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Statistical Inference. Chapter 12/13. Statistical Inference. Given a sample of observations from a population, the statistical inference consists in estimating characteristics of the population. A characteristic may be guessed to: Be a number (point estimation) Lay within an interval.

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### Statistical Inference

### Interval Estimation(Single Sample)

Chapter 12/13

Statistical Inference

- Given a sample of observations from a population, the statistical inference consists in estimating characteristics of the population.
- A characteristic may be guessed to:
- Be a number (point estimation)
- Lay within an interval

COMP 5340/6340 Statistical Inference

Random Sampling

- A sample is a subset of observations out of a population (in general, the all population is not observable)
- For correct inference, sampling must be random

COMP 5340/6340 Statistical Inference

Point Estimate

- A statistic is any function of random variables constituting one or more samples, provided that the function does not depend on any unknown parameter values
- A point estimate of a parameter is a single number that can be regarded as the most plausible value of . A point estimate is obtained by selecting a suitable statistic and computing its value from a given sample. The selected statistic is called the point estimate of .

COMP 5340/6340 Statistical Inference

Example

- We want to evaluate the packet loss rate on a given channel. 25 packets are sent. Let X = number of corrupted (lost) packets. The parameter to be estimated is p = the proportion of lost packets
- Propose an estimator

COMP 5340/6340 Statistical Inference

Example (2)

- We assume that the waiting time for a bus is uniformly distributed. However, we do not know the upper limit of the probability distribution. We want to estimate this parameter of the uniform probability distribution.
- Propose an estimator

COMP 5340/6340 Statistical Inference

Sampling Distribution

- Multiple samples can be drawn
- Each sample may yield a different estimate
- Therefore, the estimate is a random variable with a probability distribution.
- In order to “evaluate” an estimate, we want to have an idea of its:
- Central tendency
- Variability

COMP 5340/6340 Statistical Inference

Unbiased Estimators

- A point estimator ˆis said to be an unbiased estimator of if E(ˆ) = for every possible value of . If is not unbiased, the difference E(ˆ) – is called the bias.
- Intuitively, an unbiased estimator is one that can equally underestimate or overestimate a given parameter.

COMP 5340/6340 Statistical Inference

Estimators

- Mean
- Median
- [Max(samples)-Min(samples)]/2
- Trimmed mean Xtr(10)

COMP 5340/6340 Statistical Inference

Some Unbiased Estimators

- If X is a binomial random variable with parameter n and p, the sample proportion p=X/n is an unbiased estimator of p.
- Let X1, X2,…Xn be a random sample from a distribution with mean and variance . Then the estimator
is an unbiased estimator of .

COMP 5340/6340 Statistical Inference

Some Unbiased Estimators (2)

- Let X1, X2,…Xnbe a random sample from a distribution with mean and variance . Then the estimator
is an unbiased estimator of .

- If the distribution is continuous and symmetric, then any trimmed mean is also an unbiased estimator.

COMP 5340/6340 Statistical Inference

Desirable Properties of Estimators

- Unbiased
- Minimal variance
- The precision of an estimator is measured by the standard error of the estimator, i.e.

COMP 5340/6340 Statistical Inference

Methods of Point Estimation

- Method of moments
- Maximum likelihood estimation (recommended for large samples)

COMP 5340/6340 Statistical Inference

Method of Moments

- Reminder: Definition of moments
- Definition: Let X1, X2,…Xn be a random sample from a p.m.f or p.d.f f(x). For k=1,2,3… the kth population moment or kth moment of the distribution f(x) is E(Xk). The kth sample moment is

- Let X1, X2,…Xn be a random sample from a p.m.f or p.d.f f(x, 1…. m) where 1…. m are the parameters whose values are unknown. Then, the moments estimators ˆ1…. ˆm are obtained by equating the first m sample moments to the first correspondng population moments and solving for 1…. m.

COMP 5340/6340 Statistical Inference

Example 1

- Let x1, x2,…xn represent a random sample of service time of n customers at some facility, where the underlying distributionis assumed exponential with parameter .
- How many parameters need to be estimated?
- What is the 1st sample moment?
- What is the moment estimator of

COMP 5340/6340 Statistical Inference

Example 2

- Let x1, x2,…xn represent a random sample from a gamma distribution with parameters and .
- Reminder: E(X) = and V(X) = .
- How many parameters need to be estimated?
- What is the 1st sample moment?
- What is the 2nd sample moment?
- What are the estimators for and

COMP 5340/6340 Statistical Inference

Example 3/Exercise

- Let x1, x2,…xn represent a random sample from a generalized negative binomial distribution with parameters r and p.
- Reminder: E(X) = ? and V(X) = ?.
- How many parameters need to be estimated?
- What is the 1st sample moment?
- What is the 2nd sample moment?
- What are the estimators for r and p

COMP 5340/6340 Statistical Inference

Maximum Likelihood Estimation

- Example:
- 10 packets are sent over a lossy channel with packet loss rate p. The 2nd, 4th, and 8th are lost. The joint p.mf. of this sample is:
- f(x1, x2,…x10;p) = (1-p)p(1-p)p(1-p)…p(1-p)(1-p)
= p3(1-p)7

- For what value of p is the observed sample most likely to have occurred?
- In other words, for which value of p is [p3(1-p)7] maximized?
- f(x1, x2,…x10;p) is maximized for the value of p such that

COMP 5340/6340 Statistical Inference

Maximum Likelihood Estimation

Definition:

- Let X1, X2,…Xn have a joint p.m.f or p.d.f. f(x1,x2,…,xn, 1…. m) where the parameters 1…. m take unknown values.

-f(x1,x2,…,xn, ˆ1…. ˆm) can be considered as a function of the parameters ˆ1…. ˆm and is called the likelihood function.

- The maximum likelihood estimates ˆ1…. ˆm are those values for the qi that maximize the likelihood function.

COMP 5340/6340 Statistical Inference

M.L.E Example

- Let x1, x2,…xn represent a random sample from an exponential distribution with parameter . Because of the independence, the likelihood function is a product of the individual p.d.f.’s.

- To maximize products, it is better to work with the ln (natural log)

COMP 5340/6340 Statistical Inference

M.L.E Example (2)

- To find the value of that maximizes the likelihood function, we derive

- We solve the equation

COMP 5340/6340 Statistical Inference

Introduction to Confidence Interval

- Simple case. We are interested in the following parameter: the population mean .
- Assume (unrealistically) that:
- The population distribution is normal
- The value of the population standard deviation is known.

COMP 5340/6340 Statistical Inference

Introduction to Confidence Interval (2)

- Let x1, x2,…xn represent a random sample from normal distribution with mean and standard deviation . The objective is to find a confidence interval of 95% for .
- What can we say of the random variable ?

- Probability distribution?
- Mean?
- Standard deviation?
- What is the standardized variable Z for Y?
- Using the normal distribution table:

Normal

COMP 5340/6340 Statistical Inference

Introduction to Confidence Interval (3)

Manipulating

We get

Substituting with the sample values

COMP 5340/6340 Statistical Inference

1-

0

100(1- )% Confidence Interval- Definition: a 100(1- )% confidence interval for the mean of a normal population when the value of is known is given by

COMP 5340/6340 Statistical Inference

1-

0

100(1- )% Confidence IntervalCommon Values- Any confidence is achievable (need normal distribution table)
- Common values used are 90%, 95% , and 99%

COMP 5340/6340 Statistical Inference

1-

0

Confidence, Sample Size, and Interval Length- Increasing confidence increases the interval length (sample size fixed)
- To increase confidence without increasing interval length, one must increase sample size n

COMP 5340/6340 Statistical Inference

Confidence Interval for the Population Mean

- If n est sufficiently large
has approximately a standard normal distribution.

This implies that

is a large-sample confidence interval for with confidence level approximately 100(1- )%.

COMP 5340/6340 Statistical Inference

Confidence Interval for the Population Proportion

- A large-sample 100(1- )% confidence interval for a population proportion is
where , n is the sample size, and x is the number of successes.

This interval is valid whenever

COMP 5340/6340 Statistical Inference

Exercise

- 1) What should be the sample size to achieve 100(1- )% confidence interval over an interval of length L? Assume p known
- 2) What should be the sample size to achieve 100(1- )% confidence interval over an interval of length L? Assume p UNknown

COMP 5340/6340 Statistical Inference

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