1 Overview
1 is based on probability theory and mathematical statistics, based on research on experimental data collection, collation, analysis and concluded that a discipline.
2 In the framework of mathematical statistics, statistical inference is conclusions about the nature of the distribution information acquired from the information acquired data or information. In probability theory before learning, we usually start from a known distribution, understand the specific nature of this distribution, while as noted above, we are in the statistics through a set of data to look for it in line with the distribution and thus know his characteristics.
3 Specifically, statistical inference consists of two parts, which are estimated and hypothesis testing , it is estimated that is derived from a given sample set of sample data in line with what the distribution, hypothesis testing we infer that is above the estimated distribution is through a number of methods accurate.
2 basic concepts of mathematical statistics
Undergraduate mathematical statistics is the front of the base of the course. So I went back up a bit concepts.
Sample and sample distribution
Statistical model is a probability distribution, sample distribution
Probability theory is the basis of statistics, statistics is an application of probability theory
- Overall: estimating the weight of an object, with a balance weighing is repeated n times, results were recorded as X1, X2 ... Xn. This is the sample. But overall it should be understood as: "weighing all possible outcomes of an item set"
Overall Distribution: defined as when the sample size distribution of sample 1
And often the general population distribution as the same semantic statistically, but also to call general, such as "normal population" according to the type of population distribution
Statistical Inference
- Statistical Inference: From certain conditions and assumptions (samples and statistical models) and following certain methods or rules derived unknown (unknown parameters) of some kind of conclusion.
- Parameter space: according to the parameter value range where the nature of the parameter and the like, obtained
- Sample distribution family (the exact definition of the statistical model of a little of the statistical model is the sample distribution family): sample distribution contains some unknown parameters, then the sample distribution might be more than one, but a distribution family.
- Statistical Parameter Problem: parameter takes a real value, the parameter space is the Euclidean space part, the problem in this case is the statistical parameter called statistical issues.
Statistics and sampling distribution
- Statistics: the amount calculated by the sample (the sample and the information to solve the problem together), for example, in order to facilitate problem solving, calculate the arithmetic mean of the sample and so on. These are the statistics.
Statistics can only rely on the samples, can not depend on unknown parameters, because the statistic is used to infer the unknown parameters. But the amount of useful statistics must be related to the distribution of the parameters, otherwise the statistics will not contain any information about the parameters, so it can not infer the corresponding parameters
Such as: X1, ... Xn hypothesis with mean a, variance σ is too squared distribution. a is the unknown parameters. Statistics obeys X1-X2 of zero mean and variance 2σ square normal distribution, distribution of statistics X1-X2 with a nothing to do, naturally, to a meaningless inference
commonly used statistic: freedom, order statistics, extreme value, range and so on.
Distribution of sample: samples of random variables, there is a certain probability distribution, that is, the sample distribution.
Sampling Distribution: statistic because it is a known function of the sample, so he also has a probability distribution, the probability distribution of statistic called the sampling distribution of the statistics.
reference
"Probability and Statistics" with Wu Yi
Probability and Mathematical Statistics (Fourth Edition) Zhejiang University