Statistical Inference - Implement in R

1. 分布函数

1.1. Basis

R语言中提供了 4 类统计分布的函数，以下为函数和相应前缀：

• d (x, distparams, log = FALSE) ：概率密度函数，PDF，PMF
  • x：数字或向量
  • log ：logical; if TRUE, will return log(p).
    • d(x, log.p=T) = log(d(x, log.p=F))
  • distparams：分布函数的参数，见下表
• p (q, distparams, lower.tail = TRUE, log.p = FALSE)： 累计分布函数，CDF
  • q：分位数
  • lower.tail：logical; if True, \(p(x)=P(X\leq x)\), else if FALSE, \(p(x)=P(X > x)\)
  • log.p：logical; if TRUE, will return log(p).
    • p(q, log.p=T) = log(p(q, log.p=F))
• q (p, distparams, lower.tail = TRUE, log.p = FALSE) ：分位函数，Quartile
  • p：probability
  • log.p：
    • q(q, log.p=T) = q(log(q), log.p=T)
• r (n, distparams) ：随机数函数（抽样）Random
  • n：the number of sample

下表为分布函数表，加上不同的前缀表示不同的含义：

1.2. Discrete Distribution

Distribution Name	Mathematical Expression	Function Name in R	Distribution parameters
binomial	\(p(x) = {n \choose x} {p}^{x} {(1-p)}^{n-x} \quad x = 0, \ldots, n\)	binom	size：\(n\) prob：\(p\)
Poisson	\(p(x) = \dfrac{\lambda^x e^{-\lambda}}{x!} \quad x = 0, 1, 2, \ldots\)	pois	lambda：\(\lambda\)
geometric	\(p(x) = p {(1-p)}^{x}\) \(x = 0, 1, 2, \ldots; \quad 0 < p \le 1\)	geom	prob：
hypergeometric	$ p(x) = \dfrac{{m \choose x}{n \choose k-x}}{{m+n \choose k}} \quad x = 0, \ldots, k$	hyper	m: n： k：
negative binomial	\(p(x) = \dfrac{\Gamma(x+n)}{\Gamma(n) x!} p^n (1-p)^x = {x+n-1 \choose x} p^n (1-p)^x\) \(x = 0, 1, 2, \ldots, \quad n > 0, \quad 0 < p \le 1\)	nbinom	size：\(n\) prob：\(p\) mu：alternative parametrization

1.3. Continuous Distribution

Distribution Name	Mathematical Expression	Function Name in R	Distribution parameters
uniform	\(f(x) = \dfrac{1}{\max-\min} \quad \min \le x \le \max\)	unif	min： max：
normal	\(f(x) = \dfrac{1}{\sqrt{2\pi}\sigma} e^{-\frac{(x-\mu)^2}{2\sigma^2}}\)	norm	mean=0：\(\mu\) sd=1：\(\sigma\)
logistic	\(f(x)= \dfrac{1}{\sigma}\dfrac{e^{(x-\mu)/\sigma}}{(1 + e^{(x-\mu)/\sigma})^2}\) $ F(x) = \dfrac{1}{1 + e^{-(x-\mu)/\sigma}}% $	logis	location=0：\(\mu\) scale=1：\(\sigma\)
exponential	\(f(x) = \lambda {e}^{- \lambda x} \quad x \geq 0\)	exp	rate=1：\(\lambda\)
chi-squared	\(f_n(x) = \dfrac{1}{{2}^{\frac{n}{2}} \Gamma (\frac{n}{2})} {x}^{\frac{n}{2}-1} {e}^{-\frac{x}{2}} \quad x>0\)	chisq	df：\(n\) ncp=1：for non-central chi-squared distribution
Student's	\(f(x) = \dfrac{\Gamma (\frac{\nu+1}{2})}{\sqrt{\pi \nu} \Gamma (\frac{\nu}{2})} \left(1 + \dfrac{x^2}{\nu}\right)^{-\frac{\nu+1}{2}}\)	t	df：\(\nu\) ncp：for noncentral t-distribution
F	\(f(x) = \dfrac{\Gamma \left(\frac{n_1}{2} + \frac{n_2}{2} \right)}{\Gamma(\frac{n_1}{2}) \Gamma(\frac{n_2}{2})} \left(\dfrac{n_1}{n_2}\right)^{\frac{n_1}{2}} x^{\frac{n_1}{2} -1} \left(1 + \dfrac{n_1}{n_2} x \right)^{-\frac{n_1+n_2}{2}}\) \(x>0\)	f	df1：\(n_1\) df2：\(n_2\) ncp：for noncentral F-distribution
Gamma	$ f(x)= \dfrac{1}{{\sigma}^{\alpha}\Gamma(\alpha)} {x}^{\alpha-1} e^{-\frac{x}{\sigma}} ={\dfrac {\beta ^{\alpha }x^{\alpha -1}e^{-\beta x}}{\Gamma (\alpha )}}$ \(x \ge 0, \quad \alpha, \sigma, \beta> 0\)	gamma	shape：\(\alpha\) rate = 1：\(\beta\) scale = 1/rate：\(\sigma\)

2. 统计检验

2.1. Function in R

2.1.1. \(z\)-test in

z.test(x, y = NULL,
  	   alternative = "two.sided",
       mu = 0, sigma.x = NULL, sigma.y = NULL, 
       conf.level = 0.95)

parameters:

• x：samples 1

• y：samples 2. y=NULL indicates one-sample test

• alternative：string and one of"two.sided", "less", "greater". Alternative hypothesis.

  • alternative = "greater" represent two-side test

  • alternative = "less" or alternative = "greater" represents one-side test

  • alternative = "greater" is the alternative that x has a larger mean than y.

• mu：\(\mu_0\)

  • for one-sample test: \(H_0: \mu = \mu_0\)
  • for two-sample test: \(H_0: \mu_1 - \mu_2 = \mu_0\)

• sigma.x：\(\sigma_1\), the population standard deviation for x

• sigma.y：\(\sigma_2\), the population standard deviation for y

• conf.level：confidence level of the interval, equal to 1 minus significance level: \(1-\alpha\)

2.1.2. \(t\)-test in

t.test(x, y = NULL,
       alternative = "two.sided",
       mu = 0, paired = FALSE, var.equal = FALSE,
       conf.level = 0.95, …)

• mu：\(\mu_0\)

  • for one-sample test: \(H_0: \mu = \mu_0\)
  • for two-sample test: \(H_0: \mu_1 - \mu_2 = \mu_0\)

• var.equal：Only effect for two-sample test, whether the variances of two population is equal or not.

  • var.equal=TRUE：\(\sigma_1^2=\sigma_2^2\)
  • var.equal=False：\(\sigma_1^2 \neq \sigma_2^2\)

• conf.level：confidence level of the interval, equal to 1 minus significance level: \(1-\alpha\)

• paired：logical, indicating whether a paired t-test or not

2.1.3. Binomial test

binom.test(x, n, p = 0.5,
           alternative = c("two.sided", "less", "greater"),
           conf.level = 0.95)

• x：number of successes, or a vector of length 2 giving the numbers of successes and failures, respectively.
• n：number of trials; ignored if x has length 2.
• p：\(p_0\) hypothesized probability of success.

2.1.4. \(F\)-test (variances test)

var.test(x, y, ratio = 1,
         alternative = c("two.sided", "less", "greater"),
         conf.level = 0.95, …)

• ratio: \(H_0: \sigma_1^2 / \sigma_2^2 =\text{ratio}\). the hypothesized ratio of the population variances of x and y.

2.1.5. \(\chi^2\)- test

chisq.test(x, y = NULL, correct = TRUE,
           p = rep(1/length(x), length(x)), rescale.p = FALSE,
           simulate.p.value = FALSE, B = 2000)

• x: a numeric vector or matrix; x and y can also both be factors.

• y: a numeric vector or matrix

  • ignored if x is a matrix
  • If x is a factor, y should be a factor of the same length

• correct:

• p: probabilities of the same length of x.

• rescale.p：

  • if rescale.p=TRUE then p is rescaled (if necessary) to sum to 1.
  • If rescale.p=FALSE , and p does not sum to 1, an error is given.

• simulate.p.value: logical; indicating whether to compute p-values by Monte Carlo simulation.

• B: an integer specifying the number of replicates used in the Monte Carlo test.

2.2. Application

2.1.1. Inference on the mean \(\mu\) of populations

• The population mean of one sample

  • One-sample Gauss Test (\(z\)-test): test for the population mean when the population variance \(\sigma\) is known

    z.test(x, ...)

  • One-sample \(t\)-test: Test for the population mean When the population variance is Unknown

  t.test(x, var.equal=T, ...)

• One-sample binomial test for the probability \(p\)

  binom.test(x, n, p, ...)

• Comparing the population mean of two independent samples

  • Two-sample Gauss Test (\(z\)-test): The population variances are known \(\sigma_1^2\), \(\sigma_2^2\)

    z.test(x, y, ...)

  • **Pooled \(t\)-test: ** The population variances are unknown, but equal \(\sigma_1^2=\sigma_2^2\).

  t.test(x, y, var.equal=T, ...)

  • **Welch's \(t\)-test: ** The population variances are unknown and unequal

    t.test(x, y, var.equal=F, ...)

  • Paired \(t\)-test: Test for comparing the population mean of two dependent samples

    t.test(x, y, paired=T, ...)

• Testing the ratio of two population variances

  var.test(x, y, ratio, ...)

2.1.2. Statistical Test

• \(\chi^2\) Goodness-of-Fit Test

  • \(H_0\): \(F(x)=F_0(x)\)

  • \(H_1\): \(F(x)\neq F_0(x)\)

    chisq.test(x, p)

    • x: the observed absolute frequencies
    • p: are calculated from the assumed distribution of \(F_0(x)\) under \(H_0\)

• \(\chi^2\) Independence Test

  • \(H_0\): The two classification variables are statistically independent.

  • \(H_1\): The two classification variables are not statistically independent.

    chisq.test(x=matrix)