2 Law of Large Numbers and Central Limit Theorem
In this section we will:
- Introduce the Law of Large Numbers (LLN)
- We will use simulations to demonstrate the LLN
- Simulations -> Math
- Math -> Simulations
- Then we will create a simulation where the LLN doesn’t work
2.0.1 Conceptual overview
This is to illustrate the law of large numbers using some example data.
- The law of large numbers describes what happens to the empirical average as it is taken over increasing sample sizes.
Code
# read in student names
# each student is a researcher
students = read.csv('../../unshared/students2023.csv')
# read in data
puf = readRDS('../datasets/nsduh/puf.rds')
# ever tried cigarettes indicator
triedCigs = puf$cigflag
# make it a Bernoulli random variable
triedCigs = ifelse(triedCigs=='yes', 1, 0)
ns = c(10, 50, 100, 200, 300, 400, 500)
for(n in ns){
# Each person in the class is performing a study of smoking
studies = lapply(c(students$First.Name, 'Kun', 'Simon'), function(student) sample(triedCigs, size=n))
names(studies) = students$First.Name
# get the mean for each person's study
studyMeans = sapply(studies, mean)
# histogram of the study means
hist(studyMeans, xlim=c(0,1), breaks=10, main=paste0('n=', n))
abline(v=mean(triedCigs))
}
2.1 Law of Large Numbers
- The LLN is a tool we can use to say that the average of a sample is close to its expected value (and get’s closer with larger sample sizes).
- I wouldn’t consider an estimator that doesn’t satisfy the LLN
2.1.1 Simulations -> Math: illustrating the LLN with simulations
2.1.1.1 Initial simulation without convergence definition
- First, let’s choose a distribution.
- Then, let’s initialize empty vectors,
xandmeans. - For
n in 1:maxn,- Draw a sample from our distribution and add it to
x,x<-c(x,<new random sample>). - Compute the mean of
xand save it in another vectormeans[n]<- mean(x).
- Draw a sample from our distribution and add it to
- Plot the vector
means.
Code
# distribution we're sampling from
p = mean(triedCigs)
RVfunc = function(n) rbinom(n, size=1, prob=p)
# x is the observed sample
x = c()
# this is the mean of the observed sample
means = c()
# maximum sample size to get to
maxn=5000
# for loop through samples
for(n in 1:maxn){
xn = RVfunc(1)
x = c(x, xn)
means = c(means, mean(x))
}
plot(1:maxn, means, type='l', xlab='', ylab='', ylim=c(0,1))
# what is this converging to?
abline(h=p, lty=2)
What do you notice about this plot?
Let’s describe mathematically what we did.
- \(X_i \sim\), for \(n=1,2, \ldots\)
- \(\bar X_n = \frac{1}{n} \sum_{i=1}^n X_i\)
- \(\bar X_n \to\) what? What does this little arrow mean?
2.1.1.2 Convergence of random variables
Definition: \(Y_n\) converges to \(Y\) in probability if for all \(\epsilon>0\), \(\lim_{n\to\infty}\mathbb{P}(\lvert Y_n - Y \rvert \le \epsilon) = 1\) as \(n\to\infty\).
- “Eventually, every world will have \(Y_n\) within \(\epsilon\) of \(Y\).”
- “The probability that \(Y_n\) is within \(\epsilon\) of \(Y\) goes to 1.”
Definition: \(Y_n\) converges almost surely to \(Y\) if \(\mathbb{P}(\lim_{n\to \infty}\lvert Y_n - Y \rvert =0) = 1\) as \(n\to\infty\).
- “There is not a world where \(Y_n\) does not converge to \(Y\).”
- “The probability \(Y_n\) does not converge to \(Y\) is zero.”
2.1.2 Math -> Simulations: adding convergence definition
How do we incorporate one of these convergence definitions into our simulations?
- Someone in the class give
epsilon>0. - For each
sim in 1:nsim- Then, let’s initialize empty vector,
xand empty arraymeans=matrix(NA, nrow=nsim, ncol=maxn). - For
n in 1:maxn,- Draw a sample from our distribution and add it to
x,x<-c(x,<new random sample>). - Compute the mean of
xand save it in another vectormeans[sim,n]<- mean(x).
- Draw a sample from our distribution and add it to
- Then, let’s initialize empty vector,
- Use
meansto compute \(\mathbb{P}(\lvert \bar X_n - \mathbb{E}X_i \rvert < \epsilon)\). - Plot \(\mathbb{P}(\lvert \bar X_n - \mathbb{E}X_i \rvert < \epsilon)\) as a function
of
n.
Code
epsilon = 0.02
nsim = 500
# distribution we're sampling from
p = mean(triedCigs)
RVfunc = function(n) rbinom(n, size=1, prob=p)
# maximum sample size to get to
maxn=10000
# this is the mean of the observed sample
meansout = matrix(NA, nrow=nsim, ncol=maxn)
for(sim in 1:nsim){
cat(sim, '\t')
# for loop through samples
# x is the observed sample
x = c()
means = c()
x = RVfunc(maxn)
meansout[sim,] = cumsum(x)/1:maxn
# for(n in 1:maxn){
# xn = RVfunc(1)
# x = c(x, xn)
# means = c(means, mean(x))
# }
# meansout[sim,] = means
}## 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 Code
2.1.3 Formal definition of LLN
2.1.3.1 IID random variables
IID stands for independent and identically distributed.
For a sample \(X_i, \ldots, X_n\):
- Independence means that their probability distributions factor
- Identical means \(X_i \sim F_x\) (They all have the same distribution function).
2.1.3.2 Law of large numbers (Weak law)
- Weak LLN is about convergence in probability
Theorem (Durrett, pg. 61): Let \(X_1, X_2, \ldots\), be iid random variables with \(\mathbb{E}X_i = \mu\) and \(\mathbb{E}\lvert X_i \rvert <\infty\).
If \(S_n = X_1 + \ldots + X_n\), then as \(n\to \infty\), \[ S_n/n \to_p \mu. \]
Better weak laws can allow some dependence among the observations and usually just require that \(\mathbb{E}X_i^2 < \infty\) (a slightly stronger assumption than \(\mathbb{E}\lvert X_i \rvert <\infty\)). Check out google if you’re interested.
The strong law is about almost sure convergence and doesn’t require any further assumptions.
Intuition about LLN. What is the mean (expected value) of \(S_n/n\)? What is the variance of \(S_n/n\)?
- Mean
- Variance
2.1.4 Breaking the LLN: Cauchy distribution
- What is a Cauchy distribution (ratio of normal random variables; t RV on 1 DoF)
- The LLN can break if \(\mathbb{E}\lvert X_i \rvert\) doesn’t exist (is infinite). What does this mean?
It does not have any moments (\(\mathbb{E}X_i = \infty\)).
Let’s use the simulations we wrote above to show that the mean of Cauchy random variables does not converge.
Code
epsilon = 0.01
nsim = 500
# Changed this to a Cauchy random variable!
RVfunc = function(n) rcauchy(n)
# maximum sample size to get to
maxn=5000
# this is the mean of the observed sample
meansout = matrix(NA, nrow=nsim, ncol=maxn)
for(sim in 1:nsim){
cat(sim, '\t')
# for loop through samples
# x is the observed sample
x = RVfunc(maxn)
means = cumsum(x)/1:maxn
meansout[sim,] = means
}## 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 Code
Note: we’ve already taken advantage of the LLN in the class. Does anyone know how?
2.2 Central Limit Theorems (CLTs)
Today we’ll consider another theorem about a different type of convergence for random variables.
The goals for today are:
- Illustrate the CLT using Simulations -> Math
- Illustrate the CLT using the Math -> Simulation approach with the Bernoulli distribution
2.2.1 Preliminary: Normalizing sums of random variables
The CLT is about things that look like this:
- \(X_i \sim F\) (IID arbitrary distribution) with \(\mathbb{E}X_i = \mu\) and \(\text{Var}(X_i) = \sigma^2\).
- \(\bar X = n^{-1} \sum_{i=1}^n X_i\)
- \(Z = \sqrt{n} (\bar X - \mu)/\sigma\)
2.2.2 Preliminary: The normal distribution
Things about the normal distribution:
- Standard normal \(Z\sim N(0,1)\) often denoted with a \(Z\).
- PDF often denoted by \(\phi(z)\).
- CDF often denoted by \(\Phi(z)\).
- For \(Y \sim N(\mu, \sigma^2)\), \((Y-\mu)/\sigma \sim N(0, 1)\) (often called Z-normalization).
- \(\mathbb{P}(\lvert Z \rvert\le 1.96) = \Phi(1.96) - \Phi(-1.96) \approx 0.95\).
- \(\mathbb{P}( Z \le 1.64) = \Phi(1.64) \approx 0.95\).
2.2.3 Conceptual overview
This is to illustrate the CLT numbers using some example data.
Code
# read in student names
# each student is a researcher
students = read.csv('../../unshared/students2023.csv')
# read in data
puf = readRDS('../datasets/nsduh/puf.rds')
# ever tried cigarettes indicator
triedCigs = puf$cigflag
# make it a Bernoulli random variable
triedCigs = ifelse(triedCigs=='yes', 1, 0)
mu = mean(triedCigs)
sigma = sqrt(var(triedCigs))
ns = c(5, 10, 50, 100, 200, 300, 400, 500)
students = rep(c(students$First.Name, 'Kun', 'Simon'), 100)
layout(matrix(1:8, nrow=2, byrow=TRUE))
for(n in ns){
# Each person in the class is performing a study of smoking
studies = lapply(students, function(student) sample(triedCigs, size=n))
names(studies) = students
# get the mean for each person's study
studyMeans = sapply(studies, mean)
stdMeans = sqrt(n)*(studyMeans - mu)/sigma
# histogram of the study means
hist(stdMeans, xlim = c(-3,3), breaks=10, main=paste0('n=', n))
}
2.2.4 Preliminary: Convergence in distribution
Definition: A random variable \(Y_n\) with distribution function \(F_n(y)\) is said to converge in distribution to a limit \(Y\) with distribution function \(F(y)\) if, for all \(y\) that are continuity points of \(F\), \(F_n(y) \to F(y)\).
I’ll denote it \(Y_n\to_D Y\).
- Let’s unpack the notation \(F_n(y) \to F(y)\). This means for any given \(y\) and \(\epsilon>0\) there exists \(N\) such that for all \(n\ge N\), \[ \lvert F_n(y) - F(y)\rvert < \epsilon. \]
In other words, the distance between the distributions goes to zero as \(n\) gets larger.
2.2.5 The Central Limit Theorem
The Central Limit Theorem (Durrett, pg. 124): Let \(X_1, X_2, \ldots\) be iid with \(\mathbb{E}X_i = \mu\) and \(\text{Var}(X_i) = \sigma^2 \in (0, \infty)\).
If \(\bar X_n = n^{-1} \sum_{i=1}^n X_i\), then \[ n^{1/2}(\bar X_n - \mu)/\sigma \to_D X, \] where \(X \sim N(0,1)\).
Comments:
- We need the variance to be finite (stronger assumptions than LLN)
2.2.6 The Lindeberg-Feller Theorem
The Lindeberg-Feller Theorem (Wikipedia): Let \(X_i\) be independent random variables with \(\mathbb{E}X_i = \mu _i\) and variances \(\text{Var}(X_i) = \sigma^2_i \in (0, \infty)\). Let \(\sigma^2_n = \sum_{i=1}^n \sigma^2_i\).
If this sequence of random variables satisfies Lindeberg’s condition \[ \lim_{n\to\infty}\sigma^{-2}_n \sum_{i=1}^n \mathbb{E}\left\{(X_i-\mu_i)^2 I(\vert X_i - \mu_i\rvert>\epsilon \sigma_n) \right\} = 0, \] for all \(\epsilon>0\). Then \[ Z_n = \sum_{i=1}^n (X_i-\mu_i)/\sigma_n \to_D Z, \] where \(Z\sim N(0,1)\).
Comments:
Relaxes iid assumption to just independence!
CLTs are well studied, there are other ones for some what dependent variables.
Will review CLT statement
Will look at what it means for Bernoulli example and using simulations
2.2.7 Example of CLT with Bernoulli distribution
- The Bernoulli is appealing because we can assess the CLT mathematically.
- Sum of Bernoulli RV is Binomial.
- Using the CLT, for samples \(X_1, \ldots, X_n\), from this distribution the standardized mean is \[ \sqrt{n}\frac{(\bar X_n -p)}{p(1-p)} \sim N(0,1) \text{ (approximately)} \]
- \(\mathbb{P}(\sqrt{n} (\bar X - p)/\sqrt{p(1-p)} \le z) = \mathbb{P}\left(n\bar X \le \left(np(1-p)\right)^{1/2}\times z + np \right) \to \mathbb{P}(Z\le z)\),
where \(Z\sim N(0,1)\)
So how do we evaluate this:
- Pick a value for \(p\).
- Choose a vector for \(n\) and \(z\).
- Compare the CDFs by the sample size.
Note: We don’t even need to run simulations because we can do it using the CDFs in R.
Code
ns = 1:5000
p = 0.01
z = seq(-2, 2, length.out=1000)
probDiffs = rep(NA, length(ns))
for(n in ns){
probDiffs[n] = max(abs( pbinom(sqrt(n*p*(1-p))*z + n*p, size = n, prob=p) - pnorm(z) ) )
}
plot(ns, probDiffs, xlab='Sample size', ylab='Distribution error', main='CLT for Bernoulli', type='l')
abline(h=0, lty=2)
Code
2.2.8 QQ-plot!
2.2.8.1 Binomial proportion: Another look using simulations
Code
ns = seq(5, 100, by=10)
layout(matrix(1:10, ncol=5, byrow=TRUE))
p = 0.75
for(n in ns){
means = replicate(10000,
# compute mean 1000 times
sqrt(n) * (mean(rbinom(n, 1, p))-p) / sqrt(p*(1-p))
)
hist(means,main=paste('n =', n), probability = TRUE, ylab='', xlab='' )
# draw standard normal density
x = qnorm(ppoints(1000))
points(x, dnorm(x), type='l')
}
Code
2.2.10 Conclusions of CLT
- The CLT allows us to make approximate probability statements for things that can be expressed as sums.
- This is handy when we don’t know the distribution of things.
- Some form of independence is necessary.
2.2.11 Take home points of LLN and CLT
- LLN is about the convergence of the mean estimator to a constant.
- CLT is about the convergence of the mean estimator times \(\sqrt{n}\) to a normal distribution.
- CLT is used a lot in statistics to make approximate probability statements.
2.2.12 Example: HCP data set
A brain.
- Human Connectome Project data
- Study designed to understand how regions of the brain are interconnected
Code
hcp = read.csv('../datasets/hcp/hcp.csv')
corr = cor(hcp$FS_Total_GM_Vol, hcp$FS_TotCort_GM_Vol, use = 'pairwise.complete.obs')
plot(hcp$FS_Total_GM_Vol, hcp$FS_TotCort_GM_Vol, xlab='Gray Matter Vol', ylab='Cortical GM Vol', main='Cortical GM Volume vs total GM vol')
legend('topleft', legend=paste('Cor =', round(corr, 2) ))
abline(lm(FS_TotCort_GM_Vol ~ FS_Total_GM_Vol, data=hcp))
Code
Code
## [1] -0.09299307## [1] -0.09299307Code
## [1] -0.09299307Code
2.2.13 Simulated data
Code
set.seed(100)
# number of simulations per n and rho
nsim = 2
# sample sizes
ns = c(100)
# correlation
rhos = c(0.1)
resultsTab = expand.grid(n=ns, rho=rhos)
resultsTab$rhoBias = NA
# each column is a draw of x and y
for(rhoind in 1:length(rhos)){
rho = rhos[rhoind]
Sigma = matrix(c(1, rho, rho, 1), nrow=2)
sqrtSigma = svd(Sigma)
sqrtSigma = sqrtSigma$u %*% diag(sqrt(sqrtSigma$d))
for(nind in 1:length(ns)){
n = ns[nind]
# NEED TO DO SOMETHING HERE
# need a vector to store the correlation values in
simresults = rep(NA, nsim)
for(sim in 1:nsim){
xy = tcrossprod(matrix(rnorm(n*2), ncol=2), sqrtSigma)
# YOU NEED TO EDIT THE CODE HERE
simresults[sim] = cor(xy[,1], xy[,2])
}
# AND SOMETHING ELSE HERE (compute bias and assign in resultsTab)
}
}
2.2.9 Comment about dependence in the CLT
For a bunch of RVs with zero mean and nonzero covariance \[ \begin{aligned} \text{Var}(n^{-1/2}\bar X_n) & = n^{-1}\text{Cov}(S_n, S_n) \\ & = \sum_{i,j}^n \text{Cov}(X_i, X_j) \\ & = n^{-1}\sum_{i=1}^n \text{Var}(X_i) + n^{-1}\sum_{i\ne j} \text{Cov}(X_j, X_k)\\ & = \text{Var}(X_i) + (n-1)\text{Cov}(X_j, X_k)\\ &\text{(if Var and Cov are the same for all RVs)} \end{aligned} \]
Comments: