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1. Statistic

1.1 Definition

A statistic, is a single measure of some attribute of a sample (e.g. sample mean). It is calculated by applying a function to the values of the sample.

More formally, statistical theory defines a statistic as a function of a sample where the function itself is independent of the sample’s distribution; that is, the function can be stated before realization of the data. The term statistic is used both for the function and for the value of the function on a given sample.

A statistic is distinct from a statistical parameter, which is not computable because often the population is much too large to examine and measure all its items.

  • A statistic is an observable random variable, computed on a sample.
  • A parameter is a generally unobservable quantity describing a property of a statistical population, which can only be computed exactly if the entire population can be observed without error.

However, a statistic, when used to estimate a population parameter, is called an estimator. For instance, the sample mean is a statistic that estimates the population mean, which is a parameter.

1.2 Types

When a statistic (a function) is being used for a specific purpose, it may be referred to by a name indicating its purpose:

  • in descriptive statistics, a descriptive statistic is used to describe the data;
  • in estimation theory, an estimator is used to estimate a parameter of the distribution (population);
  • in statistical hypothesis testing, a test statistic is used to test a hypothesis, e.g.
    • t statistics
    • chi-squared statistics
    • f statistics

1.3 Statistical Properties

Important potential properties of statistics include

  • completeness
  • consistency
  • sufficiency
  • unbiasedness
  • minimum mean square error
  • low variance
  • robustness
  • computational convenience

2. Statistical Hypothesis Test(ing)

A statistical hypothesis test is a method of statistical inference. In statistics, a result is called statistically significant if it has been predicted as unlikely to have occurred by chance alone, according to a pre-determined threshold probability, the significance level.

Statistical hypothesis testing is sometimes called confirmatory data analysis, in contrast to EDA, which may not have pre-specified hypotheses.

简单说，Statistical hypothesis testing 就是指

  1. 提出 $ H_0 $, $ H_a $
  2. 建立 test statistic
  3. 计算是否应该 reject hypothesis

这么一套流程和方法。

3. Test Statistic

A test statistic is a statistic used in statistical hypothesis testing.

4. t-test

A t-test is a statistical hypothesis test in which the test statistic follows a Student’s t distribution if the null hypothesis is supported.

5. p-value

以 t-test 为例。

在使用 t-test 时，如果 we assume $ H_0 $ is true，然后我们用的是一个 t-statistic following a Student’s t distribution，这时，我们手头上不是有一个 sample 嘛，我们用这个 sample 来算一下这个 t-statistic 的具体值，称为 t-value.

然后 p-value 就可以用来 answers this question: If my null hypothesis were true, what is the probability of getting a t-value at least as big as mine?

也就是 $ \text{p-value} = P(\text{t-statistic} \geq \lvert \text{t-value} \rvert \mid H_0 = true) $. Obviously, the lower this value is, the less likely it is that you would find a difference like yours by chance.

结合分位数的概念来看，当 p-value 越小时，t-value 越靠近 tail，说明在 $ H_0 = true $ 时取到这个 sample 对应的 t-value 的几率越小，于是我们越有信心来 reject $ H_0 $。

一般我们会给 p-value 取个阈值，常用的是 0.05，当 p-value < 0.05 时我们判定 reject $ H_0 $。这个阈值我们称为 Significance Level。

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