MGH as a subclass of the mean-variance mixing of the multivariate normal law. The essential contribution here lies in expressing MGH densities by utilizing various integral representations of the Bessel function. Moreover, in a more convenient form these modified density representations are more advantageous for deriving limiting results. The forms are also convenient for studying the transient as well as tail behavior of MGH distributions. The results include the normal distribution as a limiting form for the MGH distribution. To support the MGH model an empirical study is conducted to demonstrate the applicability of the MGH distribution for modeling not only high frequency data but also for modeling low frequency data. This is against the currently prevailing notion that the MGH model is relevant for modeling only high frequency data. [ABSTRACT FROM AUTHOR]

The article develops a theory for the last time the subordinated Brownian motion ( SBM with negative drift reaches its supremum. The study includes obtaining expressions for the Laplace transform of the last time that the SBM reaches its supremum and also for its density. In the process, we establish that the last time that the SBM reaches its supremum is a member of the generalized gamma convolution ( GGC) family. The theoretical results for the general case have been explicitly derived for some well-known subordinators. Numerical investigations show close agreement between the theoretical derivations and empirical computations. [ABSTRACT FROM AUTHOR]

This paper analyzes purchasing strategies for retailers regarding the best timing and amount of purchases when operating under combined timing and quantity flexibility contracts in an environment of uncertain prices. To decrease the computational complexity and make the procedure adaptable to the case of multiple suppliers, we introduce, analyze, and compare a 'time strategy' and a 'target strategy' and then develop a hybrid 'adaptive target strategy' to facilitate and improve the purchasing decision for the case of option contracts with generally rising prices. The adaptive target strategy is simpler and more intuitive than the traditional binomial lattice method, while the risk of failing to meet a target profit can also easily be calculated. We then extend the solution procedure to maximize expected profits in an environment of selecting among multiple suppliers with potentially different price processes, and we further provide risk analysis to help determine a good estimate for the number of option contracts from different suppliers to generate in order to create adequate risk protection. Numerical analysis demonstrates how the number of candidate suppliers impacts the expected profit and the risk. Monte Carlo simulation results demonstrate that the developed solution procedures provide satisfying outcomes and that the calculation is fast, even for multiple-dimension and multiple-supplier cases. [ABSTRACT FROM AUTHOR]

To date, three major articles have analyzed the use of coefficient alpha on scales published in the marketing literature, These studies have improved our understanding of the scale development process and the behavior of coefficient alpha. One consistent result is that coefficient alpha produces an R-square of approximately 10 when regressed against scale length. In the most recent of these three meta-analyses, Peterson (1994) suggested that alpha does not behave as suggested by theory. We reexamine the behavior of alpha in relation to scale length, scale width, and response centrality. Our results suggest different conclusions from those previously reached. [ABSTRACT FROM AUTHOR]

Let X= A1/2 G be a scale mixture of a multivariate normal distribution with X, G∈ ℝ n, Gis a multivariate normal vector, and A is a positive random variable independent of the multivariate random vector G. This study presents asymptotic results of the conditional variance-covariance, Cov( X2| X1), X1∈ ℝ m, m < n, under some moment expressions. A new representation form is also presented for conditional expectation of the scale variable on the random vector X1∈ ℝ m, m < n. Both the asymptotic expression and the representation are manageable and in computable form. Finally, an example is presented to illustrate how the computations are carried out. [ABSTRACT FROM AUTHOR]