Statistical Inference

Transcription

1 Statistical Inference Classical and Bayesian Methods Class 5 AMS-UCSC Tue 24, 2012 Winter Session 1 (Class 5) AMS-132/206 Tue 24, / 11

2 Topics Topics We will talk about... 1 Confidence Intervals Winter Session 1 (Class 5) AMS-132/206 Tue 24, / 11

3 Definition of Confidence Intervals Example 1: CI for the mean of a Normal Distribution Confidence Intervals for the mean of a Normal Distribution In the example about the rain from seeded clouds we would like to know how much confidence can we place on the estimator of the mean of log-rainfalls. The sample mean of the n = 26 is X n. The standard deviation of X n is σ/(26) 1/2. We can use σ n i=1 = ( (X i X n) 2 n 1 ) 1/2 to estimate σ. We can use the statistic: U = n 1/2 ( X n µ)/σ to address this question. U has a t distribution with n 1 degrees of freedom. We can calculate the cdf of U or quantiles of U from tables or statistical software. In particular we can choose a value of γ and find the value of c such that Pr( c < U < c) = γ. Winter Session 1 (Class 5) AMS-132/206 Tue 24, / 11

4 Confidence Intervals for the mean of a Normal Distribution Confidence Interval for the mean of a Normal distribution Example 2: Rain from seeded clouds Suppose γ =.95. We need c such that Pr( c < U < c) = 0.95 Let T n 1 be the cdf of a t distribution with n 1 degrees of freedom. Then Pr( c < U < c) = γ = T n 1 (c) T n 1 ( c) = T n 1 (c) (1 T n 1 (c)) = 2T n 1 (c) 1 This means c is the (1 + γ)/2 quantile of the t distribution with n 1 degrees of freedom. We need c to be the 1.95/2 = quantile of the t distribution with 25 degrees of freedom. This values is c = Since c < U < c is equivalent to X n cσ < µ < X n 1/2 n + cσ, there n 1/2 is a.95 probability that µ lies between the two random variables A = X n (2.060/26 1/2 )σ and B = X n (2.060/26 1/2 )σ Winter Session 1 (Class 5) AMS-132/206 Tue 24, / 11

5 Confidence Intervals for the mean of a Normal Distribution Definition of Confidence Interval The interval (A, B) calculated in the previous example is called a confidence interval. Confidence Interval Let X = (X 1,..., X n ) be a random sample from a distribution that depends of a parameter (or paramater vector) θ. Let g(θ) a real valued function of θ. Let A and B two statistics such that A B and for all values of θ: Pr(A < g(θ) < B) γ. Then the random interval (A, B) is called a coefficient γ confidence interval for g(θ) or a 100γ percent confidence interval for g(θ). When the random variables X 1, X 2,..., X n are observed, we get A = a and B = b and (a, b) is the observed value of the confidence interval. When the inequality γ becomes an equality the confidence interval is exact. Winter Session 1 (Class 5) AMS-132/206 Tue 24, / 11

6 Confidence Intervals for the mean of a Normal Distribution Theorem Confidence interval for the mean of a Normal distribution Theorem Let X = (X 1,..., X n ) be a random sample from a Normal distribution with parameters µ and σ 2.For each γ in (0, 1), the interval with the following end points is an exact coefficient γ confidence interval for µ: A = X n T 1 n 1 ((γ + 1)/2)σ /n 1/2 B = X n + T 1 n 1 ((γ + 1)/2)σ /n 1/2 Note: Before observing the data we can be 95% confident that the interval (A, B) will contain µ. However after observing the data we are not sure whether (a, b) will contain µ. The safest interpretation is to say that (a, b) is simply an observed value of the random interval (A, B). Winter Session 1 (Class 5) AMS-132/206 Tue 24, / 11

7 Confidence Intervals for other parameters Confidence Intervals from a Pivotal Quantity Example of lifetime electronic components Example of lifetime electronic components: We used the fact that the statistic T = n i=1 X i has a Gamma distribution with parameters 3 and θ. We can use the distribution of T to construct confidence intervals on θ. Please recall also that θt will have a Gamma distribution with parameters n and 1 (Try to prove this!.) Let G be the cdf of the Gamma distribution. The company would like to have an upper limit B such that the probability of θ being less than B is 98%. In this case: Pr(θT < c) = Pr(θ < c/t ) = 0.98 where c = G 1 (0.98). This value is So the upper limit B is B = 7.156/T. Note: In this case the quantity θt has the property that its distribution is the same for all values of θ. In the previous example the quantity U has the same distribution for all values of µ and σ 2. These quantities are called pivotal quantities and they are very useful to build confidence intervals. Winter Session 1 (Class 5) AMS-132/206 Tue 24, / 11

8 Confidence Intervals for other parameters Confidence Intervals from a Pivotal Quantity Definition Pivotal Quantity Let X = (X 1,..., X n ) be a random sample from a distribution that depends on a parameter (or paramater vector) θ. Let V (X, θ) be a random variable whose distribution is the same for all values of θ. Then V is called a pivotal quantity. Note: In order to use the pivotal quantity to find a confidence interval for g(θ), we need to invert the pivotal. This means, we need a function r(v, x) such that r(v (X, θ), X) = g(θ) Winter Session 1 (Class 5) AMS-132/206 Tue 24, / 11

9 Confidence Intervals for other parameters Confidence Intervals from a Pivotal Quantity Example: Pivotal for estimating the variance of a Normal distribution Let X 1, X 2,..., X n be a random sample from a Normal distribution with parameters µ and σ 2. In a previous theorem we had that the random variable V (X, θ) = n i=1 (X i X n ) 2 /σ 2 has a χ 2 distribution with n 1 degrees of freedom for all θ = (µ, σ 2 ). This makes V a pivotal quantity. V can be used to find a confidence interval for g(θ) = σ 2. Problem: Suppose X 1, X 2,..., X n is a random sample from the normal distribution with unknown mean and unknown variance. Describe a method to construct a confidence interval for σ 2 with a specified confidence coefficient γ (0 < γ < 1). Winter Session 1 (Class 5) AMS-132/206 Tue 24, / 11

10 Limitation of Confident Intervals Interpretation of Confident Intervals Limitations It is not correct to say that θ lies in the interval (a, b) with probability γ. It is not possible to assign a probability to the event that θ lies in the interval (a, b) without assuming θ as a random variable. It is necessary to assign a prior distribution to θ and then use the resulting posterior. Instead of assigning a prior distribution to θ many statisticians say there is a confidence γ rather than a probability γ, that θ lies in the interval (a, b) This distinction between confidence and probability is somewhat a controversial topic. Winter Session 1 (Class 5) AMS-132/206 Tue 24, / 11

11 Limitation of Confident Intervals Thanks for your attention... Winter Session 1 (Class 5) AMS-132/206 Tue 24, / 11