Air Force Academy, EE 810
Excerpt: ... EE810: Communication Theory I Instructor: Ha H. Nguyen EE810: Communication Theory I (Part II) Outline: The second part of this course presents an introduction to the principles and applications of detection and estimation theories. Topics covered include hypothesis testing, Bayes and the Neyman-Pearson criteria, sucient statistics, maximum likelihood (ML) estimation, minimum mean-square error (MMSE) and maximum a posteriori (MAP) estimation, detection of known and random signals, composite hypothesis testing, generalized likelihood ratio test . Discussion of the applications of these theories will concentrate on digital communications (signal constellations, optimum receivers and error probabilities). Text: H. L. Van Trees, Detection, Estimation and Modulation Theory: Part I, Wiley, 1968. References: S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory, Prentice-Hall, 1993. S. M. Kay, Fundamentals of Statistical Signal Processing: Detection Theory, Prentice-Hall, 1998. ...
East Los Angeles College, EC 968
Excerpt: ... d functions using non-parametric estimators (Lesson 3) do these provide clues about whether e.g. the hazard is monotonically increasing or decreasing or what? Note too the informal graphical checks for the proportional hazards and log-logistic models. (Related) The shape of the hazard may also be explored in a regression framework using piecewise constant exponential models or their discrete time counterparts, using judiciously chosen partitions of the survival time axis. (Of course, if you are not interested in estimating how the hazard varies with survival time, you could simply use the Cox model for continuous time data, or the corresponding nonparametric model for grouped/discrete data.) (Related) Use a general-to-specific modeling strategy: start with a parametric model that characterizes the hazard in a relatively flexible way (e.g. generalized Gamma model for continuous time data), and then test whether the specifications nested within this are appropriate using likelihood ratio test s. (For non-nes ...
UWO, STAT 3858
Excerpt: ... g Rootograms. In section 9.8 we only discussed QQ plots. Note also that we have not covered Bayesian methods, so in any parts of questions below omit the parts about Bayes calculations. 2. Section 9.12 : 1-23, 25-36, 40-45, 54, Remark: These questions begin with the material from starting from likelihood ratio test s. 3. Chapter 11 deals with two sample problems. Sections 11.2.1, except box plots, 11.2.2, and 11.3.1 are valid sections, as far as working with likelihood ratio test s and generalized likelihood ratios. One assignment question dealt with this material. A possible exam question related to this could deal with a two sample generalized likelihood ratio. For example, suppose that X1 , . . . , Xn are iid Poisson, 1 and Y1 , . . . , Ym are iid 2 , and the X's and Y 's are independent. Consider the hypotheses H0 : 1 = 2 versus HA : 1 = 2 1 What is the generalized likelihood ratio, and what is the form of the rejection region? Suppose n = m = 100 (something large) find the size = .05 rejection region. C ...
UPenn, STAT 512
Excerpt: ... Homework 8, Statistics 512, Spring 2005 This homework is due Thursday, March 24th at the beginning of class. 1. Let X 1 ,K , X n be iid Bernoulli ( p ) random variables. (a) Find the size 0.05 Wald test of H 0 : p = 0.5 versus H a : p X 0.5 . (b) Find the size 0.05 likelihood ratio test of H 0 : p = 0.5 versus H a : p X 0.5 . (c) Conduct a Monte Carlo simulation study in R to compare the actual size of the tests in (a) and (b) for n = 20 , p = 0.5 . (d) Conduct a Monte Carlo simulation study in R to compare the power of the tests in (a) and (b) for n = 20 , p = 0.4 . 2. Hogg, McKean and Craig, Exercise 6.3.1. Hint: Review Section 3.3, particularly Theorem 3.3.2. 3. Hogg, McKean and Craig, Exercise 6.4.7 4. (a) Hogg, McKean and Craig, Exercise 6.4.8. (b) Hogg, McKean and Craig, Exercise 6.4.9. Note: Location-scale families are commonly used in statistics. Some examples are the following: (i) normal distribution (ii) double exponential distribution with a scale 1 -| x -a|/ b e parameter f ( x; a, b) = ; (iii) e ...
Texas A&M, CS 790
Excerpt: ... Lecture 4: Decision theory g The Likelihood Ratio Test g The Probability of Error g The Bayes Risk g Bayes, MAP and ML Criterion g Multi-class problems g Discriminant Functions Introduction to Pattern Recognition Ricardo Gutierrez-Osuna Wright State University 1 Likelihood Ratio Test (LRT) g Given the problem of classifying a given measurement x, a reasonable heuristic decision rule would be: n Choose the class that is more likely given the evidence provided by the measured feature x g More formally: Evaluate the posterior probability of each class P(i|x) and choose the class with largest P(i|x) g Let us examine the consequences of this decision rule for a 2-class problem n In this case the decision rule becomes if P( g Or, in a more compact form 1 | x ) > P( 2 | x ) choose else choose 1 1 2 > P( 1 | x ) < P( 2 2 | x) n Applying Bayes Rule P( x | )P( P( x ) 1 1 ) >1 P( x | < 2 )P( P( x ) 2 2 ) n P(x) does not affect the decision rule so it can be eliminated*. Rearranging t ...
N.C. State, ST 745
Excerpt: ... nformation matrix from this model and then evaluate them under the hypothesis that the data are from an exponential distribution. (c) Perform the score test to test whether or not the survival times are from an exponential distribution. 5. Consider the AZT cohort study data of the textbook in problem 4.7.7 (Page 122, the data is also available from the online link of the book). Do the following by using statistical software (such as SAS or R): (a) Fit a Weibull model to the censored survival data (b) Perform Wald test to test whether or not the survival times are from an exponential distribution. (c) Perform likelihood ratio test to test whether or not the survival times are from an exponential distribution. (d) Suggest ways to check the Weibull model assumption and conduct the diagnostics. 2 ...
Washington University in St. Louis, MATH 5062
Excerpt: ... Washington University Math5062 (Spring 2007) 1 Homework 5: Due Wednesday March 28, 2006 1. Problem 6.3.1 on page 428 in BD. 2. Problem 6.3.6 on page 430 in BD. 3. Let X1 , . . . , Xn be i.i.d. N (, 1) with 0. And we want to test H0 : = 0 versus H1 : > 0. Note that since the parameter space is not open, the regular asymptotics do not apply. (a) Show that the MLE of is (b) If > 0, show that ^ n = Xn 1{Xn >0} . d ^ n(n - ) N (0, 1) ^ d ^ (c) If = 0, the probability is 1/2 that n = 0 and 1/2 that n(n - ) N (0, 1). (d) Derive the maximum likelihood ratio test at asymptotic level . ...
UCSB, PSTAT 120c
Excerpt: ... PSTAT 120C: Assignment # 1 These problems are due at the start of lecture on Thursday. Due April 13, 2006 1. Show that if T is a sufficient statistic for estimating from the data X1 , . . . , Xn then the Generalized Likelihood Ratio Test (GLRT) statistic for testing H0 : 0 versus Ha : 0 is a function of T . (Hint: use the factorization theorem.) 2. Suppose that X is a binomial random variable from n trials with probability p. We want to 1 test H0 : p = 2 versus Ha : p = 1 . 2 (a) Find , the GLRT statistic. (b) Argue that the critical region from the GLRT is of the same form as X- 3. Exercise 10.93 on page 523 in the textbook. 4. Exercises 8.81 and 8.87 on pages 409410 in the textbook. 5. Exercise 10.67 on pages 505506 in the textbook. n k 2 1 ...
UPenn, STAT 434
Excerpt: ... Statistics 434: Bullet Points for Day 15 Time Series Regression, CAPM, and the Three Factor Model For us, time series regressions are typied by the Capital Asset Pricing Model (CAPM) and the Fama French Three Factor Model (FF3FM). Well also add to your tool kit by discussing the likelihood function and its major applications: Maximum likelihood estimation and the likelihood ratio test s. Youve actually been using these for all of your statistical career. Finally, well take a more extensive look at the Akaike Information Criterion, which you may have seen in your regression course. Here the AIC also helps us to select models, but it can also be used to combine forecasts. It is the second kind of application that seems to oer the most promise in nancial times eries. Discussion of the In-coming HW Time Series Regression 1. CAPM 2. Three Factor Model 3. OLS in S-Plus (CAPM Examples) Probability Densities and the Likelihood Function 1. Densities and Likelihood 2. Maximum Likelihood Est ...
Sveriges lantbruksuniversitet, STAT 402
Excerpt: ... Tutorial 4 - Wald Statistics & Likelihood Ratio Test (last updated February 4, 2009) 1. Based on Question 2 from last week tutorial, carry out a likelihood ratio test to assess the evidence against this hypothesis H0 : = 2 , 2 (1 - ) , (1 - ) 2 T 2. The following data are the remission times (in weeks) for a group of 25 leukemia patients undergoing chemotherapy: 1,1,2,4,4,6,6,6,7,8,9,9,10,12,13,14,18,19,24,26,29,42,57,60,91 These times represent the time from entry into a study until the patients experienced a relapse. In addition to these patients, five patients were observed to remain in remission (ie. no relapse was observed) for the following number of weeks 31, 45, 50, 71 and 85. Suppose the remission are exponentially distributed with rate . (a) Write down the likelihood function for the data above. (b) Derive the score and estimated information function. (c) Construct Wald based 95% confidence intervals for . 3. Consider the data in the following table which describes the relationship between the ...
Columbia, P 8400
Excerpt: ... this result with the one you did in part 1. Are they the same? Perform a likelihood ratio test to examine whether the model includes `alcgrp' improves the predicting ability of the model. Interpret. Model 1. intercept only 2. intercept & alcgrp -2 Log Likelihood 990.864 901.036 DF -1 In model 1, intercept () = -1.3584, -1.3584 is the log odds of cancer for total sample. The odds (e) is 0.2571. Y = exp(-1.3584) / [1 + exp(-1.3584)] =0.2045 =200/(200+778) In model 2, intercept () = -2.5991, -2.5911 is the log odds of cancer for light drinkers (alcgrp=0). Log odds of cancer for heavy drinkers (alcgrp=1) is 0.827 (-2.5911 + 1.7641). Y = 0.0697 for light drinkers, and 0.3043 for heavy drinkers. Beta=1.76 OR=5.84, 95% CI: 3.84-8.86 Heavy drinkers are about 6 times more likely to get esophageal cancer than light drinkers. This crude OR is the same as the one we did in part 1. The likelihood ratio test is 89.828 (990.864-901.036) of df=1 which is statistically significant (as compared to chi-square statistic of df ...
Washington, B 515
Excerpt: ... ample h(t|rx, age) = h(t) exp(1 rx + 2 age) > cph1=coxph(Surv(futime, fustat)~rx+age > summary(cph1) , ovarian) coef exp(coef) se(coef) z p rx -0.804 0.448 0.6320 -1.27 0.2000 age 0.147 1.159 0.0461 3.19 0.0014 exp(coef) exp(-coef) lower .95 upper .95 rx 0.448 2.234 0.130 1.54 age 1.159 0.863 1.059 1.27 Rsquare= 0.457 (max possible= Likelihood ratio test = 15.9 on Wald test = 13.5 on Score (logrank) test = 18.6 on 0.932 ) 2 df, p=0.000355 2 df, p=0.00119 2 df, p=9.34e-05 BIOST 515, Lecture 17 7 Interpreting the output from R This is actually quite easy. The coxph() function gives you the hazard ratio for a one unit change in the predictor as well as the 95% condence interval. Also given is the Wald statistic for each parameter as well as overall likelihood ratio, wald and score tests. What if we wanted to estimate hr(rx = 1, age = 50 : rx = 2, age = 60)? The point estimate is easily obtained as exp{0.804(1 2) + 0.147(50 60)} = 0.514. How do we interpret this quantity? BIOST 515, Le ...
University of Illinois, Urbana Champaign, ECE 313
Excerpt: ... ) = f1 (x) for the observation x, < x < . Simplify your answer. f0 (x) (b) The maximum likelihood decision rule is a likelihood ratio test with threshold = 1: it decides that H 1 is true if (X) 1, and decides H0 is true otherwise. Show that this rule is equivalent to deciding H1 is true if |X| K for some threshold K, and nd K in terms of a2 and b2 . (c) If prior probabilities 0 and 1 are given for the hypotheses, the MAP decision rule is a likelihood ratio test with threshold = 0 . Show that the MAP rule is equivalent to deciding H1 is true if |X| K for 1 some threshold K, and nd K in terms of a2 , b2 , and the prior probabilities. (d) Find the conditional error probabilities, pf alse alarm and pmiss for either of the rules above. Express your answers in terms of a, b, K, and either the cumulative distribution function for a standard normal variate or the Q function dened by Q(u) = 1 (u). (Here K depends on which rule is used, but the value of K need not ...
S.F. State, BIOL 710
Excerpt: ... Logistic Regression Logistic regression is part of a category of statistical models called generalized linear models. This broad class of models includes ordinary regression and ANOVA, as well as multivariate statistics such as ANCOVA and loglinear r ...