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ite. Let f be a density on the interval 230 Section 4.5: Monte Carlo Approximations a b, such that f x algorithm for generating X1 X2 0 for every x a b and suppose we have a convenient i.i.d. with distribution given by f. We have that b a g x dx b a g x f x f x dx E g X f X when X is distributed with density f. So we c... |
can be used to approximate complicated sums, in tegrals, and sampling distributions, all by choosing the random experiment ap propriately. EXERCISES 4.5.1 Describe a Monte Carlo approximation of cos2 x e x 2 2 dx 4.5.2 Describe a Monte Carlo approximation of the Binomial m 2 3 distribution.) 4.5.3 Describe a Monte C... |
n such that we can be sure that Var Y regardless of the value of 4.5.11 Suppose X and Y are random variables with joint density given by f X Y x y C g x y for 0 stant C, where 1 is unknown. Suppose we repeat the experiment n times, and let Y be the fraction 0 for other x y), for appropriate con is Var Y the largest? ... |
. 105, if possible). Assess the error in the approximation. N 0 1. Use a Monte Carlo algorithm to approximate P X 2 105, if possible). Assess the error in the approximation. 105, if possible). Assess the error in 0 based on a large sample (take n 3 55 j based on a j 0 j 2 2 PROBLEMS where 1 4 for all x 2 Suppose we kno... |
n i 1 a s g Xi f Xi b a g x dx (We refer to f as an importance sampler and note this shows that every f satisfying the above conditions, provides a consistent estimator Mn f of (b) Prove that b a g x dx ) Var Mn f 1 n b g2 x f x a dx 2 g x dx b a (c) Suppose that g x tance sampling with respect to f leads to the estim... |
) Explain how this experiment could be used to obtain a Monte Carlo approximation for the value of 4.5.26 (Optimal importance sampling) Consider importance sampling as described in Problem 4.5.21. (a) Prove that Var Mn f is minimized by taking. f x g x b a g x dx a b. Calculate the minimum variance and show that the mi... |
Xi independent random variables. If X 2 X1 for i 1 2 n and that they are Xn n, then X N 2 n A more subtle property of normal distributions is the following. Theorem 4.6.2 Suppose Xi are independent. Let U and bi. Then Cov U V if U and V are independent. N i 2 i for i n i 1 ai Xi and V 1 2 n and also that the Xi n i 1 ... |
6.2 can be interpreted as saying that if U and V are given by (4.6.1), then U and V are independent if and only if the rows of A are orthogonal. Linear algebra is used extensively in more advanced treatments of these ideas. i ai bi i ai bi. Hence, if i 4.6.1 The ChiSquared Distribution We now introduce another distrib... |
1: Plot of the 2 1 density. Theorem 4.6.5 Let Z 2 n. Then Z Gamma n 2 1 2. That is, f Z z 1 2n 2 n 2 z n 2 1e z 2 for z 0, with f Z z 0 for z 0. 238 Section 4.6: Normal Distribution Theory 2 n, we can write Z PROOF Because Z are i.i.d. N 0 1. But this means that X 2 we have X 2 i i.i.d. Gamma 1 2 1 2 for i independent ... |
result. Because the 2 n 1 distribution has mean n 1, we obtain the following. Corollary 4.6.2 E S2 2. PROOF Theorems 4.6.6 and 4.6.3 imply that E n E S2 2. 1 S2 2 n 1 and that Theorem 4.6.6 will find extensive use in Chapter 6. For example, this result, to gether with Corollary 4.6.1, gives us the joint sampling distr... |
n converges with probability 1 to the constant 1. Hence, the 4.6.2) converges to the distribution of Z, which is the standard normal distribution. By Defi nition 4.6.2, we have that (4.6.2) is distributed t n In Figure 4.6.3, we have plotted several t densities. Notice that the densities of the t distributions are sym... |
necessary to carry out computations with the F m n distribution. Theorem 4.6.10 If Z F m n then 1 Z F n m 242 Section 4.6: Normal Distribution Theory PROOF Using Definition 4.6.3, we have and the result is immediate from the definition. Therefore, if Z and P 1 Z F m n, then is the cdf of the F n m distribution evaluated... |
t n distribution, then t 2 is distrib uted F 1 n. EXERCISES N 8 52 be independent. Let U X1 5X2 N 7 2 be independent. N 3 22 and X2 N 3 5 and Y 6X1 C X2, where C is a constant. 4.6.1 Let X1 and V (a) What are the distributions of U and V? (b) What value of C makes U and V be independent? 4.6.2 Let X (a) What is the d... |
40] 3. X 2 40]. 244 Section 4.6: Normal Distribution Theory (e) Compute the distribution of 30 70 X 2 1 X 2 71 X 2 2 X 2 72 X 2 70 X 2 100 4.6.11 Let X1 X2 1 61 X1 X2 X61 be independent, each distributed as N X61 and 2. Set X S2 1 60 X1 X 2 X2 X 2 X61 X 2 y K X 0 05. 2 S2 has S2 has a t as usual. (a) For what values o... |
) Prove that for t n. Prove that P Z F n 2n for n 1 2 3 P Z x. Prove that Xn x for x f z 1 2 2 2 z 2 1e z 2 R1, namely, prove 1 in probability and 0 the function Chapter 4: Sampling Distributions and Limits 245 defines a probability distribution on 0 distribution, i.e., it generalizes the distribution in Section 4.6.2 b... |
ES 4.6.21 Following Problem 4.6.19, prove that the mean of X does not exist whenever 1 1. Further prove that the variance of X does not exist whenever 0 0 and is infinite when 1 2. 4.6.22 Prove the identity (4.7.1) in Section 4.7, which arises as part of the proof of Theorem 4.6.6. 246 Section 4.7: Further Proofs (Advan... |
limn x. P Xn x is “sandwiched” between P X x and P X But because P X P Xn x limn x P X 0, we must have P X x, as required. x P X x. Hence, Chapter 4: Sampling Distributions and Limits 247 Proof of Theorem 4.4.3 (The central limit theorem) We must prove the following. LetX1 X2 variance N 0 1. Set Sn 2. Let Z X1 be i.i.... |
where o s2 stands for a quantity that, as s namely, o s2 0 as s 0. This means that s 0, goes to 0 faster than s2 does — mY s2 2n o 1 n where now o 1 n stands for a quantity that, as n does., goes to 0 faster than 1 n 248 Section 4.7: Further Proofs (Advanced) Finally, we recall from calculus that, for any real number ... |
1 2 But b2u a2 2 a1 b1u 2 b2 1 2 u2 b2 a2 1 2 a2 2 2 a1b1 a2b2 u and Cov U V a1b1 a2b2. Hence, if Cov U V 0, then b2u a2 2 a1 b1u 2 b2 1 2 u2 b2 a2 1 2 a2 2 Chapter 4: Sampling Distributions and Limits 249 and fU V u exp b2 1 exp b2 1 2 u2 2 a1b2 b2 b2 2 u2 a2 1 2 a1b2 b1a2 2 a1b2 2 2 a1b2 a2 2 b1a2 b1a2 2 2 exp a2 1 ... |
U t n, then fU u n 1 2 n 2 1 u2 n n 1 2 1 n for all u R1. 250 Section 4.7: Further Proofs (Advanced) Because U t n, we can write U Y n, where X and Y are independent 2 n. It follows that X and Y have joint density given by X with X N 0 1 and Y f X Y x y e x2 2 y n 2 2 2n 2 1e y 2 n 2 when y 0 (with f X Y x y 0 for y 0... |
it follows that X m n U V and Y 0 for x 0 or y 0). n my 0 1 m n J x y det u x u y x y det n my n X mY 2 Hence, fU 1e 2 1 e m n u m 2 2n 2 m 2 2m 1e 2 1 mu n for u 0 (with fU V u 0 for u 0 or 0). Finally, we compute the marginal density of U as fU u fU 1e 2 1 mu 1e d m n u n m 2 m n where we have used the substitution ... |
functions are described in Chapter 8. 253 254 Section 5.1: Why Do We Need Statistics? 5.1 Why Do We Need Statistics? While we will spend much of our time discussing the theory of statistics, we should always remember that statistics is an applied subject. By this we mean that ultimately statistical theory will be appl... |
). Then we consider the lifelength of a patient who received a transplant as a random observation from fT and the lifelength of a patient who did not receive a transplant as a random observation from fC We want to compare fT and fC in some fashion, to determine whether or not the transplant treatment is working. For ex... |
the heart transplant until the termination date of the study, were both recorded. The Z These survival times for the treatment group are then given by the values of Y data, together with an indicator for the status of the patient at the termination date of the study, are presented in Table 5.2. We cannot compare fT an... |
) in the treatment group. The previous example provides some evidence that questions of great practical im portance require the use of statistical thinking and methodology. There are many sit uations in the physical and social sciences where statistics plays a key role, and the reasons are just like those found in Ex... |
, the student forgot which value of was used, so we are uncertain about the correct probability distribution to use to describe the variation in the data? Explain your reasoning. Can you suggest a plausible value for 5.1.6 Suppose you are interested in determining the average age of all male students at a particular co... |
5.1.12 Sometimes it is claimed that all uncertainties can and should be modeled using probability. Discuss this issue in the context of Example 5.1.1, namely, indicate all the things you are uncertain about in this example and how you might propose probability distributions to quantify these uncertainties. 5.2 Inferen... |
Figure 5.2.1: Plot of the Exponential(1) density f. Then for a new machine, we might predict its lifelength by E X 1 year. Further more, from the graph of the Exponential 1 density, it is clear that the smallest interval containing 95% of the probability for X is 0 c where c satisfies 0 95 c 0 e x dx 1 e c ln 0 05 2 9... |
special characteristic of the Exponential distribution. This will not be true in general (see Exercise 5.2.4). The tail probability measuring the plausibility of the value x0 5 is given by P X 5 X 1 e x 1 dx e 4 0 0183, 5 which indicates that x0 5 is a little more plausible in light of the fact that the machine has al... |
2 when the lifelength of a machine is known to be distributed as Y 5.2.5 Suppose that X future value of X? How would you justify your choice? 5.2.6 Suppose that X probability for a future response. (Hint: Consider a plot of the density.) 5.2.7 Suppose that X of a future value of X? How would you justify your choice? 5.... |
the most accurate predictor of a 8 is plausible. 262 Section 5.3: Statistical Models future X when using meansquared error, i.e., the expected squared distance between X and the prediction. 5.2.15 Suppose that a response X is distributed N 0 1 and that we have decided to predict a future value using the mean of the d... |
about it. The variable is called the parameter of the model, and the set indexes the probability measures in the model, i.e., P 1 2 If the probability measures P can all be presented via probability functions or (for convenience we will not distinguish between the discrete and density functions f is called the paramet... |
belled the possible probability measures as P1 and P2 respectively. The parameter is in essence only a label that allows us to distinguish amongst the possible candidates for the true probability measure. It is typical, however, to choose this label conveniently so that it means something in the problem under discussi... |
the parameter to be 1 2 Clearly, longer observed lifelengths favor 2. For example, if x1 x5 then intuitively we are more certain that 2 than if x1 x5 The subject of statistical inference is concerned with making statements like this more precise and quantifying our uncertainty concerning the validity of such assertion... |
1 xi 1 1 xi nx 1 n 1 x. This specifies the model for a sample. Note that we could parameterize this model by any 1–1 function of. 2 would work (as it is 1–1 on ), as would ln 1 For example, R unknown, where R EXAMPLE 5.3.4 LocationScale Normal Model 2 distribution with Suppose that x1 R1. For example, we may have obse... |
true value of the parameter for a model, but it is somewhat imprecise, e.g., we also probably have 100 300. In Chapter 7, we will discuss ways of incorporating such information into our analysis. but the statistical model allows for any value for 0 300 Where does the model information P : come from in an application? ... |
on population II. A sam ple X1 is generated from one of the populations; you are not told which population the sample came from, but you are required to draw inferences about the true distribution based on the sample. Describe the statistical model for this problem. Could you parameterize this model by the population... |
distribution of a response s after you have observed a single observation. 5.3.9 Suppose you know that the probability distribution of a variable X is either P1 or P2 If you observe X 0 001, then what would you guess as the true distribution of X? Give your reasoning for this conclusion. 5.3.10 Suppose you are told th... |
X Y If Y is not observed, describe the statistical model for X 5.3.17 Suppose we have a statistical model P1 P2, where P1 is an N 10 1 distrib ution while P2 is an N 0 1 distribution. (a) Is it possible to make any kind of reliable inference about the true distribution based on a single observation? Why or why not? (... |
, we must be aware of their limitations when interpreting their results. 270 Section 5.4: Data Collection 5.4.1 Finite Populations For example, we could take X of objects, called the population, and a realvalued we Suppose we have a finite set function X (sometimes called a measurement) defined on have a realvalued qua... |
or even desirable, due to the costs involved in the accurate accumulation of all the measurements — think of how difficult it might be to collect the heights of all the students at your school. for every While sometimes a census is necessary, even mandated by law, often a very accu rate approximation to FX can be obta... |
data are generated by some rule, typically 272 Section 5.4: Data Collection unknown to the statistician; this means that any conclusions drawn based on the data X 1 X n may not be valid for the full population. 1 There seems to be only one way to guarantee that selection effects are avoided, n must be selected using r... |
sampling from the population, the random variables X 1 X n Chapter 5: Statistical Inference 273 are approximately i.i.d. and with distribution given by FX So we will treat the observed values x1 as a sample (in the sense of Definition 2.8.6) from FX In this text, unless we indicate otherwise, we will always assume that... |
a sample as possible. On the other hand, there are always costs associated with sampling, and sometimes each sample value is very expensive to obtain. Furthermore, often the more data we collect, the more difficulty we have in making sure that the data are not corrupted by various errors that can arise in the collectio... |
N hi 1 hi x hi hi 1] otherwise and refer to h X as a density histogram function. Here, h X x is the proportion of in the interval hi hi 1] population elements containing x divided by the length of the interval. that have their measurement X In Figure 5.4.1, we have plotted a density histogram based on a sample of 10,0... |
.4 0.3 0.2 0.1 0.0 -4.5 -3.0 -1.5 0.0 x 1.5 3.0 4.5 Figure 5.4.2: Density histogram function for a sample of 10,000 from an N 0 1 distribution using the values h1 5 h2 4 75 h41 5. 5.4.4 Survey Sampling Finite population sampling provides the formulation for a very important application of statistics, namely, survey sam... |
will vote by X1 0 meaning no. For those voting, we denote with X1 by X2 the response concerning which candidate they will vote for, with X2 i indicating candidate i Finally, the age in years of the respondent is denoted by X3 In addition to the distributions of X1 and X2 the pollster is also interested in the joint di... |
Table D.1.) (c) Using the method you outlined in part (b), generate three samples of size n calculate X for each sample. 5.4.3 Suppose you take a sample of n Exercise 5.4.1. (a) Can you consider this as an approximate i.i.d. sample from the population distribu tion? Why or why not? (b) Explain how you would actually ... |
a computer at a particular time. is the type of file as indicated by its extension, e.g.,.mp3. Is X a 5.4.7 Consider the population Suppose that X categorical or quantitative variable? 5.4.8 Suppose that you are asked to estimate the proportion of students in a college of 15 000 students who intend to work during the s... |
(b) It is common to treat a variable such as X as a quantitative variable. Do you think this is correct? Would it be correct to treat X as a categorical variable? (c) A common criticism of using such a scale is that the interpretation of a statement such as 3 “I’m somewhat dissatisfied” varies from one person to anothe... |
a simple random sample of size n from where (a) Prove that the mean of f X 0 is given by f X 0 n f X 0 and b) Prove that the variance of f X 0 is given by (Hint: Note that we can write ) 2 over all population elements i.e., the average of X x x equals x x f X x Bernoulli 2 f X x i.e., the equals 5.4.1) (Hint: Use the ... |
Under i.i.d. sampling, prove that as n (Hint: f X x CHALLENGES i 1 1 and into two subpopulations and that we can partition 5.4.20 (Stratified sampling) Suppose that X is a quantitative variable defined on a pop ulation 2, such that a proportion p of the full population is in 1 Let fi X denote the conditional population... |
by an expert. Comment on this assertion. 5.4.22 Suppose it is claimed that a quantitative measurement X defined on a finite population is approximately distributed according to a normal distribution with un known mean and unknown variance. Explain fully what this claim means. 5.5 Some Basic Inferences Now suppose we ar... |
values In this case, f X x 0 1 whenever x is a data value and is 0 otherwise. To compute FX x we simply count how many sample values are less than or equal to x and divide by n 0 2, and FX 4 10. For example, FX 0 FX 0 9 10 0 10 2 10 0 9. 3 An important class of characteristics of the distribution of a quantitative var... |
5.2 Estimating Quantiles A natural estimate of a population quantile x p p. Note, however, that FX is not continuous, so there may not be a solution to (5.5.1) using FX. Applying Definition 5.5.1, however, leads to the following estimate. First, order the x n (see is the i n th quantile of the empirical distribution, x... |
2 75 So in this case, we estimate that 25% of the population under study has an X measure ment less than 0.05, etc. EXAMPLE 5.5.3 Measuring Location and Scale of a Population Distribution Often we are asked to make inferences about the value of the population mean and the population variance X 1 2 X 1 X X 2 X where i... |
in an area are well known to exhibit a rightskewed distribution. A few houses selling for very high prices will not change the median price but could result in a big change in the mean price. When we have a symmetric distribution, the mean and median will agree (provided the mean exists). The greater the skewness in ... |
less than or equal to the upper limit (the third quartile plus 1.5 times the I Q R) and by the least value greater than or equal to the lower limit (the first quartile minus 1.5 times the I Q R). Values beyond the adjacent values, when these exist, are plotted with a ; in this case, there are none. If we changed x 10 1... |
.22 0.08 Coding Chocolate as 1, Vanilla as 2, Butterscotch as 3, and Strawberry as 4, Figure 5.5.6 presents a bar chart of these data. It is typical for the bars in these charts not to touch. Chapter 5: Statistical Inference 289.4 0.3 0.2 0.1 1 2 3 4 Flavor Figure 5.5.6: A bar chart for the data of Example 5.5.5. 5.5.3... |
as its mean, median, or the, and we is correct, or, equiva : f f 290 Section 5.5: Some Basic Inferences value of the true distribution function F at a specified value. We will denote this characteristic of interest by. For example, when the characteristic of interest is the mean of the true distribution of a continuou... |
where ingly, the statistical model is given by the family of N 2 R1 R is unknown. Does this statistical model make sense, i.e., is the assumption of normality appro priate for this situation? The density histogram (based on 12 equallength intervals from 59.5 to 71.5) in Figure 5.5.7 looks very roughly normal, but th... |
. Notice that x is at the center of the interval. The 0 3734 leads to what is theory in Chapter 6 will show that, in this case, choosing c known as a 0 95confidence interval for We then take the halflength of this interval, namely, sc 2 379 0 373 4 0 888, 292 Section 5.5: Some Basic Inferences as a measure of the accu... |
an automatic banking machine during 15 successive oneminute time intervals and f X 4 (a) Record estimates of f X 0 (b) Record estimates of FX 0 FX 1 FX 2 FX 3 and FX 4 (c) Plot f X. (d) Record the mean and variance. f X 1 f X 2 Chapter 5: Statistical Inference 293 (e) Record the median and IQR and provide a boxplot. ... |
0 distributions where If our interest is in making inferences about the third moment of the distribution, then determine 2 0 distributions If our interest is in making R1 is unknown, while R1 is unknown, while 2 0 is known. 2 0 is known. 294 Section 5.5: Some Basic Inferences 2 2 2 R1 R1 2 0 is known. R1 is unknown, w... |
(b) Calculate the empirical distribution function at the data points. Chapter 5: Statistical Inference 295 (c) Calculate the sample mean and the sample standard deviation. (d) Obtain the sample median and the sample interquartile range. (e) Based on the histograms obtained in Exercise 5.4.5, which set of descriptive s... |
. 1 are unknown. (a) Compute an estimate of x0 9 based on the appropriate order statistic. (b) Compute an estimate based on the fact that x0 9 percentile of the N 0 1 distribution. (c) If you knew, or at least were willing to assume, that the sample came from a normal distribution, which of the estimates in parts (a) o... |
. The likelihood function is one of the most basic concepts in statistical inference. Entire theories of inference have been constructed based on it. We discuss likeli hood methods in Sections 6.1, 6.2, 6.3, and 6.5. In Section 6.4, we introduce some distributionfree methods of inference. These are not really example... |
observe s 10. Then L 1 10 1 106. Both values are quite small, but note that the likelihood sup 1 a thousand times more than it supports 106 1 2 2. Accordingly, we are only interested in likelihood ratios L L 1 s 2 s 1 2 based on the likelihood for when it comes to determining inferences for i.e., function This implie... |
Binomial 4, and the likelihood specified by the observed data is L 6 9 6 4 1 6 Note that this likelihood function is a positive multiple of (6.1.1). So the likelihood principle asserts that these two model and data combinations must yield the same inferences about the unknown. In effect, the likelihood principle says ... |
We interpret this to mean that the probability of s occurring when 1 is true is greater than the probability of s occurring when 2 is true. So the data s support 1 more than 2 A similar interpretation applies when s 1 and V is a region containing s Rn for n s and interpret the ordering this imposes on the values of Th... |
applications, as it assumes that the variance is known, while the mean is unknown. For example, if we are interested in the distribution of heights in a population, it seems unlikely that we will know the population variance but not know the population mean. Still, it is an important statis tical model, as it is a co... |
fficiency, discussed in the next section. 1 6.1.1 Sufficient Statistics The equivalence for inference of positive multiples of the likelihood function leads to a useful equivalence amongst possible data values coming from the same model. For s2 for some example, suppose data values s1 and s2 are such that L c 0 From the ... |
that, when T s1 T s2 we have L s1 h s1 g T s1 h s1 g T s1 h s2 g T s2 h s2 g T s2 h s1 h s2 g T s2 because g T s1 h s2 g T s2 c s1 s2 L s2 Note that the name of this result is motivated by the fact that we have factored f as a product of two functions. The important point about a sufficient statistic T is that we are i... |
By the factorization theorem we see immediately, from the discussion in Example 6.1.4, that x is a sufficient statistic. Now any likelihood function for this model is a positive multiple of exp n 2 2 0 x 2. Notice that any such function of its maximum, namely, at function for this model, and it is therefore a minimal s... |
have the same likelihood, so the counts x1 x2 x3 comprise a sufficient statistic. Now it turns out that this likelihood function is maximized by taking 1 2 3 x1 n x2 n x3 n So, given the likelihood, we can compute the counts (the sample size n is assumed known). Therefore, x1 x2 x3 is a minimal sufficient statistic. Sum... |
2 x3 34 44 22. Determine the form of the likelihood function for the unknown proportions of students in the population that are in these categories. 6.1.5 Determine the constant that makes the likelihood functions in Examples 6.1.2 and 6.1.3 equal. 6.1.6 Suppose that x1 distribution, where xn is a sample from the Berno... |
0 1] while another statistician records a likelihood function as 100 2 for [0 1] Explain why these likelihood functions are effectively the same. PROBLEMS 6.1.15 Show that T defined in Example 6.1.6 is a minimal sufficient statistic. (Hint: Show that once you know the likelihood function, you can determine which of the t... |
point will not maximize the likelihood when x 0 5, so x cannot be obtained from the likelihood by maximization, as in Exercise 6.1.6. In general, consider the second derivative of the log of the likelihood at any point 2 2 is the maximum.) distribution where 308 Section 6.2: Maximum Likelihood 0 0 5 and note that know... |
5.5.3 and start with estimation. s as a basis for making We now begin to consider the specific When we are interested in a point estimate of then a value s that maximizes L s is a sensible choice, as this value is the best supported by the data, i.e., L s s L s (6.2.1) for every Definition 6.2.1 We call likelihood estim... |
where g, we use 1 such that 2 f only the way they are labelled. We then have the following result. Theorem 6.2.1 If 1–1 function defined on ization. s is an MLE for the original parameterization and, if is a is an MLE in the new parameter, then s s PROOF If we select the likelihood function for the new parameterizatio... |
this exists. So when onedimensional realvalued parameter, then S s l s provided this partial derivative exists (see Appendix A.5 for a definition of partial deriv ative). We restrict our attention now to the situation in which is onedimensional. To obtain the MLE, we must then solve the score equation S s 0 (6.2.2)... |
able to derive simple formulas for the MLE. This is not always possible. Consider the following example. EXAMPLE 6.2.4 Consider a population in which individuals are classified according to one of three types labelled 1, 2, and 3, respectively. Further suppose that the proportions of individuals falling in these catego... |
roots are 0 47847 We can see this graphically in the plot of the loglikelihood provided in Fig ure 6.2.1. 1 28616 and 0 47847 so it is immediate that the MLE is 5 and x3 70 x2 x1 0.4 0.45 0.5 0.55 0.6 theta theta 90 95 100 105 110 115 120 lnL lnL Figure 6.2.1: The loglikelihood function in Example 6.2.4 when... |
Model Suppose that x1 xn is a sample from an N and R1 0 are unknown. The parameter in this model is twodimensional, given by 2 2 distribution, where The likelihood function is then given by R1 0 L 2 x1 xn 2 2 n 2 exp n 2 2 x 2 exp n 1 2 2 s2 as shown in Example 6.1.8. The loglikelihood function is given by l 2 x1 xn... |
establishes that s is a true MLE but not otherwise. is a 1–1 function defined on for some function If s If is not 1–1, then we can often find a complementing function defined on so that is a 1–1 function of. Then, by Theorem 6.2.1, s s s s is the joint MLE, but perform badly, as it ignores the information in example illu... |
numerical methods. 1 A necessary and sufficient condition for to be a local maximum, when the loglikelihood has continuous second partial derivatives, is that the matrix of second partial derivatives of the loglikelihood, evaluated at k, must be negative definite (equivalently, all of its eigenvalues must be negative)... |
, where [0 1] is distribution, where [0 1] is 318 Section 6.2: Maximum Likelihood xn is a sample from a Poisson 6.2.4 If x1 unknown, then determine the MLE of 6.2.5 If x1 0 xn is a sample from a Gamma 0 is unknown, then determine the MLE of. distribution, where 0 is distribution, where 0 0 and. xn is the result of inde... |
5 3 2 for 0 is 2. How does this MLE differ is a sample from an N 0 2 0 distribution, where 2 distribution, where R1 xn 2. the MLE. 6.2.15 If two functions of are equivalent versions of the likelihood when one is a positive multiple of the other, then when are two loglikelihood functions equivalent? 6.2.16 Suppose you... |
un 1. Justify is a sample from an N 1 distribution where 1 distribution where Multinomial n 0 is un xn and we observe X1 X2 X3 (a) Determine the MLE of 1 (b) What is the plugin MLE of 6.2.24 If x1 x1 x2 x3. 2 1 3. 2 2 2 3? xn is a sample from a Uniform[ 1 2] distribution with 1 2 R2 : 1 2 determine the MLE of deter... |
son and S. Newcomb in 1882. 28 22 36 26 28 28 26 24 32 30 27 24 33 21 36 32 31 25 24 25 28 36 27 32 34 30 25 26 26 25 44 23 21 30 33 29 27 29 28 22 26 27 16 31 29 36 32 28 40 19 37 23 32 29 2 24 25 27 24 16 29 20 28 27 39 23 Table 6.3.1: Speed of light measurements. Figure 6.3.1 is a boxplot of these data with the vari... |
of interest (e.g., s s quartile or the variance). s to be close to the true value of In an application, we want to know how reliable the estimate is. In other, or is there a reasonable words, can we expect s is far from the true value? This leads us to consider the sampling chance that distribution of s under repeated... |
given by E T Note that when the bias in an estimator is 0, then the MSE is just the variance. Unbiasedness tells us that, in a sense, the sampling distribution of the estimator is centered on the true value. For unbiased estimators, MSE s T Var s T Chapter 6: Likelihood Inference 323 and Sd s T Var s T is an estimate ... |
.1. EXAMPLE 6.3.3 Application of the Bernoulli Model A polling organization is asked to estimate the proportion of households in the pop ulation in a specific district who will participate in a proposed recycling program by separating their garbage into various components. The pollsters decided to take a sam ple of n ... |
ollary 4.6.2) E S2 2 i.e., S2 is an unbiased estimator of error of the estimate x 2. The quantity s n is referred to as the standard EXAMPLE 6.3.5 Application of the LocationScale Normal Model In Example 5.5.6, we have a sample of n calculated x obtained the estimate s 64 517 is s interpreting exactly what this number... |
2 We see immediately that this gives for the consistency of some of the estimators discussed in this section. In fact, Theorem 6.5.2 gives the consistency of the MLE in very general circumstances. Furthermore, Accordingly, we the plugin MLE will also be consistent under weak restrictions on can think of maximum likeli... |
the true value. But this tells us nothing we did not already know. So the idea is to try to make use of the information in the data to construct an interval such that we have a high confidence, say, 0 99 that it contains the true value and is not any longer than necessary. We then interpret the length of the interval a... |
2) (6.3.3) 328 Section 6.3: Inferences Based on the MLE for every equality in (6.3.3) whenever R1 where is the N 0 1 cumulative distribution function. We have 1 k 2 th quantile of the N 0 1 distribution. and so k This is the smallest constant k satisfying (6.3.3). 2 where z denotes the z 1 We have shown that the likeli... |
For ex 0 9974), dard error controls the lengths of the confidence intervals for the unknown ample, we know that with probability approximately equal to 1 (actually the interval [x n] contains the true value of. 3 0 Example 6.3.6 serves as a standard example for how confidence intervals are often constructed in statist... |
79 1 96 0 79 1 0 79 1000 [0 76475 0 81525] The margin of error in this case equals 0 025245 so we can conclude that we know the true proportion with reasonable accuracy based on our sample. Actually, it may be that this accuracy is not good enough or is even too good. We will discuss methods for ensuring that we achie... |
1 is the distribution function of Now, by Theorem 4.6.66.3.6) independent of n 1 S2 2 2 n 1. Therefore, by Definition 4.6.2, X T n n 1 S2 2 X S n t n 1 So if we take where t t 1 is the th quantile of the t k 1 2 n distributionconfidence interval for The quantiles of the t distributions are available is an exact from a s... |
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