Some Estimating Problems Based On Grouped Data

Sometimes we may get the grouped data in the statistical analysis.The grouped data is a kind of uncompleted data. We could not get the values of the variable and what we could know is the intervals which include the values.Recently, how to estimate the parameters based on grouped data becomes a hot problem.We mainly discuss several estimating issues from grouped data. In the second part of the paper, we focus on the question that if the population distribution was known, the number of samples falling into each interval follows the multinomial distribution and the moment method of estimating the unknown parameters will be applied, after the introduction of maximum likelihood estimation based on grouped data. However, while the data are grouped data, we encounter the trouble that the explicit solution of the unknown parameter in the distribution function can be not expressed by the acquired estimating equation. Hence, Newton-Raphson method is introduced and regarded as a suitable choice, which can get the numerical solution of unknown parameter by iterations.However, we often do not know the distribution of the samples. So we prefer nonparametric methods in the third part of this paper. Empirical likelihood is a nonparametric statistical inference method, which does not depend on the specific distribution of population and completely relied on the likelihood ratio function which is determined by the sample observations. In this text, we mainly consider the empirical likelihood estimation of the mean based on the grouped data without the assumption of distribution. We will calculate the confidence interval or approximate confidence interval of the mean by empirical likelihood method, and demonstrate the feasibility of this method by means of simulations.