Model Assisted Statistics and Applications

In the absence of uniformly most powerful (UMP) tests or uniformly most powerful invariant (UMPI) tests, King [80] suggested the use of Point Optimal (PO) tests, which are most powerful at a chosen point under the alternative hypothesis. This paper surveys the literature and major developments on point optimal testing since 1987 and suggests some areas for future research. Topics include tests for which all nuisance parameters have been eliminated and dealing with nuisance parameters via (i) a weighted average of p values, (ii) approximate point optimal tests, (iii) plugging in estimated parameter values, (iv) using asymptotics and (v) integration. Progress on using point-optimal testing principles for two-sided testing and multi-dimensional alternatives is also reviewed. The paper concludes with thoughts on how best to deal with nuisance parameters under both the null and alternative hypotheses, as well as the development of a new class of point optimal tests for multi-dimensional testing.

In the context of a general regression model in which some regression coefficients are of interest and others are purely nuisance parameters, Bhowmik and King [6] constructed the locally best invariant (LBI) test against one-sided alternatives. This paper investigates whether this LBI test is uniformly most powerful invariant (UMPI) or not through simulation results. A test that is locally best invariant against one-sided alternative hypotheses is found to be uniformly most powerful invariant (UMPI) in a wider class of tests than the invariant tests for the standard F test. The results of a simulation study conducted to prove that the LBI test is UMPI are presented. Through simulation results this study proves that the LBI test is UMPI.

Over the last few decades there has been a growing literature on diagnostic tests of regression disturbances. Tests that have been constructed based on marginal likelihood methods have been found to do well when the disturbances are normally distributed. This paper investigates the small-sample size and power properties of marginal likelihood based tests when testing for random regression coefficients in the presence of first-order autoregressive disturbances. We find test sizes are less robust to non-normality as the sample size increases and the relative power performance of the various tests hardly changes as non-normality is introduced. Consequently marginal likelihood score based tests typically have the best power properties with a particular approximate point optimal invariant test providing some exceptions.

Mahmood and King [17] revealed that the marginal likelihood has the inherent property of unbiased estimating equations amongst a range of modified likelihood methods. In this paper we extend our investigation to deriving the mean squared error of the scores of the profile likelihood, the marginal likelihood, the conditional profile likelihood and the conditional profile restricted likelihood. In terms of minimum mean squared error, the estimating equation from the conditional profile restricted likelihood emerged as the preferred method. This provides further support to the implications of the findings of poor small-sample properties of Lagrange Multiplier (LM) tests in the literature which are based on biased estimating equations or having a larger mean squared error of the scores. We demonstrate that the relative error of the mean squared error between the conditional profile restricted and the marginal likelihood methods is negligible for increasingly larger samples. Amongst the unbiased estimating equations the minimum mean squared error criteria provides a clear choice of selecting the estimating equation for the purpose of estimation and testing.

The log-normal distribution arises in many different domains such as finance, stock prices, risk assessments, in motor and health insurance analysis, soil aggregate size distributions (in agriculture), lifetime distributions (quality engineering and survival analysis), telecommunications and traffic engineering, and was introduced to model inherently positive, continuous random phenomena. Positive observations from a continuous random variable (with constant variance) are generally analyzed either by log-normal or gamma models. However, in practice, the variance is not constant always. For handling non-constant variance in the log-normal process random variable distribution, some concomitant variables are included as regressor variables. In the present article, the response distribution is assumed to be log-normal, and the errors under the process are assumed to have a first-order autocorrelated structure. A log-linear composite autocorrelated errors regression model has been developed. The best linear unbiased estimators of all the regression coefficients have been derived except for the intercept which is often unimportant in practice. Autocorrelation coefficient has been estimated by iterative method. A testing procedure for any set of linear hypotheses regarding the unknown regression coefficients has been developed. Confidence intervals of an estimable function and confidence ellipsoids of a set of estimable functions of regression coefficients have been developed. An index of fit for the fitted regression model has also been developed. An example (with simulated data) illustrates the results derived in this report.

Autoregressive integrated moving average with exogenous variable-Generalized autoregressive conditional heteroscedastic (ARIMAX-GARCH) model is employed for describing volatile data by incorporating the exogenous variables in the mean-model. Brief description of this model along with its estimation procedure is discussed. For computing out-of-sample forecast using ARIMAX-GARCH model, one need to compute the out-of-sample forecast of exogenous variable first. In the present investigation, the forecasts for exogenous variable have been obtained by using ARIMA methodology as well as by wavelet analysis in frequency domain. As an illustration, wheat yield in Kanpur district of Uttar Pradesh, India with an exogenous variable as maximum temperature at critical root initiation (CRI) stage of wheat crop during 1972 to 2013 have been considered. The forecast of maximum temperature have been obtained using ARIMA and wavelet methodology. The forecast performance has been compared with respect to relative mean absolute prediction error (RMAPE). Finally forecast of wheat yield has been obtained by ARIMAX, ARIMAX-GARCH and ARIMAX-GARCH-WAVELET models. To this end comparison of forecast performance among above three models was carried out using Diebold-Mariano test along with mean absolute prediction error(MAPE), RMAPE and root mean squares error (RMSE). It is found that ARIMAX-GARCH-WAVELET model outperforms other models as far as modelling and forecasting is concerned.

The main objective of this paper is to outline the estimation and initialization procedures for the exponential smoothing with regressors forecasting approach which was recently introduced. The paper also discusses what restrictions need to be imposed during the estimation process so that the algorithm satisfies the forecastability conditions. An empirical study using real non-seasonal data shows that the new approach sometimes has the ability to produce better forecasts than the existing exponential smoothing methods without regressors.

The Central Limit Theorem (CLT) is an important result in statistics and econometrics and econometricians often rely on the CLT for inference in practice. Even though different conditions apply to different kinds of data, the CLT results are believed to be generally available for a range of situations. This paper illustrates the use of the Kullback-Leibler Information (KLI) measure to assess how close an approximating distribution is to a true distribution in the context of investigating how different population distributions affect convergence in the CLT. For this purpose, three different non-parametric methods for estimating the KLI are proposed and investigated. The main findings of this paper are 1) the distribution of the sample means better approximates the normal distribution as the sample size increases, as expected, 2) for any fixed sample size, the distribution of means of samples from skewed distributions converges faster to the normal distribution as the kurtosis increases, 3) at least in the range of values of kurtosis considered, the distribution of means of small samples generated from symmetric distributions is well approximated by the normal distribution, and 4) among the nonparametric methods used, Vasicek's [33] estimator seems to be the best for the purpose of assessing asymptotic approximations. Based on the results of this paper, recommendations on minimum sample sizes required for an accurate normal approximation of the true distribution of sample means are made.