The paper was to reduce biased estimation using new approach (Penalized Maximum Likelihood Estimation (PMLE) Method) in Logistic Regression. For this aim, unreal four small data sets were randomly generated. Maximum Likelihood Estimation (MLE) and PMLE Methods were applied and compared for separation case including biased estimation in Logistic Regression when one of the cells in 2 x 2 tables becomes equal to zero (separation problem). Parameters1 and their standard error obtained by using MLE for four data sets were 12.56±257.8, 13.46±264.3, 13.42±210.3, and 13.41±180.4, respectively, meaning that MLE’s are biased estimates. Corresponding values for PMLE method were found 2.28 ± 1.81, 3.05 ± 1.59, 3.45 ± 1.53, and 3.45 ± 1.53, respectively, meaning that PMLE’s was unbiased estimates. It is clear that standard error value for data set 1 reduced from 257.8 to 1.81 when using PMLE method for separation problem. According to PMLE Method, the odds of being coronary heart disease risk for smokers were increased 21.08 times than that for non-smokers smoking in data set 2, which is significant at 1% level. The odds of being coronary heart disease risk for smokers were increased 31.63 times than that for non-smokers in data set 3 (P < 0.001). The odds of being coronary heart disease risk for smokers were increased 41.93 times than that for non-smokers in data set 4. When one of the cells in 2 x 2 contingency tables becomes equal to zero, PMLE was more superior to MLE Method because PMLE Method may be performed unbiased (reliable) estimation.

When modeling time series data using autoregressive-moving average processes, it is a common practice to presume that the residuals are normally distributed. However, sometimes we encounter non-normal residuals and asymmetry of data marginal distribution. Despite widespread use of pure autoregressive processes for modeling non-normal time series, the autoregressive-moving average models have less been used. The main reason is the difficulty in estimating the autoregressive-moving average model parameters. The purpose of this study is to address this intricacy by approximating maximum likelihood estimators, which is particularly important from model selection perspective. Accordingly, the coefficients and residual distribution parameters of the first-order stationary autoregressive-moving average model with residuals that follow exponential and Weibull families, were estimated. Then based on the simulation study, the obtained theoretical results were investigated and it was shown that the modified maximum likelihood estimators were suitable estimators to estimate the first-order autoregressive-moving average model parameters in non-normal mode. In a numerical example positive skewness of obtained residuals from fitting the first-order autoregressive-moving average model was shown. Following that, the parameters of candidate residual distributions estimated by modified maxim-um likelihood estimators and one of the estimated models for modeling the data was selected.

The present work focuses on the second order Markov chain model which arises in a variety of settings and is well-suited to be modeled in many applications. The efficiency of the maximum quasi-likelihood estimators with the full maximum likelihood estimators for second order Markov chain models are given, besides the limiting normality results on the asymptotic properties of the associated estimates. Some efficiency calculations are also given to discuss the feasibility and computational complexity of the QL approach relative to the full likelihood approach.

The information content of high frequency data has made them the main instruments for studying market microstructure. However, the noise content of this data may negatively affect the results of studies on market microstructure. Using maximum likelihood methodology, we disentangle from high frequency observations on the transaction prices of a sample of Tehran Stock Exchange stocks a fundamental component and a microstructure noise component. We then relate these statistical measurements of market microstructure noise to different financial measures of their market depth. We find that stocks with higher market depth have higher noise measured from their high frequency returns. This is in accordance with Fischer Black’s hypothesis that the existence of noise traders and the noise, which can be caused by the activities of this group of traders, to be the vital condition of a liquid market. We also find that pre-trade depth measures are the most powerful depth measure in explaining the noise in the market.