1. Statistical Inference Chapter 12/13

2. COMP 5340/6340 Statistical Inference2 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

3. Point Estimation

4. COMP 5340/6340 Statistical Inference4 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

5. COMP 5340/6340 Statistical Inference5 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 .

6. COMP 5340/6340 Statistical Inference6 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

7. COMP 5340/6340 Statistical Inference7 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

8. COMP 5340/6340 Statistical Inference8 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

9. COMP 5340/6340 Statistical Inference9 Unbiased Estimators 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.

10. COMP 5340/6340 Statistical Inference10 Estimators Mean Median [Max(samples)-Min(samples)]/2 Trimmed mean Xtr(10)

11. COMP 5340/6340 Statistical Inference11 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 X 1, X 2,…X n be a random sample from a distribution with mean  and variance  . Then the estimator is an unbiased estimator of  .

12. COMP 5340/6340 Statistical Inference12 Some Unbiased Estimators (2) Let X 1, X 2,…X n be 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.

13. COMP 5340/6340 Statistical Inference13 Desirable Properties of Estimators Unbiased Minimal variance –The precision of an estimator is measured by the standard error of the estimator, i.e.

14. Methods of Point Estimation

15. COMP 5340/6340 Statistical Inference15 Methods of Point Estimation Method of moments Maximum likelihood estimation (recommended for large samples)

16. COMP 5340/6340 Statistical Inference16 Method of Moments Reminder: Definition of moments Definition: Let X 1, X 2,…X n be a random sample from a p.m.f or p.d.f f(x). For k=1,2,3… the k th population moment or k th moment of the distribution f(x) is E(X k ). The k th sample moment is Let X 1, X 2,…X n 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.

17. COMP 5340/6340 Statistical Inference17 Example 1 Let x 1, x 2,…x n 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 1 st sample moment? What is the moment estimator of 

18. COMP 5340/6340 Statistical Inference18 Example 2 Let x 1, x 2,…x n 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 1 st sample moment? What is the 2 nd sample moment? What are the estimators for  and  

19. COMP 5340/6340 Statistical Inference19 Example 3/Exercise Let x 1, x 2,…x n 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 1 st sample moment? What is the 2 nd sample moment? What are the estimators for r and p 

20. COMP 5340/6340 Statistical Inference20 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(x 1, x 2,…x 10 ;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(x 1, x 2,…x 10 ;p) is maximized for the value of p such that

21. COMP 5340/6340 Statistical Inference21 Maximum Likelihood Estimation Definition: - Let X 1, X 2,…X n have a joint p.m.f or p.d.f. f(x 1,x 2,…,x n,  1 ….  m ) where the parameters  1 ….  m take unknown values. -f(x 1,x 2,…,x n, 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.

22. COMP 5340/6340 Statistical Inference22 M.L.E Example Let x 1, x 2,…x n 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)

23. COMP 5340/6340 Statistical Inference23 M.L.E Example (2) To find the value of that maximizes the likelihood function, we derive We solve the equation

24. Interval Estimation (Single Sample)

25. COMP 5340/6340 Statistical Inference25 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.

26. COMP 5340/6340 Statistical Inference26 Introduction to Confidence Interval (2) Let x 1, x 2,…x n 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 

27. COMP 5340/6340 Statistical Inference27 Introduction to Confidence Interval (3) Manipulating We get Substituting with the sample values

28. COMP 5340/6340 Statistical Inference28 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 1-  0

29. COMP 5340/6340 Statistical Inference29 100(1-  )% Confidence Interval Common Values 1-  0 Any confidence is achievable (need normal distribution table) Common values used are 90%, 95%, and 99%

30. COMP 5340/6340 Statistical Inference30 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 1-  0

31. Large Sample Confidence Intervals -Population mean - Proportion

32. COMP 5340/6340 Statistical Inference32 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-  )%.

33. COMP 5340/6340 Statistical Inference33 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

34. COMP 5340/6340 Statistical Inference34 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