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Mathematical statistics is the application of mathematical theory, especially probability theory to statistical problem solving. Many mathematical methods including stochastic analysis, linear algebra, mathematical analysis, differential equation and the theory of measure are frequently used in mathematical statistics. It is devoted to the study of mathematical methods for organizations, and to process and utilize statistical data in order to derive practical conclusions.
Mathematical statistics in this regard differs from the techniques of collecting statistical data. Statistical data here refers to the collection of objects which have some specific characteristics defined for the purpose of the study. The collection of statistical data involves the planning of studies—mostly the designing of randomized experiments, and the planning of random sampling and surveys. Primary analysis is usually based on the study protocol defined at the beginning of the proposed study. Data from such studies and their initial results can also be analyzed in the consideration of secondary test hypothesis. The process of using data analysis tools to perform secondary analysis of the study is called mathematical statistics.
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The description of data or the collection of objects has an intermediary position in a statistical study. It is positioned between the discrete descriptions of each individual object in the collection and the combined description of the entire collection by their common characteristics without breaking it down to separate objects. In comparison with the first method, statistical data are always collective, at least to some extent and are of very limited value to the study where the focus is on gaining insights into individual objects. Also, when compared to the second method which involves studying the collective data from the outside and summarizing the results on the basis of common characteristics, statistical data provide a much deeper insight into the crux of the matter.
Statistical method is the method of research identified as the thorough discussion of statistical data on different collections of objects. It can be applied in several and diverse areas of knowledge. However, the attributes of statistical method when applying to different types of objects are highly specific, making it meaningless to integrate for example, physical statistics, social statistics, economic statistics and many other types of statistics into one science. The common attributes of statistical method in all areas of knowledge come down to mathematical calculations, discussion and distribution of quantitative properties, the use of probability theory in estimating the adequacy of the number of observations to draw conclusions and the application of sampling methods among others. This conventional mathematical aspect of statistical research remains unaffected by the distinct nature of objects under study and forms the basis for mathematical statistics
The application of probability theory is one of the fundamental aspects of mathematical statistics. It is used to study random or to say, probabilistically random phenomena and not simply any mass phenomena. It is therefore best suited for phenomena or cases in which it is logical and coherent to discuss associated probability distributions. This is not to say that probability theory cannot be applied to other kinds of mass phenomena. On the contrary, it can be applied to the statistical study of even those phenomena which are significantly different to probabilistically random phenomena. Such application can be seen is the use of theories of error and sampling method. It should be noted however that in such a case, the subject of the laws of probability is the means of investigation and not the phenomenon itself.
Still, probability theory plays a much more important role in the study of probabilistically random phenomena where the investigator can apply in full measure the probabilistically based concepts of mathematical statistics such as hypothesis testing, statistical estimation of probability distributions, their parameters, etc. The scope of application of these extensive statistical methods is quite limited given the requirement that probabilistically random phenomena must be subjected to reasonably defined laws of probability. The statistical expression of probability laws is based on the law of large numbers for the fact that probabilities are obtained and described approximately as frequencies and expectations as averages.
Probability distribution is the function of probability theory which assigns a specific probability to every measurable subset of all possible outcomes of a random study or experiment, survey or the statistical inference procedure. Some common examples include experiments in which the sample space is non-numerical and the distribution is categorical; experiments in which the sample space is concealed by discrete random variables and the distribution is denoted by a probability mass function; and experiments in which the sample space is concealed by continuous random variables and the distribution is denoted by a probability density function. Complex experiments on the other hand may require the use of many other general measures of probability.
A key example is the experiment involving stochastic processes which are defined in continuous time. Probability distributions can be univariate or multivariate. A univariate distribution describes the probabilities of a single random variable acquiring different alternative values. Examples of commonly seen and important univariate distributions include the normal distribution, the binomial distribution and the hyper-geometric distribution. A multivariate distribution describes the probabilities of two or more random variables (also called a random vector) taking on different combinations of values. Important example here is the multivariate normal distribution.
Regression analysis in statistics is a process for estimating or measuring the relationship among variables. It comprises of various methods of modeling and analyzing multiple variables. The focus of this analysis is the relationship of a dependent variable with two or more independent variables. It helps the investigator in understanding how the normal value of a dependent variable changes when the value of an independent variable is changed while the other independent variables remain unchanged. It is commonly used to estimate the conditional expectation of the dependent variable with respect to independent variables. In some cases, the focus is on identifying the location parameters of the dependent variables’ conditional distribution with respect to independent variables. In each case the target for estimation is the regression function which is a function of independent variables.
While performing the regression analysis, investigators are also interested in characterizing the variations in the value of the dependent variable on the basis of the regression function described by a probability distribution. Most common methods of regression analysis such as linear regression are usually parametric. This means that regression function is defined in relation to a finite number of unspecified parameters which need to be estimated using the given data. In contrast, nonparametric regression comprises of techniques in which the regression function can lie in a specified set of functions with an infinite number of parameters.
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