DOCSLIB.ORG

Lecture 14 – Statistical Inference, Point Estimation, Hypothesis Testing

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lecture 14 – Statistical Inference, Point Estimation, Hypothesis Testing Lecture 14 – Statistical Inference, Point Estimation, Hypothesis Testing Statistical inference is the process of using the information contained in random samples from a population to make inferences about the population. This can be further divided into parameter estimation and hypothesis testing. Parameter Estimation The different kinds of population parameters one might be interested in are population mean, population variance, population proportion, difference between two population means, and difference between two population proportions. The last two are useful to compare two populations. The corresponding sample statistics for each of the above population parameters are sample mean, sample variance, sample proportion, difference between two samples means, and difference between two sample proportions, respectively. Each of these statistics has estimates (values) that can be gotten from the random sample collected from the population. Inferences on the mean of a population, variance known Let’s say, we are interested in questions about the population mean. Recall for large samples using central limit theorem, 휎2 푋̅ ~ 푁 (휇, ) 푛 Let us first find a μl such that, 훼 푃(푋̅ < 휇 ) = 푙 2 This is the same as, 휇푙 − 휇 훼 푃 (푍 < 휎 ) = ⁄ 2 √푛 And, 휇푙 − 휇 휎 = 푍훼 ⁄ 2 √푛 휎 휇푙 = 휇 + 푍훼 ⁄ 2 √푛 Similarly, we can find a μu such that, 훼 푃(푋̅ < 휇 ) = 1 − 푢 2 훼 휎 휇푙 = 휇 + 푍1− ⁄ 2 √푛 Now the probability that 푋̅ lies between μl and μu is given by 1-α. Recognizing that 훼 훼 푍1− = −푍 , if we ignore the sign for the Z values, we have, 2 2 휎 휎 푃 (휇 − 푍훼 < 푋̅ < 휇 + 푍훼 ) = 1 − 훼 2 √푛 2 √푛 Rearranging this to find an interval for μ we have, 휎 휎 푃 (푋̅ − 푍훼 < 휇 < 푋̅ + 푍훼 ) = 1 − 훼 2 √푛 2 √푛 Recognize that μ actually is a number and not a variable, the correct way to express the above is, 휎 휇 ∈ (푋̅ ± 푍훼 ) 푤푡ℎ 푎 (1 − 훼) 푐표푛푓푑푒푛푐푒 2 √푛 This is a 2 sided confidence interval. Similarly, the 1-sided confidence intervals can be shown as, 휎 휇 ∈ (푋̅ − 푍훼 , ∞) 푙표푤푒푟 푏표푢푛푑 푤푡ℎ 푎 (1 − 훼) 푐표푛푓푑푒푛푐푒 √푛 휎 휇 ∈ (−∞, 푋̅ + 푍훼 ) 푢푝푝푒푟 푏표푢푛푑 푤푡ℎ 푎 (1 − 훼) 푐표푛푓푑푒푛푐푒 √푛 Example 28 observations have a sample mean of 0.0328. If the experimenter knows the standard deviation to be 0.015, find the 95% 2-sided confidence interval for the population mean. Solution 0.015 0.015 푃 (0.0328 − 푍0.05 < 휇 < 0.0328 − 푍0.05 ) = 95% 2 √28 2 √28 0.015 0.015 휇 ∈ (0.0328 − 푍0.05 , 0.0328 − 푍0.05 ) 2 √28 2 √28 휇 ∈ (0.0272,0.0384) Hypothesis Testing Many problems require whether to accept or reject a statement about some population parameter. Example – is the strength of the column strong enough to withstand the load on it? The population parameter is the mean carrying capacity of the column. The sample parameter is the average of the carrying capacities of n randomly selected columns. The statement is called a hypothesis, and the decision making procedure is called hypothesis testing. A statistical hypothesis, therefore, is a statement about the parameters of one or more populations. There are two basic kinds of hypothesis tests. You might want to compare the population parameter (say the population mean) to a single value, and may want to determine whether the parameter is equal to that value. In this case, the hypotheses may be stated as H0 – μ = μ0 H1 - μ ≠ μ0 H0 is called the null hypothesis and H1 is called the alternate hypothesis. The above is a two sided test because in the alternative hypothesis you are allowing the population mean to be either less than or greater than the value used for comparison (μ0). However, if you want to check whether your parameter is less than or greater than a particular value, then you can have either of the following sets of hypotheses, H0 – μ ≤ μ0 H1 - μ > μ0 Or H0 – μ ≥ μ0 H1 - μ < μ0 These are one sided tests. Testing Hypotheses – Truth Table Conclusion Fact H0 is true H0 is false H0 is true Correct Action Type 2 Error H0 is false Type 1 Error Correct Action So you make a type 1 error if you reject H0 when in fact it is true and a type 2 error if you don’t reject it when in fact it is false. The type 1 error is denoted by α which is also called the significance level. The type 2 error is denoted by β. Because we work with random variables which have probability distributions, we can associate probabilities for each type of error. Our focus mostly would be on α for decision making i.e. we will want to minimize the possibility of erroneously rejecting H0. In other words, when we favor H1, there is very little chance that we made a wrong decision. So our possible conclusions are, 1) We favor H1 against H0 2) We cannot favor H1 against H0 i.e. either of them can be true. The standard values of α are 0.01 or 0.05. .
Load more

Recommended publications
  • Multiple-Change-Point Detection for Auto-Regressive Conditional Heteroscedastic Processes
    J. R. Statist. Soc. B (2014) Multiple-change-point detection for auto-regressive conditional heteroscedastic processes P.Fryzlewicz London School of Economics and Political Science, UK and S. Subba Rao Texas A&M University, College Station, USA [Received March 2010. Final revision August 2013] Summary.The emergence of the recent financial crisis, during which markets frequently under- went changes in their statistical structure over a short period of time, illustrates the importance of non-stationary modelling in financial time series. Motivated by this observation, we propose a fast, well performing and theoretically tractable method for detecting multiple change points in the structure of an auto-regressive conditional heteroscedastic model for financial returns with piecewise constant parameter values. Our method, termed BASTA (binary segmentation for transformed auto-regressive conditional heteroscedasticity), proceeds in two stages: process transformation and binary segmentation. The process transformation decorrelates the original process and lightens its tails; the binary segmentation consistently estimates the change points. We propose and justify two particular transformations and use simulation to fine-tune their parameters as well as the threshold parameter for the binary segmentation stage. A compara- tive simulation study illustrates good performance in comparison with the state of the art, and the analysis of the Financial Times Stock Exchange FTSE 100 index reveals an interesting correspondence between the estimated change points
    [Show full text]
  • Chapter 6 Point Estimation
    Chapter 6 Point estimation This chapter deals with one of the central problems of statistics, that of using a sample of data to estimate the parameters for a hypothesized population model from which the data are assumed to arise. There are many approaches to estimation, and several of these are mentioned; but one method has very wide acceptance, the method of maximum likelihood. So far in the course, a major distinction has been drawn between sample quan- tities on the one hand-values calculated from data, such as the sample mean and the sample standard deviation-and corresponding population quantities on the other. The latter arise when a statistical model is assumed to be an ad- equate representation of the underlying variation in the population. Usually the model family is specified (binomial, Poisson, normal, . ) but the index- ing parameters (the binomial probability p, the Poisson mean p, the normal variance u2, . ) might be unknown-indeed, they usually will be unknown. Often one of the main reasons for collecting data is to estimate, from a sample, the value of the model parameter or parameters. Here are two examples. Example 6.1 Counts of females in queues In a paper about graphical methods for testing model quality, an experiment Jinkinson, R.A. and Slater, M. is described to count the number of females in each of 100 queues, all of length (1981) Critical discussion of a ten, at a London underground train station. graphical method for identifying discrete distributions. The The number of females in each queue could theoretically take any value be- Statistician, 30, 239-248.
    [Show full text]
  • THE THEORY of POINT ESTIMATION a Point Estimator Uses the Information Available in a Sample to Obtain a Single Number That Estimates a Population Parameter
    THE THEORY OF POINT ESTIMATION A point estimator uses the information available in a sample to obtain a single number that estimates a population parameter. There can be a variety of estimators of the same parameter that derive from different principles of estimation, and it is necessary to choose amongst them. Therefore, criteria are required that will indicate which are the acceptable estimators and which of these is the best in given circumstances. Often, the choice of an estimate is governed by practical considerations such as the ease of computation or the ready availability of a computer program. In what follows, we shall ignore such considerations in order to concentrate on the criteria that, ideally, we would wish to adopt. The circumstances that affect the choice are more complicated than one might imagine at first. Consider an estimator θˆn based on a sample of size n that purports to estimate a population parameter θ, and let θ˜n be any other estimator based on the same sample. If P (θ − c1 ≤ θˆn ≤ θ + c2) ≥ P (θ − c1 ≤ θ˜n ≤ θ + c2) for all values of c1,c2 > 0, then θˆn would be unequivocally the best estimator. However, it is possible to show that an estimator has such a property only in very restricted circumstances. Instead, we must evaluate an estimator against a list of partial criteria, of which we shall itemise the principal ones. The first is the criterion of unbiasedness. (a) θˆ is an unbiased estimator of the population parameter θ if E(θˆ)=θ. On it own, this is an insufficient criterion.
    [Show full text]
  • Point Estimation Decision Theory
    Point estimation Suppose we are interested in the value of a parameter θ, for example the unknown bias of a coin. We have already seen how one may use the Bayesian method to reason about θ; namely, we select a likelihood function p(D j θ), explaining how observed data D are expected to be generated given the value of θ. Then we select a prior distribution p(θ) reecting our initial beliefs about θ. Finally, we conduct an experiment to gather data and use Bayes’ theorem to derive the posterior p(θ j D). In a sense, the posterior contains all information about θ that we care about. However, the process of inference will often require us to use this posterior to answer various questions. For example, we might be compelled to choose a single value θ^ to serve as a point estimate of θ. To a Bayesian, the selection of θ^ is a decision, and in dierent contexts we might want to select dierent values to report. In general, we should not expect to be able to select the true value of θ, unless we have somehow observed data that unambiguously determine it. Instead, we can only hope to select an estimate that is “close” to the true value. Dierent denitions of “closeness” can naturally lead to dierent estimates. The Bayesian approach to point estimation will be to analyze the impact of our choice in terms of a loss function, which describes how “bad” dierent types of mistakes can be. We then select the estimate which appears to be the least “bad” according to our current beliefs about θ.
    [Show full text]
  • Chapter 7 Point Estimation Method of Moments
    Introduction Method of Moments Procedure Mark and Recapture Monsoon Rains Chapter 7 Point Estimation Method of Moments 1 / 23 Introduction Method of Moments Procedure Mark and Recapture Monsoon Rains Outline Introduction Classical Statistics Method of Moments Procedure Mark and Recapture Monsoon Rains 2 / 23 Introduction Method of Moments Procedure Mark and Recapture Monsoon Rains Parameter Estimation For parameter estimation, we consider X = (X1;:::; Xn), independent random variables chosen according to one of a family of probabilities Pθ where θ is element from the parameter spaceΘ. Based on our analysis, we choose an estimator θ^(X ). If the data x takes on the values x1; x2;:::; xn, then θ^(x1; x2;:::; xn) is called the estimate of θ. Thus we have three closely related objects. 1. θ - the parameter, an element of the parameter space, is a number or a vector. 2. θ^(x1; x2;:::; xn) - the estimate, is a number or a vector obtained by evaluating the estimator on the data x = (x1; x2;:::; xn). 3. θ^(X1;:::; Xn) - the estimator, is a random variable. We will analyze the distribution of this random variable to decide how well it performs in estimating θ. 3 / 23 Introduction Method of Moments Procedure Mark and Recapture Monsoon Rains Classical Statistics In classical statistics, the state of nature is assumed to be fixed, but unknown to us. Thus, one goal of estimation is to determine which of the Pθ is the source of the data. The estimate is a statistic θ^ : data ! Θ: For estimation procedures, the classical approach to statistics is based
    [Show full text]
  • Point Estimation
    6.1 Point Estimation What is an estimate? • Want to study a population • Ideally the knowledge of the distribution, • Some parameters of the population may be the first consideration: θ • Use the information from a sample to estimate • Estimator θˆ : defines the procedure (the formula) • Different estimators, each is a rv. Examples of estimators • want to estimate the mean µ of a normal dist • Possibilities of estimators: X sample mean X¯ = i • n ! • sample median • average of the smallest and the largest • 10% trimmed mean Which one is best (better)? • Error in estimation θˆ = θ + error of estimation • Error is random, depending on the sample • We would like to control the error • mean 0 (unbiased) • smallest variance Unbiased Estimator • estimator: a particular statistic • different estimators: some tend to over (under) estimate • unbiased: E(θˆ) = θ • difference is called the bias • examples: sample mean (yes), sample proportion (yes) if X is a binomial rv. • If possible, we should always choose an unbiased Sample Variance • Need an estimator for the variance • Natural candidate: sample variance • Why divide by n-1, not n? • Make sure the estimator is unbiased • We can not say the same thing for sample standard deviation Unbiased Estimators for Population Mean • several choices (make things complicated) • sample mean • if distribution is continuous and symmetric, sample median and any trimmed mean Minimum Variance • suppose we have two estimators, both unbiased, which one do we prefer? • pick the one with smaller variance • minimum variance unbiased estimator (MVUE) • for normal distribution, the sample mean X ¯ is the MVUE for the population mean.
    [Show full text]
  • Point Estimators Math 218, Mathematical Statistics
    that has \central tendencies." Which of these is best, and why? Desirable criteria for point estimators. The MSE. Well, of course, we want our point estima- ^ Point estimators tor θ of the parameter θ to be close to θ. We can't Math 218, Mathematical Statistics expect them to be equal, of course, because of sam- D Joyce, Spring 2016 pling error. How should we measure how far off the random variable θ^ is from the unknown constant θ? The main job of statistics is to make inferences, One standard measure is what is called the mean specifically inferences bout parameters based on squared error, abbreviated MSE and defined by data from a sample. MSE(θ^) = E((θ^ − θ)2) We assume that a sample X1;:::;Xn comes from a particular distribution, called the population dis- the expected square of the error. If we have two dif- tribution, and although that particular distribution ferent estimators of θ, the one that has the smaller is not known, it is assumed that it is one of a fam- MSE is closer to the actual parameter (in some ily of distributions parametrized by one or more sense of \closer"). parameters. For example, if there are only two possible out- Variance and bias. The MSE of an estimator θ^ comes, then the distribution is a Bernoulli distri- can be split into two parts, the estimator's variance bution parametrized by one parameter p, the prob- and a \bias" term. We're familiar with the variance ability of success. of a random variable X; it's For another example, many measurements are as- sumed to be normally distributed, and for a normal Var(X) = σ2 = E((X − µ )2) = E(X2) − µ2 : distribution, there are two parameters µ and σ2.
    [Show full text]
  • Point Estimation Statistics (OA3102)
    Module 4: Point Estimation Statistics (OA3102) Professor Ron Fricker Naval Postgraduate School Monterey, California Reading assignment: WM&S chapter 8.1-8.4 Revision: 1-12 1 Goals for this Module • Define and distinguish between point estimates vs. point estimators • Discuss characteristics of good point estimates – Unbiasedness and minimum variance – Mean square error – Consistency, efficiency, robustness • Quantify and calculate the precision of an estimator via the standard error – Discuss the Bootstrap as a way to empirically estimate standard errors Revision: 1-12 2 Welcome to Statistical Inference! • Problem: – We have a simple random sample of data – We want to use it to estimate a population quantity (usually a parameter of a distribution) – In point estimation, the estimate is a number • Issue: Often lots of possible estimates – E.g., estimate E(X) with x , x , or ??? • This module: What’s a “good” point estimate? – Module 5: Interval estimators – Module 6: Methods for finding good estimators Revision: 1-12 3 Point Estimation • A point estimate of a parameter q is a single number that is a sensible value for q – I.e., it’s a numerical estimate of q – We’ll use q to represent a generic parameter – it could be m, s, p, etc. • The point estimate is a statistic calculated from a sample of data x – The statistic is called a point estimator – Using “hat” notation, we will denote it as qˆ – For example, we might use to estimate m, so in this case mˆ x Revision: 1-12 4 Definition: Estimator An estimator is a rule, often expressed as
    [Show full text]
  • Outline Estimation Point Estimation
    1/22 Outline Basic Econometrics in Transportation PrProblemoblem ooff EEstimationstimation Point Interval Estimation Methods Statistical Inference Statistical Properties Small sample Large sample Amir Samimi Hypothesis Testing Civil Engineering Department Sharif University of Technology Primary Sources: Basic Econometrics, by Gujarati and lecture notes of Professor Greene 2/22 3/22 Estimation Point Estimation The popu l atio n has c ha racte rist ics: For s im pli ci ty, assum e th at th er e i s onl y on e unkn own Mean, variance parameter (θ) in the PDF. Median Develop a function of the sample values such that: Percentiles is a statistic, or an estimator and is a random variable. Get a slice (random sample) of the population. A particular numerical value of the estimator is known as an Hldbihlfi?How would you obtain the value of its parameters? estimate. This is known as the problem of estimation: is known as a point estimator because it provides a single Point estimation (point) estimate of θ. Interval estimation 4/22 5/22 Example Interval Estimation LLetet An inintervalterval betweebetweenn two vavalueslues tthathat mmayay inincludeclude tthehe ttruerue θ. The key concept is the of probability distribution of an estimator. is the sample mean, an estimator of the true mean value (μ). If in a specific case = 50, this provides an estimate of μ. Will the interval contain the true value? The interval is a “Confidence Interval” The degree of certainty is the degree of confidence or level of significance. 6/22 7/22 Example Estimation Methods Suppose t hat t he d ist ribut io n o f he ig ht o f me n in a popu lat io n Least Squar es (L S) is normally distributed with mean = μ inches and σ = 2.5 A considerable time will be devoted to illustrate the LS method.
    [Show full text]
  • Inferential Stats 1: Point Estimation Copyright © 2000, 2011, 2016, J
    Inferential Stats 1: Point Estimation Copyright © 2000, 2011, 2016, J. Toby Mordkoff The purpose of an inferential statistic is to make a statement about the population from which a sample was taken. Assume, for example, that you are interested in the mean level of anxiety suffered by graduate students. You could, I suppose, measure the anxiety level of every grad from every department at every university and thereby learn the true value, but this is impractical. So, instead, you take a sample of grads, measure only their anxiety levels, and then make an inference (i.e., a best guess) about the entire population. Clearly the quality of this inferential process depends on (at least) two things: how well the sample represents the target group, and how well one can calculate an estimate based on a sample. Here we will focus on the second part. The question of external validity -- which is the “how well the sample represents the target group” part -- is not really a statistical issue. In general, all inferences start with a “best estimate” of the population characteristic of interest. In most cases, the formula for the best guess of something is the same as the formula for the same characteristic of a sample. For example, the best guess for the mean of the population, μ (mu), uses the same formula as that for the mean of a sample, -- i.e., ΣX/N. In other words, the best guess for the population mean is the sample mean. But this is not always the case; there are attributes of populations that are best estimated by something other than the value for the sample.
    [Show full text]
  • 10 Point Estimation Using Maximum Likelihood
    B.Sc./Cert./M.Sc. Qualif. - Statistics: Theory and Practice 10 Point Estimation using Maximum Likelihood 10.1 The method of maximum likelihood De¯nition For each x 2 X , let θ^(x) be such that, with x held ¯xed, the likelihood function fL(θ; x): θ 2 £g attains its maximum value as a function of θ at θ^(x). The estimator θ^(X) is then said to be a maximum likelihood estimator (MLE) of θ. ² Given any observed value x, the value of a MLE, θ^(x), can be interpreted as the most likely value of θ to have given rise to the observed x value. ² To ¯nd a MLE, it is often more convenient to maximize the log likelihood function, ln L(θ; x), which is equivalent to maximizing the likelihood function. ² It should be noted that a MLE may not exist | there may be an x 2 X such that there is no θ that maximizes the likelihood function fL(θ; x): θ 2 £g. However, in practice, it has been found that such cases will be exceptional. ² The MLE, if it does exist, may not be unique | given x 2 X , there may be more than one value of θ that maximizes the likelihood function fL(θ; x): θ 2 £g. Again, in practice, such cases are found only very exceptionally. ² There will be simple cases in which the MLE may be found analytically, possibly just using the standard methods of calculus. In other cases, especially if the dimension q of the vector of parameters is large, the maximization will have to be carried out using numerical methods within computer packages.
    [Show full text]
  • STAT 830 Bayesian Point Estimation
    STAT 830 Bayesian Point Estimation In this section I will focus on the problem of estimation of a 1 dimen- sional parameter, θ. Earlier we discussed comparing estimators in terms of Mean Squared Error. In the language of decision theory Mean Squared Error corresponds to using L(d; θ) = (d − θ)2 which is called squared error loss. The multivariate version would be L(d; θ) = jjd − θjj2 or possibly the more general formula L(d; θ) = (d − θ)T Q(d − θ) for some positive definite symmetric matrix Q. The risk function of a pro- cedure (estimator) θ^ is ^ 2 Rθ^(θ) = Eθ[(θ − θ) ]: Now consider prior with density π(θ). The Bayes risk of θ^ is Z rπ = Rθ^(θ)π(θ)dθ ZZ = (θ^(x) − θ)2f(x; θ)π(θ)dxdθ ^ For a Bayesian the problem is then to choose θ to minimize rπ? This problem will turn out to be analogous to the calculations I made when I minimized β + λα in hypothesis testing. First recognize that f(x; θ)π(θ) is really a joint density ZZ f(x; θ)π(θ)dxdθ = 1 For this joint density: conditional density of X given θ is just the model f(x; θ). This justifies the standard notation f(xjθ) for f(; θ)> Now I will compute rπ a different way by factoring the joint density a different way: f(xjθ)π(θ) = π(θjx)f(x) 1 where now f(x) is the marginal density of x and π(θjx) denotes the condi- tional density of θ given X.
    [Show full text]