# A Nonparametric Estimator of Species Overlap

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 A Nonparametric Estimator of Species Overlap Jack C. Yue 1, Murray K. Clayton 2, and Feng-Chang Lin 1 1 Department of Statistics, National Chengchi University, Taipei, Taiwan 11623, R.O.C. and 2 Department of Statistics, University of Wisconsin-Madison, Madison, WI , U.S.A. SUMMARY For two communities, species overlap has been defined by Smith et al. (1996) as the probability that a randomly selected species is present in both communities, given that it is present in at least one community. Species overlap can thus be used to describe the similarity of two communities. In contrast to the parametric estimator of Smith et al., we propose a Nonparametric Maximum Likelihood Estimator (NPMLE). We prove that the NPMLE is consistent and asymptotically normally distributed, and show that computation of the NPMLE and its standard error is straightforward. We also compare the NPMLE and the estimator of Smith et al. for a variety of situations. Key Words: Bootstrap; Jaccard s index; Maximum likelihood estimator; Similarity index; Species diversity. 1

2 1. Introduction In ecology, the comparison of two or more populations and the evaluation of a population s change over time are often of interest. The number of species and proportions of species, and functions of these, are often used to measure species diversity. Shannon s index (or the Shannon-Wiener index) and Simpson s index are two well-known measures used to describe community structure. The comparisons of populations are based on the indices calculated separately for each population. Jaccard s index is another way to compare populations, specifically for describing their similarity. It is defined as the ratio of the number of common species to the number of distinct species in two populations, i.e. Jaccard s index is given by θ J = c/(s 1 + s 2 c), where s i is the number of species in population i, i = 1, 2 and c is the number of common (i.e., shared ) species. The Jaccard index does take the species common to both populations into account and is easy to compute. However, all species have equal weight and species proportion information is not used in the Jaccard index. Because all species are equally weighted, it is possible that the similarity of two communities would be underestimated by the Jaccard index. Smith et al. (1996) proposed a new species overlap measure, defined as the probability that, given a randomly selected species is present in at least one of the two communities, it is present in both communities. This measure takes into account the number of species and puts larger weight on those species which appear more frequently in the sample. When Smith et al. estimated this measure for data of Abele (1979), their estimate of community species overlap was 50% larger than Jaccard s index. 2

3 The species overlap measure proposed by Smith et al. seems more appropriate than Jaccard s measure of species overlap, since more information is used in the new measure. Moreover, it has an interpretation that is intuitive, and expressible as a probability. In this paper, we propose a Nonparametric Maximum Likelihood Estimator (NPMLE) of species overlap, given that the species proportions are fixed. The NPMLE is easy to compute (similar to the Jaccard index) and its standard error can be computed via bootstrapping or via an asymptotic expression. We first introduce the NPMLE in Section 2, followed by some theoretical results in Section 3. We use two examples to compare the estimate of Jaccard s index, the estimate proposed by Smith et al., and the NPMLE in Section 4. Further comparisons among these estimates, based on computer simulation, are in Section 5. In Section 6 we make some concluding remarks, paying special attention to the context in which our evaluations and comparisons are made. 2. Notation Smith et al. (1996) first proposed describing the similarity of two communities using the probability that a randomly selected species is present in both communities, given that it is present in at least one community. Their approach is based on a parametric assumption, which they called the delta-beta-binomial model. Let N i1 and N i2 be the numbers of individuals of species i in the sample from population 1 and 2, let B(, ) be the beta function, and let α and β be parameters. Also, let p be the probability of observing species that are in population 1 but not in population 2, and let q be the probability of observing species that are in population 2 but not in population 1. Smith et al. assume that the 3

4 probability of observing j individuals of species i in population 1, conditional on N i1 + N i2 = n i, is d(j; n i, α, β, p, q) = δ(j)q + δ(n i j)p + (1 p q)b(j; n i, α, β), (1) where and b(j; n i, α, β) = 1 if j = 0 δ(j) = 0 if j 0 n i j B(j + α, n i j + β). (2) B(α, β) Based on this model, 1 p q is the corresponding overlap index. An estimate of the species overlap can be obtained by numerical maximum likelihood estimation, and Smith et al. approximate its standard error by inverting the observed information matrix. To introduce an NPMLE of species overlap, consider an experiment where a species is selected at random. For this experiment, define two events: A = {Species is observed in population 1} and B = {Species is observed in population 2}. Then A B is the event that the species selected is in both populations 1 and 2. Let θ n denote the probability that a randomly selected species is present in both populations, given that it is present in at least one population. Then using the notation defined above, θ n = P (A B)/P (A B). We will not evaluate P (A B) and P (A B) directly. Instead, since P (A) + P (B) = P (A B) + P (A B), we can express θ n as θ n = P (A B) P (A B) = a b a + b a b, (3) where a = P (A B A) and b = P (A B B), i.e. a and b are the probabilities of observing shared species in populations 1 and 2, respectively. Note that, 4

5 similar to Smith et al. θ n is defined conditionally on A B. In other words, we do not explicitly define a mechanism for sampling species directly; given any such mechanism that follows the usual rules of probability, our definition is logically consistent. When any species in population 1 is equally likely to be selected, i.e. population 1 is uniform, then P (A B A) = c/s 1. Similarly, P (A B B) = c/s 2 under the uniform distribution assumption for population 2. Then it is straightforward to show that θ n = θ J, and thus the Jaccard index can be treated as a special case of θ n if sampling of species is uniform within each population. However, as pointed out by Smith et al., the Jaccard index is likely to underestimate the true percentage of overlap since the species proportions are not included in the overlap measure. In order to reduce the underbias of the Jaccard index, in this study, we assume that the probability of observing a certain species in a population is equal to its species proportion. Then a is the sum of species proportions for all shared species in population 1, and b is the sum of species proportions for all shared species in population 2. To gain some intuition for our model, note that, based on our definition, the probability that a randomly sampled species is from population 1 is given by P (A)/P (A B) = b/(a + b ab) if a, b 0. Then, for example, suppose that a = 1, i.e. all species in population 1 are shared species, and so population 1 can be treated as a sub-population of population 2. Then P (A)/P (A B) = P (A B)/P (B) = b and P (B)/P (A B) = 1. These results make intuitive sense under the assumption a = 1. Likewise, if a = 0, i.e. there are no shared species and b = 0 as well, then A B = and P (A B)/P (A B) = 0. We can use Maximum Likelihood to find estimates of a and b, and then find the MLE of θ n, i.e. ˆθn = âˆb/(â + ˆb âˆb) where â and ˆb are the MLE s 5

6 of a and b, respectively. For example, suppose that 6 of the observed species, which account for 20 of 100 individuals in the sample of population 1, are also observed in the sample of population 2. Then â equals 20/100 = The standard errors of â, ˆb, and ˆθ n can be obtained from a bootstrap simulation. The computation of the NPMLE will be discussed in detail in Section 4. First, we show some theoretical results in the following section. 3. Theoretical Results Let p i be the proportion of species i (i = 1,..., s 1 ) in population 1, and q j the proportion of species j (j = 1,..., s 2 ) in population 2. We assume that the p i s and q j s are fixed. Let ξ X and ξ Y be the observed species counts for populations 1 and 2, respectively. Finally, let n 1 and n 2 be the numbers of observations from population 1 and 2, and let x i and y i be the numbers of occurrences of the ith species in populations 1 and 2, respectively. We now proceed to introduce an NPMLE of a. Note that a can be expressed as a = i p i I{i C} where C is the index set of the shared species. A natural choice of the estimate for p i is ˆp i = x i /n 1, and a natural estimate of I{i C} is I{x i > 0}I{y i > 0}, i.e. the ith species is in both populations if we observe it from the sample at least once in each population. Then the NPMLE of a can be expressed as â = i x i /n 1 I{x i > 0} I{y i > 0} = i x i /n 1 I{y i > 0}. The last equality holds because x i = 0 implies ˆp i = 0. Finally, without loss of generality we assume that species 1 to c are species common to both populations. Then a = c i=1 p i, â = c i=1 x i /n 1 I{y i > 0}, b = c i=1 q i, and ˆb = c i=1 y i /n 2 I{x i > 0}. 6

7 Intuitively, â is close to a when the number of observations taken from population 2 is sufficiently large. In fact, E(â) = E[E( 1 i c c i=1 x i n 1 I{y i > 0} ξ X ) ξ Y ] = p i = a, as n 2. 1 i c We can show a similar property for ˆb = c i=1 y i /n 2 I{x i > 0}. p i (1 (1 q i ) n 2 ) It is straightforward to show that the variances of â and ˆb are V ar(â) = and V ar(ˆb) = c c p 2 i [1 (1 q i ) n 2 ](1 q i ) n 2 + i=1 i=1 +2 p i p j [(1 q i q j ) n 2 (1 q i ) n 2 (1 q j ) n 2 ] 1 i<j c 2 1 i<j c p i (1 p i ) n 1 [1 (1 q i ) n 2 ] p i p j n 1 [1 (1 q i ) n 2 (1 q j ) n 2 + (1 q i q j ) n 2 ] (4) c c qi 2 [1 (1 p i) n 1 ](1 p i ) n 1 + i=1 i=1 +2 q i q j [(1 p i p j ) n 1 (1 p i ) n 1 (1 p j ) n 1 ] 1 i<j c 2 1 i<j c q i (1 q i ) n 2 [1 (1 p i ) n 1 ] q i q j n 2 [1 (1 p i ) n 1 (1 p j ) n 1 + (1 p i p j ) n 1 ] (5) respectively, which can be computed via V ar(x) = V ar[e(x Y )] + E[V ar(x Y )]. Therefore, V ar(â) a(1 a)/n 1 as n 2 and V ar(ˆb) b(1 b)/n 2 as n 1. Similarly, the covariance of â and ˆb is given by Cov(â, ˆb) = c p i q i [(1 q i ) n 2 + (1 p i ) n 1 (1 p i ) n 1 (1 q i ) n 2 ] i=1 + p i q j [p j (1 p j ) n1 1 + q i (1 q i ) n i j c (1 p j ) n1 1 (1 q i ) n2 1 (p j + q i p j q i )] (6) 7

8 and Cov(â, ˆb) 0 as min{n 1, n 2 }. From direct calculation, we can show that the moment generating function of n 1 â = c i=1 x i I{y i > 0} satisfies E[e c i=1 x it I{y 1 > 0,..., y c > 0}] E[e c i=1 x ii{y i >0}t ] E[e c i=1 x it ]. Note that the left term in the preceding inequality E[e c i=1 x it I{y 1 > 0,..., y c > 0}] [(1 a) + ae t ] n 1 [1 [ (1 a) + ae t] n 1 c (1 q i ) n 2 ] i=1 as n 2, and the right term also converges to the same limit. In other words, n 1 â converges to a binomial random variable if n 2 is sufficiently large. Thus, â converges to a in probability and â is asymptotically normally distributed if min{n 1, n 2 }. ˆb behaves similarly, as does the joint distribution of (â, ˆb). Since (â, ˆb) are asymptotically jointly normally distributed, applying Cramer s delta theorem, ˆθ n is also asymptotically normally distributed, and ˆθ n converges to θ n in probability. 4. Examples In this section, two examples are used to compare the performance of different estimates of species overlap: one example was originally introduced in Abele (1979), and the other is from Chao (1995). Let ˆθ J be the estimate of the Jaccard index, let ˆθ b the estimate of θ n proposed by Smith et al., and let ˆθ n the NPMLE of θ n. Example 1. Abele describes the species abundance distribution of decapod crustacean (crab) communities at two locations in Panama (data shown in Smith et al.). Table 1 shows the estimate of Jaccard s index, the estimate 8

9 of Smith et al., and the NPMLE. Note that the estimate of Jaccard s index is usually calculated by plugging in the numbers of observed species, yielding ˆθ J = 31/ The tabulated estimate of the species overlap proposed by Smith et al. is from their paper in The standard errors of the estimate for Jaccard s index, the estimator of Smith et al., and the NPMLE are from 1,000 bootstrap simulations. (Note that Smith et al. calculate the standard error of their estimator to be.100. This results from a different sampling model, as discussed in Section 6.) As mentioned previously, the estimate by Smith et al. is about 50% larger than that of Jaccard s index. The NPMLE is also about 50% larger, but is slightly smaller than (and within 2 standard errors of) that of Smith et al. Also, the standard error of the NPMLE is the smallest among these estimates, and is about one-half of that for Jaccard s index. The standard error of the estimate of Smith et al. is the largest. The variances of â and ˆb can be estimated from (4) and (5), yielding and , respectively. Similarly, the covariance of â and ˆb estimated from (6) equals Applying the delta method, the standard error of ˆθ n is approximately , which is very close to the standard error obtained via bootstrapping (0.0150). Table 1 Species Overlap Estimates Decapod crustaceans Wild birds ˆθ J ˆθb ˆθn ˆθJ ˆθb ˆθn Estimate s.e Example 2. Chao (1995) and Chao et al. (2000) describe the species abun- 9

10 dance of wild bird communities at two heavily polluted river estuaries (the Ke- Yar River and the Chung-Kang River) of north-western Taiwan. Bird counts were collected by the Wild Bird Society of Hsin-Chu on a weekly basis for one year. Species overlap is of interest here because the two estuaries are environmentally similar. Table 2 shows the numbers of individuals observed for different species (with ranks representing different species) of birds in these two estuaries. The standard errors of all three estimates are from 1,000 bootstrap simulations. These and the estimates ˆθ J, ˆθ n, and ˆθ b are listed in Table 1. The NPMLE is 58% larger than that of Jaccard s index, and the estimate by Smith et al. is about 40% larger, about 12% smaller than the NPMLE. Similar to the previous example, the standard error of the NPMLE is the smallest and is one-half the size of the estimate of Jaccard s index. The standard error of the estimator of Smith et al. again is the largest, about three times that for the NPMLE, similar to the result in the previous example. However, based on the standard errors of ˆθ n and ˆθ b, and since ˆθ n is asymptotically unbiased, ˆθ b appears to be significantly underbiased. From (4), (5), and (6), we have V ar(â) = , V ar(ˆb) = , and Cov(â, ˆb) = The standard error of ˆθ n via the delta method thus equals and again is very close to that obtained via bootstrapping (0.0083). 5. Simulation In this section, we use simulation to compare the performance of an estimate of Jaccard s index, the estimate proposed by Smith et al., and the NPMLE. Two types of species proportion distribution are considered: balanced and unbal- 10

11 anced. In a balanced population, every species is equally likely to be observed, while in an unbalanced population some species are dominant. In particular, we assume that the species proportions follow a geometric distribution, i.e. p i α i for 1 i s 1, and similarly for q j. Computations and simulations in this report were based on a Pentium II IBM compatible PC. The simulations were based on S-Plus statistical software, version 4.5. We model three different types of species overlap between the two populations: Type 1: The shared species are dominant in both populations; Type 2: The shared species have low abundance in both populations; Type 3: The shared species are dominant in one population, but have low abundance in the other. When every species has the same species proportion, i.e. the balanced population case, these 3 types of overlap are the same. But in the unbalanced population case, if all other conditions are the same θ n has its maximum value in the Type 1 case since common species are dominant in both populations. Note that, because the true population structure and type of overlap is known for these simulations, it is possible to calculate the true values of θ J and θ n, which we present in Table 3. In addition to comparing the accuracy of the estimates to the real species overlap, we also use SD (standard deviation) to measure the precision of the estimates. We define n SD = i=1 (ˆθ i ˆθi ) 2 n 1 where ˆθ i is the estimate of species overlap in the ith simulation run, and ˆθi is 11

12 the average of the estimates in n simulations. The results shown in this section are all based on 500 simulation runs. Table 3(a) shows the simulation results for the balanced population cases (thus θ J = θ n ), where both populations have 20 species, and 5 and 15 species are in common, respectively. In both cases, ˆθJ appears to have the fastest convergence rate and is quite accurate even when the number of observations is small. The NPMLE appears to have the slowest convergence rate but is generally good compared to ˆθ b since the overestimation by ˆθ b is large when n = 50. When the number of observations is large, all three estimates perform well. Tables 3(b) to 3(d) show the simulation results for unbalanced population cases with s 1 = s 2 = 20, and when the species proportions in each population are geometric with α = 0.8. Table 3(b) shows the results of the Type 1 case, where the common species are dominant. The NPMLE and the estimate of Jaccard s index are both very accurate when the number of observations is large. The SD s of the NPMLE decrease in proportion to the inverse of the square root of the sample size, while the SD s of the estimate of Jaccard s index decrease much faster. The estimates of Smith et al. are significantly different from ˆθ n and ˆθ J even when the number of observations is 1000, and ˆθ b is actually closer to θ J, instead of θ n. The SD s of ˆθ b decrease faster than the inverse of the square root of the sample size when the number of common species is 5, but do not decrease monotonically as the sample size increases when the number of species is 15. Similar patterns appear in the Type 2 and Type 3 cases as well. The NPMLE is the most accurate estimate of θ n in both Type 2 (Table 3(c)) and Type 3 (Table 3(d)) cases. But unlike the Type 1 case where the species overlap is the largest, it takes more observations to have the NPMLE close to θ n. The 12

13 SD s of the NPMLE are also the smallest among these three estimates. The estimate of Smith et al. is closer to θ J in the Type 2 case, but does not seem to converge to θ n or θ J in the Type 3 case. Note that the SD s of ˆθ b in the Type 3 cases do not decrease monotonically as the sample size increases, suggesting that θ b may be slow to converge, if it converges at all. 6. Conclusion Our paper deals with the notion of species overlap as defined by Smith et al. We find this definition appealing: it makes natural reference to a probabilistic interpretation that is intuitive and straightforward. In this paper we have compared a nonparametric maximum likelihood estimate of this quantity with the estimator of Smith et al. It is important to delineate the context in which this comparison has taken place. We have focused on models for which the species proportions are fixed, similar to the notion of fixed populations defined by Engen (1978). This is different from the setting in Smith et al., which is based on a superpopulation model. Superpopulation models may be viewed in some sense as similar to the random populations of Engen, or, put more broadly, the comparison may be viewed as analogous to the difference between fixed and random effects in ANOVA. Although we have focused on fixed populations, we believe that fixed and superpopulation models are equally valid, and their specific use depends on the problem at hand. Engen notes a number of reasons for considering populations as fixed, and numerous authors use such models in the ecological setting. (See, for example, Engen, 1974, and the references cited therein.) So, for example, consider an ecologist who wishes to study and compare two 13

14 populations that are relatively fixed in their composition at the time of sampling. The ecologist samples these by sampling individuals from them. The ecologist has at hand two possible estimators: the NPMLE and the estimator of Smith et al. How would the ecologist expect those two estimators to behave? Our paper provides some insights into this question. It is important to emphasize that, regardless of the genesis of the estimators, the comparison of the estimators remains valid. (To make a broad analogy, it is appropriate to ask what the frequentist properties are for a given Bayesian estimator, even if that estimator was not conceived with frequentist issues in mind.) In this context, then, our proposed NPMLE generally provides a good estimate to the new index if the sample size is not too small. Also, ˆθ n has good theoretical and empirical properties, and it is easy to compute, similar to the estimate of Jaccard s index. The variances of ˆθ n derived from the delta method and bootstrapping are very close in the two examples shown. The decreasing rate of the SD s in ˆθ n is approximately proportional to 1/ (Sample Size) from our simulation. The parametric estimate by Smith et al. was designed to estimate their new similarity index. However, their estimate seems to be strongly influenced (or dictated) by its parametric (species structure) assumption. If the parametric assumption can describe well the species structure, e.g., the balanced population case, ˆθ b performs well when the sample size is fairly large. In particular, in the balanced population case and when the sample size is large, all species would have about the same number of occurrences. As a result, d(j; n i, α, β, p, q) approximately equals p, q, and 1 p q if species i appears only in population 1, 2, and both populations, respectively. Applying maximum likelihood estimation, we have, approximately, ˆp = ˆq = (s c)/(2s c) and ˆθ b = c/(2s c) = θ J = θ n 14

15 when s 1 = s 2 = s. This gives a heuristic sense of why ˆθ b converges faster than ˆθ n in the balanced population case of our simulation. When the underlying species structure is not equiprobable, ˆθ b may not be a good estimate of θ n (or θ J ). For example, in the Type 1 case, the number of occurrences for the common species in our model follows a geometric pattern and b(n i ; j, α, β) would be small when n i is large. Since fewer observations have a large weight, using the MLE would produce an inappropriate ˆθ b. Also, when the number of observations is very large, d(j; n i, α, β, p, q) would have a form similar to that in the balanced population case. For example, suppose that s 1 = s 2 = 2, c = 1, and α = 0.8 in the Type 1 unbalanced population case, and suppose the sample size is large. Again, arguing heuristically, we would expect that ˆθ b = 1/3(= θ J ), instead of θ n = 5/13. This might explain why ˆθ b is actually closer to the Jaccard index instead of the index proposed by Smith et al. Therefore, if the new index of overlap is to be used, we would recommend the NPMLE as its estimate. Although it is not a specific focus of our study, we note that the plug-in estimate of Jaccard s index also performs well when the sample size is not too small. In particular, ˆθ J appears to have the fastest convergence rate in the balanced population case, and is also quite comparable in unbalanced population cases when the sample size is big. Except in the Type 2 unbalanced population cases, the decreasing rate of SD s in ˆθ J appears to be faster than 1/ (Sample Size). Note that McCormick et al. (1992) proved the asymptotic normality of ˆθ J in the balanced population case, but they require that the sample size and the number of species increase at the same rate. In this paper, we modified the conditional probability definition of species overlap proposed by Smith et al. and generalized the setting to include the 15

16 Jaccard index as a special case. All three estimates of species overlap measures considered perform well in various situations, and we think that the NPMLE has properties that make it appealing. In particular, the NPMLE has good theoretical properties and is easy to compute. As a result, it provides a good estimate to the similarity index proposed by Smith et al. under the setting of fixed populations. However, although the overlap measure defined via the conditional probability setting can reduce the underbias of the Jaccard index in describing the species overlap, neither Smith et al. nor we provide a concrete plan for sampling species from two populations. In the future, we will continue working on searching a species overlap measure which not only reflect the true species overlap and has a probability interpretation, but also comes with a firm sampling plan. ACKNOWLEDGEMENTS The authors are grateful for the S-Plus program from Professor W. Smith, Taiwan wild bird data from Professor A. Chao, and insightful comments from the associate editor and an anonymous reviewer which helped us to clarify the context of our work. REFERENCES Abele, L. G. (1979). The Community Structure of Coral-associated Decapod Crustaceans in a Variable Environment. In Ecological Processes in Coastal Marine Systems: Marine Science 10, , Florida State University. New York: Plenum Press. 16

17 Chao, A. (1995). How Many Classes? (in Chinese). Communications in Mathematics 19, 1-7. Chao, A., Hwang, W., Chen, Y., and Kuo, C. (2000). Estimating the Number of Shared Species in Two Communities. Statistica Sinica 10, Engen, S. (1974). On Species Frequency Models. Biometrika 61, Engen, S. (1978). Stochastic Abundance Model. London: Chapman and Hall. McCormick, W. P., Lyons, N. I., and Hutcheson, K. (1992). Distributional Properties of Jaccard s Index of Similarity. Communications in Statistics: Theory and Methods 21, Smith, W., Solow, A. R., and Preston, P. E. (1996). An Estimator of Species Overlap Using a Modified Beta-binomial Model. Biometrics 52,

### i=1 h n (ˆθ n ) = 0. (2)

Stat 8112 Lecture Notes Unbiased Estimating Equations Charles J. Geyer April 29, 2012 1 Introduction In this handout we generalize the notion of maximum likelihood estimation to solution of unbiased estimating

### Chapter 3: Maximum Likelihood Theory

Chapter 3: Maximum Likelihood Theory Florian Pelgrin HEC September-December, 2010 Florian Pelgrin (HEC) Maximum Likelihood Theory September-December, 2010 1 / 40 1 Introduction Example 2 Maximum likelihood

### Diversity partitioning without statistical independence of alpha and beta

1964 Ecology, Vol. 91, No. 7 Ecology, 91(7), 2010, pp. 1964 1969 Ó 2010 by the Ecological Society of America Diversity partitioning without statistical independence of alpha and beta JOSEPH A. VEECH 1,3

### Statistics - Lecture One. Outline. Charlotte Wickham 1. Basic ideas about estimation

Statistics - Lecture One Charlotte Wickham wickham@stat.berkeley.edu http://www.stat.berkeley.edu/~wickham/ Outline 1. Basic ideas about estimation 2. Method of Moments 3. Maximum Likelihood 4. Confidence

### One-Sample Numerical Data

One-Sample Numerical Data quantiles, boxplot, histogram, bootstrap confidence intervals, goodness-of-fit tests University of California, San Diego Instructor: Ery Arias-Castro http://math.ucsd.edu/~eariasca/teaching.html

### ORDER RESTRICTED STATISTICAL INFERENCE ON LORENZ CURVES OF PARETO DISTRIBUTIONS. Myongsik Oh. 1. Introduction

J. Appl. Math & Computing Vol. 13(2003), No. 1-2, pp. 457-470 ORDER RESTRICTED STATISTICAL INFERENCE ON LORENZ CURVES OF PARETO DISTRIBUTIONS Myongsik Oh Abstract. The comparison of two or more Lorenz

### A process capability index for discrete processes

Journal of Statistical Computation and Simulation Vol. 75, No. 3, March 2005, 175 187 A process capability index for discrete processes MICHAEL PERAKIS and EVDOKIA XEKALAKI* Department of Statistics, Athens

### How likely is Simpson s paradox in path models?

How likely is Simpson s paradox in path models? Ned Kock Full reference: Kock, N. (2015). How likely is Simpson s paradox in path models? International Journal of e- Collaboration, 11(1), 1-7. Abstract

### A Bias Correction for the Minimum Error Rate in Cross-validation

A Bias Correction for the Minimum Error Rate in Cross-validation Ryan J. Tibshirani Robert Tibshirani Abstract Tuning parameters in supervised learning problems are often estimated by cross-validation.

### Statistical Applications in Genetics and Molecular Biology

Statistical Applications in Genetics and Molecular Biology Volume 5, Issue 1 2006 Article 28 A Two-Step Multiple Comparison Procedure for a Large Number of Tests and Multiple Treatments Hongmei Jiang Rebecca

### Discussion of the paper Inference for Semiparametric Models: Some Questions and an Answer by Bickel and Kwon

Discussion of the paper Inference for Semiparametric Models: Some Questions and an Answer by Bickel and Kwon Jianqing Fan Department of Statistics Chinese University of Hong Kong AND Department of Statistics

### Seminar über Statistik FS2008: Model Selection

Seminar über Statistik FS2008: Model Selection Alessia Fenaroli, Ghazale Jazayeri Monday, April 2, 2008 Introduction Model Choice deals with the comparison of models and the selection of a model. It can

### A Course in Applied Econometrics Lecture 14: Control Functions and Related Methods. Jeff Wooldridge IRP Lectures, UW Madison, August 2008

A Course in Applied Econometrics Lecture 14: Control Functions and Related Methods Jeff Wooldridge IRP Lectures, UW Madison, August 2008 1. Linear-in-Parameters Models: IV versus Control Functions 2. Correlated

### Linear Regression. Junhui Qian. October 27, 2014

Linear Regression Junhui Qian October 27, 2014 Outline The Model Estimation Ordinary Least Square Method of Moments Maximum Likelihood Estimation Properties of OLS Estimator Unbiasedness Consistency Efficiency

### Methods of Estimation

Methods of Estimation MIT 18.655 Dr. Kempthorne Spring 2016 1 Outline Methods of Estimation I 1 Methods of Estimation I 2 X X, X P P = {P θ, θ Θ}. Problem: Finding a function θˆ(x ) which is close to θ.

### An Encompassing Test for Non-Nested Quantile Regression Models

An Encompassing Test for Non-Nested Quantile Regression Models Chung-Ming Kuan Department of Finance National Taiwan University Hsin-Yi Lin Department of Economics National Chengchi University Abstract

### Using Estimating Equations for Spatially Correlated A

Using Estimating Equations for Spatially Correlated Areal Data December 8, 2009 Introduction GEEs Spatial Estimating Equations Implementation Simulation Conclusion Typical Problem Assess the relationship

### large number of i.i.d. observations from P. For concreteness, suppose

1 Subsampling Suppose X i, i = 1,..., n is an i.i.d. sequence of random variables with distribution P. Let θ(p ) be some real-valued parameter of interest, and let ˆθ n = ˆθ n (X 1,..., X n ) be some estimate

### 1. Fisher Information

1. Fisher Information Let f(x θ) be a density function with the property that log f(x θ) is differentiable in θ throughout the open p-dimensional parameter set Θ R p ; then the score statistic (or score

### Hypothesis Testing. 1 Definitions of test statistics. CB: chapter 8; section 10.3

Hypothesis Testing CB: chapter 8; section 0.3 Hypothesis: statement about an unknown population parameter Examples: The average age of males in Sweden is 7. (statement about population mean) The lowest

### A NOTE ON ROBUST ESTIMATION IN LOGISTIC REGRESSION MODEL

Discussiones Mathematicae Probability and Statistics 36 206 43 5 doi:0.75/dmps.80 A NOTE ON ROBUST ESTIMATION IN LOGISTIC REGRESSION MODEL Tadeusz Bednarski Wroclaw University e-mail: t.bednarski@prawo.uni.wroc.pl

### Topic 12 Overview of Estimation

Topic 12 Overview of Estimation Classical Statistics 1 / 9 Outline Introduction Parameter Estimation Classical Statistics Densities and Likelihoods 2 / 9 Introduction In the simplest possible terms, the

### CS 630 Basic Probability and Information Theory. Tim Campbell

CS 630 Basic Probability and Information Theory Tim Campbell 21 January 2003 Probability Theory Probability Theory is the study of how best to predict outcomes of events. An experiment (or trial or event)

### Basic math for biology

Basic math for biology Lei Li Florida State University, Feb 6, 2002 The EM algorithm: setup Parametric models: {P θ }. Data: full data (Y, X); partial data Y. Missing data: X. Likelihood and maximum likelihood

### Lecture Notes 15 Prediction Chapters 13, 22, 20.4.

Lecture Notes 15 Prediction Chapters 13, 22, 20.4. 1 Introduction Prediction is covered in detail in 36-707, 36-701, 36-715, 10/36-702. Here, we will just give an introduction. We observe training data

### IEOR 165 Lecture 7 1 Bias-Variance Tradeoff

IEOR 165 Lecture 7 Bias-Variance Tradeoff 1 Bias-Variance Tradeoff Consider the case of parametric regression with β R, and suppose we would like to analyze the error of the estimate ˆβ in comparison to

### What s New in Econometrics? Lecture 14 Quantile Methods

What s New in Econometrics? Lecture 14 Quantile Methods Jeff Wooldridge NBER Summer Institute, 2007 1. Reminders About Means, Medians, and Quantiles 2. Some Useful Asymptotic Results 3. Quantile Regression

### Investigation of goodness-of-fit test statistic distributions by random censored samples

d samples Investigation of goodness-of-fit test statistic distributions by random censored samples Novosibirsk State Technical University November 22, 2010 d samples Outline 1 Nonparametric goodness-of-fit

### Information Measure Estimation and Applications: Boosting the Effective Sample Size from n to n ln n

Information Measure Estimation and Applications: Boosting the Effective Sample Size from n to n ln n Jiantao Jiao (Stanford EE) Joint work with: Kartik Venkat Yanjun Han Tsachy Weissman Stanford EE Tsinghua

### Statistics II Lesson 1. Inference on one population. Year 2009/10

Statistics II Lesson 1. Inference on one population Year 2009/10 Lesson 1. Inference on one population Contents Introduction to inference Point estimators The estimation of the mean and variance Estimating

### A Simulation Comparison Study for Estimating the Process Capability Index C pm with Asymmetric Tolerances

Available online at ijims.ms.tku.edu.tw/list.asp International Journal of Information and Management Sciences 20 (2009), 243-253 A Simulation Comparison Study for Estimating the Process Capability Index

### 14.30 Introduction to Statistical Methods in Economics Spring 2009

MIT OpenCourseWare http://ocw.mit.edu 4.0 Introduction to Statistical Methods in Economics Spring 009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.

### Basics on Probability. Jingrui He 09/11/2007

Basics on Probability Jingrui He 09/11/2007 Coin Flips You flip a coin Head with probability 0.5 You flip 100 coins How many heads would you expect Coin Flips cont. You flip a coin Head with probability

### STA 4273H: Statistical Machine Learning

STA 4273H: Statistical Machine Learning Russ Salakhutdinov Department of Statistics! rsalakhu@utstat.toronto.edu! http://www.utstat.utoronto.ca/~rsalakhu/ Sidney Smith Hall, Room 6002 Lecture 7 Approximate

### Characterizing Forecast Uncertainty Prediction Intervals. The estimated AR (and VAR) models generate point forecasts of y t+s, y ˆ

Characterizing Forecast Uncertainty Prediction Intervals The estimated AR (and VAR) models generate point forecasts of y t+s, y ˆ t + s, t. Under our assumptions the point forecasts are asymtotically unbiased

### STATISTICAL INFERENCE FOR SURVEY DATA ANALYSIS

STATISTICAL INFERENCE FOR SURVEY DATA ANALYSIS David A Binder and Georgia R Roberts Methodology Branch, Statistics Canada, Ottawa, ON, Canada K1A 0T6 KEY WORDS: Design-based properties, Informative sampling,

### A nonparametric two-sample wald test of equality of variances

University of Wollongong Research Online Faculty of Informatics - Papers (Archive) Faculty of Engineering and Information Sciences 211 A nonparametric two-sample wald test of equality of variances David

### Frailty Models and Copulas: Similarities and Differences

Frailty Models and Copulas: Similarities and Differences KLARA GOETHALS, PAUL JANSSEN & LUC DUCHATEAU Department of Physiology and Biometrics, Ghent University, Belgium; Center for Statistics, Hasselt

### Chapter 15 Confidence Intervals for Mean Difference Between Two Delta-Distributions

Chapter 15 Confidence Intervals for Mean Difference Between Two Delta-Distributions Karen V. Rosales and Joshua D. Naranjo Abstract Traditional two-sample estimation procedures like pooled-t, Welch s t,

### Maximum Likelihood, Logistic Regression, and Stochastic Gradient Training

Maximum Likelihood, Logistic Regression, and Stochastic Gradient Training Charles Elkan elkan@cs.ucsd.edu January 17, 2013 1 Principle of maximum likelihood Consider a family of probability distributions

### Probabilistic Graphical Models

Parameter Estimation December 14, 2015 Overview 1 Motivation 2 3 4 What did we have so far? 1 Representations: how do we model the problem? (directed/undirected). 2 Inference: given a model and partially

### Probabilistic Graphical Models

2016 Robert Nowak Probabilistic Graphical Models 1 Introduction We have focused mainly on linear models for signals, in particular the subspace model x = Uθ, where U is a n k matrix and θ R k is a vector

### Introduction: MLE, MAP, Bayesian reasoning (28/8/13)

STA561: Probabilistic machine learning Introduction: MLE, MAP, Bayesian reasoning (28/8/13) Lecturer: Barbara Engelhardt Scribes: K. Ulrich, J. Subramanian, N. Raval, J. O Hollaren 1 Classifiers In this

### Generalized Linear Models (GLZ)

Generalized Linear Models (GLZ) Generalized Linear Models (GLZ) are an extension of the linear modeling process that allows models to be fit to data that follow probability distributions other than the

### Modified Simes Critical Values Under Positive Dependence

Modified Simes Critical Values Under Positive Dependence Gengqian Cai, Sanat K. Sarkar Clinical Pharmacology Statistics & Programming, BDS, GlaxoSmithKline Statistics Department, Temple University, Philadelphia

### Model Combining in Factorial Data Analysis

Model Combining in Factorial Data Analysis Lihua Chen Department of Mathematics, The University of Toledo Panayotis Giannakouros Department of Economics, University of Missouri Kansas City Yuhong Yang

### A Note on Bayesian Inference After Multiple Imputation

A Note on Bayesian Inference After Multiple Imputation Xiang Zhou and Jerome P. Reiter Abstract This article is aimed at practitioners who plan to use Bayesian inference on multiplyimputed datasets in

### Akaike Information Criterion to Select the Parametric Detection Function for Kernel Estimator Using Line Transect Data

Journal of Modern Applied Statistical Methods Volume 12 Issue 2 Article 21 11-1-2013 Akaike Information Criterion to Select the Parametric Detection Function for Kernel Estimator Using Line Transect Data

### Estimation of Conditional Kendall s Tau for Bivariate Interval Censored Data

Communications for Statistical Applications and Methods 2015, Vol. 22, No. 6, 599 604 DOI: http://dx.doi.org/10.5351/csam.2015.22.6.599 Print ISSN 2287-7843 / Online ISSN 2383-4757 Estimation of Conditional

### Bias-corrected Estimators of Scalar Skew Normal

Bias-corrected Estimators of Scalar Skew Normal Guoyi Zhang and Rong Liu Abstract One problem of skew normal model is the difficulty in estimating the shape parameter, for which the maximum likelihood

### General Linear Model: Statistical Inference

Chapter 6 General Linear Model: Statistical Inference 6.1 Introduction So far we have discussed formulation of linear models (Chapter 1), estimability of parameters in a linear model (Chapter 4), least

### Nonparametric Drift Estimation for Stochastic Differential Equations

Nonparametric Drift Estimation for Stochastic Differential Equations Gareth Roberts 1 Department of Statistics University of Warwick Brazilian Bayesian meeting, March 2010 Joint work with O. Papaspiliopoulos,

### Chapter 4: Constrained estimators and tests in the multiple linear regression model (Part III)

Chapter 4: Constrained estimators and tests in the multiple linear regression model (Part III) Florian Pelgrin HEC September-December 2010 Florian Pelgrin (HEC) Constrained estimators September-December

### An EM-Algorithm Based Method to Deal with Rounded Zeros in Compositional Data under Dirichlet Models. Rafiq Hijazi

An EM-Algorithm Based Method to Deal with Rounded Zeros in Compositional Data under Dirichlet Models Rafiq Hijazi Department of Statistics United Arab Emirates University P.O. Box 17555, Al-Ain United

### Noninformative Priors for the Ratio of the Scale Parameters in the Inverted Exponential Distributions

Communications for Statistical Applications and Methods 03, Vol. 0, No. 5, 387 394 DOI: http://dx.doi.org/0.535/csam.03.0.5.387 Noninformative Priors for the Ratio of the Scale Parameters in the Inverted

### f(x θ)dx with respect to θ. Assuming certain smoothness conditions concern differentiating under the integral the integral sign, we first obtain

0.1. INTRODUCTION 1 0.1 Introduction R. A. Fisher, a pioneer in the development of mathematical statistics, introduced a measure of the amount of information contained in an observaton from f(x θ). Fisher

### Particle Filters. Outline

Particle Filters M. Sami Fadali Professor of EE University of Nevada Outline Monte Carlo integration. Particle filter. Importance sampling. Degeneracy Resampling Example. 1 2 Monte Carlo Integration Numerical

### A Method for Inferring Label Sampling Mechanisms in Semi-Supervised Learning

A Method for Inferring Label Sampling Mechanisms in Semi-Supervised Learning Saharon Rosset Data Analytics Research Group IBM T.J. Watson Research Center Yorktown Heights, NY 1598 srosset@us.ibm.com Hui

### Physics 403. Segev BenZvi. Parameter Estimation, Correlations, and Error Bars. Department of Physics and Astronomy University of Rochester

Physics 403 Parameter Estimation, Correlations, and Error Bars Segev BenZvi Department of Physics and Astronomy University of Rochester Table of Contents 1 Review of Last Class Best Estimates and Reliability

### σ(a) = a N (x; 0, 1 2 ) dx. σ(a) = Φ(a) =

Until now we have always worked with likelihoods and prior distributions that were conjugate to each other, allowing the computation of the posterior distribution to be done in closed form. Unfortunately,

### The University of Hong Kong Department of Statistics and Actuarial Science STAT2802 Statistical Models Tutorial Solutions Solutions to Problems 71-80

The University of Hong Kong Department of Statistics and Actuarial Science STAT2802 Statistical Models Tutorial Solutions Solutions to Problems 71-80 71. Decide in each case whether the hypothesis is simple

EECE 574 - Adaptive Control Basics of System Identification Guy Dumont Department of Electrical and Computer Engineering University of British Columbia January 2010 Guy Dumont (UBC) EECE574 - Basics of

### How to select a good vine

Universitetet i Oslo ingrihaf@math.uio.no International FocuStat Workshop on Focused Information Criteria and Related Themes, May 9-11, 2016 Copulae Regular vines Model selection and reduction Limitations

### P Values and Nuisance Parameters

P Values and Nuisance Parameters Luc Demortier The Rockefeller University PHYSTAT-LHC Workshop on Statistical Issues for LHC Physics CERN, Geneva, June 27 29, 2007 Definition and interpretation of p values;

### Analysis of 2 n Factorial Experiments with Exponentially Distributed Response Variable

Applied Mathematical Sciences, Vol. 5, 2011, no. 10, 459-476 Analysis of 2 n Factorial Experiments with Exponentially Distributed Response Variable S. C. Patil (Birajdar) Department of Statistics, Padmashree

### Inverse Sampling for McNemar s Test

International Journal of Statistics and Probability; Vol. 6, No. 1; January 27 ISSN 1927-7032 E-ISSN 1927-7040 Published by Canadian Center of Science and Education Inverse Sampling for McNemar s Test

### Lawrence D. Brown* and Daniel McCarthy*

Comments on the paper, An adaptive resampling test for detecting the presence of significant predictors by I. W. McKeague and M. Qian Lawrence D. Brown* and Daniel McCarthy* ABSTRACT: This commentary deals

### Chapter 5 Prediction of Random Variables

Chapter 5 Prediction of Random Variables C R Henderson 1984 - Guelph We have discussed estimation of β, regarded as fixed Now we shall consider a rather different problem, prediction of random variables,

### A TEST OF FIT FOR THE GENERALIZED PARETO DISTRIBUTION BASED ON TRANSFORMS

A TEST OF FIT FOR THE GENERALIZED PARETO DISTRIBUTION BASED ON TRANSFORMS Dimitrios Konstantinides, Simos G. Meintanis Department of Statistics and Acturial Science, University of the Aegean, Karlovassi,

### Statistical Methods in HYDROLOGY CHARLES T. HAAN. The Iowa State University Press / Ames

Statistical Methods in HYDROLOGY CHARLES T. HAAN The Iowa State University Press / Ames Univariate BASIC Table of Contents PREFACE xiii ACKNOWLEDGEMENTS xv 1 INTRODUCTION 1 2 PROBABILITY AND PROBABILITY

### Confidence and prediction intervals for. generalised linear accident models

Confidence and prediction intervals for generalised linear accident models G.R. Wood September 8, 2004 Department of Statistics, Macquarie University, NSW 2109, Australia E-mail address: gwood@efs.mq.edu.au

### ITERATED BOOTSTRAP PREDICTION INTERVALS

Statistica Sinica 8(1998), 489-504 ITERATED BOOTSTRAP PREDICTION INTERVALS Majid Mojirsheibani Carleton University Abstract: In this article we study the effects of bootstrap iteration as a method to improve

### Estimation, Inference, and Hypothesis Testing

Chapter 2 Estimation, Inference, and Hypothesis Testing Note: The primary reference for these notes is Ch. 7 and 8 of Casella & Berger 2. This text may be challenging if new to this topic and Ch. 7 of

### Estimation of Treatment Effects under Essential Heterogeneity

Estimation of Treatment Effects under Essential Heterogeneity James Heckman University of Chicago and American Bar Foundation Sergio Urzua University of Chicago Edward Vytlacil Columbia University March

### Bayesian Nonparametrics

Bayesian Nonparametrics Lorenzo Rosasco 9.520 Class 18 April 11, 2011 About this class Goal To give an overview of some of the basic concepts in Bayesian Nonparametrics. In particular, to discuss Dirichelet

### EXACT AND ASYMPTOTICALLY ROBUST PERMUTATION TESTS. Eun Yi Chung Joseph P. Romano

EXACT AND ASYMPTOTICALLY ROBUST PERMUTATION TESTS By Eun Yi Chung Joseph P. Romano Technical Report No. 20-05 May 20 Department of Statistics STANFORD UNIVERSITY Stanford, California 94305-4065 EXACT AND

### BIAS OF MAXIMUM-LIKELIHOOD ESTIMATES IN LOGISTIC AND COX REGRESSION MODELS: A COMPARATIVE SIMULATION STUDY

BIAS OF MAXIMUM-LIKELIHOOD ESTIMATES IN LOGISTIC AND COX REGRESSION MODELS: A COMPARATIVE SIMULATION STUDY Ingo Langner 1, Ralf Bender 2, Rebecca Lenz-Tönjes 1, Helmut Küchenhoff 2, Maria Blettner 2 1

### A Course in Applied Econometrics Lecture 18: Missing Data. Jeff Wooldridge IRP Lectures, UW Madison, August Linear model with IVs: y i x i u i,

A Course in Applied Econometrics Lecture 18: Missing Data Jeff Wooldridge IRP Lectures, UW Madison, August 2008 1. When Can Missing Data be Ignored? 2. Inverse Probability Weighting 3. Imputation 4. Heckman-Type

### BIOS 312: Precision of Statistical Inference

and Power/Sample Size and Standard Errors BIOS 312: of Statistical Inference Chris Slaughter Department of Biostatistics, Vanderbilt University School of Medicine January 3, 2013 Outline Overview and Power/Sample

### Lecture 21: Spectral Learning for Graphical Models

10-708: Probabilistic Graphical Models 10-708, Spring 2016 Lecture 21: Spectral Learning for Graphical Models Lecturer: Eric P. Xing Scribes: Maruan Al-Shedivat, Wei-Cheng Chang, Frederick Liu 1 Motivation

### Parameter Estimation

Parameter Estimation Chapters 13-15 Stat 477 - Loss Models Chapters 13-15 (Stat 477) Parameter Estimation Brian Hartman - BYU 1 / 23 Methods for parameter estimation Methods for parameter estimation Methods

### Preliminaries The bootstrap Bias reduction Hypothesis tests Regression Confidence intervals Time series Final remark. Bootstrap inference

1 / 171 Bootstrap inference Francisco Cribari-Neto Departamento de Estatística Universidade Federal de Pernambuco Recife / PE, Brazil email: cribari@gmail.com October 2013 2 / 171 Unpaid advertisement

### The Impact of Organizer Market Structure on Participant Entry Behavior in a Multi-Tournament Environment

The Impact of Organizer Market Structure on Participant Entry Behavior in a Multi-Tournament Environment Timothy Mathews and Soiliou Daw Namoro Abstract. A model of two tournaments, each with a field of

### Statistics 135 Fall 2008 Final Exam

Name: SID: Statistics 135 Fall 2008 Final Exam Show your work. The number of points each question is worth is shown at the beginning of the question. There are 10 problems. 1. [2] The normal equations

### 12 Generalized linear models

12 Generalized linear models In this chapter, we combine regression models with other parametric probability models like the binomial and Poisson distributions. Binary responses In many situations, we

### Chapter 8: Estimation 1

Chapter 8: Estimation 1 Jae-Kwang Kim Iowa State University Fall, 2014 Kim (ISU) Ch. 8: Estimation 1 Fall, 2014 1 / 33 Introduction 1 Introduction 2 Ratio estimation 3 Regression estimator Kim (ISU) Ch.

### Metropolis-Hastings Algorithm

Strength of the Gibbs sampler Metropolis-Hastings Algorithm Easy algorithm to think about. Exploits the factorization properties of the joint probability distribution. No difficult choices to be made to

### A Very Brief Summary of Statistical Inference, and Examples

A Very Brief Summary of Statistical Inference, and Examples Trinity Term 2009 Prof. Gesine Reinert Our standard situation is that we have data x = x 1, x 2,..., x n, which we view as realisations of random

### Ridge regression. Patrick Breheny. February 8. Penalized regression Ridge regression Bayesian interpretation

Patrick Breheny February 8 Patrick Breheny High-Dimensional Data Analysis (BIOS 7600) 1/27 Introduction Basic idea Standardization Large-scale testing is, of course, a big area and we could keep talking

### STAT 730 Chapter 5: Hypothesis Testing

STAT 730 Chapter 5: Hypothesis Testing Timothy Hanson Department of Statistics, University of South Carolina Stat 730: Multivariate Analysis 1 / 28 Likelihood ratio test def n: Data X depend on θ. The

### Estimation of parametric functions in Downton s bivariate exponential distribution

Estimation of parametric functions in Downton s bivariate exponential distribution George Iliopoulos Department of Mathematics University of the Aegean 83200 Karlovasi, Samos, Greece e-mail: geh@aegean.gr

### ZIPF S LAW FOR CITIES: AN EMPIRICAL EXAMINATION 1

gabaixcm14.tex ZIPF S LAW FOR CITIES: AN EMPIRICAL EXAMINATION 1 Yannis M. Ioannides Department of Economics Tufts University Medford, Massachusetts 2155, U.S.A. 1-617-627-3294 yioannid@tufts.edu Henry

### Machine Learning

Machine Learning 10-701 Tom M. Mitchell Machine Learning Department Carnegie Mellon University January 13, 2011 Today: The Big Picture Overfitting Review: probability Readings: Decision trees, overfiting

### A Bootstrap Test for Conditional Symmetry

ANNALS OF ECONOMICS AND FINANCE 6, 51 61 005) A Bootstrap Test for Conditional Symmetry Liangjun Su Guanghua School of Management, Peking University E-mail: lsu@gsm.pku.edu.cn and Sainan Jin Guanghua School

### Approximating likelihoods for large spatial data sets

Approximating likelihoods for large spatial data sets By Michael Stein, Zhiyi Chi, and Leah Welty Jessi Cisewski April 28, 2009 1 Introduction Given a set of spatial data, often the desire is to estimate

### Political Science 236 Hypothesis Testing: Review and Bootstrapping

Political Science 236 Hypothesis Testing: Review and Bootstrapping Rocío Titiunik Fall 2007 1 Hypothesis Testing Definition 1.1 Hypothesis. A hypothesis is a statement about a population parameter The

### The Bayes classifier

The Bayes classifier Consider where is a random vector in is a random variable (depending on ) Let be a classifier with probability of error/risk given by The Bayes classifier (denoted ) is the optimal

### Graph Detection and Estimation Theory

Introduction Detection Estimation Graph Detection and Estimation Theory (and algorithms, and applications) Patrick J. Wolfe Statistics and Information Sciences Laboratory (SISL) School of Engineering and