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# Statistical Inference - PowerPoint PPT Presentation

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

Chapter 12/13

• 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

### Point Estimation

• 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

• 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

• 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

• 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

• 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

• A point estimator ˆis 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

• Mean

• Median

• [Max(samples)-Min(samples)]/2

• Trimmed mean Xtr(10)

COMP 5340/6340 Statistical Inference

• 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

• 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

• 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

• 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 ˆ1…. ˆm 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

• 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

• 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

• 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

• 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

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, ˆ1…. ˆm) can be considered as a function of the parameters ˆ1…. ˆm and is called the likelihood function.

- The maximum likelihood estimates ˆ1…. ˆm are those values for the qi that maximize the likelihood function.

COMP 5340/6340 Statistical Inference

• 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

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

• We solve the equation

COMP 5340/6340 Statistical Inference

### Interval Estimation(Single Sample)

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

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Introduction to Confidence Interval (3)

Manipulating

We get

Substituting with the sample values

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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

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1-

0

100(1- )% Confidence IntervalCommon Values

• Any confidence is achievable (need normal distribution table)

• Common values used are 90%, 95% , and 99%

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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

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### Large Sample Confidence Intervals

-Population mean

- Proportion

• 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

• 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

• 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