> For the complete documentation index, see [llms.txt](https://zavalab.gitbook.io/statschemebook/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://zavalab.gitbook.io/statschemebook/chapter-4-estimation-for-random-variables/05-data-collection-and-asymptotic-properties.md).

# Data Collection and Asymptotic Properties

* So far, we have assumed that we have data $$x\_\omega,; \omega \in \mathcal{S}$$.
* However, we have not discussed how this data is being *collected*.
* We also want to know how sample approximations and estimates $$\hat{\theta}$$ behave as we accumulate data.

We make following observations:

{% hint style="info" %}

* Data $$x\_\omega,; \omega \in \mathcal{S}$$ is a set of observations of $$X$$ collected from a population $$\Omega$$ by a defined procedure; i.e., sampling is a data collection procedure.

* A data sample sequence $$x\_\omega,; \omega \in \mathcal{S}$$ is called *random* if each sample $$x\_\omega$$ is drawn from same underlying pdf $$f\_X(x)$$ and if it is drawn *independently* from others; i.e., samples are independent and identically distributed (i.i.d.).

* If sample $$x\_\omega$$ is selected at random, the sample itself is an RV. Consequently, sometimes we denote a data sample sequence as a sequence of RVs $$X\_\omega,; \omega \in \mathcal{S}$$.
  {% endhint %}

* Random sample $$X\_\omega$$ has same probability $$1/S$$ of being selected and is *unbiased*.

* Lack of bias indicates that $$\mathbb{E}\[X\_\omega]=\mathbb{E}\[X]$$ (drawing sample many times and averaging results gives expected value of underlying RV $$X$$).

* Random samples can be used to construct *approximation* techniques with powerful asymptotic properties.

* Collecting data at random is not as easy as it sounds; one must ensure that there is no bias in selecting a sample (i.e., there is no hidden mechanism).

* As humans, it is difficult to select something randomly (due to inherent biases).

## Example: Random and Non-Random Samples

{% hint style="danger" icon="flask" %}

* Random sequence $$X\_\omega,,\omega \in \mathcal{S}$$ generated by drawing random samples from $$X\sim \mathcal{N}(0,1)$$. Samples distribute evenly around mean $$\mathbb{E}\[X]=0$$ and there is no bias.
* Nonrandom sequence $$X\_\omega,,\omega \in \mathcal{S}$$ is generated by drawing a random sample from $$\mathcal{N}(0,1)$$ and discarding it if it is above a value of one.
* This introduces a discrimination mechanism that biases sample. Note samples do not distribute evenly around $$\mathbb{E}\[X]=0$$ and there is a bias.
  {% endhint %}

<figure><img src="https://1753452229-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FakKE106umUCSWFtTnl31%2Fuploads%2Fgit-blob-cc4f351ed83c35e2803f0247349e1ca2b58c4b9c%2Fch4_random_sample.png?alt=media" alt="" width="350"><figcaption></figcaption></figure>

#### Python implementation

{% hint style="success" %}
Notebook [`ch4_random_sample.ipynb`](https://colab.research.google.com/github/vzavala/StatsBook/blob/master/Python/ch4_random_sample.ipynb) — open it in Colab and run the cells alongside this section.

MATLAB live script: [`ch4_random_sample.mlx`](https://github.com/vzavala/StatsBook/blob/master/Matlab-Live/ch4_random_sample.mlx)
{% endhint %}

Random and non-random samples (Victor Z, UW-Madison, 2024). A random sequence $$X\_\omega$$ from $$\mathcal{N}(0,1)$$ (`x_random`) and a non-random one that discards draws above one (`x_biased`). The bias is the average deviation from $$\mathbb{E}\[X]=0$$ (`E_X`).

**Note:** This notebook reproduces Example 4.5 (Section 4.3) of *Statistics for Chemical Engineers*: a random sample of N(0,1) and a sample selected with a hidden mechanism.

```python
import numpy as np
import matplotlib.pyplot as plt

np.random.seed(0)
S = 100
E_X = 0
x_random = np.random.normal(0, 1, S)
draws = np.random.normal(0, 1, 2 * S)
x_biased = draws[draws <= 1][:S]           # discrimination mechanism: discard x > 1

print(f"bias (random)     = {np.mean(x_random - E_X):+.3f}")
print(f"bias (non-random) = {np.mean(x_biased - E_X):+.3f}")

fig, axs = plt.subplots(2, 1, figsize=(10, 7))
for ax, x_omega, title in ((axs[0], x_random, 'random'), (axs[1], x_biased, 'non-random')):
    ax.plot(np.arange(1, S + 1), x_omega, 'o', markerfacecolor='w', markeredgecolor='black')
    ax.axhline(E_X, color='k', linewidth=1.5)
    ax.set_title(title)
    ax.axis([0, S, -4, 4])
    ax.set_xlabel(r'$\omega$')
    ax.set_ylabel(r'$x_\omega$')
    ax.grid(True)
plt.tight_layout()
plt.show()
```

```
bias (random)     = +0.060
bias (non-random) = -0.214
```

<figure><img src="https://1753452229-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FakKE106umUCSWFtTnl31%2Fuploads%2Fgit-blob-c16a8dd7c4a544d891120029cd9531a8ba38a85f%2Fch4_random_sample-03-1.png?alt=media" alt="" width="520"><figcaption></figcaption></figure>

**Note:** As in the book, discarding observations above a threshold biases the sample: its mean is shifted toward negative values (bias of about -0.43).