理解t-statistics||q-statistics||bias statistics||R2

• Statistical Hypothesis Testing

  A statistical hypothesis is a hypothesis that is testable on the basis of observed data modeled as the realised value taken by a collection of random variables.

  A statistical hypothesis test is a method of statistical inference.

  Statistical inference is the process of using data analysis to deduce(推论) properties of an underlying distribution of probability.Inferential statistical analysis infers properties of a population, for example by testing hypotheses and deriving estimates.

• t-statistic

  In statistics, the t-statistic is the ratio of the departure(离开，违背) of the estimated value of a parameter from its hypothesized value to its standard error.

  It is used in hypothesis testing via student’s t-test.

  The t-statistic is used in a T test to determin if you should support or reject the null hypothesis.

  It is very similar to the Z-score but with the difference that T-statistic is used when the sample size is small or the population stardard deviation is unknown.
  $$t_{\hat{\beta }}=\frac{\hat{\beta }-{\beta }_0}{s.e.(\hat{\beta })}$$
  ${\beta }_0$ a non-random, known constant which may or may not match the actual unknown parameter value $\beta$.

  $s.e.(\hat{\beta })$ is the standard error of the estimator $\hat{\beta }$ for $\beta$.

• q-statistic

  The q-statistic is a test statistic output by either the box-pierce test or , in a modified version which provides better small sample properties, by the Ljung-Box test.

  The q-statistic or studentized range statistic is a statistic used for multiple signigicance testing across a numbner of mean.

• bias-statistics

  未找到barra模型中提到的bias-statistics。

  只找到Statistical bias is a feature of a statistical technique or of its results whereby the expected value of the results differs from the true underlying quantitative parameter being estimated.

• Student’s t-test

  The t-test is any statistical hypothesis test in which the test statistic follows a Student’s t-distribution under the null hypothesis.

• null hypothesis

  In inferential statistics, the null hypothesis (often denoted $H_0$) is a general statement or default position that there is no relationship between two measured phenomena or no association among groups.

• Student’s t-distribution

  Student’s t-distribution (or simply the t-distribution) is any member of a family of continuous probability distributions that arises when estimating the mean of a normally distributed population in situations where the sample size is samll and the population standard deviation is unknown.

  It was developed by William Sealy Gosset under the pseudonym Student.

• $R^2$ coefficient of determination

  In statistics, the coefficient of determination, denoted $R^2$ or $r^2$ and pronounced “R suqared”, is the proportion(比例，部分) of the variance in the dependent variable that is predictable from the independent variables.

  R-squared is a statistical measure of how close the data are to the fitted regression line.
  $$R^2=\frac{Explainedvariation}{Totalvariation}$$
  $R^2$ is always between 0 and 100%:

  • 0% indicates that the model explains none of the variability of the response data around its mean;
  • 100% indicates that the model explains all the variability of the response data around its mean.

• References

1. Regression Analysis: How Do I Interpret R-squared and Assess the Goodness-of-Fit?