Loglikelihood and Confidence Intervals


 Laura O’Brien’
 1 years ago
 Views:
Transcription
1 Stat 504, Lecture 3 Stat 504, Lecture 3 2 Review (contd.): Loglikelihood and Confidence Intervals The likelihood of the samle is the joint PDF (or PMF) L(θ) = f(x,.., x n; θ) = ny f(x i; θ) i= Review: Let X, X 2,..., X n be a simle random samle from a robability distribution f(x; θ). A arameter θ of f(x; θ) is a variable that is characteristic of f(x; θ). A statistic T is any quantity that can be calculated from a samle; it s a function of X,..., X n. An estimate ˆθ for θ is a single number that is a reasonable value for θ. The maximum likelihood estimate (MLE) ˆθ MLE maximizes L(θ): L(ˆθ MLE) L(θ), If we use X i s instead of x i s then ˆθ is the maximum likelihood estimator. Usually, MLE: is unbiased, E(ˆθ) = θ is consistent, ˆθ θ, as n θ An estimator ˆθ for θ is a statistic that gives the formula for comuting the estimate ˆθ. is efficient, has small SE(ˆθ) as n is asymtotically normal, (ˆθ θ) N(0, ) SE(ˆθ) Stat 504, Lecture 3 3 Stat 504, Lecture 3 4 The loglikelihood function is defined to be the natural logarithm of the likelihood function, l(θ ; x) = log L(θ ; x). For a variety of reasons, statisticians often work with the loglikelihood rather than with the likelihood. One reason is that l(θ ; x) tends to be a simler function than L(θ ; x). When we take logs, roducts are changed to sums. If X = (X, X 2,..., X n) is an iid samle from a robability distribution f(x; θ), the overall likelihood is the roduct of the likelihoods for the individual X i s: L(θ ; x) = f(x ; θ) ny = f(x i ; θ) = i= ny L(θ ; x i) i= The loglikelihood, however, is the sum of the individual loglikelihoods: l(θ ; x) = log f(x ; θ) ny = log f(x i ; θ) = = i= nx log f(x i ; θ) i= nx l(θ ; x i). i= Below are some examles of loglikelihood functions.
2 Stat 504, Lecture 3 5 Stat 504, Lecture 3 6 Binomial. Suose X Bin(n, ) where n is known. The likelihood function is L( ; x) = n! x! (n x)! x ( ) n x and so the loglikelihood function is l( ; x) = k + x log + (n x) log( ), where k is a constant that doesn t involve the arameter. In the future we will omit the constant, because it s statistically irrelevant. Poisson. Suose X = (X, X 2,..., X n) is an iid samle from a Poisson distribution with arameter λ. The likelihood is L(λ ; x) = and the loglikelihood is l(λ ; x) = ny i= λ x i e λ x i! = λpn i= x i e nλ x! x 2! x n!,! nx x i log λ nλ, i= ignoring the constant terms that don t deend on λ. As the above examles show, l(θ ; x) often looks nicer than L(θ ; x) because the roducts become sums and the exonents become multiliers. Stat 504, Lecture 3 7 Stat 504, Lecture 3 8 Asymtotic confidence intervals Loglikelihood forms the basis for many aroximate confidence intervals and hyothesis tests, because it behaves in a redictable manner as the samle size grows. The following examle illustrates what haens to l(θ ; x) as n becomes large. InClass Exercise: Suose that we observe X = from a binomial distribution with n = 4 and unknown. Calculate the loglikelihood. What does the grah of likelihood look like? Find the MLE (do you understand the difference between the estimator and the estimate)? Locate the MLE on the grah of the likelihood.
3 Stat 504, Lecture 3 9 Stat 504, Lecture 3 0 The MLE is ˆ = /4 =.25. Ignoring constants, the loglikelihood is which looks like this: l(;x) l( ; x) = log + 3 log( ), Here is a samle code for lotting this function in R: (For clarity, I omitted from the lot all values of beyond.8, because for >.8 the loglikelihood dros down so low that including these values of would distort the lot s aearance. When lotting loglikelihoods, we don t need to include all θ values in the arameter sace; in fact, it s a good idea to limit the domain to those θ s for which the loglikelihood is no more than 2 or 3 units below the maximum value l(ˆθ; x) because, in a singlearameter roblem, any θ whose loglikelihood is more than 2 or 3 units below the maximum is highly imlausible.) <seq(from=.0,to=.80,by=.0) loglik<log() + 3*log() lot(,loglik,xlab="",ylab="",tye="l",xlim=c(0,)) Stat 504, Lecture 3 Now suose that we observe X = 0 from a binomial distribution with n = 40. The MLE is again ˆ = 0/40 =.25, but the loglikelihood is l(;x) l( ; x) = 0 log + 30 log( ), Finally, suose that we observe X = 00 from a binomial with n = 400. The MLE is still ˆ = 00/400 =.25, but the loglikelihood is now l(;x) l( ; x) = 00 log log( ), Stat 504, Lecture 3 2 As n gets larger, two things are haening to the loglikelihood. First, l( ; x) is becoming more sharly eaked around ˆ. Second, l( ; x) is becoming more symmetric about ˆ. The first oint shows that as the samle size grows, we are becoming more confident that the true arameter lies close to ˆ. If the loglikelihood is highly eaked that is, if it dros sharly as we move away from the MLE then the evidence is strong that is near ˆ. A flatter loglikelihood, on the other hand, means that is not well estimated and the range of lausible values is wide. In fact, the curvature of the loglikelihood (i.e. the second derivative of l(θ ; x) with resect to θ) is an imortant measure of statistical information about θ. The second oint, that the loglikelihood function becomes more symmetric about the MLE as the samle size grows, forms the basis for constructing asymtotic (largesamle) confidence intervals for the unknown arameter. In a wide variety of roblems, as the samle size grows the loglikelihood aroaches a quadratic function (i.e. a arabola) centered at the MLE.
4 Stat 504, Lecture 3 3 The arabola is significant because that is the shae of the loglikelihood from the normal distribution. Stat 504, Lecture 3 4 If we had a random samle of any size from a normal distribution with known variance σ 2 and unknown mean µ, the loglikelihood would be a erfect arabola centered at the MLE ˆµ = x = P n i= xi/n. From elementary statistics, we know that if we have a samle from a normal distribution with known variance σ 2, a 95% confidence interval for the mean µ is x ±.96 σ n. () The confidence interval () is valid because over reeated samles the estimate x is normally distributed about the true value µ with a standard deviation of σ/ n. The quantity σ/ n is called the standard error; it measures the variability of the samle mean x about the true mean µ. The number.96 comes from a table of the standard normal distribution; the area under the standard normal density curve between.96 and.96 is.95 or 95%. Stat 504, Lecture 3 5 Stat 504, Lecture 3 6 There is much confusion about how to interret a confidence interval (CI). A CI is NOT a robability statement about θ since θ is a fixed value, not a random variable. One interretation: if we took many samles, most of our intervals would cature true arameter (e.g. 95% of out intervals will contain the true arameter). Examle: The nationwide telehone oll was conducted by NY Times/CBS News between Jan with 8 adults. About 58% of resondents feel otimistic about next four years. The results are reorted with a margin of error of 3%. In Stat 504, the arameter of interest will not be the mean of a normal oulation, but some other arameter θ ertaining to a discrete robability distribution. We will often estimate the arameter by its MLE ˆθ. But because in large samles the loglikelihood function l(θ ; x) aroaches a arabola centered at ˆθ, we will be able to use a method similar to () to form aroximate confidence intervals for θ.
5 Stat 504, Lecture 3 7 Stat 504, Lecture 3 8 Just as x is normally distributed about µ, ˆθ is aroximately normally distributed about θ in large samles. This roerty is called the asymtotic normality of the MLE, and the technique of forming confidence intervals is called the asymtotic normal aroximation. This method works for a wide variety of statistical models, including all the models that we will use in this course. The asymtotic normal 95% confidence interval for a arameter θ has the form ˆθ ±.96 q, (2) l (ˆθ; x) Of course, we can also form intervals with confidence coefficients other than 95%. All we need to do is to relace.96 in (2) by z, a value from a table of the standard normal distribution, where ±z encloses the desired level of confidence. If we wanted a 90% confidence interval, for examle, we would use.645. where l (ˆθ; x) is the second derivative of the loglikelihood function with resect to θ, evaluated at θ = ˆθ. Stat 504, Lecture 3 9 Observed and exected information The quantity l (ˆθ; x) q is called the observed information, and / l (ˆθ; x) is an aroximate standard error for ˆθ. As the loglikelihood becomes more sharly eaked about the MLE, the second derivative dros and the standard error goes down. When calculating asymtotic confidence intervals, statisticians often relace the second derivative of the loglikelihood by its exectation; that is, relace l (θ; x) by the function I(θ) = E ˆl (θ; x), which is called the exected information or the Fisher information. In that case, the 95% confidence interval would become ˆθ ±.96 q I(ˆθ). (3) Stat 504, Lecture 3 20 When the samle size is large, the two confidence intervals (2) and (3) tend to be very close. In some roblems, the two are identical. Now we give a few examles of asymtotic confidence intervals. Bernoulli. If X is Bernoulli with success robability, the loglikelihood is l( ; x) = x log + ( x) log ( ), the first derivative is l ( ; x) = and the second derivative is l ( ; x) = x ( ) (x )2 2 ( ) 2 (to derive this, use the fact that x 2 = x). Because E ˆ(x ) 2 = V (x) = ( ), the Fisher information is I() = ( ).
6 Stat 504, Lecture 3 2 Of course, a single Bernoulli trial does not rovide enough information to get a reasonable confidence interval for. Let s see what haens when we have multile trials. Binomial. If X Bin(n, ), then the loglikelihood is Stat 504, Lecture 3 22 Notice that the Fisher information for the Bin(n, ) model is n times the Fisher information from a single Bernoulli trial. This is a general rincile; if we observe a samle size n, X = (X, X 2,..., X n), l( ; x) = x log + (n x) log ( ), the first derivative is l ( ; x) = the second derivative is x n ( ), where X, X 2,..., X n are indeendent random variables, then the Fisher information from X is the sum of the Fisher information functions from the individual X i s. If X, X 2,..., X n are iid, then the Fisher information from X is n times the Fisher information from a single observation X i. l ( ; x) = and the Fisher information is I() = x 2x + n2 2 ( ) 2, n ( ). Thus an aroximate 95% confidence interval for based on the Fisher information is r ˆ( ˆ) ˆ ±.96, (4) n where ˆ = x/n is the MLE. What haens if we use the observed information rather than the exected information? Evaluating the second derivative l ( ; x) at the MLE ˆ = x/n gives l n (ˆ ; x) = ˆ( ˆ), so the 95% interval based on the observed information is identical to (4). Unfortunately, Agresti (2002,. 5) oints out that the interval (4) erforms oorly unless n is very large; the actual coverage can be considerably less than the nominal rate of 95%. Stat 504, Lecture 3 23 The confidence interval (4) has two unusual features: The endoints can stray outside the arameter sace; that is, one can get a lower limit less than 0 or an uer limit greater than. If we haen to observe no successes (x = 0) or no failures (x = n) the interval becomes degenerate (has zero width) and misses the true arameter. This unfortunate event becomes quite likely when the actual is close to zero or one. A variety of fixes are available. One ad hoc fix, which can work surrisingly well, is to relace ˆ by = x +.5 n +, which is equivalent to adding half a success and half a failure; that kees the interval from becoming degenerate. To kee the endoints within the arameter sace, we can exress the arameter on a different scale, such as the logodds θ = log which we will discuss later., Stat 504, Lecture 3 24 Poisson. If X = (X, X 2,..., X n) is an iid samle from a Poisson distribution with arameter λ, the loglikelihood is! nx l(λ ; x) = x i log λ nλ, the first derivative is i= l (λ ; x) = the second derivative is l (λ ; x) = and the Fisher information is I(λ) = n λ. P i xi n, λ P i xi λ 2, An aroximate 95% interval based on the observed or exected information is s ˆλ ˆλ ±.96 n, (5) where ˆλ = P i xi/n is the MLE.
7 Stat 504, Lecture 3 25 Stat 504, Lecture 3 26 Suose we observe X = 2 from a binomial distribution Bin(20, ). The MLE is ˆ = 2/20 =.0 and the loglikelihood is not very symmetric: One again, this interval may not erform well in some circumstances; we can often get better results by changing the scale of the arameter. Alternative arameterizations Statistical theory tells us that if n is large enough, the true coverage of the aroximate intervals (2) or (3) will be very close to 95%. How large n must be in ractice deends on the articulars of the roblem. Sometimes an aroximate interval erforms oorly because the loglikelihood function doesn t closely resemble a arabola. If so, we may be able to imrove the quality of the aroximation by alying a suitable rearameterization, a transformation of the arameter to a new scale. Here is an examle. l(;x) This asymmetry arises because ˆ is close to the boundary of the arameter sace. We know that must lie between zero and one. When ˆ is close to zero or one, the loglikelihood tends to be more skewed than it would be if ˆ were near.5. The usual 95% confidence interval is r ˆ( ˆ) ˆ ±.96 = 0.00 ± 0.3 n or (.03,.23), which strays outside the arameter sace. Stat 504, Lecture 3 27 The logistic or logit transformation is defined as «φ = log. (6) The logit is also called the log odds, because /( ) is the odds associated with. Whereas is a roortion and must lie between 0 and, φ may take any value from to +, so the logit transformation solves the roblem of a shar boundary in the arameter sace. Solving (6) for roduces the backtransformation = eφ + e φ. (7) Let s rewrite the binomial loglikelihood in terms of φ: l(φ ; x) = x log + (n x) log( ) «= x log + n log( ) «= xφ + n log. + e φ Stat 504, Lecture 3 28 Now let s grah the loglikelihood l(φ; x) versus φ: loglik It s still skewed, but not quite as sharly as before. This lot strongly suggests that an asymtotic confidence interval constructed on the φ scale will be more accurate in coverage than an interval constructed on the scale. An aroximate 95% confidence interval for φ is hi ˆφ ±.96 q I( ˆφ) where ˆφ is a the MLE of φ, and I(φ) is the Fisher information for φ. To find the MLE for φ, all we need to do is aly the logit transformation to ˆ: «0. ˆφ = log =
8 Stat 504, Lecture 3 29 Stat 504, Lecture 3 30 The general method for rearameterization is as follows. First, we choose a transformation φ = φ(θ) for which we think the loglikelihood will be symmetric. Assuming for a moment that we know the Fisher information for φ, we can calculate this 95% confidence interval for φ. Then, because our interest is not really in φ but in, we can transform the endoints of the confidence interval back to the scale. This new confidence interval for will not be exactly symmetric i.e. ˆ will not lie exactly in the center of it but the coverage of this rocedure should be closer to 95% than for intervals comuted directly on the scale. Then we calculate ˆθ, the MLE for θ, and transform it to the φ scale, ˆφ = φ(ˆθ). Next we need to calculate I( ˆφ), the Fisher information for φ. It turns out that this is given by I( ˆφ) = I(ˆθ) [ φ (ˆθ) ] 2, (8) where φ (θ) is the first derivative of φ with resect to θ. Then the endoints of a 95% confidence interval for φ are: s φ low = ˆφ.96 I( ˆφ) φ high = ˆφ +.96 s I( ˆφ) Stat 504, Lecture 3 3 Stat 504, Lecture 3 32 Table : Some common transformations, their back transformations, and derivatives. transformation back derivative The aroximate 95% confidence interval for φ is [φ low, φ high ]. The corresonding confidence interval for θ is obtained by transforming φ low and φ high back to the original θ scale. A few common transformations are shown in Table, along with their backtransformations and derivatives. «θ φ = log θ θ = e φ + e φ φ (θ) = φ = log θ θ = e φ φ (θ) = θ φ = θ θ = φ 2 φ (θ) = θ( θ) 2 θ φ = θ /3 θ = φ 3 φ (θ) = 3 θ 2/3
9 Stat 504, Lecture 3 33 Stat 504, Lecture 3 34 Going back to the binomial examle with n = 20 and X = 2, let s form a 95% confidence interval for φ = log /( ). The MLE for is ˆ = 2/20 =.0, so the MLE for φ is ˆφ = log(./.9) = Using the derivative of the logit transformation from Table, the Fisher information for φ is I(φ) = = I(θ) [ φ (θ) ] 2 n ( )» = n( ). Evaluating it at the MLE gives ( ) I( ˆφ) = =.8 2 The endoints of the 95% confidence interval for φ are interval for are low = high = e e = 0.025, e e = The MLE ˆ =.0 is not exactly in the middle of this interval, but who says that a confidence interval must be symmetric about the oint estimate? r φ low = = 3.658, r φ high = = 0.736, and the corresonding endoints of the confidence Stat 504, Lecture 3 35 Stat 504, Lecture 3 36 Intervals based on the likelihood ratio Another way to form a confidence interval for a single arameter is to find all values of θ for which the loglikelihood l(θ ; x) is within a given tolerance of the maximum value l(ˆθ ; x). Statistical theory tells us that, if θ 0 is the true value of the arameter, then the likelihoodratio statistic 2 log L(ˆθ ; x) h i L(θ 0 ; x) = 2 l(ˆθ ; x) l(θ 0 ; x) (9) is aroximately distributed as χ 2 when the samle size n is large. This gives rise to the well known likelihoodratio (LR) test. In the LR test of the null hyothesis H 0 : θ = θ 0 versus the twosided alternative H : θ θ 0, we would reject H 0 at the αlevel if the LR statistic (9) exceeds the 00( α)th ercentile of the χ 2 distribution. That is, for an α =.05level test, we would reject H 0 if the LR statistic is greater than 3.84.
10 Stat 504, Lecture 3 37 Stat 504, Lecture 3 38 The LR testing rincile can also be used to construct confidence intervals. An aroximate 00( α)% confidence interval for θ consists of all the ossible θ 0 s for which the null hyothesis H 0 : θ = θ 0 would not be rejected at the α level. For a 95% interval, the interval would consist of all the values of θ for which h i 2 l(ˆθ ; x) l(θ ; x) 3.84 or l(θ ; x) l(ˆθ ; x).92. In other words, the 95% interval includes all values of θ for which the loglikelihood function dros off by no more than.92 units. Returning to our binomial examle, suose that we observe X = 2 from a binomial distribution with n = 20 and unknown. The grah of the loglikelihood function looks like this, l(;x) l( ; x) = 2 log + 8 log( ) the MLE is ˆ = x/n =.0, and the maximized loglikelihood is l(ˆ ; x) = 2 log. + 8 log.9 = Let s add a horizontal line to the lot at the loglikelihood value = 8.42: Stat 504, Lecture 3 39 Stat 504, Lecture 3 40 l(;x) The horizontal line intersects the loglikelihood curve at =.08 and =.278. Therefore, the LR confidence interval for is (.08,.278). When n is large, the LR method will tend to roduce intervals very similar to those based on the observed or exected information. Unlike the informationbased intervals, however, the LR intervals are scaleinvariant. That is, if we find the LR interval for a transformed version of the arameter such as φ = log /( ) and then transform the endoints back to the scale, we get exactly the same answer as if we aly the LR method directly on the scale. For that reason, statisticians tend to like the LR method better.
11 Stat 504, Lecture 3 4 If the loglikelihood function exressed on a articular scale is nearly quadratic, then a informationbased interval calculated on that scale will agree closely with the LR interval. Therefore, if the informationbased interval agrees with the LR interval, that rovides some evidence that the normal aroximation is working well on that articular scale. If the informationbased interval is quite different from the LR interval, the aroriateness of the normal aroximation is doubtful, and the LR aroximation is robably better.
where a, b, c, and d are constants with a 0, and x is measured in radians. (π radians =
Introduction to Modeling 3.61 3.6 Sine and Cosine Functions The general form of a sine or cosine function is given by: f (x) = asin (bx + c) + d and f(x) = acos(bx + c) + d where a, b, c, and d are constants
More informationMonitoring Frequency of Change By Li Qin
Monitoring Frequency of Change By Li Qin Abstract Control charts are widely used in rocess monitoring roblems. This aer gives a brief review of control charts for monitoring a roortion and some initial
More informationBeyond the F Test: Effect Size Confidence Intervals and Tests of Close Fit in the Analysis of Variance and Contrast Analysis
Psychological Methods 004, Vol. 9, No., 164 18 Coyright 004 by the American Psychological Association 108989X/04/$1.00 DOI: 10.1037/108989X.9..164 Beyond the F Test: Effect Size Confidence Intervals
More informationPoint Location. Preprocess a planar, polygonal subdivision for point location queries. p = (18, 11)
Point Location Prerocess a lanar, olygonal subdivision for oint location ueries. = (18, 11) Inut is a subdivision S of comlexity n, say, number of edges. uild a data structure on S so that for a uery oint
More informationEffect Sizes Based on Means
CHAPTER 4 Effect Sizes Based on Means Introduction Raw (unstardized) mean difference D Stardized mean difference, d g Resonse ratios INTRODUCTION When the studies reort means stard deviations, the referred
More informationChapter 9, Part B Hypothesis Tests. Learning objectives
Chater 9, Part B Hyothesis Tests Slide 1 Learning objectives 1. Able to do hyothesis test about Poulation Proortion 2. Calculatethe Probability of Tye II Errors 3. Understand ower of the test 4. Determinethe
More informationChapter 6: Point Estimation. Fall 2011.  Probability & Statistics
STAT355 Chapter 6: Point Estimation Fall 2011 Chapter Fall 2011 6: Point1 Estimat / 18 Chap 6  Point Estimation 1 6.1 Some general Concepts of Point Estimation Point Estimate Unbiasedness Principle of
More information1 Gambler s Ruin Problem
Coyright c 2009 by Karl Sigman 1 Gambler s Ruin Problem Let N 2 be an integer and let 1 i N 1. Consider a gambler who starts with an initial fortune of $i and then on each successive gamble either wins
More information6.042/18.062J Mathematics for Computer Science December 12, 2006 Tom Leighton and Ronitt Rubinfeld. Random Walks
6.042/8.062J Mathematics for Comuter Science December 2, 2006 Tom Leighton and Ronitt Rubinfeld Lecture Notes Random Walks Gambler s Ruin Today we re going to talk about onedimensional random walks. In
More informationPOISSON PROCESSES. Chapter 2. 2.1 Introduction. 2.1.1 Arrival processes
Chater 2 POISSON PROCESSES 2.1 Introduction A Poisson rocess is a simle and widely used stochastic rocess for modeling the times at which arrivals enter a system. It is in many ways the continuoustime
More informationSufficient Statistics and Exponential Family. 1 Statistics and Sufficient Statistics. Math 541: Statistical Theory II. Lecturer: Songfeng Zheng
Math 541: Statistical Theory II Lecturer: Songfeng Zheng Sufficient Statistics and Exponential Family 1 Statistics and Sufficient Statistics Suppose we have a random sample X 1,, X n taken from a distribution
More informationPrice Elasticity of Demand MATH 104 and MATH 184 Mark Mac Lean (with assistance from Patrick Chan) 2011W
Price Elasticity of Demand MATH 104 and MATH 184 Mark Mac Lean (with assistance from Patrick Chan) 2011W The rice elasticity of demand (which is often shortened to demand elasticity) is defined to be the
More informationAn Introduction to Risk Parity Hossein Kazemi
An Introduction to Risk Parity Hossein Kazemi In the aftermath of the financial crisis, investors and asset allocators have started the usual ritual of rethinking the way they aroached asset allocation
More informationMultiperiod Portfolio Optimization with General Transaction Costs
Multieriod Portfolio Otimization with General Transaction Costs Victor DeMiguel Deartment of Management Science and Oerations, London Business School, London NW1 4SA, UK, avmiguel@london.edu Xiaoling Mei
More information4. Introduction to Statistics
Statistics for Engineers 41 4. Introduction to Statistics Descriptive Statistics Types of data A variate or random variable is a quantity or attribute whose value may vary from one unit of investigation
More informationRisk and Return. Sample chapter. e r t u i o p a s d f CHAPTER CONTENTS LEARNING OBJECTIVES. Chapter 7
Chater 7 Risk and Return LEARNING OBJECTIVES After studying this chater you should be able to: e r t u i o a s d f understand how return and risk are defined and measured understand the concet of risk
More informationCharacterizing and Modeling Network Traffic Variability
Characterizing and Modeling etwork Traffic Variability Sarat Pothuri, David W. Petr, Sohel Khan Information and Telecommunication Technology Center Electrical Engineering and Comuter Science Deartment,
More informationJoint Distributions. Lecture 5. Probability & Statistics in Engineering. 0909.400.01 / 0909.400.02 Dr. P. s Clinic Consultant Module in.
3σ σ σ +σ +σ +3σ Joint Distributions Lecture 5 0909.400.01 / 0909.400.0 Dr. P. s Clinic Consultant Module in Probabilit & Statistics in Engineering Toda in P&S 3σ σ σ +σ +σ +3σ Dealing with multile
More informationA Multivariate Statistical Analysis of Stock Trends. Abstract
A Multivariate Statistical Analysis of Stock Trends Aril Kerby Alma College Alma, MI James Lawrence Miami University Oxford, OH Abstract Is there a method to redict the stock market? What factors determine
More informationNormally Distributed Data. A mean with a normal value Test of Hypothesis Sign Test Paired observations within a single patient group
ANALYSIS OF CONTINUOUS VARIABLES / 31 CHAPTER SIX ANALYSIS OF CONTINUOUS VARIABLES: COMPARING MEANS In the last chater, we addressed the analysis of discrete variables. Much of the statistical analysis
More informationA Simple Model of Pricing, Markups and Market. Power Under Demand Fluctuations
A Simle Model of Pricing, Markus and Market Power Under Demand Fluctuations Stanley S. Reynolds Deartment of Economics; University of Arizona; Tucson, AZ 85721 Bart J. Wilson Economic Science Laboratory;
More informationA MOST PROBABLE POINTBASED METHOD FOR RELIABILITY ANALYSIS, SENSITIVITY ANALYSIS AND DESIGN OPTIMIZATION
9 th ASCE Secialty Conference on Probabilistic Mechanics and Structural Reliability PMC2004 Abstract A MOST PROBABLE POINTBASED METHOD FOR RELIABILITY ANALYSIS, SENSITIVITY ANALYSIS AND DESIGN OPTIMIZATION
More informationThe Lognormal Distribution Engr 323 Geppert page 1of 6 The Lognormal Distribution
Engr 33 Geert age 1of 6 The Lognormal Distribution In general, the most imortant roerty of the lognormal rocess is that it reresents a roduct of indeendent random variables. (Class Handout on Lognormal
More informationA 60,000 DIGIT PRIME NUMBER OF THE FORM x 2 + x Introduction StarkHeegner Theorem. Let d > 0 be a squarefree integer then Q( d) has
A 60,000 DIGIT PRIME NUMBER OF THE FORM x + x + 4. Introduction.. Euler s olynomial. Euler observed that f(x) = x + x + 4 takes on rime values for 0 x 39. Even after this oint f(x) takes on a high frequency
More informationRisk in Revenue Management and Dynamic Pricing
OPERATIONS RESEARCH Vol. 56, No. 2, March Aril 2008,. 326 343 issn 0030364X eissn 15265463 08 5602 0326 informs doi 10.1287/ore.1070.0438 2008 INFORMS Risk in Revenue Management and Dynamic Pricing Yuri
More informationOn the predictive content of the PPI on CPI inflation: the case of Mexico
On the redictive content of the PPI on inflation: the case of Mexico José Sidaoui, Carlos Caistrán, Daniel Chiquiar and Manuel RamosFrancia 1 1. Introduction It would be natural to exect that shocks to
More informationStatistiek (WISB361)
Statistiek (WISB361) Final exam June 29, 2015 Schrijf uw naam op elk in te leveren vel. Schrijf ook uw studentnummer op blad 1. The maximum number of points is 100. Points distribution: 23 20 20 20 17
More informationTworesource stochastic capacity planning employing a Bayesian methodology
Journal of the Oerational Research Society (23) 54, 1198 128 r 23 Oerational Research Society Ltd. All rights reserved. 165682/3 $25. www.algravejournals.com/jors Tworesource stochastic caacity lanning
More informationMeasuring relative phase between two waveforms using an oscilloscope
Measuring relative hase between two waveforms using an oscilloscoe Overview There are a number of ways to measure the hase difference between two voltage waveforms using an oscilloscoe. This document covers
More informationStatistics  Written Examination MEC Students  BOVISA
Statistics  Written Examination MEC Students  BOVISA Prof.ssa A. Guglielmi 26.0.2 All rights reserved. Legal action will be taken against infringement. Reproduction is prohibited without prior consent.
More informationVariations on the Gambler s Ruin Problem
Variations on the Gambler s Ruin Problem Mat Willmott December 6, 2002 Abstract. This aer covers the history and solution to the Gambler s Ruin Problem, and then exlores the odds for each layer to win
More informationLargeScale IP Traceback in HighSpeed Internet: Practical Techniques and Theoretical Foundation
LargeScale IP Traceback in HighSeed Internet: Practical Techniques and Theoretical Foundation Jun Li Minho Sung Jun (Jim) Xu College of Comuting Georgia Institute of Technology {junli,mhsung,jx}@cc.gatech.edu
More informationBinomial Random Variables. Binomial Distribution. Examples of Binomial Random Variables. Binomial Random Variables
Binomial Random Variables Binomial Distribution Dr. Tom Ilvento FREC 8 In many cases the resonses to an exeriment are dichotomous Yes/No Alive/Dead Suort/Don t Suort Binomial Random Variables When our
More informationPRIME NUMBERS AND THE RIEMANN HYPOTHESIS
PRIME NUMBERS AND THE RIEMANN HYPOTHESIS CARL ERICKSON This minicourse has two main goals. The first is to carefully define the Riemann zeta function and exlain how it is connected with the rime numbers.
More informationc 2009 Je rey A. Miron 3. Examples: Linear Demand Curves and Monopoly
Lecture 0: Monooly. c 009 Je rey A. Miron Outline. Introduction. Maximizing Pro ts. Examles: Linear Demand Curves and Monooly. The Ine ciency of Monooly. The Deadweight Loss of Monooly. Price Discrimination.
More informationCBus Voltage Calculation
D E S I G N E R N O T E S CBus Voltage Calculation Designer note number: 3121256 Designer: Darren Snodgrass Contact Person: Darren Snodgrass Aroved: Date: Synosis: The guidelines used by installers
More informationThe Delta Method and Applications
Chapter 5 The Delta Method and Applications 5.1 Linear approximations of functions In the simplest form of the central limit theorem, Theorem 4.18, we consider a sequence X 1, X,... of independent and
More informationUniversiteitUtrecht. Department. of Mathematics. Optimal a priori error bounds for the. RayleighRitz method
UniversiteitUtrecht * Deartment of Mathematics Otimal a riori error bounds for the RayleighRitz method by Gerard L.G. Sleijen, Jaser van den Eshof, and Paul Smit Prerint nr. 1160 Setember, 2000 OPTIMAL
More information(This result should be familiar, since if the probability to remain in a state is 1 p, then the average number of steps to leave the state is
How many coin flis on average does it take to get n consecutive heads? 1 The rocess of fliing n consecutive heads can be described by a Markov chain in which the states corresond to the number of consecutive
More informationCSI:FLORIDA. Section 4.4: Logistic Regression
SI:FLORIDA Section 4.4: Logistic Regression SI:FLORIDA Reisit Masked lass Problem.5.5 2 .5  .5 .5  .5.5.5 We can generalize this roblem to two class roblem as well! SI:FLORIDA Reisit Masked lass
More informationTRANSCENDENTAL NUMBERS
TRANSCENDENTAL NUMBERS JEREMY BOOHER. Introduction The Greeks tried unsuccessfully to square the circle with a comass and straightedge. In the 9th century, Lindemann showed that this is imossible by demonstrating
More informationPinhole Optics. OBJECTIVES To study the formation of an image without use of a lens.
Pinhole Otics Science, at bottom, is really antiintellectual. It always distrusts ure reason and demands the roduction of the objective fact. H. L. Mencken (18801956) OBJECTIVES To study the formation
More informationFREQUENCIES OF SUCCESSIVE PAIRS OF PRIME RESIDUES
FREQUENCIES OF SUCCESSIVE PAIRS OF PRIME RESIDUES AVNER ASH, LAURA BELTIS, ROBERT GROSS, AND WARREN SINNOTT Abstract. We consider statistical roerties of the sequence of ordered airs obtained by taking
More informationIndex Numbers OPTIONAL  II Mathematics for Commerce, Economics and Business INDEX NUMBERS
Index Numbers OPTIONAL  II 38 INDEX NUMBERS Of the imortant statistical devices and techniques, Index Numbers have today become one of the most widely used for judging the ulse of economy, although in
More informationMaximum Likelihood Estimation
Math 541: Statistical Theory II Lecturer: Songfeng Zheng Maximum Likelihood Estimation 1 Maximum Likelihood Estimation Maximum likelihood is a relatively simple method of constructing an estimator for
More informationSampling Distribution of a Normal Variable
Ismor Fischer, 5/9/01 5.1 5. Formal Statement and Examples Comments: Sampling Distribution of a Normal Variable Given a random variable. Suppose that the population distribution of is known to be normal,
More informationOPTIMAL EXCHANGE BETTING STRATEGY FOR WINDRAWLOSS MARKETS
OPTIA EXCHANGE ETTING STRATEGY FOR WINDRAWOSS ARKETS Darren O Shaughnessy a,b a Ranking Software, elbourne b Corresonding author: darren@rankingsoftware.com Abstract Since the etfair betting exchange
More informationFrequentist vs. Bayesian Statistics
Bayes Theorem Frequentist vs. Bayesian Statistics Common situation in science: We have some data and we want to know the true hysical law describing it. We want to come u with a model that fits the data.
More informationMachine Learning with Operational Costs
Journal of Machine Learning Research 14 (2013) 19892028 Submitted 12/11; Revised 8/12; Published 7/13 Machine Learning with Oerational Costs Theja Tulabandhula Deartment of Electrical Engineering and
More informationRNAseq. Quantification and Differential Expression. Genomics: Lecture #12
(2) Quantification and Differential Expression Institut für Medizinische Genetik und Humangenetik Charité Universitätsmedizin Berlin Genomics: Lecture #12 Today (2) Gene Expression per Sources of bias,
More informationImplementation of Statistic Process Control in a Painting Sector of a Automotive Manufacturer
4 th International Conference on Industrial Engineering and Industrial Management IV Congreso de Ingeniería de Organización Donostia an ebastián, etember 8 th  th Imlementation of tatistic Process Control
More informationJoint Probability Distributions and Random Samples (Devore Chapter Five)
Joint Probability Distributions and Random Samples (Devore Chapter Five) 101634501 Probability and Statistics for Engineers Winter 20102011 Contents 1 Joint Probability Distributions 1 1.1 Two Discrete
More informationLab O3: Snell's Law and the Index of Refraction
O3.1 Lab O3: Snell's Law and the Index of Refraction Introduction. The bending of a light ray as it asses from air to water is determined by Snell's law. This law also alies to the bending of light by
More informationThe Online Freezetag Problem
The Online Freezetag Problem Mikael Hammar, Bengt J. Nilsson, and Mia Persson Atus Technologies AB, IDEON, SE3 70 Lund, Sweden mikael.hammar@atus.com School of Technology and Society, Malmö University,
More informationProbabilistic models for mechanical properties of prestressing strands
Probabilistic models for mechanical roerties of restressing strands Luciano Jacinto a, Manuel Pia b, Luís Neves c, Luís Oliveira Santos b a Instituto Suerior de Engenharia de Lisboa, Rua Conselheiro Emídio
More informationReDispatch Approach for Congestion Relief in Deregulated Power Systems
ReDisatch Aroach for Congestion Relief in Deregulated ower Systems Ch. Naga Raja Kumari #1, M. Anitha 2 #1, 2 Assistant rofessor, Det. of Electrical Engineering RVR & JC College of Engineering, Guntur522019,
More informationEffects of Math Tutoring
Requestor: Math Deartment Researcher(s): Steve Blohm Date: 6/30/15 Title: Effects of Math Tutoring Effects of Math Tutoring The urose of this study is to measure the effects of math tutoring at Cabrillo
More informationSimple thermodynamic systems
Simle thermodynamic systems Asaf Pe er 1 October 30, 2013 1. Different statements of the second law We have seen that Clausius defined the second law as the entroy of an isolated system cannot decrease
More informationBinomial distribution From Wikipedia, the free encyclopedia See also: Negative binomial distribution
Binomial distribution From Wikipedia, the free encyclopedia See also: Negative binomial distribution In probability theory and statistics, the binomial distribution is the discrete probability distribution
More informationSoftmax Model as Generalization upon Logistic Discrimination Suffers from Overfitting
Journal of Data Science 12(2014),563574 Softmax Model as Generalization uon Logistic Discrimination Suffers from Overfitting F. Mohammadi Basatini 1 and Rahim Chiniardaz 2 1 Deartment of Statistics, Shoushtar
More informationManaging specific risk in property portfolios
Managing secific risk in roerty ortfolios Andrew Baum, PhD University of Reading, UK Peter Struemell OPC, London, UK Contact author: Andrew Baum Deartment of Real Estate and Planning University of Reading
More informationDouble Integrals in Polar Coordinates
Double Integrals in Polar Coorinates Part : The Area Di erential in Polar Coorinates We can also aly the change of variable formula to the olar coorinate transformation x = r cos () ; y = r sin () However,
More informationStability Improvements of Robot Control by Periodic Variation of the Gain Parameters
Proceedings of the th World Congress in Mechanism and Machine Science ril ~4, 4, ianin, China China Machinery Press, edited by ian Huang. 868 Stability Imrovements of Robot Control by Periodic Variation
More information4. Discrete Probability Distributions
4. Discrete Probabilit Distributions 4.. Random Variables and Their Probabilit Distributions Most of the exeriments we encounter generate outcomes that can be interreted in terms of real numbers, such
More informationFinding a Needle in a Haystack: Pinpointing Significant BGP Routing Changes in an IP Network
Finding a Needle in a Haystack: Pinointing Significant BGP Routing Changes in an IP Network Jian Wu, Zhuoqing Morley Mao University of Michigan Jennifer Rexford Princeton University Jia Wang AT&T Labs
More informationFactoring Variations in Natural Images with Deep Gaussian Mixture Models
Factoring Variations in Natural Images with Dee Gaussian Mixture Models Aäron van den Oord, Benjamin Schrauwen Electronics and Information Systems deartment (ELIS), Ghent University {aaron.vandenoord,
More informationPythagorean Triples and Rational Points on the Unit Circle
Pythagorean Triles and Rational Points on the Unit Circle Solutions Below are samle solutions to the roblems osed. You may find that your solutions are different in form and you may have found atterns
More informationComparing Dissimilarity Measures for Symbolic Data Analysis
Comaring Dissimilarity Measures for Symbolic Data Analysis Donato MALERBA, Floriana ESPOSITO, Vincenzo GIOVIALE and Valentina TAMMA Diartimento di Informatica, University of Bari Via Orabona 4 76 Bari,
More informationStatic and Dynamic Properties of Smallworld Connection Topologies Based on Transitstub Networks
Static and Dynamic Proerties of Smallworld Connection Toologies Based on Transitstub Networks Carlos Aguirre Fernando Corbacho Ramón Huerta Comuter Engineering Deartment, Universidad Autónoma de Madrid,
More informationAn important observation in supply chain management, known as the bullwhip effect,
Quantifying the Bullwhi Effect in a Simle Suly Chain: The Imact of Forecasting, Lead Times, and Information Frank Chen Zvi Drezner Jennifer K. Ryan David SimchiLevi Decision Sciences Deartment, National
More informationLarge firms and heterogeneity: the structure of trade and industry under oligopoly
Large firms and heterogeneity: the structure of trade and industry under oligooly Eddy Bekkers University of Linz Joseh Francois University of Linz & CEPR (London) ABSTRACT: We develo a model of trade
More informationSummary of Formulas and Concepts. Descriptive Statistics (Ch. 14)
Summary of Formulas and Concepts Descriptive Statistics (Ch. 14) Definitions Population: The complete set of numerical information on a particular quantity in which an investigator is interested. We assume
More informationOn Software Piracy when Piracy is Costly
Deartment of Economics Working aer No. 0309 htt://nt.fas.nus.edu.sg/ecs/ub/w/w0309.df n Software iracy when iracy is Costly Sougata oddar August 003 Abstract: The ervasiveness of the illegal coying of
More informationOn tests for multivariate normality and associated simulation studies
Journal of Statistical Comutation & Simulation Vol. 00, No. 00, January 2006, 1 14 On tests for multivariate normality and associated simulation studies Patrick J. Farrell Matias SalibianBarrera Katarzyna
More informationAlpha Channel Estimation in High Resolution Images and Image Sequences
In IEEE Comuter Society Conference on Comuter Vision and Pattern Recognition (CVPR 2001), Volume I, ages 1063 68, auai Hawaii, 11th 13th Dec 2001 Alha Channel Estimation in High Resolution Images and Image
More informationA Note on the Stock Market Trend Analysis Using MarkovSwitching EGARCH Models
( 251 ) Note A Note on the Stock Market Trend Analysis Using MarkovSwitching EGARCH Models MITSUI, Hidetoshi* 1) 1 Introduction The stock marketʼs fluctuations follow a continuous trend whether or not
More informationThe impact of metadata implementation on webpage visibility in search engine results (Part II) q
Information Processing and Management 41 (2005) 691 715 www.elsevier.com/locate/inforoman The imact of metadata imlementation on webage visibility in search engine results (Part II) q Jin Zhang *, Alexandra
More informationIEEM 101: Inventory control
IEEM 101: Inventory control Outline of this series of lectures: 1. Definition of inventory. Examles of where inventory can imrove things in a system 3. Deterministic Inventory Models 3.1. Continuous review:
More informationInvestigation of Variance Estimators for Adaptive Cluster Sampling with a Single Primary Unit
Thail Statistician Jul 009; 7( : 09 www.statassoc.or.th Contributed aer Investigation of Variance Estimators for Adative Cluster Samling with a Single Primar Unit Urairat Netharn School of Alied Statistics
More informationTHE MULTINOMIAL DISTRIBUTION. Throwing Dice and the Multinomial Distribution
THE MULTINOMIAL DISTRIBUTION Discrete distribution  The Outcomes Are Discrete. A generalization of the binomial distribution from only 2 outcomes to k outcomes. Typical Multinomial Outcomes: red A area1
More informationWeb Application Scalability: A ModelBased Approach
Coyright 24, Software Engineering Research and Performance Engineering Services. All rights reserved. Web Alication Scalability: A ModelBased Aroach Lloyd G. Williams, Ph.D. Software Engineering Research
More information4 Perceptron Learning Rule
Percetron Learning Rule Objectives Objectives  Theory and Examles  Learning Rules  Percetron Architecture 3 SingleNeuron Percetron 5 MultileNeuron Percetron 8 Percetron Learning Rule 8 Test Problem
More informationSECOND PART, LECTURE 4: CONFIDENCE INTERVALS
Massimo Guidolin Massimo.Guidolin@unibocconi.it Dept. of Finance STATISTICS/ECONOMETRICS PREP COURSE PROF. MASSIMO GUIDOLIN SECOND PART, LECTURE 4: CONFIDENCE INTERVALS Lecture 4: Confidence Intervals
More informationProbability and Statistics Lecture 9: 1 and 2Sample Estimation
Probability and Statistics Lecture 9: 1 and Sample Estimation to accompany Probability and Statistics for Engineers and Scientists Fatih Cavdur Introduction A statistic θ is said to be an unbiased estimator
More informationX How to Schedule a Cascade in an Arbitrary Graph
X How to Schedule a Cascade in an Arbitrary Grah Flavio Chierichetti, Cornell University Jon Kleinberg, Cornell University Alessandro Panconesi, Saienza University When individuals in a social network
More informationStat 134 Fall 2011: Gambler s ruin
Stat 134 Fall 2011: Gambler s ruin Michael Lugo Setember 12, 2011 In class today I talked about the roblem of gambler s ruin but there wasn t enough time to do it roerly. I fear I may have confused some
More informationComputational Statistics and Data Analysis
Computational Statistics and Data Analysis 53 (2008) 17 26 Contents lists available at ScienceDirect Computational Statistics and Data Analysis journal homepage: www.elsevier.com/locate/csda Coverage probability
More information4: Probability. What is probability? Random variables (RVs)
4: Probability b binomial µ expected value [parameter] n number of trials [parameter] N normal p probability of success [parameter] pdf probability density function pmf probability mass function RV random
More informationChapter 14: 16, 9, 12; Chapter 15: 8 Solutions When is it appropriate to use the normal approximation to the binomial distribution?
Chapter 14: 16, 9, 1; Chapter 15: 8 Solutions 141 When is it appropriate to use the normal approximation to the binomial distribution? The usual recommendation is that the approximation is good if np
More informationNOISE ANALYSIS OF NIKON D40 DIGITAL STILL CAMERA
NOISE ANALYSIS OF NIKON D40 DIGITAL STILL CAMERA F. Mojžíš, J. Švihlík Detartment of Comuting and Control Engineering, ICT Prague Abstract This aer is devoted to statistical analysis of Nikon D40 digital
More informationWe are going to delve into some economics today. Specifically we are going to talk about production and returns to scale.
Firms and Production We are going to delve into some economics today. Secifically we are going to talk aout roduction and returns to scale. firm  an organization that converts inuts such as laor, materials,
More informationJoint Production and Financing Decisions: Modeling and Analysis
Joint Production and Financing Decisions: Modeling and Analysis Xiaodong Xu John R. Birge Deartment of Industrial Engineering and Management Sciences, Northwestern University, Evanston, Illinois 60208,
More informationThe risk of using the Q heterogeneity estimator for software engineering experiments
Dieste, O., Fernández, E., GarcíaMartínez, R., Juristo, N. 11. The risk of using the Q heterogeneity estimator for software engineering exeriments. The risk of using the Q heterogeneity estimator for
More informationCompensating Fund Managers for RiskAdjusted Performance
Comensating Fund Managers for RiskAdjusted Performance Thomas S. Coleman Æquilibrium Investments, Ltd. Laurence B. Siegel The Ford Foundation Journal of Alternative Investments Winter 1999 In contrast
More informationThe Changing Wage Return to an Undergraduate Education
DISCUSSION PAPER SERIES IZA DP No. 1549 The Changing Wage Return to an Undergraduate Education Nigel C. O'Leary Peter J. Sloane March 2005 Forschungsinstitut zur Zukunft der Arbeit Institute for the Study
More informationAlgorithms for Constructing ZeroDivisor Graphs of Commutative Rings Joan Krone
Algorithms for Constructing ZeroDivisor Grahs of Commutative Rings Joan Krone Abstract The idea of associating a grah with the zerodivisors of a commutative ring was introduced in [3], where the author
More informationPrecalculus Prerequisites a.k.a. Chapter 0. August 16, 2013
Precalculus Prerequisites a.k.a. Chater 0 by Carl Stitz, Ph.D. Lakeland Community College Jeff Zeager, Ph.D. Lorain County Community College August 6, 0 Table of Contents 0 Prerequisites 0. Basic Set
More informationStatistical Inference
Statistical Inference Idea: Estimate parameters of the population distribution using data. How: Use the sampling distribution of sample statistics and methods based on what would happen if we used this
More informationExamination 110 Probability and Statistics Examination
Examination 0 Probability and Statistics Examination Sample Examination Questions The Probability and Statistics Examination consists of 5 multiplechoice test questions. The test is a threehour examination
More informationBuffer Capacity Allocation: A method to QoS support on MPLS networks**
Buffer Caacity Allocation: A method to QoS suort on MPLS networks** M. K. Huerta * J. J. Padilla X. Hesselbach ϒ R. Fabregat O. Ravelo Abstract This aer describes an otimized model to suort QoS by mean
More information