Page 1. Statistical Inference. Second Edition. George Casella. Roger L. Berger. D U X B U R Y A D V A N C E D S E R I ES. Page 2. Page 3. Page 4. Page 5. Ap. 57) 5. ros. CĄS. Statistical Inference. Second Edition. George Casella. University of Florida. Roger L. Berger. North Carolina State University. DUXERU RY. George Casella. University of Roger L. Berger Of the exercises in Statistical Inference, Second Edition, a. f(x) is a pdf since it is positive and.

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George Casella. Roger l. Berger Statistical inference / George Casella, Roger L. Bergernd ed. .. Histogram of exponential pdf. Solutions Manual for Statistical Inference, Second Edition George Casella Roger L. Berger University of Florida North Carolina State University Damaris. Statistical Inference-Second Edition ( Alexander Villar Espinoza. A. Villar Espinoza. Loading Preview. Sorry, preview is currently unavailable.

Chapter 7 has been expanded and updated, and includes a new section on the EM algorithm.

Chapter 8 has also received minor editing and updating, and ""'11 '- f. Unfortunately, coverage of randomized block designs has been eliminated for space reasons.

Chapter 12 covers regression with errors-in-variables and contains new material on robust and logistic regression. After teaching from the first edition for a number of years, we know approximately what can be covered in a one-year course.

Finally, it is almost impossible to thank all of the people who have contributed in some way to making the second edition a reality and help us correct the mistakes in the first edition. To all of our students, friends, and colleagues who took the time to send us a note or an e-mail, we thank you. A number of people made key suggestions that led to substantial changes in presentation.

Sometimes these suggestions were just short notes or comments, and some were longer reviews. Some were so long ago that their authors may have forgotten, but we haven't. We also owe much to Jay Beder, who has sent us numerous comments and suggestions over the years and possibly knows the first edition better than we do, and to Michael Perlman and his class, who are sending comments and corrections even as we write this.

This book has seen a number of editors. We thank Alex Kugashev, who in the mids first suggested doing a second edition, and our editor, Carolyn Crockett, who constantly encouraged us.

Perhaps the one person other than us who is most responsible for this book is our first editor, John Kimmel, who encouraged, published, and marketed the first edition. Thanks, John. George Casella Roger L. The first is "Why are you writing a book? You are writing a book because you are not entirely satisfied with the available texts. The second question is harder to answer. The answer can't be put in a few sentences so, in order not to bore your audience who may be asking the question only out of politeness , you try to say something quick and witty.

It usually doesn't work. Logical development, proofs, ideas, themes, etc. When this endeavor was started, we were not sure how well it would work. The final judgment of our success is, of course, left to the reader. The book is intended for first-year graduate students majoring in statistics or in a field where a statistics concentration is desirable. The prerequisite is one year of calculus. Under H0 , the scale parameters of W and V are equal.

Then, a simple generalization of Exercise 4. The size is. P From Corollary 8. By Corollary 8. By Exercise 8. By the Neyman-Pearson Lemma, the most powerful test of H0: So this family has MLR. Thus the ratio is increasing in x, and the family has MLR. We will prove the result for continuous distributions. But it is also true for discrete MLR families. From Exercise 3. The family does not have MLR.

Statistical Inference

This family has MLR. From part a , this ratio is increasing 0 in x. Thus this. Thus, the family does not have MLR. Thus, the given test is UMP of its size. The test is not UMP for testing H0: Hence the ratio is not monotone. Hence the family has MLR. This is Example 8.

Statistical Inference - (Casella & Berger)

From Theorems 5. Y1 , Yn are sufficient statistics. So we can attempt to find a UMP test using Corollary 8. Thus the given test is UMP by Corollary 8. So these conditions are satisfied for any n.

This is Exercise 3. This is Exercise 8. We will use the equality in Exercise 3. The argument that the noncentral t has an MLR is fairly involved. It may be found in Lehmann , p. The proof that the one-sided t test is UMP unbiased is rather involved, using the bounded completeness of the normal distribution and other facts. See Chapter 5 of Lehmann for a complete treatment.

Again, see Chapter 5 of Lehmann From Exercise 4. The Wi s are independent because the pairs Xi , Yi are. The hypotheses are equivalent to Hp 0: Hence, the LRT is the two-sample t-test. The two-sample t test is UMP unbiased, but the proof is rather involved. See Chapter 5 of Lehmann So there is no evidence that the mean age differs between the core and periphery.

Using the values in Exercise 8. So the p-value is. There is no evidence that the mean age differs between the core and periphery.

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So there is some slight evidence that the variance differs between the core and periphery. Note, early printings had a typo with the numerator and denominator degrees of freedom switched. That is, Test 3 is unbiased. Second Edition 8. Example 8. Hence, the tests are all unbiased. This is very similar to the argument for Exercise 8. Use Theorem 8. By Theorem 8. First calculate the posterior density. The following table illustrates this. So the indicated equality is true. Chapter 9 Interval Estimation 9.

From 7. We now must establish that this set is indeed an interval. To do this, we establish that the function on the left hand side of the inequality has only an interior maximum. That is, it looks like an upside-down bowl. We make some further simplifications.

Since this is the sign change of the derivative, the function must increase then decrease. Hence, the function is an upside-down bowl, and the set is an interval. Analogous to Example 9. Since k p is nondecreasing, this gives an upper bound on p.

This is clearly a highest density region. The interval of Typically, the second degree and n degree polynomials will not have the same roots. Therefore, the two intervals are different.

Then the two intervals are the same. Using the result of Exercise 8. Using the results of Exercise 8. Second Edition 9. The interval in part a is a special case of the one in part b.

Thus the interval in part a is nonoptimal. A shorter interval with confidence coefficient.

Recall the Bonferroni Inequality 1. Use the interval 9. Use the interval after 9. This will happen if the test of H0: The LRT see Example 8. There are only two discrepancies. This is Exercise 2.

The LRT statistic for H0: The values a y and b y are not expressible in closed form. This is an example of the effect of the imposition of one type of inference frequentist on another theory likelihood. For the confidence interval in Example 9. This confidence interval is The two confidence intervals are virtually the same. The LRT method derives its interval from the test of H0: To compare the intervals we compare their lengths.

We know from Example P 7.


So no values of a and b will make the intervals match. To evaluate this probability we have two cases: So the conditions of Theorem 9. So moving the interval toward zero increases the probability, and it is therefore maximized by moving a all the way to zero. Using Theorem 8. For Exercise 9. The inequality follows directly from Definition 8. The solution for the lower confidence interval is similar.

Start with the hypothesis test H0: Arguing as in Example 8. The LRT of H0: Thus, the values of a and b that give the minimum length interval must satisfy this along with the probability constraint. The confidence interval, say I s2 will be unbiased if Definition 9. Some algebra will establish!

For those values of K, C 0 dominates C. Chapter 10 Asymptotic Evaluations So by Theorem Applying the formulas of Example 5. The integral of ETn2 is unbounded near zero. Then we apply Theorem 5. It is easiest to use the Mathematica code in Example A. The MLE comes from differentiating the log likelihood!

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The approximate variance is quite a pain to calculate. Now using Example 5. There are some entries that are less than one - this is due to using an approximation for the MOM variance. For part e ,verifying the bootstrap identities can involve much painful algebra, but it can be made easier if we understand what the bootstrap sample space the space of all nn bootstrap samples looks like.

Given a sample x1 , x2 ,. The first column is 9 x1 s followed by 9 x2 s followed by 9 x3 s, the second column is 3 x1 s followed by 3 x2 s followed by 3 x3 s, then repeated, etc. The general result should now be clear.

The correlation is. Here is R code R is available free at http: The output is V1 V2 V1 1. The bootstrap standard deviation is 0. The histogram looks similar to the nonparametric bootstrap histogram, displaying a skewness left. Also, the approximate pdf of r will be normal, hence symmetric. The variance of X! Write n n! Again the heavier tails favor the median. From the discussion preceding Example The other limit can be calculated in a similar manner.

One might argue that in hypothesis testing, the first one should be used, since under H0 , it provides a better estimator of variance. If interest is in finding the confidence interval, however, we are making inference under both H0 and H1 , and the second one is preferred.

Now the hypothesis is about conditional probabilities is given by H0: The information number is P i xi! We test the equivalent hypothesis H0: The likelihood is the same as Exercise We assume that the underlying distribution is normal, and use that for all score calculations.

The actual data is generated from normal, logistic, and double exponential. The sample size is 15, we use simulations and draw 20 bootstrap samples. Boot 0. Median 0. Logistic Naive 0. Normal Naive 0. Here is Mathematica code: This program calculates size and power for Exercise We test H0: But it is still difficult to analytically compare lengths with the non-corrected interval - we will do a numerical comparison. Chapter 11 Analysis of Variance and Regression From Exercise Second Edition Similarly for H0: To see this, suppose H04 and H05 are rejected.

In part a all of the contrasts are orthogonal. Now, from Lemma 5. However, each contrast can be interpreted meaningfully in the context of the experiment.

For example, a1 tests the effect of potassium alone, while a5 looks at the effect of adding zinc to potassium. This is a special case of From Exercise 5. This is true for equal sample sizes. Then from Exercise This produces the F statistic. In fact, the rejection region is contained in the t rejection region. So the t is more powerful. The experimentwise error rate is preserved. The pooled standard deviation is 2.

The marginal distributions of the Yi P are somewhat straightforward to derive.

We have to make a multivariate change of variable. This is made a bit more palatable if we do it in two steps. The joint density of the Wi is n Y 1 f w1 , w2 ,. ANOVA table: Completing the proof of Finally, this gives n! So we need to minimize the last term. From Start from the display following Sxx For the least squares estimate: Thus hX.

Statistical Inference, 2nd Edition, by G. Casella and R. Berger. Solutions.pdf

The expected complete-data log likelihood is n! The EM calculations are simple here. So the MLE s are the same as those without the extra xn. Now we use the bivariate normal density see Definition 4. We update the other t estimates as follows.

Chapter 12 Regression Models The answer is The resulting function must be the joint pdf of X and Y. The double integral is infinite, however. From the last two equations in Sxx Sxy Note that we did not need the normality assumption, just the moments. Exercise There are several different justifications for using the Bayesian approach. Examples of Bayesian inference[ edit ] Bayes factors for model comparison Bayesian inference, subjectivity and decision theory[ edit ] Many informal Bayesian inferences are based on "intuitively reasonable" summaries of the posterior.

For example, the posterior mean, median and mode, highest posterior density intervals, and Bayes Factors can all be motivated in this way. While a user's utility function need not be stated for this sort of inference, these summaries do all depend to some extent on stated prior beliefs, and are generally viewed as subjective conclusions.

Methods of prior construction which do not require external input have been proposed but not yet fully developed. Formally, Bayesian inference is calibrated with reference to an explicitly stated utility, or loss function; the 'Bayes rule' is the one which maximizes expected utility, averaged over the posterior uncertainty.

Formal Bayesian inference therefore automatically provides optimal decisions in a decision theoretic sense. Given assumptions, data and utility, Bayesian inference can be made for essentially any problem, although not every statistical inference need have a Bayesian interpretation. Analyses which are not formally Bayesian can be logically incoherent ; a feature of Bayesian procedures which use proper priors i.Boot 0.

In the passage from the first to the second edition, problems were shuffled with no attention paid to numbering hence no attention paid to minimize the new effort , but rather we tried to put the problems in logical order. The fourth moment is not easy to get, one way to do it is to get the mgf of X. So this sufficient statistic is not complete. It does not fit the definition of either one.

Similar to part a. We will do this in the standard case. In fact, all odd moments of X are 0. Thus hX. Let n a, b denote the pdf of a normal distribution with mean a and variance b.

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