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Most real data sets from many sources such as medical sciences, quality engineering, environmental, econometrics etc. are correlated in nature.  The present article aims to derive the necessary regression analysis techniques for a correlated data set with a special form of compound symmetry error structure with two sets of observations such that the first set contains only the first observation, and the other set contains the remaining (N – 1) observations, where N is the total number of observations. A constant correlation (????1) is assumed between the first and anyone of the remaining observation, and for the second set, a constant correlation (????) is assumed between any two observations within themselves. The variance is assumed constant for all the observations. Correlation structural form is known, but the parameters involved in it are always unknown. In the article, we have derived a robust estimating method for the best linear unbiased estimators (BLUE) of all the regression parameters except the intercept, which is often unimportant. In addition, we have developed a robust testing procedure for any set of linear hypotheses regarding the unknown regression coefficients, and along with a confidence ellipsoid for a set of estimable functions of regression coefficients. Index of fit for the fitted regression equation has also been developed. An example with simulated data illustrates all the developed theories in the article.

Keywords: Confidence ellipsoid; Correlated error; Index of fit; Linear hypothesis; Regression analysis; Robust estimation.

The partially linear single-index spatial autoregressive model is a new class of semiparametric spatial autoregressive models, which achieves both dimension reduction and nice model interpretation. In this paper, we propose a new estimation method for the partially linear single-index spatial autoregressive model by combining local linear smoothing approach and quasi-maximum likelihood method. Compared to existing estimation method, the proposed method does not need to select instrumental variables. Furthermore, we propose a generalized likelihood ratio test to check the parametric form of the nonparametric component, in which a residual-based bootstrap procedure is suggested to calculate p-value of the proposed test. Some simulation studies are conducted to assess the performance of the proposed estimation and test methods and simulation results show that both methods perform well in finite samples. A real data example is provided to illustrate the proposed estimation and test methods.

Keywords: Spatial dependence; Single-index modeling; Quasi-maximum likelihood method; Local linear smoothing method; Bootstrap.

In the world economy, silver has acted as a significant character as a universally expensive commodity and, in most countries, a currency. In early and modern economics, out of the most widely-traded commodities, silver is an exemplary commodity for which comprehensive documentation appears feasible. The early history of silver production data was not consistent, as they were recorded in many diaries, or books, based on the author ’s own estimation, or obtained from some unreliable sources. It is well-known that any historical data set is not accurate as physical measurement, or scientific experimental data. It is always necessary to study history with unbiased or robust historical data which can only be derived by using some scientific modeling method from the raw available data. The present paper focuses on studying the history of new world- Spanish and Portuguese colonies’ silver production trends from 1521 to 1810 statistically, using probabilistic parametric and cubic spline models. The report not only derives the robust estimates of silver productions during this period, but also focuses on many historical events such as the early industrial advancement, industrial revolution, and depression status in silver mining during the period. All these above historical events are identified from the derived probabilistic parametric and cubic spline models. In addition, a probabilistic parametric model gives better estimates than a cubic spline model within this period.

Keywords: History of silver; Cubic spline; Gamma & Lognormal models; Joint generalized linear models; Silver production trend; Spanish and Portuguese colonies.

With only two parameters the New Distribution (ND) introduced recently by Doostmoradi (2018) is very flexible for modeling litetime data because the failure rate function can have different shapes (increasing, decreasing and unimodal). This work is devoted to the maximum likelihood estimation of the unknown parameters and the construction of goodness-offittest statistics for this model when data are right censored. Also, a comparative study is provided to distinguish between this model (ND) and the competing distributions, namely the exponentiated Weibull (EW), modified Weibull (MW), extended generalized Lindley distribution (EGL), generalized Lindley (GL), power Lindley (PL), and the inverse Lindley (IL). An important simulation study was carried out and theoretical results obtained through this study are applied to real data sets from reliability and survival analysis.

Keywords: Censored data, Fisher information matrix, maximum likelihood estimation, modified goodness-of-fit test.

JEL Classification: 62F03-62G05-62G10-62N05

In this study, crude oil production in Nigeria was constructed using SARIMA model. The study was carried out to estimate the parameters of the model, to identify a model that best fits the production of crude oil, and to check for the model adequacy on the quarterly production of crude oil. Test of normality of the data was done via Anderson Darling test statistics. Augmented Dickey-Fuller test was employed to test for stationarity of thedata. Adequacy of the model was carried out using Ljung-Box chi-square amongst others. Finally, the results show that Seasonal Autoregressive Integrated Moving Average (SARIMA) model best fits the production of crude oil in Nigeria.

Keywords: Autoregressive, stationarity, normality, seasonal and invertibility.

In this paper, exponentiated generalized inverted exponential distribution is considered for Bayesian analysis. The expressions for Bayes estimators of the parameter have been derived under squared error, precautionary, entropy, K-loss, and Al-Bayyati’s loss functions by using quasi and gamma priors..

Keywords: Bayesian method, exponentiated generalized inverted exponential distribution, quasi and gamma priors, squared error, precautionary, entropy, K-loss, and Al-Bayyati’s loss functions.