# Part 4 The James-Stein Estimator

## 4.1 James-Stein Estimator

In the recent book Computer Age Statistical Inference The author gives the following introduction to the James-Stein Estimator.

… maxiumum likelihood estimation has shown itself to be an inadequate and dangerous tool in many twenty-first-centry applications. Again speaking roughly, unbiasedness can be an unafforable luxury when there are hundreds or thousands of parameters to estimate at the same time

The James-Stein estimator made this point dramatically in 1961, and made is in the context of just a few unknown parameters, not hundreds or thousands. It begins the story of shrinkage estimation, in which deliberate biases are introduced to improve overall performance, at a possible danger to individual estimates.

### 4.1.1 Assertion

The assertion is that the James-Stein estimator shrinks estimates closer to the truth than the mean! This is a bold claim, one that we will show by simulation.

Of course, when you read about a new method in a paper the assertion, may not be written in simple text like above. You will usually see an equation of some sort. There is a good reason for this. Theorems are clear, they remove the ambiguity. Though daunting at first, it is good to practice with mapping equations into code.

### 4.1.2 James-Stein Theorem

For $$N \geq 3$$, the James-Stein estimator dominates the MLE in terms of expected total squared error; that is

$E\left[\left\lVert\hat{\mu}^{JS}-\mu\right\rVert^{2}\right]<E\left[\left\lVert\hat{\mu}^{MLE}-\mu\right\rVert^{2}\right]$

where $$x_i|\mu_i$$ is drawn from a distribution as follows

$x_i|\mu_i\sim N(\mu_i,1)$

Up to this point we have not covered a conditional draw from a distribution as stated in the above theorem, but you have already coded it in the previous chapter. The difference between $$x_i|\mu_i\sim N(\mu_i,1)$$ and what we have seen so far $$x\sim N(0,1)$$ is that we are allowing for the mean to vary. We see this in R as follows

# Conditional Normal
set.seed(42)
mu <- list(5,10,0)
mu %>%
map(rnorm,n=5) %>%
str()
List of 3
$: num [1:5] 6.37 4.44 5.36 5.63 5.4$ : num [1:5] 9.89 11.51 9.91 12.02 9.94
N = length(obs$MLE) obs %>% summarise(median(AB_total)) # A tibble: 1 x 1 median(AB_total) <dbl> 1 1624.5 df <- obs %>% mutate(sigma2 = (p_*(1-p_))/1624.5, JS=p_+(1-((N-3)*sigma2/(sum((MLE-p_)^2))))*(MLE-p_)) %>% select(-AB_total,-sigma2) head(df) # A tibble: 6 x 4 playerID MLE TRUTH JS <chr> <dbl> <dbl> <dbl> 1 bostoda01 0.2589053 0.2491442 0.2607254 2 burroje01 0.2718579 0.2606575 0.2709029 3 carrach01 0.2648840 0.2581826 0.2654231 4 cartwed01 0.2954784 0.2954784 0.2894627 5 eisenji01 0.3111380 0.2903630 0.3017672 6 flintsi01 0.2476435 0.2358393 0.2518764 In the plot below the mean (MLE), the James-Stein Estimator, and the True values are plotted. One thing to notice is that the James-Stein (JS) estimator shrinks estimates. Also, notice that some points are way off from the truth. Individually, the MLE seems to do better, but we are curious to know if the JS is better over all. This can be tested by calculating the predictive squared error. $\sum_{i=1}^{30}(MLE_i-TRUTH_i)^2 =$ errors <- df %>% mutate(mle_pred_error_i = (MLE-TRUTH)^2, js_pred_error_i = (JS-TRUTH)^2) %>% summarise(js_pred_error = sum(js_pred_error_i), mle_pred_error = sum(mle_pred_error_i)) Thus by example we prove the James-Stein Theorem $E\left[\left\lVert\hat{\mu}^{JS}-\mu\right\rVert^{2}\right]<E\left[\left\lVert\hat{\mu}^{MLE}-\mu\right\rVert^{2}\right]$ $3.0\times10^{-3}<4.4\times10^{-3}$ In this example the JS estimator beat the MLE by $$31\%$$! This is remarkable, but this was just one example. We ought to run our own simulation on the method, replicate the simulation on the method, then stretch the estimator to its limits with data tailored to be a challenge for it. ### 4.1.5 Simulation This example is nice because I only have to worry about having more than 3 samples, as stated in James-Steins theorem and we need to draw from the following distribution. $x_i|\mu_i\sim N(\mu_i,1)$ You will notice that in the baseball example we did above we used an approximation because we had binomial data. We will stick to this in out simulation, but you may want to run an analogous simulation using the parameters for the normal. #### 4.1.5.1 Binomial Approximation $x_i|p_in\sim N(p_in,p_in(1-p_i))$ $p_i\sim Bin(n,p_i)$ # build data with MLE estiamte set.seed(42) sim <- tibble(player = letters, AB_total=300, TRUTH = rbeta(100,300,n=26), hits_list = map(.f=rbinom,TRUTH,n=300,size=1) ) bats_sim <- sim %>% unnest(hits_list) %>% group_by(player) %>% summarise(Hits = sum(hits_list)) %>% inner_join(sim, by = "player") %>% mutate(MLE = Hits/AB_total) %>% select(-hits_list) bats_sim # A tibble: 26 x 5 player Hits AB_total TRUTH MLE <chr> <int> <dbl> <dbl> <dbl> 1 a 65 300 0.2362555 0.2166667 2 b 71 300 0.2590379 0.2366667 3 c 82 300 0.2660844 0.2733333 4 d 78 300 0.2600898 0.2600000 5 e 62 300 0.2474129 0.2066667 6 f 92 300 0.2929336 0.3066667 7 g 80 300 0.2476917 0.2666667 8 h 98 300 0.3126689 0.3266667 9 i 82 300 0.2484695 0.2733333 10 j 90 300 0.2859101 0.3000000 # ... with 16 more rows #calculate variables for JS estimator p_=mean(bats_sim$MLE)
N = length(bats_sim\$MLE)

# add JS to simulated baseball data
bats_sim <- bats_sim %>%
mutate(sigma2 = (p_*(1-p_))/300,
JS=p_+(1-((N-3)*sigma2/(sum((MLE-p_)^2))))*(MLE-p_)) %>%
select(-AB_total,-Hits,-sigma2)
# plot
bats_sim %>%
gather(type,value,2:4) %>%
mutate(is_truth=+(type=="TRUTH")) %>%
mutate(type = factor(type, levels = c("TRUTH","JS","MLE"))) %>%
arrange(player, type) %>%
ggplot(aes(x=value,y=type))+
geom_path(aes(group=player),lty=2,color="lightgrey")+
scale_color_manual(values=c("lightgrey", "skyblue"))+ #truth==skyblue
geom_point(aes(color=factor(is_truth)))+
guides(color=FALSE)+ #remove legend
labs(title="The James-Stein Estimator Shrinks the MLE")+
theme_minimal()

errors <- bats_sim %>%
mutate(mle_pred_error_i = (MLE-TRUTH)^2,
js_pred_error_i = (JS-TRUTH)^2) %>%
summarise(js_pred_error = sum(js_pred_error_i),
mle_pred_error = sum(mle_pred_error_i))
errors
# A tibble: 1 x 2
js_pred_error mle_pred_error
<dbl>          <dbl>
1    0.00752319     0.01685781

### 4.1.6 Excercise

Replicating Simulations: Using simulation test the James Stein estimator with the following sample sizes 2,3,10,30,100,1000.

This concludes our James-Stein estimator discussion, my hope is that the reader has gained some appreciation for the James-Stein estimator and learned a little about analyzing properties of statistical methods.