Applied Statistics I
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1 Applied Statistics I Liang Zhang Department of Mathematics, University of Utah July 14, 2008 Liang Zhang (UofU) Applied Statistics I July 14, / 18
2 Point Estimation Liang Zhang (UofU) Applied Statistics I July 14, / 18
3 Point Estimation Problem: when there are more then one point estimator for parameter θ, which one of them should we use? Liang Zhang (UofU) Applied Statistics I July 14, / 18
4 Point Estimation Problem: when there are more then one point estimator for parameter θ, which one of them should we use? There are a few criteria for us to select the best point estimator: Liang Zhang (UofU) Applied Statistics I July 14, / 18
5 Point Estimation Problem: when there are more then one point estimator for parameter θ, which one of them should we use? There are a few criteria for us to select the best point estimator: unbiasedness, Liang Zhang (UofU) Applied Statistics I July 14, / 18
6 Point Estimation Problem: when there are more then one point estimator for parameter θ, which one of them should we use? There are a few criteria for us to select the best point estimator: unbiasedness, minimum variance, Liang Zhang (UofU) Applied Statistics I July 14, / 18
7 Point Estimation Problem: when there are more then one point estimator for parameter θ, which one of them should we use? There are a few criteria for us to select the best point estimator: unbiasedness, minimum variance, and mean square error. Liang Zhang (UofU) Applied Statistics I July 14, / 18
8 Point Estimation Liang Zhang (UofU) Applied Statistics I July 14, / 18
9 Point Estimation Definition A point estimator ˆθ is said to be an unbiased estimator of θ if E(ˆθ) = θ for every possible value of θ. If ˆθ is not unbiased, the difference E(ˆθ) θ is called the bias of ˆθ. Liang Zhang (UofU) Applied Statistics I July 14, / 18
10 Point Estimation Definition A point estimator ˆθ is said to be an unbiased estimator of θ if E(ˆθ) = θ for every possible value of θ. If ˆθ is not unbiased, the difference E(ˆθ) θ is called the bias of ˆθ. Principle of Unbiased Estimation When choosing among several different estimators of θ, select one that is unbiased. Liang Zhang (UofU) Applied Statistics I July 14, / 18
11 Point Estimation Liang Zhang (UofU) Applied Statistics I July 14, / 18
12 Point Estimation Proposition Let X 1, X 2,..., X n be a random sample from a distribution with mean µ and variance σ 2. Then the estimators ˆµ = X = n i=1 X i n and ˆσ 2 = S 2 = n i=1 (X i X ) 2 n 1 are unbiased estimator of µ and σ 2, respectively. If in addition the distribution is continuous and symmetric, then X and any trimmed mean are also unbiased estimators of µ. Liang Zhang (UofU) Applied Statistics I July 14, / 18
13 Point Estimation Liang Zhang (UofU) Applied Statistics I July 14, / 18
14 Point Estimation Principle of Minimum Variance Unbiased Estimation Among all estimators of θ that are unbiased, choose the one that has minimum variance. The resulting ˆθ is called the minimum variance unbiased estimator ( MVUE) of θ. Liang Zhang (UofU) Applied Statistics I July 14, / 18
15 Point Estimation Principle of Minimum Variance Unbiased Estimation Among all estimators of θ that are unbiased, choose the one that has minimum variance. The resulting ˆθ is called the minimum variance unbiased estimator ( MVUE) of θ. Theorem Let X 1, X 2,..., X n be a random sample from a normal distribution with mean µ and variance σ 2. Then the estimator ˆµ = X is the MVUE for µ. Liang Zhang (UofU) Applied Statistics I July 14, / 18
16 Point Estimation Liang Zhang (UofU) Applied Statistics I July 14, / 18
17 Point Estimation Definition Let ˆθ be a point estimator of parameter θ. Then the quantity E[(ˆθ θ) 2 ] is called the mean square error (MSE) of ˆθ. Liang Zhang (UofU) Applied Statistics I July 14, / 18
18 Point Estimation Definition Let ˆθ be a point estimator of parameter θ. Then the quantity E[(ˆθ θ) 2 ] is called the mean square error (MSE) of ˆθ. Proposition MSE = E[(ˆθ θ) 2 ] = V (ˆθ) + [E(ˆθ) θ] 2 Liang Zhang (UofU) Applied Statistics I July 14, / 18
19 Point Estimation Liang Zhang (UofU) Applied Statistics I July 14, / 18
20 Point Estimation Definition The standard error of an estimator ˆθ is its standard deviation σ ˆθ = V (ˆθ). If the standard error itself involves unknown parameters whose values can be estimated, substitution of these estimates into σˆθ yields the estimated standard error (estimated standard deviation) of the estimator. The estimated standard error can be denoted either by ˆσ ˆθ or by s ˆθ. Liang Zhang (UofU) Applied Statistics I July 14, / 18
21 Methods of Point Estimation Liang Zhang (UofU) Applied Statistics I July 14, / 18
22 Methods of Point Estimation The Invariance Principle Let ˆθ be the mle of the parameter θ. Then the mle of any function h(θ) of this parameter is the function h(ˆθ). Liang Zhang (UofU) Applied Statistics I July 14, / 18
23 Methods of Point Estimation The Invariance Principle Let ˆθ be the mle of the parameter θ. Then the mle of any function h(θ) of this parameter is the function h(ˆθ). Proposition Under very general conditions on the joint distribution of the sample, when the sample size n is large, the maximum likelihood estimator of any parameter θ is approximately unbiased [E(ˆθ) θ] and has variance that is nearly as small as can be achieved by any estimator. Stated another way, the mle ˆθ is approximately the MVUE of θ. Liang Zhang (UofU) Applied Statistics I July 14, / 18
24 Liang Zhang (UofU) Applied Statistics I July 14, / 18
25 Example (a variant of Problem 62, Ch5) The total time for manufacturing a certain component is known to have a normal distribution. However, the mean µ and variance σ 2 for the normal distribution are unknown. After an experiment in which we manufactured 10 components, we recorded the sample time which is given as follows: time time Liang Zhang (UofU) Applied Statistics I July 14, / 18
26 Example (a variant of Problem 62, Ch5) The total time for manufacturing a certain component is known to have a normal distribution. However, the mean µ and variance σ 2 for the normal distribution are unknown. After an experiment in which we manufactured 10 components, we recorded the sample time which is given as follows: time time We know that both MME and MLE for the population mean µ is the sample mean X, i.e. ˆµ = X = How accurate is this estimation? Liang Zhang (UofU) Applied Statistics I July 14, / 18
27 Liang Zhang (UofU) Applied Statistics I July 14, / 18
28 Assume the other parameter σ is known, e.g. σ = 2.7 Liang Zhang (UofU) Applied Statistics I July 14, / 18
29 Assume the other parameter σ is known, e.g. σ = 2.7 X is normally distributed with mean µ and variance σ 2 /n. Therefore, Z = X µ σ/ is a standard normal random variable. n Liang Zhang (UofU) Applied Statistics I July 14, / 18
30 Assume the other parameter σ is known, e.g. σ = 2.7 X is normally distributed with mean µ and variance σ 2 /n. Therefore, Z = X µ σ/ is a standard normal random variable. n For the interval [ A, A], how large should A be such that with 95% confidence we are sure Z falls in that interval? Liang Zhang (UofU) Applied Statistics I July 14, / 18
31 Assume the other parameter σ is known, e.g. σ = 2.7 X is normally distributed with mean µ and variance σ 2 /n. Therefore, Z = X µ σ/ is a standard normal random variable. n For the interval [ A, A], how large should A be such that with 95% confidence we are sure Z falls in that interval? P( A < Z < A) =.95 Liang Zhang (UofU) Applied Statistics I July 14, / 18
32 Assume the other parameter σ is known, e.g. σ = 2.7 X is normally distributed with mean µ and variance σ 2 /n. Therefore, Z = X µ σ/ is a standard normal random variable. n For the interval [ A, A], how large should A be such that with 95% confidence we are sure Z falls in that interval? P( A < Z < A) =.95 A is the 97.5the percentle, which is Liang Zhang (UofU) Applied Statistics I July 14, / 18
33 Assume the other parameter σ is known, e.g. σ = 2.7 X is normally distributed with mean µ and variance σ 2 /n. Therefore, Z = X µ σ/ is a standard normal random variable. n For the interval [ A, A], how large should A be such that with 95% confidence we are sure Z falls in that interval? P( A < Z < A) =.95 A is the 97.5the percentle, which is ( ) P 1.96 < X µ σ/ n < 1.96 =.95 Liang Zhang (UofU) Applied Statistics I July 14, / 18
34 Assume the other parameter σ is known, e.g. σ = 2.7 X is normally distributed with mean µ and variance σ 2 /n. Therefore, Z = X µ σ/ is a standard normal random variable. n For the interval [ A, A], how large should A be such that with 95% confidence we are sure Z falls in that interval? P( A < Z < A) =.95 A is the 97.5the percentle, which is ( ) P 1.96 < X µ σ/ n < 1.96 =.95 ) σ P (X 1.96 n σ < µ < X n =.95 Liang Zhang (UofU) Applied Statistics I July 14, / 18
35 Assume the other parameter σ is known, e.g. σ = 2.7 X is normally distributed with mean µ and variance σ 2 /n. Therefore, Z = X µ σ/ is a standard normal random variable. n For the interval [ A, A], how large should A be such that with 95% confidence we are sure Z falls in that interval? P( A < Z < A) =.95 A is the 97.5the percentle, which is ( ) P 1.96 < X µ σ/ n < 1.96 =.95 ) σ P (X 1.96 n σ < µ < X n =.95 ) σ The interval (X 1.96 n σ, X n is called the 95% confidence interval for µ. Liang Zhang (UofU) Applied Statistics I July 14, / 18
36 Assume the other parameter σ is known, e.g. σ = 2.7 X is normally distributed with mean µ and variance σ 2 /n. Therefore, Z = X µ σ/ is a standard normal random variable. n For the interval [ A, A], how large should A be such that with 95% confidence we are sure Z falls in that interval? P( A < Z < A) =.95 A is the 97.5the percentle, which is ( ) P 1.96 < X µ σ/ n < 1.96 =.95 ) σ P (X 1.96 n σ < µ < X n =.95 ) σ The interval (X 1.96 n σ, X n is called the 95% confidence interval for µ. In our case, 95% confidence interval for µ is (63.28, 66.62). Liang Zhang (UofU) Applied Statistics I July 14, / 18
37 Liang Zhang (UofU) Applied Statistics I July 14, / 18
38 Interpretation of Confidence Interval Liang Zhang (UofU) Applied Statistics I July 14, / 18
39 Interpretation of Confidence Interval The 95% confidence interval for µ (63.28, 66.62) doesn t mean P(µ falls in the interval(63.28, 66.62)) =.95 Liang Zhang (UofU) Applied Statistics I July 14, / 18
40 Interpretation of Confidence Interval The 95% confidence interval for µ (63.28, 66.62) doesn t mean P(µ falls in the interval(63.28, 66.62)) =.95 It is a long-run effect: if we have 1000 random samples, then for approximately 950 of them, µ falls in the interval (X 1.96 σ n, X σ n ). Liang Zhang (UofU) Applied Statistics I July 14, / 18
41 Liang Zhang (UofU) Applied Statistics I July 14, / 18
42 Example (a variant of Problem 62, Ch5) The total time for manufacturing a certain component is known to have a normal distribution. However, the mean µ for the normal distribution is unknown. After an experiment in which we manufactured 10 components, we recorded the sample time which is given as follows: time time We know that both MME and MLE for the population mean µ is the sample mean X, i.e. ˆµ = X = We further assume the standard deviation is known to be σ = 2.7. What is the 99% confidence interval for µ? Liang Zhang (UofU) Applied Statistics I July 14, / 18
43 Liang Zhang (UofU) Applied Statistics I July 14, / 18
44 Definition A 100(1 α)% confidence interval for the mean µ of a normal population when the value of σ is known is given by ( x z α/2 σ n, x + z α/2 ) σ n or, equivalently, by x z α/2 σ n Liang Zhang (UofU) Applied Statistics I July 14, / 18
45 Liang Zhang (UofU) Applied Statistics I July 14, / 18
46 Graphically interpretation: Liang Zhang (UofU) Applied Statistics I July 14, / 18
47 Liang Zhang (UofU) Applied Statistics I July 14, / 18
48 Example (a variant of Problem 62, Ch5) The total time for manufacturing a certain component is known to have a normal distribution. However, the mean µ for the normal distribution is unknown. Thus we decide to do an experiment in which we manufacture n components to estimate the population mean µ. We know that both MME and MLE for the population mean µ is the sample mean X, i.e. ˆµ = X. We further assume the standard deviation is known to be σ = 2.7. If we want a 99% confidence interval for µ with width 3.34, how large should n be? Liang Zhang (UofU) Applied Statistics I July 14, / 18
49 Liang Zhang (UofU) Applied Statistics I July 14, / 18
50 Proposition To obtain a 100(1 α)% confidence interval with width w for the mean µ of a normal population when the value of σ is known, we need a random sample of size at least ( n = 2z α/2 σ ) 2 w Liang Zhang (UofU) Applied Statistics I July 14, / 18
51 Proposition To obtain a 100(1 α)% confidence interval with width w for the mean µ of a normal population when the value of σ is known, we need a random sample of size at least ( n = 2z α/2 σ ) 2 w Remark: The half-width w 2 of the 100(1 α)% CI is called the bound on the error of estimation associated with a 100(1 α)% confidence level. Liang Zhang (UofU) Applied Statistics I July 14, / 18
52 Liang Zhang (UofU) Applied Statistics I July 14, / 18
53 Example: Extensive experience with fans of a certain type used in diesel engines has suggested that the exponential distribution provides a good model for time until failure. However, the parameter λ is unknown. The following table records the data for a size 10 sample: time time What is a 95% confidence interval for λ? Liang Zhang (UofU) Applied Statistics I July 14, / 18
54 Liang Zhang (UofU) Applied Statistics I July 14, / 18
55 Proposition Let X 1, X 2,..., X n i.i.d random variables from an expentional distribution with parameter λ. Then the random variable Y = 2λ n i=1 X i has the chi-squared distribution with 2n degrees of freedom, i.e., Y χ 2 (2n) Liang Zhang (UofU) Applied Statistics I July 14, / 18
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