1. Point and interval estimations of parameters of the normally up-diffused sign. Concept of statistical evaluation.

2. Estimation of Population Parameters
Statistical inference refers to making inferences about a population parameter through the use of sample information
The sample statistics summarize sample information and can be used to make inferences about the population parameters
Two approaches to estimate population parameters
Point estimation: Obtain a value estimate for the population parameter
Interval estimation: Construct an interval within which the population parameter will lie with a certain probability

3. Point Estimation
In attempting to obtain point estimates of population parameters, the following questions arise
What is a point estimate of the population mean?
How good of an estimate do we obtain through the methodology that we follow?
Example: What is a point estimate of the average yield on ten-year Treasury bonds?
To answer this question, we use a formula that takes sample information and produces a number

4. Point Estimation
A formula that uses sample information to produce an estimate of a population parameter is called an estimator
A specific value of an estimator obtained from information of a specific sample is called an estimate
Example: We said that the sample mean is a good estimate of the population mean
The sample mean is an estimator
A particular value of the sample mean is an estimate

5. Point Estimation
Note: An estimator is a random variable that takes many possible values (estimates)
Question: Is there a unique estimator for a population parameter? For example, is there only one estimator for the population mean?
The answer is that there may be many possible estimators
Those estimators must be ranked in terms of some desirable properties that they should exhibit

6. Properties of Point Estimators
The choice of point estimator is based on the following criteria
Unbiasedness
Efficiency
Consistency
A point estimator is said to be an unbiased estimator of the population parameter  if its expected value (the mean of its sampling distribution) is equal to the population parameter it is trying to estimate

7. Properties of Point Estimators
Interesting Results on Unbiased Estimators
The sample mean, variance and proportion are unbiased estimators of the corresponding population parameters
Generally speaking, the sample standard deviation is not an unbiased estimator of the population standard deviation
We can also define the bias of an estimator as follows

8. Properties of Point Estimators
It is usually the case that, if there is an unbiased estimator of a population parameter, there exist several others, as well
To select the “best unbiased” estimator, we use the criterion of efficiency
An unbiased estimator is efficient if no other unbiased estimator of the particular population parameter has a lower sampling distribution variance

9. Properties of Point Estimators
If and are two unbiased estimators of the population parameter , then is more efficient than if
The unbiased estimator of a population parameter with the lowest variance out of all unbiased estimators is called the most efficient or minimum variance unbiased estimator
In some cases, we may be interested in the properties of an estimator in large samples, which may not be present in the case of small samples

10. Properties of Point Estimators
We say that an estimator is consistent if the probability of obtaining estimates close to the population parameter increases as the sample size increases
The problem of selecting the most appropriate estimator for a population parameter is quite complicated
In some occasions, we may prefer to have some bias of the estimator at the gain of increases efficiency

11. Properties of Point Estimators
One measure of the expected closeness of an estimator to the population parameter is its mean squared error

12. Interval Estimation
Point estimates of population parameters are prone to sampling error and are not likely to equal the population parameter in any given sample
Moreover, it is often the case that we are interested not in a point estimate, but in a range within which the population parameter will lie
An interval estimator for a population parameter is a formula that determines, based on sample information, a range within which the population parameter lies with certain probability

13. Interval Estimation The estimate is called an interval estimate
The probability that the population parameter will lie within a confidence interval is called the level of confidence and is given by 1 - 
Two interpretations of confidence intervals
Probabilistic interpretation
Practical interpretation

14. Interval Estimation In the probabilistic interpretation, we say that
A 95% confidence interval for a population parameter means that, in repeated sampling, 95% of such confidence intervals will include the population parameter
In the practical interpretation, we say that
We are 95% confident that the 95% confidence interval will include the population parameter

15. Constructing Confidence Intervals
Confidence intervals have similar structures
Point Estimate  Reliability Factor  Standard Error
Reliability factor is a number based on the assumed distribution of the point estimate and the level of confidence
Standard error of the sample statistic providing the point estimate

16. Confidence Interval for Mean of a Normal Distribution with Known Variance
Suppose we take a random sample from a normal distribution with unknown mean, but known variance
We are interested in obtaining a confidence interval such that it will contain the population mean 90% of times
The sample mean will follow a normal distribution and the corresponding standardized variable will follow a standard normal distribution

17. Confidence Interval for Mean of a Normal Distribution with Known Variance
If is the sample mean, then we are interested in the confidence interval, such that the following probability is .9

18. Confidence Interval for Mean of a Normal Distribution with Known Variance
Following the above expression for the structure of a confidence interval, we rewrite the confidence interval as follows
Note that from the standard normal density

19. Confidence Interval for Mean of a Normal Distribution with Known Variance
Given this result and that the level of confidence for this interval (1-) is .90, we conclude that
The area under the standard normal to the left of –1.65 is 0.05
The area under the standard normal to the right of 1.65 is 0.05
Thus, the two reliability factors represent the cutoffs -z/2 and z/2 for the standard normal

20. Confidence Interval for Mean of a Normal Distribution with Known Variance
In general, a 100(1-)% confidence interval for the population mean  when we draw samples from a normal distribution with known variance 2 is given by
where z/2 is the number for which

21. Confidence Interval for Mean of a Normal Distribution with Known Variance
Note: We typically use the following reliability factors when constructing confidence intervals based on the standard normal distribution
90% interval: z0.05 = 1.65
95% interval: z0.025 = 1.96
99% interval: z0.005 = 2.58
Implication: As the degree of confidence increases the interval becomes wider

22. Confidence Interval for Mean of a Normal Distribution with Known Variance
Example: Suppose we draw a sample of 100 observations of returns on the Nikkei index, assumed to be normally distributed, with sample mean 4% and standard deviation 6%
What is the 95% confidence interval for the population mean?
The standard error is .06/ = .006
The confidence interval is .04  1.96(.006)

23. Confidence Interval for Mean of a Normal Distribution with Unknown Variance
In a more typical scenario, the population variance is unknown
Note that, if the sample size is large, the previous results can be modified as follows
The population distribution need not be normal
The population variance need not be known
The sample standard deviation will be a sufficiently good estimator of the population standard deviation
Thus, the confidence interval for the population mean derived above can be used by substituting s for 

24. Confidence Interval for Mean of a Normal Distribution with Unknown Variance
However, if the sample size is small and the population variance is unknown, we cannot use the standard normal distribution
If we replace the unknown  with the sample st. deviation s the following quantity
follows Student’s t distribution with (n – 1) degrees of freedom

25. Confidence Interval for Mean of a Normal Distribution with Unknown Variance
The t-distribution has mean 0 and (n – 1) degrees of freedom
As degrees of freedom increase, the t-distribution approaches the standard normal distribution
Also, t-distributions have fatter tails, but as degrees of freedom increase (df = 8 or more) the tails become less fat and resemble that of a normal distribution

26. Confidence Interval for Mean of a Normal Distribution with Unknown Variance
In general, a 100(1-)% confidence interval for the population mean  when we draw small samples from a normal distribution with an unknown variance 2 is given by
where tn-1,/2 is the number for which

27. Confidence Interval for Mean of a Normal Distribution with Unknown Variance
Example: Suppose we want to estimate a 95% confidence interval for the average quarterly returns of all fixed-income funds in the US
We assume that those returns are normally distributed with an unknown variance
We draw a sample of 150 observations and calculate the sample mean to be .05 and the standard deviation .03

28. Confidence Interval for Mean of a Normal Distribution with Unknown Variance
To find the confidence interval, we need tn-1,/2 = t149,0.025
From the tables of the t-distribution, this is equal to 1.96
The confidence interval is

29. Confidence Interval for the Population Variance of a Normal Population
Suppose we have obtained a random sample of n observations from a normal population with variance 2 and that the sample variance is s2. A 100(1 - )% confidence interval for the population variance is

30. Confidence Interval for the Population Variance of a Normal Population
The values of the chi-squared distribution with n-1,/2 and
n-1,1-/2 are determined as follows
where follows the chi-squared distribution with (n-1) degrees of freedom

31. Good luck