Improved estimation of the mean vector for Student-t model.

Improved James-Stein type estimation of the mean vector \mbox{\boldmath \mu} of a multivariate Student-t population of dimension p with $\nu$ degrees of freedom is considered. In addition to the sample data, uncertain prior information on the value of the mean vector, in the form of a null hypothesis, is used for the estimation. The usual maximum likelihood estimator (mle) of \mbox{\boldmath \mu} is obtained and a test statistic for testing H_0: \mbox{\boldmath \mu} = \mbox{\boldmath \mu}_0 is derived. Based on the mle of \mbox{\boldmath \mu} and the test statistic the preliminary test estimator (PTE), Stein-type shrinkage estimator (SE) and positive-rule shrinkage estimator (PRSE) are defined. The bias and the quadratic risk of the estimators are evaluated. The relative performances of the estimators are investigated by analyzing the risks under different conditions. It is observed that the PRSE dominates over the other three estimators, regardless of the validity of the null hypothesis and the value $\nu.

Prediction distribution for linear regression model with multivariate Student-t errors under the Bayesian approach

[Abstract]: Prediction distribution is a basis for predictive inferences applied in many real world situations. It is a distribution of the unobserved future response(s) conditional on a set of realized responses from an informative experiment. Various statistical approaches can be used to obtain prediction distributions for different models. This study derives the prediction distribution(s) for multiple linear regression model using the Bayesian method when the error components of both the performed and future models have a multivariate Student-t distribution. The study observes that the prediction distribution(s) of future response(s) has a multivariate Student-t distribution whose degrees of freedom depends on the size of the realized sample and the dimension of the regression parameters’ vector but does not depend on the degrees of freedom of the errors distribution

Prediction distribution of generalized geometric series distribution and its different forms

The prediction distribution of generalized geometric series distribution (GGSD) and of its truncated and size-biased forms is derived and studied under the non-informative and beta prior distributions. The prediction distributions for all the models are beta distribution, but the parameters of the prediction distributions depend on the choice of the prior distribution as well as the model under consideration

Predictive Inference For The Multilinear Models With Multivariate Student-t Errors

The prediction distributions of the future responses, conditional on the observed responses, from the multilinear models with multivariate Student-t and matrix-T errors are derived. These distributions provide the basis for the inference about the future responses. The cases of (i) dependent but uncorrelated and (ii) dependent and correlated responses are considered. The distribution of the Wishart matrix, and the prediction distribution of the regression and Wishart matrices have also been obtained for the generalized multilinear model with matrix-T errors.;For the multilinear models with first order auto-correlation, the marginal likelihood function of the auto-correlation coefficient has been obtained and used for the derivation of the prediction distributions. A simulation study has been conducted to illustrate the results.;As an application of the prediction distribution, the {dollar}\beta{dollar}-expectation tolerance regions are obtained for multilinear models by using the prediction distributions. First, a general procedure has been developed for the {dollar}\beta{dollar}-expectation tolerance region based on dependent responses under the framework of a structural model. Then it has been applied for the construction of the {dollar}\beta{dollar}-expectation tolerance regions for different multilinear models.;Structural inference procedure for the parameters of heteroscedastic simultaneous equation model has also been proposed for the general error distribution. The general results are then applied to the case of the multivariate normal and multivariate Student-t errors. The prediction distribution for the linear combinations of the future responses is also derived for the multivariate Student-t error

New weighted geometric mean method to estimate the slope of measurement error model

This paper introduces a new weighted geometric mean (WG) estimator to fit regression line when both the response and explanatory variables are subject to measurement errors. The proposed estimator is based on the mathematical relationship between the vertical and orthogonal distances of the observed points and the regression line (cf. Saqr and Khan, 2012). It minimizes the orthogonal distance of the observed points from the unfitted line. The WG estimator is less sensitive to the ratio of error variances. It is a better alternative than the currently used geometric mean (GM) and OLS-bisector estimators. Extensive simulation results show that the proposed WG estimator is much more stable than the geometric mean and OLS-bisector estimators. The mean absolute error of the WG estimator is consistently smaller than the geometric mean and OLS-bisector estimators

Testing equality of two intercepts for the parallel regression model with non-sample prior information

This paper proposes tests for equality of intercepts of two simple regression models when non-sample prior information (NSPI) is available on the equality of two slopes. For three different scenarios on the values of the slope, namely (i) unknown (unspecified), (ii) known (specified), and (iii) suspected, we derive the unrestricted test (UT), restricted test (RT) and pre-test test (PTT) for testing equality of intercepts. The test statistics, their sampling distributions, and power functions of the tests are obtained. Comparison of power function and size of the tests reveal that the PTT has a reasonable dominance over the UT and RT

Bi-directional grid absorption barrier constrained stochastic processes with applications in finance and investment

Whilst the gambler’s ruin problem (GRP) is based on martingales and the established probability theory proves that the GRP is a doomed strategy, this research details how the semimartingale framework is required for the grid trading problem (GTP) of financial markets, especially foreign exchange (FX) markets. As banks and financial institutions have the requirement to hedge their FX exposure, the GTP can help provide a framework for greater automation of the hedging process and help forecast which hedge scenarios to avoid. Two theorems are adapted from GRP to GTP and prove that grid trading, whilst still subject to the risk of ruin, has the ability to generate significantly more profitable returns in the short term. This is also supported by extensive simulation and distributional analysis. We introduce two absorption barriers, one at zero balance (ruin) and one at a specified profit target. This extends the traditional GRP and the GTP further by deriving both the probability of ruin and the expected number of steps (of reaching a barrier) to better demonstrate that GTP takes longer to reach ruin than GRP. These statistical results have applications into finance such as multivariate dynamic hedging (Noorian, Flower, & Leong, 2016), portfolio risk optimization, and algorithmic loss recovery

Logistic regression for circular data

This paper considers the relationship between a binary response and a circular predictor. It develops the logistic regression model by employing the linear-circular regression approach. The maximum likelihood method is used to estimate the parameters. The Newton-Raphson numerical method is used to find the estimated values of the parameters. A data set from weather records of Toowoomba city is analysed by the proposed methods. Moreover, a simulation study is considered. The R software is used for all computations and simulations