Problem: 1) Show that is a set of sufficient statistics 2) Being location and scale parameters, take as (improper) prior and show that inferences on ……

Slides:

Advertisements

Similar presentations

Pattern Recognition and Machine Learning

Advertisements

The Simple Linear Regression Model Specification and Estimation Hill et al Chs 3 and 4.

Functions of Random Variables. Method of Distribution Functions X 1,…,X n ~ f(x 1,…,x n ) U=g(X 1,…,X n ) – Want to obtain f U (u) Find values in (x 1,…,x.

Let X 1, X 2,..., X n be a set of independent random variables having a common distribution, and let E[ X i ] = . then, with probability 1 Strong law.

Point Estimation Notes of STAT 6205 by Dr. Fan.

Exponential Distribution. = mean interval between consequent events = rate = mean number of counts in the unit interval > 0 X = distance between events.

Bayesian inference “Very much lies in the posterior distribution” Bayesian definition of sufficiency: A statistic T (x 1, …, x n ) is sufficient for 

Bayesian Estimation in MARK

Week11 Parameter, Statistic and Random Samples A parameter is a number that describes the population. It is a fixed number, but in practice we do not know.

Sampling: Final and Initial Sample Size Determination

Use of moment generating functions. Definition Let X denote a random variable with probability density function f(x) if continuous (probability mass function.

Review of Basic Probability and Statistics

Descriptive statistics Experiment  Data  Sample Statistics Sample mean Sample variance Normalize sample variance by N-1 Standard deviation goes as square-root.

Simulation Modeling and Analysis

Statistical Inference Chapter 12/13. COMP 5340/6340 Statistical Inference2 Statistical Inference Given a sample of observations from a population, the.

Descriptive statistics Experiment  Data  Sample Statistics Experiment  Data  Sample Statistics Sample mean Sample mean Sample variance Sample variance.

Presenting: Assaf Tzabari

Parametric Inference.

2. Point and interval estimation Introduction Properties of estimators Finite sample size Asymptotic properties Construction methods Method of moments.

Visual Recognition Tutorial1 Random variables, distributions, and probability density functions Discrete Random Variables Continuous Random Variables.

Some standard univariate probability distributions

A random variable that has the following pmf is said to be a binomial random variable with parameters n, p The Binomial random variable.

The moment generating function of random variable X is given by Moment generating function.

Syllabus for Engineering Statistics Probability: rules, cond. probability, independence Summarising and presenting data Discrete and continuous random.

Some standard univariate probability distributions

Using ranking and DCE data to value health states on the QALY scale using conventional and Bayesian methods Theresa Cain.

Copyright © 2014 by McGraw-Hill Higher Education. All rights reserved.

5-3 Inference on the Means of Two Populations, Variances Unknown

Some standard univariate probability distributions

1 Confidence Intervals for Means. 2 When the sample size n< 30 case1-1. the underlying distribution is normal with known variance case1-2. the underlying.

Normal and Sampling Distributions A normal distribution is uniquely determined by its mean, , and variance,  2 The random variable Z = (X-  /  is.

Standard error of estimate & Confidence interval.

Distribution Function properties. Density Function – We define the derivative of the distribution function F X (x) as the probability density function.

Review of Lecture Two Linear Regression Normal Equation

Binary Variables (1) Coin flipping: heads=1, tails=0 Bernoulli Distribution.

Moment Generating Functions 1/33. Contents Review of Continuous Distribution Functions 2/33.

The paired sample experiment The paired t test. Frequently one is interested in comparing the effects of two treatments (drugs, etc…) on a response variable.

Model Inference and Averaging

Moment Generating Functions

Some standard univariate probability distributions Characteristic function, moment generating function, cumulant generating functions Discrete distribution.

Bayesian Inference Ekaterina Lomakina TNU seminar: Bayesian inference 1 March 2013.

Random Sampling, Point Estimation and Maximum Likelihood.

1 G Lect 10a G Lecture 10a Revisited Example: Okazaki’s inferences from a survey Inferences on correlation Correlation: Power and effect.

Functions of Random Variables. Methods for determining the distribution of functions of Random Variables 1.Distribution function method 2.Moment generating.

Bayesian Analysis and Applications of A Cure Rate Model.

Chapter 7 Sampling and Sampling Distributions ©. Simple Random Sample simple random sample Suppose that we want to select a sample of n objects from a.

Lecture 4: Statistics Review II Date: 9/5/02  Hypothesis tests: power  Estimation: likelihood, moment estimation, least square  Statistical properties.

Stats Probability Theory Summary. The sample Space, S The sample space, S, for a random phenomena is the set of all possible outcomes.

Chapter 7 Point Estimation of Parameters. Learning Objectives Explain the general concepts of estimating Explain important properties of point estimators.

Confidence Interval & Unbiased Estimator Review and Foreword.

IE 300, Fall 2012 Richard Sowers IESE. 8/30/2012 Goals: Rules of Probability Counting Equally likely Some examples.

The generalization of Bayes for continuous densities is that we have some density f(y|  ) where y and  are vectors of data and parameters with  being.

Brief Review Probability and Statistics. Probability distributions Continuous distributions.

Point Estimation of Parameters and Sampling Distributions Outlines:  Sampling Distributions and the central limit theorem  Point estimation  Methods.

Statistics 2: generalized linear models. General linear model: Y ~ a + b 1 * x 1 + … + b n * x n + ε There are many cases when general linear models are.

Beginning Statistics Table of Contents HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2008 by Hawkes Learning Systems/Quant Systems, Inc.

Statistics Sampling Distributions and Point Estimation of Parameters Contents, figures, and exercises come from the textbook: Applied Statistics and Probability.

Week 21 Order Statistics The order statistics of a set of random variables X 1, X 2,…, X n are the same random variables arranged in increasing order.

Parameter Estimation. Statistics Probability specified inferred Steam engine pump “prediction” “estimation”

Week 21 Statistical Model A statistical model for some data is a set of distributions, one of which corresponds to the true unknown distribution that produced.

Sampling and Sampling Distributions

ASV Chapters 1 - Sample Spaces and Probabilities

Modeling and Simulation CS 313

Parameter Estimation 主講人：虞台文.

Chapter 7: Sampling Distributions

Parameter, Statistic and Random Samples

OVERVIEW OF BAYESIAN INFERENCE: PART 1

STAT 5372: Experimental Statistics

Moment Generating Functions

Further Topics on Random Variables: 1

Presentation transcript:

Problem: 1) Show that is a set of sufficient statistics 2) Being location and scale parameters, take as (improper) prior and show that inferences on ……

Let and be independent r.q. Problem: Comparison of means and variances and the sufficient statistics 1) Show that for the priors (improper) Consider the samplings H1: same (but unknown) variances: Comparison of variances: Comparison of means: H2: unknown and different variances: (Behrens-Fisher problem)

1)Show that a probability matching prior with the parameter of interest is given by 2) Show that the posterior for the correlation coefficient is: PROBLEM: Correlation Coefficient of the Bivariate Normal Model sample correlation is a sufficient statistic for

PROBLEM: Poisson Distribution Consider and Show that and, in consequence PROBLEM: Binomial Distribution Consider and Show that Hint: Analize the behaviour of expanding around and considering the asymptotic behaviour of the Polygamma Function, the moments of the Distribution,…

PROBLEM: Negative Binomial a: number of failures until experiment is stopped (fixed) X: number of successes observed θ: probability of failure 1) Find the transformations and such that the new parameters are location and scale parameters and transform them back to get the corresponding (improper) prior 2) Obtain the Fisher’s matrix and the Jeffrey’s prior 3) Find the reference prior 4) Show that it is a Probability Matching Prior PROBLEM: Weibull Distribution

Data: Problem: Linear Regression Linear Model: Assume precisions are known and show that: Model: Take and obtain (with uncertainty in x and y)

4) If and Problem: 1) Generate a sample : 2) Get for each case the sampling distribution of 3) Discuss the sampling distribution of in connection with the Law of Large Numbers and the Central Limit Theorem How is distributed? 5) If How is distributed? 6) If (assumed to be independent random quantities) How is distributed?

PROBLEMS: 1) Show that if then 2) Show that if (  generate exponentials) 1) Show that if then Gamma Distribution: Beta Distribution:

(3,1,0) (3,2,±1) Problem 3D: Sampling of Hidrogen atom wave functions Evaluate the energy using Virial Theorem (3,2,0)