03/31/2021
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The following topics will be covered in this lecture:
We have now learned about the fundamentals of theoretical probabilistic models.
Particularly, we have learned about the:
for discrete and continuous random variables.
We have also learned about several fundamental probability distributions:
We will begin to combine these models with data to produce statistical inference.
Our goal in this course is to use statistics from a small, representative sample to say something general about the larger, unobservable population or phenomena.
The process of saying something general from the smaller representative sample, while qualifying our uncertainty, is what we mean by statistical inference.
The link between the probability models in the earlier chapters and the data is made as follows.
Suppose we take a sample of \( n = 10 \) observations \( \{x_{1,i}\}_{i=1}^{10} \) from a population and compute the sample average,
\[ \overline{x}_1 = \frac{1}{n} \sum_{i=1}^n x_{1,i} = \frac{1}{10}\sum_{i=1}^{10} x_{1,i} \]
getting the result \( \overline{x}_1 = 10.2 \).
Now we repeat this process, taking a second sample of \( n = 10 \) observations from the same population,
\[ \{x_{2,i}\}_{i=1}^{10} \]
and the resulting sample average is \( \overline{x}_2=10.4 \).
This discrepancy is what we call sampling error, in which the random variation in a sample of a fixed size \( n \) upon replication produces differences in the computation of a statistic.
The sample average depends on the observations in the sample, which differ from sample to sample because they are random variables.
Consequently, the sample average (or any other function of the sample data) is a random variable.
Because a statistic is a random variable, it has a probability distribution.
Specifically, suppose that we want to obtain an estimate of a population parameter, where the population is modeled with a random variable \( X \).
We know that before the data are collected, the observations are considered to be random variables,
\[ X_1, X_2, \cdots , X_n \]
Random sample
The random variables \( X_1 , X_2, \cdots , X_n \) are a random sample of size \( n \) if the \( X_i \)’s are independent random variables and every \( X_i \) has the same probability distribution.
We then say that the measurements we obtain are possible outcomes of the sample variables \( \{X_i\}_{i=1}^n \); particularly, if we make a computation of the sample mean,
\[ \overline{X} = \frac{1}{n} \sum_{i=1}^n X_i \]
the above is treated as a random variable (a linear combination of random variables) which has a random outcome, dependent on the realizations of the \( X_i \).
More generally, any function of the observations, i.e., any statistic, is also modeled as a random variable.
If \( h \) is a general function used to compute some statistic, we thus define
\[ \tilde{X} = h(X_1, \cdots, X_n) \]
to be a random variable that will depend on the particular realizations of \( X_1,\cdots, X_n \).
We call the probability distribution of a statistic a sampling distribution.
Sampling Distribution
The probability distribution of a statistic is called a sampling distribution.
Given particular realizations of the sample random variables, we obtain a fixed numerical value.
Each numerical value in a data set is treated as the observed realization of a random variable.
Given particular realizations \( x_1,\cdots,x_n \) of the random variables \( X_1, \cdots, X_n \), the value
\[ \overline{x} = \frac{1}{n}\sum_{i=1}^n x_i \]
is not a random variable, as this is a fixed numerical value.
Given some particular, observed realizations \( x_1, \cdots,x_n \),
\[ \tilde{x} = h(x_1, \cdots, x_n) \]
is a fixed numerical value, based on the fixed, observed data values \( x_1, \cdots, x_n \).
When discussing inference problems, it is convenient to have a general symbol to represent the parameter of interest – we use the Greek symbol \( \theta \) (theta) to represent the parameter.
The symbol \( \theta \) can represent the mean \( \mu \), the variance \( \sigma^2 \), or any parameter of interest to us.
The objective of point estimation is to estimate a single number based on sample data that is the most plausible value for \( \theta \).
The numerical value of a sample statistic is used as the point estimate.
Once we describe the process of point estimation, the next step is to describe how we quantify the uncertainty of the estimate.
If \( X \) is a random variable with probability distribution \( F(x) \), characterized by the unknown parameter \( \theta \),
and if \( X_1 , X_2, \cdots , X_n \) is a random sample of size \( n \) from \( X \),
the statistic \( \hat{\Theta} = h(X_1 , X_2 , ... , X_n ) \) given as a function of the sample is called a point estimator of \( \theta \).
Note that \( \hat{\Theta} \) is a random variable because it is a function of random variables.
After the sample has been selected, \( \hat{\Theta} \) takes on a particular numerical value \( \hat{\Theta} \) called the point estimate of \( \theta \).
The uncertainty of the point estimate \( \hat{\Theta} \) can be understood as how much will the sampling error cause a discrepancy between \( \hat{\Theta} \) and the true \( \theta \).
Point estimators
A point estimate of some population parameter \( \theta \) is a single numerical value \( \hat{\Theta} \) of a statistic \( \hat{\Theta} \). This is a particular realization of the random variable \( \hat{\Theta} \), viewed as a random variable; \( \hat{\Theta} \) is called the point estimator.
Estimation problems modeled as above occur frequently in engineering.
We often need to estimate
Reasonable point estimates of these parameters are as follows:
Generally, however, we may have several different choices for the point estimator of a parameter.
To decide which point estimator of a particular parameter is the best one to use, we need to examine their statistical properties and develop some criteria for comparing estimators.
Let's consider a simple argument for the sampling distribution of the sample mean \( X \).
Suppose that a random sample of size \( n \) is taken from a normal population with mean \( \mu \) and variance \( \sigma^2 \).
By definition of a random sample each observation in this sample, say, \( X_1, X_2, \cdots, X_n \), is a normally and independently distributed random variable with mean \( \mu \) and variance \( \sigma^2 \).
A special property of the normal distribution is that it can be translated and rescaled while remaining normal;
We conclude that the sample mean
\[ \overline{X}= \frac{X_1 + X_2 + \cdots + X_n}{n} \]
has a normal distribution with mean
\[ \mu_\overline{X} = \frac{\mu + \mu + \cdots + \mu}{n} = \mu \]
\[ \sigma^2_\overline{X} = \frac{\sigma^2 + \sigma^2 + \cdots + \sigma^2}{n^2} = \frac{\sigma^2}{n} \]
More generally, if we are sampling from a population that has an unknown probability distribution, the sampling distribution of the sample mean will still be approximately normal with mean \( \mu \) and variance \( \frac{\sigma^2}{n} \) if the sample size \( n \) is large.
This is one of the most useful theorems in statistics, called the central limit theorem:
The central limit theorem
Let \( X_1 , X_2 , \cdots , X_n \) be a random sample of size \( n \) taken from a population with mean \( \mu \) and finite variance \( \sigma^2 \) and \( \overline{X} \) be the sample mean. Then the limiting form of the distribution of \[ Z = \frac{X - \mu}{\frac{\sigma}{\sqrt{n}}} \] as \( n \rightarrow \infty \) is the standard normal distribution.
Put another way, for \( n \) sufficiently large, \( \overline{X} \) has approximately a \( N\left(\mu, \frac{\sigma^2}{n}\right) \) distribution – this says the following.