It should be unbiased, or approximately unbiased. Variance of the variance estimator should be small. That is, the variance estimator is stable.
|
|
- Augustus Nash
- 6 years ago
- Views:
Transcription
1 Chapter 10 Variace Estimatio 10.1 Itroductio Variace estimatio is a importat practical problem i survey samplig. Variace estimates are used i two purposes. Oe is the aalytic purpose such as costructig cofidece itervals or performig hypothesis testigs. The other is descriptive purposed to evaluate the efficiecy of the survey desigs or estimates for plaig surveys. Desirable variace estimates should satisfy the followig properties: It should be ubiased, or approximately ubiased. Variace of the variace estimator should be small. That is, the variace estimator is stable. It should ot take egative values. The computatio should be simple. The HT variace estimator is ubiased but it ca take egative values. Also, computig the joit iclusio probabilities π i j ca be cumbersome whe the sample size is large. 1
2 2 CHAPTER 10. VARIANCE ESTIMATION Example Cosider a fiite populatio of size N = 3 with y 1 = 16,y 2 = 21 ad y 3 = 18 ad cosider the followig samplig desig. Table 10.1: A samplig desig Sample A Pr A HT estimator HT variace estimator A 1 = {1,2} A 2 = {1,3} A 3 = {2,3} A 4 = {1,2,3} The samplig variace of HT estimator is 85. Note that HT variace estimator has expectatio = 85 but it ca take egative values i some samples. The variace estimator uder PPS samplig is always oegative ad ca be computed without computig the joit iclusio probability. I practice, PPS samplig variace estimator is ofte applied as a alterative variace estimator eve for without-replacemet samplig. The resultig variace estimator ca be writte 1 ˆV 0 = 1 2 yi Ŷ HT = i A p i 1 yi 1 2, 10.1 i A π i ŶHT where p i = π i /. The followig theorem express the bias of the simplified variace estimator i 10.1 as a estimator of the variace of HT estimator. Theorem The simplified variace estimator i 10.1 satisfies where E ˆV 0 Var ŶHT = Var 1 Ŷ PPS = { } Var ŶPPS Var ŶHT 1 N 2 yi p i Y p i 10.2
3 10.1. INTRODUCTION 3 ad p i = π i /. Proof. Note that ˆV 0 satisfies { E = E k A { k A The first term is ad, usig yk Y +Y Ŷ HT p k } 2 yk Y + 2E p k E 2 } { } 2 yk Y k A p k yk Y = k A p k k A { yk Y } { } 2 Y ŶHT + E k A p k Y ŶHT. k A N 2 yk = Y π k p k = 2 Var Ŷ PPS y k π k Y = Ŷ HT Y, the secod term equals to 2Var Ŷ HT ad the last term is equal to Var ŶHT, which proves the result. I may cases, the bias term i 10.2 is positive ad the simplified variace estimator is coservatively estimate the variace. Uder SRS, the relative bias of the simplified variace estimator 10.1 is ˆV 0 Var Ŷ HT Var Ŷ HT = N ad the relative bias is egligible whe /N is egligible The simplified variace estimator ca be directly applicable to multistage samplig desigs. Uder multistage samplig desig, the HT estimator of the total ca be writte Ŷ HT = i A I where Ŷ i is the estimated total for the PSU i. The simplified variace estimator is the give by ˆV 0 = Ŷ i π Ii 1 Ŷi 1 2. i A I π Ii ŶHT
4 4 CHAPTER 10. VARIANCE ESTIMATION Uder stratified multistage cluster samplig, the simplified variace estimator ca be writte ˆV 0 = H h=1 h h 1 h w hi Ŷ hi 1 h where w hi is the samplig weight for cluster i i stratum h. 2 h w h j Ŷ h j 10.4 j= Liearizatio approach to variace estimatio Whe the poit estimator is a oliear estimator such as ratio estimator or regressio estimator, exact variace estimatio for such estimates is very difficult. Istead, we ofte rely o liearizatio methods for estimatig the samplig variace of such estimators. Roughly speakig, if y is a p-dimesioal vector ad whe ȳ = Ȳ N + O p 1/2 holds, the Taylor liearizatio of gȳ ca be writte as gȳ = g Ȳ p g Ȳ + ȳ j Ȳ j + O p 1. j=1 y j Thus, the variace of the liearized term of gȳ ca be writte V {gȳ } =. p p g Ȳ Ȳ g Cov { } ȳ i,ȳ j y i y j ad it is estimated by ˆV {gȳ }. = j=1 p p j=1 gȳ y i gȳ y j Ĉ { ȳ i,ȳ j } I summary, the liearizatio method for variace estimatio ca be described as follows: 1. Apply Taylor liearizatio ad igore the remaider terms. 2. Compute the variace ad covariace terms of each compoet of ȳ usig the stadard variace estimatio formula. 3. Estimate the partial derivative terms g/ y from the sample observatio.
5 10.2. LINEARIZATION APPROACH TO VARIANCE ESTIMATION 5 Example If the parameter of iterest is R = Ȳ / X ad we use ˆR = ȲHT X HT to estimate R, we ca apply the Taylor liearizatio to get ˆR = R + X 1 Ȳ HT R X HT + O p 1 ad the variace estimatio formula i 10.5 reduces to ˆV ˆR. = X 2 HT ˆV Ȳ HT + X 2 HT ˆR 2 ˆV X HT 2 X 2 HT ˆRĈ X HT,Ȳ HT If the variaces ad covariaces of X HT ad Ȳ HT are estimated by HT variace estimatio formula, 10.6 ca be estimated by where e i = y i ˆRx i. ˆV ˆR. = 1 ˆX 2 HT i A j A π i j π i π j π i j e i π i e j π j Usig the result i Example 10.2, the variace of the ratio estimator Ŷ r = X ˆR is estimated by ˆV. X 2 π i j π i π j e i e j Ŷ r = ˆX HT, 10.7 i A j A π i j π i π j which is obtaied by multiplyig ˆX 2 HT X 2 to the variace formula i Recall that the liearized variace estimator of the ratio estimator is give by The variace estimator i 10.7 is asymptotically equivalet to the liearizatio variace estimator i 8.13, but it give more adequate measure of the coditioal variace of the ratio estimator, as advocated by Royall ad Cumberlad More geerally, Särdal, Swesso, ad Wretma 1989 proposed usig ˆV Ŷ GREG = i A j A π i j π i π j π i j g i e i π i g j e j π j 10.8 as the coditioal variace estimator of the GREG estimator of the form Ŷ GREG = i A πi 1 g i y i, where g i = X i A x i x i x i 10.9 π i c i c i
6 6 CHAPTER 10. VARIANCE ESTIMATION ad e i = y i x i i A 1 1 x i x 1 i π i c i x i y i. π i c i i A The g i i 10.9 is the factor to applied to the desig weight to satisfy the calibratio costrait. Example We ow cosider variace estimatio of the poststratificatio estimator i To estimate the variace, the ucoditioal variace estimator is give by ˆV u = N2 1 G g 1 N g=1 1 s2 g where s 2 g = g 1 1 i Ag y i ȳ g 2. O the other had, the coditioal variace estimator i 10.8 is give by ˆV c = 1 N 1 G g=1 N 2 g g g 1 g s 2 g Note that the coditioal variace formula i is very similar to the variace estimatio formula uder the stratified samplig Replicatio approach to variace estimatio We ow cosider a alterative approach to variace estimatio that is based o several replicates of the origial sample estimator. Such replicatio approach does ot use Taylor liearizatio, but istead geerates several resamples ad compute a replicate to each resample. The variability amog the replicates is used to estimate the samplig variace of the poit estimator. Such replicatio approach ofte uses repeated computatio of the replicates usig a computer. Replicatio methods iclude radom group method, jackkife, balaced repeated replicatio, ad bootstrap method. More details of the replicatio methods ca be foud i Wolter 2007.
7 10.3. REPLICATION APPROACH TO VARIANCE ESTIMATION Radom group method Radom group method was first used i jute acreage surveys i Begal Mahalaobis; Radom group method is useful i uderstadig the basic idea for replicatio approach to variace estimatio. I the radom group method, G radom groups are costructed from the sample ad the poit estimate is computed for each radom group sample ad the combied to get the fial poit estimate. Oce the fial poit estimate is costructed, its variace estimate is computed usig the variability of the G radom group estimates. There are two versios of the radom group method. Oe is idepedet radom group method ad the other is o-idepedet radom group method. We first cosider idepedet radom group method. The idepedet radom group method ca be described as follows. [Step 1] Usig the give samplig desig, select the first radom group sample, deoted by A 1, ad compute ˆθ 1, a ubiased estimator of θ from the first radom group sample. [Step 2] Usig the same samplig desig, select the secod radom group sample, idepedetly from the first radom group sample, ad compute ˆθ 2 from the secod radom group sample. [Step 3] Usig the same procedure, obtai G idepedet estimate ˆθ 1,, ˆθ G from the G radom group sample. [Step 4] The fial estimator of θ is ˆθ RG = 1 G ad its ubiased variace estimator is ˆV 1 1 ˆθ RG = G G 1 G G ˆθ k ˆθ k ˆθ RG
8 8 CHAPTER 10. VARIANCE ESTIMATION Sice ˆθ 1,, ˆθ G are idepedetly ad idetically distributed, the variace estimator i is ubiased for the variace of ˆθ RG i Such idepedet radom group method is very easy to uderstad ad is applicable for complicated samplig desigs for selectig A g, but the sample allows for duplicatio of sample elemets ad the variace estimator may be ustable whe G is small. We ow cosider o-idepedet radom group method, which does ot allow for duplicatio of the sample elemets. I the o-idepedet radom group method, the sample is partitioed ito G groups, exhaustive ad mutually exclusive, deoted by A = G g=1 A g ad the apply the sample estimatio method for idepedet radom group method, by treatig the o-idepedet radom group samples as if they are idepedet. The followig theorem expresses the bias of the variace estimator for this case. Theorem Let ˆθ g be a ubiased estimator of θ, computed from the g-th radom group sample A g for g = 1,,G. The the radom group variace estimator?? satisfies E { ˆV ˆθ RG } V ˆθ RG = 1 GG 1 i j Cov ˆθ i, ˆθ j Proof. By 10.12, we have ad, sice V { 1 G ˆθ RG = G 2 V ˆθ i + i j Cov } ˆθ i, ˆθ j ad usig E ˆθ i = θ, E { G ˆθ i ˆθ RG 2 } G G 2 ˆθ i ˆθ RG = ˆθ i θ 2 G ˆθ RG θ 2 = G V ˆθ i G V ˆθ RG = 1 G 1 G V ˆθ i G 1 i j Cov ˆθ i, ˆθ j
9 10.3. REPLICATION APPROACH TO VARIANCE ESTIMATION 9 which implies E { ˆV } ˆθ RG = G 2 G V ˆθ i G 2 G 1 1 i j Cov ˆθ i, ˆθ j. Thus, usig 10.15, we have The right side of is the bias of ˆV ˆθ RG as a estimator for the variace of ˆθ RG. Such bias becomes zero if the samplig for radom groups is a withreplacemet samplig. For without-replacemet samplig, the covariace betwee the two differet replicates is egative ad so the bias term becomes positive. The relative amout of the bias is ofte egligible. The followig example compute the amout of the relative bias. Example Cosider a sample of size obtaied from the simple radom samplig from a fiite populatio of size N. Let b = /G be a iteger value which is the size of A g such that A = G g=1 A g. The sample mea of y obtaied from A g is deoted by ȳ g ad the overall mea of y is give by ˆθ = 1 G G g=1 ȳ g. I this case, ȳ 1,,ȳ G are ot idepedet distributed but idetically distributed with the same mea. By 10.15, we have Var ˆθ = 1 G V ȳ Cov ȳ G 1,ȳ 2 ad sice V ȳ 1 = 1 b 1 N S 2, we have Thus, the bias i reduces to Cov ȳ 1,ȳ 2 = 1 N S2. Bias { ˆV ˆθ RG } = Cov ȳ1,ȳ 2 = 1 N S2
10 10 CHAPTER 10. VARIANCE ESTIMATION which is ofte egligible. Thus, the radom group variace estimator ca be safely used to estimate the variace of ȳ i the simple radom samplig. Such radom group method provides a useful computatioal tool for estimatig the variace of the poit estimators. However, the radom group method is applicable oly whe the samplig desig for A g has the same structure as the origial sample A. The partitio A = G g=1 A g that leads to ubiasedess of ˆθ g is ot always possible for complex samplig desigs Jackkife method Jackkife was first itroduced by Queouille 1949 to reduce the bias of ratio estimator ad the was suggested by Tukey 1958 to be used for variace estimatio. Jackkife is very popular i practice as a tool for variace estimatio. To itroduce the idea of Queouille 1949, suppose that idepedet observatios of x i,y i are available that are geerated from a distributio with mea µ x, µ y. If the parameter of iterest is θ = µ y /µ x, the the sample ratio ˆθ = x 1 ȳ has bias of order O 1. That is, we have E ˆθ = θ + C + O 2. If we delete the k-th observatio ad recompute the ratio 1 ˆθ k = x i y i, i k i k we obtai E ˆθ k = θ + C 1 + O 2. Thus, the jackkife pseudo value defied by ˆθ k = ˆθ 1 ˆθ k ca be used to compute ˆθ. = 1 ˆθ k which has bias of order O 2. Thus, the jackkife ca be used to reduce the bias of oliear estimators.
11 10.4. JACKKNIFE METHOD 11 Note that if ˆθ = ȳ the ˆθ k = y k. Tukey 1958 suggested usig ˆθ k as approximate IID observatio to get the followig jackkife variace estimator. ˆV. 1 1 JK ˆθ = 1 = 1 For the special case of ˆθ = ȳ, we obtai ˆV JK = ˆθ k θ. 2 ˆθ k θ. 2 y i ȳ 2 = 1 s2 y ad the jackkife variace estimator is algebraically equivalet to the usual variace estimator of the sample mea uder simple radom samplig igorig the fiite populatio correctio term. If we are iterested i estimatig the variace of ˆθ = f x,ȳ, usig ˆθ k = f x k,ȳ k, we ca costruct ˆV JK ˆθ = 1 2 ˆθ k ˆθ as the jackkife variace estimator of ˆθ. The jackkife replicate ˆθ k is computed by usig the same formula for ˆθ usig the jackkife weight w k i istead of the origial weight w i = 1/, where w k i = { 1 1 if i k 0 if i = k. To discuss the asymptotic property of the jackkife variace estimator i 10.17, we use the followig Taylor expasio, which is ofte called secod-type Taylor expasio. Lemma Let {X,W } be a sequece of radom variables such that X = W + O p r
12 12 CHAPTER 10. VARIANCE ESTIMATION where r 0 as. If gx is a fuctio with s-th cotiuous derivatives i the lie segmet joiig X ad W ad the s-th order partial derivatives are bouded, the s 1 gx = gw + 1 k! gk W X W k + O p r s where g k a is the k-th derivative of gx evaluated at x = a. Now, sice ȳ k ȳ = 1 1 ȳ y k, we have ȳ k ȳ = O p 1. For the case of ˆθ = f x,ȳ, we ca apply the above lemma to get ˆθ k ˆθ = f x x,ȳ x k x + f y x,ȳ ȳ k ȳ + o p 1 so that 1 2 ˆθ k ˆθ { } f 2 = x x,ȳ 1 { }{ } f f 1 +2 x x,ȳ y x,ȳ { 2 f x k x + } 2 y x,ȳ 1 2 ȳ k ȳ x k x ȳ k ȳ + o p 1. Thus, the jackkife variace estimator is asymptotically equivalet to the liearized variace estimator. That is, the secod-type Taylor liearizatio leads to ˆV { } f 2 { } f 2 JK ˆθ = x x,ȳ ˆV x + y x,ȳ ˆV ȳ { }{ } f f +2 x x,ȳ y x,ȳ Ĉ x,ȳ + o p 1. The above jackkife method is costructed uder simple radom samplig. For stratified multistage samplig, jackkife replicates ca be costructed by deletig a cluster for each replicate. Let Ŷ HT = H h h=1 w hi Ŷ hi
13 10.4. JACKKNIFE METHOD 13 be the HT estimator of Y uder the stratified multistage cluster samplig. The jackkife weights are costructed by deletig a cluster sequetially as 0 if h = g ad i = j g j w hi = h 1 1 h w hi if h = g ad i j w hi otherwise ad the jackkife variace estimator is computed by where ˆV JK ŶHT = H h=1 h 1 h h H h g j Ŷ HT = h=1 Ŷ hi HT 1 h h w g j hi j=1 Ŷ hi. 2 h j Ŷ HT The followig theorem presets the algebraic property of the jackkife variace estimator i Theorem The jackkife variace estimator i is algebraically equivalet to the simplified variace estimator i If ˆθ = f ˆX HT,Ŷ HT the the jackkife replicates are costructed as ˆθ g j = g j g j f ˆX. I this case, the jackkife variace estimator is computed as HT,Ŷ HT or, more simply, ˆV JK ˆθ = H h=1 ˆV JK ˆθ = h 1 h H h=1 h h 1 h ˆθ hi 1 h h 2 h ˆθ h j j=1 2 ˆθ hi ˆθ The asymptotic properties of the above jackkife variace estimator uder stratified multistage samplig ca be established. For refereces, see Krewski ad Rao 1981 or Shao ad Tu Example We ow revisit Example 10.3 to estimate the variace of poststratificatio estimator usig the jackkife. For simplicity, assume SRS for the sample.
14 14 CHAPTER 10. VARIANCE ESTIMATION The poststratificatio estimator is computed as Ŷ post = G g=1 N g ȳ g where ȳ g = 1 g i Ag y i. Now, the k-th replicate of Ŷ post is where ȳ k h ad Ŷ k post = h 1 1 h ȳ h y k. Thus, = N g ȳ g + N h ȳ k h g h Ŷ k post Ŷ post = N h ȳ k h ȳ h ˆV JK Ŷpost = 1 = N h h 1 1 ȳ h y k = 1 G g=1 Ŷ k post Ŷ post 2 N 2 g g 1 1 s 2 g, which, igorig /N term, is asymptotically equivalet to the coditioal variace estimator i Other methods Skip Referece Krewski, D. ad Rao, J.N.K Iferece from stratified samples: properties of the liearizatio, jackkife ad balaced repeated replicatio methods. Aals of Statistics 9, Queouille, M. H Notes o bias i estimatio. Biometrika 43,
15 10.5. OTHER METHODS 15 Royall, R. M. ad Cumberlad, W. G A empirical study of the ratio estimator ad estimators of its variace. Joural of the America Statistical Associatio 76, Särdal, C. E., Swesso, B., ad Wretma, J. H The weighted residual techique for estimatio the variace of the geeral regressio estimator of the fiite populatio total, Biometrika 76, Tukey, J. W Bias ad cofidece i ot-quite large samples abstract. Aals of Mathematical Statistics 29, 614. Wolter, K. M Itroductio to Variace Estimatio: Secod Editio. New York: Spriger-Verlag.
Element sampling: Part 2
Chapter 4 Elemet samplig: Part 2 4.1 Itroductio We ow cosider uequal probability samplig desigs which is very popular i practice. I the uequal probability samplig, we ca improve the efficiecy of the resultig
More information5. Fractional Hot deck Imputation
5. Fractioal Hot deck Imputatio Itroductio Suppose that we are iterested i estimatig θ EY or eve θ 2 P ry < c where y fy x where x is always observed ad y is subject to missigess. Assume MAR i the sese
More informationResampling Methods. X (1/2), i.e., Pr (X i m) = 1/2. We order the data: X (1) X (2) X (n). Define the sample median: ( n.
Jauary 1, 2019 Resamplig Methods Motivatio We have so may estimators with the property θ θ d N 0, σ 2 We ca also write θ a N θ, σ 2 /, where a meas approximately distributed as Oce we have a cosistet estimator
More informationInvestigating the Significance of a Correlation Coefficient using Jackknife Estimates
Iteratioal Joural of Scieces: Basic ad Applied Research (IJSBAR) ISSN 2307-4531 (Prit & Olie) http://gssrr.org/idex.php?joural=jouralofbasicadapplied ---------------------------------------------------------------------------------------------------------------------------
More informationLecture 33: Bootstrap
Lecture 33: ootstrap Motivatio To evaluate ad compare differet estimators, we eed cosistet estimators of variaces or asymptotic variaces of estimators. This is also importat for hypothesis testig ad cofidece
More information4.5 Multiple Imputation
45 ultiple Imputatio Itroductio Assume a parametric model: y fy x; θ We are iterested i makig iferece about θ I Bayesia approach, we wat to make iferece about θ from fθ x, y = πθfy x, θ πθfy x, θdθ where
More informationDS 100: Principles and Techniques of Data Science Date: April 13, Discussion #10
DS 00: Priciples ad Techiques of Data Sciece Date: April 3, 208 Name: Hypothesis Testig Discussio #0. Defie these terms below as they relate to hypothesis testig. a) Data Geeratio Model: Solutio: A set
More informationProperties and Hypothesis Testing
Chapter 3 Properties ad Hypothesis Testig 3.1 Types of data The regressio techiques developed i previous chapters ca be applied to three differet kids of data. 1. Cross-sectioal data. 2. Time series data.
More informationRandom Variables, Sampling and Estimation
Chapter 1 Radom Variables, Samplig ad Estimatio 1.1 Itroductio This chapter will cover the most importat basic statistical theory you eed i order to uderstad the ecoometric material that will be comig
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2016 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More information3 Resampling Methods: The Jackknife
3 Resamplig Methods: The Jackkife 3.1 Itroductio I this sectio, much of the cotet is a summary of material from Efro ad Tibshirai (1993) ad Maly (2007). Here are several useful referece texts o resamplig
More informationMATH 320: Probability and Statistics 9. Estimation and Testing of Parameters. Readings: Pruim, Chapter 4
MATH 30: Probability ad Statistics 9. Estimatio ad Testig of Parameters Estimatio ad Testig of Parameters We have bee dealig situatios i which we have full kowledge of the distributio of a radom variable.
More informationStatistical Inference (Chapter 10) Statistical inference = learn about a population based on the information provided by a sample.
Statistical Iferece (Chapter 10) Statistical iferece = lear about a populatio based o the iformatio provided by a sample. Populatio: The set of all values of a radom variable X of iterest. Characterized
More informationLecture 2: Monte Carlo Simulation
STAT/Q SCI 43: Itroductio to Resamplig ethods Sprig 27 Istructor: Ye-Chi Che Lecture 2: ote Carlo Simulatio 2 ote Carlo Itegratio Assume we wat to evaluate the followig itegratio: e x3 dx What ca we do?
More informationChapter 11 Output Analysis for a Single Model. Banks, Carson, Nelson & Nicol Discrete-Event System Simulation
Chapter Output Aalysis for a Sigle Model Baks, Carso, Nelso & Nicol Discrete-Evet System Simulatio Error Estimatio If {,, } are ot statistically idepedet, the S / is a biased estimator of the true variace.
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationEconomics 241B Relation to Method of Moments and Maximum Likelihood OLSE as a Maximum Likelihood Estimator
Ecoomics 24B Relatio to Method of Momets ad Maximum Likelihood OLSE as a Maximum Likelihood Estimator Uder Assumptio 5 we have speci ed the distributio of the error, so we ca estimate the model parameters
More informationEstimation of the Mean and the ACVF
Chapter 5 Estimatio of the Mea ad the ACVF A statioary process {X t } is characterized by its mea ad its autocovariace fuctio γ ), ad so by the autocorrelatio fuctio ρ ) I this chapter we preset the estimators
More informationSimulation. Two Rule For Inverting A Distribution Function
Simulatio Two Rule For Ivertig A Distributio Fuctio Rule 1. If F(x) = u is costat o a iterval [x 1, x 2 ), the the uiform value u is mapped oto x 2 through the iversio process. Rule 2. If there is a jump
More informationLecture 3. Properties of Summary Statistics: Sampling Distribution
Lecture 3 Properties of Summary Statistics: Samplig Distributio Mai Theme How ca we use math to justify that our umerical summaries from the sample are good summaries of the populatio? Lecture Summary
More informationA General Family of Estimators for Estimating Population Variance Using Known Value of Some Population Parameter(s)
Rajesh Sigh, Pakaj Chauha, Nirmala Sawa School of Statistics, DAVV, Idore (M.P.), Idia Floreti Smaradache Uiversity of New Meico, USA A Geeral Family of Estimators for Estimatig Populatio Variace Usig
More informationFirst Year Quantitative Comp Exam Spring, Part I - 203A. f X (x) = 0 otherwise
First Year Quatitative Comp Exam Sprig, 2012 Istructio: There are three parts. Aswer every questio i every part. Questio I-1 Part I - 203A A radom variable X is distributed with the margial desity: >
More informationEcon 325 Notes on Point Estimator and Confidence Interval 1 By Hiro Kasahara
Poit Estimator Eco 325 Notes o Poit Estimator ad Cofidece Iterval 1 By Hiro Kasahara Parameter, Estimator, ad Estimate The ormal probability desity fuctio is fully characterized by two costats: populatio
More informationLecture 22: Review for Exam 2. 1 Basic Model Assumptions (without Gaussian Noise)
Lecture 22: Review for Exam 2 Basic Model Assumptios (without Gaussia Noise) We model oe cotiuous respose variable Y, as a liear fuctio of p umerical predictors, plus oise: Y = β 0 + β X +... β p X p +
More informationSection 14. Simple linear regression.
Sectio 14 Simple liear regressio. Let us look at the cigarette dataset from [1] (available to dowload from joural s website) ad []. The cigarette dataset cotais measuremets of tar, icotie, weight ad carbo
More informationStat 421-SP2012 Interval Estimation Section
Stat 41-SP01 Iterval Estimatio Sectio 11.1-11. We ow uderstad (Chapter 10) how to fid poit estimators of a ukow parameter. o However, a poit estimate does ot provide ay iformatio about the ucertaity (possible
More informationConfidence Interval for Standard Deviation of Normal Distribution with Known Coefficients of Variation
Cofidece Iterval for tadard Deviatio of Normal Distributio with Kow Coefficiets of Variatio uparat Niwitpog Departmet of Applied tatistics, Faculty of Applied ciece Kig Mogkut s Uiversity of Techology
More informationEstimation for Complete Data
Estimatio for Complete Data complete data: there is o loss of iformatio durig study. complete idividual complete data= grouped data A complete idividual data is the oe i which the complete iformatio of
More informationSimple Random Sampling!
Simple Radom Samplig! Professor Ro Fricker! Naval Postgraduate School! Moterey, Califoria! Readig:! 3/26/13 Scheaffer et al. chapter 4! 1 Goals for this Lecture! Defie simple radom samplig (SRS) ad discuss
More informationLecture 12: September 27
36-705: Itermediate Statistics Fall 207 Lecturer: Siva Balakrisha Lecture 2: September 27 Today we will discuss sufficiecy i more detail ad the begi to discuss some geeral strategies for costructig estimators.
More information7-1. Chapter 4. Part I. Sampling Distributions and Confidence Intervals
7-1 Chapter 4 Part I. Samplig Distributios ad Cofidece Itervals 1 7- Sectio 1. Samplig Distributio 7-3 Usig Statistics Statistical Iferece: Predict ad forecast values of populatio parameters... Test hypotheses
More information32 estimating the cumulative distribution function
32 estimatig the cumulative distributio fuctio 4.6 types of cofidece itervals/bads Let F be a class of distributio fuctios F ad let θ be some quatity of iterest, such as the mea of F or the whole fuctio
More informationExpectation and Variance of a random variable
Chapter 11 Expectatio ad Variace of a radom variable The aim of this lecture is to defie ad itroduce mathematical Expectatio ad variace of a fuctio of discrete & cotiuous radom variables ad the distributio
More informationOutput Analysis and Run-Length Control
IEOR E4703: Mote Carlo Simulatio Columbia Uiversity c 2017 by Marti Haugh Output Aalysis ad Ru-Legth Cotrol I these otes we describe how the Cetral Limit Theorem ca be used to costruct approximate (1 α%
More information4.1 Non-parametric computational estimation
Chapter 4 Resamplig Methods 4.1 No-parametric computatioal estimatio Let x 1,...,x be a realizatio of the i.i.d. r.vs X 1,...,X with a c.d.f. F. We are iterested i the precisio of estimatio of a populatio
More informationIntroductory statistics
CM9S: Machie Learig for Bioiformatics Lecture - 03/3/06 Itroductory statistics Lecturer: Sriram Sakararama Scribe: Sriram Sakararama We will provide a overview of statistical iferece focussig o the key
More informationStatistical Inference Based on Extremum Estimators
T. Rotheberg Fall, 2007 Statistical Iferece Based o Extremum Estimators Itroductio Suppose 0, the true value of a p-dimesioal parameter, is kow to lie i some subset S R p : Ofte we choose to estimate 0
More informationStatistical inference: example 1. Inferential Statistics
Statistical iferece: example 1 Iferetial Statistics POPULATION SAMPLE A clothig store chai regularly buys from a supplier large quatities of a certai piece of clothig. Each item ca be classified either
More informationImproved Class of Ratio -Cum- Product Estimators of Finite Population Mean in two Phase Sampling
Global Joural of Sciece Frotier Research: F Mathematics ad Decisio Scieces Volume 4 Issue 2 Versio.0 Year 204 Type : Double Blid Peer Reviewed Iteratioal Research Joural Publisher: Global Jourals Ic. (USA
More information1 Inferential Methods for Correlation and Regression Analysis
1 Iferetial Methods for Correlatio ad Regressio Aalysis I the chapter o Correlatio ad Regressio Aalysis tools for describig bivariate cotiuous data were itroduced. The sample Pearso Correlatio Coefficiet
More informationA statistical method to determine sample size to estimate characteristic value of soil parameters
A statistical method to determie sample size to estimate characteristic value of soil parameters Y. Hojo, B. Setiawa 2 ad M. Suzuki 3 Abstract Sample size is a importat factor to be cosidered i determiig
More informationA Question. Output Analysis. Example. What Are We Doing Wrong? Result from throwing a die. Let X be the random variable
A Questio Output Aalysis Let X be the radom variable Result from throwig a die 5.. Questio: What is E (X? Would you throw just oce ad take the result as your aswer? Itroductio to Simulatio WS/ - L 7 /
More informationECE 901 Lecture 12: Complexity Regularization and the Squared Loss
ECE 90 Lecture : Complexity Regularizatio ad the Squared Loss R. Nowak 5/7/009 I the previous lectures we made use of the Cheroff/Hoeffdig bouds for our aalysis of classifier errors. Hoeffdig s iequality
More informationProbability and statistics: basic terms
Probability ad statistics: basic terms M. Veeraraghava August 203 A radom variable is a rule that assigs a umerical value to each possible outcome of a experimet. Outcomes of a experimet form the sample
More informationExponential Families and Bayesian Inference
Computer Visio Expoetial Families ad Bayesia Iferece Lecture Expoetial Families A expoetial family of distributios is a d-parameter family f(x; havig the followig form: f(x; = h(xe g(t T (x B(, (. where
More informationLecture 19: Convergence
Lecture 19: Covergece Asymptotic approach I statistical aalysis or iferece, a key to the success of fidig a good procedure is beig able to fid some momets ad/or distributios of various statistics. I may
More information17. Joint distributions of extreme order statistics Lehmann 5.1; Ferguson 15
17. Joit distributios of extreme order statistics Lehma 5.1; Ferguso 15 I Example 10., we derived the asymptotic distributio of the maximum from a radom sample from a uiform distributio. We did this usig
More informationDouble Stage Shrinkage Estimator of Two Parameters. Generalized Exponential Distribution
Iteratioal Mathematical Forum, Vol., 3, o. 3, 3-53 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.9/imf.3.335 Double Stage Shrikage Estimator of Two Parameters Geeralized Expoetial Distributio Alaa M.
More informationAsymptotics. Hypothesis Testing UMP. Asymptotic Tests and p-values
of the secod half Biostatistics 6 - Statistical Iferece Lecture 6 Fial Exam & Practice Problems for the Fial Hyu Mi Kag Apil 3rd, 3 Hyu Mi Kag Biostatistics 6 - Lecture 6 Apil 3rd, 3 / 3 Rao-Blackwell
More informationECONOMETRIC THEORY. MODULE XIII Lecture - 34 Asymptotic Theory and Stochastic Regressors
ECONOMETRIC THEORY MODULE XIII Lecture - 34 Asymptotic Theory ad Stochastic Regressors Dr. Shalabh Departmet of Mathematics ad Statistics Idia Istitute of Techology Kapur Asymptotic theory The asymptotic
More informationThe variance of a sum of independent variables is the sum of their variances, since covariances are zero. Therefore. V (xi )= n n 2 σ2 = σ2.
SAMPLE STATISTICS A radom sample x 1,x,,x from a distributio f(x) is a set of idepedetly ad idetically variables with x i f(x) for all i Their joit pdf is f(x 1,x,,x )=f(x 1 )f(x ) f(x )= f(x i ) The sample
More informationReview Questions, Chapters 8, 9. f(y) = 0, elsewhere. F (y) = f Y(1) = n ( e y/θ) n 1 1 θ e y/θ = n θ e yn
Stat 366 Lab 2 Solutios (September 2, 2006) page TA: Yury Petracheko, CAB 484, yuryp@ualberta.ca, http://www.ualberta.ca/ yuryp/ Review Questios, Chapters 8, 9 8.5 Suppose that Y, Y 2,..., Y deote a radom
More informationIsmor Fischer, 1/11/
Ismor Fischer, //04 7.4-7.4 Problems. I Problem 4.4/9, it was show that importat relatios exist betwee populatio meas, variaces, ad covariace. Specifically, we have the formulas that appear below left.
More informationLecture Overview. 2 Permutations and Combinations. n(n 1) (n (k 1)) = n(n 1) (n k + 1) =
COMPSCI 230: Discrete Mathematics for Computer Sciece April 8, 2019 Lecturer: Debmalya Paigrahi Lecture 22 Scribe: Kevi Su 1 Overview I this lecture, we begi studyig the fudametals of coutig discrete objects.
More informationo <Xln <X2n <... <X n < o (1.1)
Metrika, Volume 28, 1981, page 257-262. 9 Viea. Estimatio Problems for Rectagular Distributios (Or the Taxi Problem Revisited) By J.S. Rao, Sata Barbara I ) Abstract: The problem of estimatig the ukow
More information1 Introduction to reducing variance in Monte Carlo simulations
Copyright c 010 by Karl Sigma 1 Itroductio to reducig variace i Mote Carlo simulatios 11 Review of cofidece itervals for estimatig a mea I statistics, we estimate a ukow mea µ = E(X) of a distributio by
More informationJournal of Multivariate Analysis. Superefficient estimation of the marginals by exploiting knowledge on the copula
Joural of Multivariate Aalysis 102 (2011) 1315 1319 Cotets lists available at ScieceDirect Joural of Multivariate Aalysis joural homepage: www.elsevier.com/locate/jmva Superefficiet estimatio of the margials
More informationSTATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS. Comments:
Recall: STATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS Commets:. So far we have estimates of the parameters! 0 ad!, but have o idea how good these estimates are. Assumptio: E(Y x)! 0 +! x (liear coditioal
More informationMATH 472 / SPRING 2013 ASSIGNMENT 2: DUE FEBRUARY 4 FINALIZED
MATH 47 / SPRING 013 ASSIGNMENT : DUE FEBRUARY 4 FINALIZED Please iclude a cover sheet that provides a complete setece aswer to each the followig three questios: (a) I your opiio, what were the mai ideas
More informationChi-Squared Tests Math 6070, Spring 2006
Chi-Squared Tests Math 6070, Sprig 2006 Davar Khoshevisa Uiversity of Utah February XXX, 2006 Cotets MLE for Goodess-of Fit 2 2 The Multiomial Distributio 3 3 Applicatio to Goodess-of-Fit 6 3 Testig for
More informationESTIMATION AND PREDICTION BASED ON K-RECORD VALUES FROM NORMAL DISTRIBUTION
STATISTICA, ao LXXIII,. 4, 013 ESTIMATION AND PREDICTION BASED ON K-RECORD VALUES FROM NORMAL DISTRIBUTION Maoj Chacko Departmet of Statistics, Uiversity of Kerala, Trivadrum- 695581, Kerala, Idia M. Shy
More informationHypothesis Testing. Evaluation of Performance of Learned h. Issues. Trade-off Between Bias and Variance
Hypothesis Testig Empirically evaluatig accuracy of hypotheses: importat activity i ML. Three questios: Give observed accuracy over a sample set, how well does this estimate apply over additioal samples?
More informationStat 319 Theory of Statistics (2) Exercises
Kig Saud Uiversity College of Sciece Statistics ad Operatios Research Departmet Stat 39 Theory of Statistics () Exercises Refereces:. Itroductio to Mathematical Statistics, Sixth Editio, by R. Hogg, J.
More informationKLMED8004 Medical statistics. Part I, autumn Estimation. We have previously learned: Population and sample. New questions
We have previously leared: KLMED8004 Medical statistics Part I, autum 00 How kow probability distributios (e.g. biomial distributio, ormal distributio) with kow populatio parameters (mea, variace) ca give
More informationLECTURE 8: ASYMPTOTICS I
LECTURE 8: ASYMPTOTICS I We are iterested i the properties of estimators as. Cosider a sequece of radom variables {, X 1}. N. M. Kiefer, Corell Uiversity, Ecoomics 60 1 Defiitio: (Weak covergece) A sequece
More informationA Note on Effi cient Conditional Simulation of Gaussian Distributions. April 2010
A Note o Effi ciet Coditioal Simulatio of Gaussia Distributios A D D C S S, U B C, V, BC, C April 2010 A Cosider a multivariate Gaussia radom vector which ca be partitioed ito observed ad uobserved compoetswe
More informationA NEW METHOD FOR CONSTRUCTING APPROXIMATE CONFIDENCE INTERVALS FOR M-ESTU1ATES. Dennis D. Boos
.- A NEW METHOD FOR CONSTRUCTING APPROXIMATE CONFIDENCE INTERVALS FOR M-ESTU1ATES by Deis D. Boos Departmet of Statistics North Carolia State Uiversity Istitute of Statistics Mimeo Series #1198 September,
More informationMath 155 (Lecture 3)
Math 55 (Lecture 3) September 8, I this lecture, we ll cosider the aswer to oe of the most basic coutig problems i combiatorics Questio How may ways are there to choose a -elemet subset of the set {,,,
More informationStudy the bias (due to the nite dimensional approximation) and variance of the estimators
2 Series Methods 2. Geeral Approach A model has parameters (; ) where is ite-dimesioal ad is oparametric. (Sometimes, there is o :) We will focus o regressio. The fuctio is approximated by a series a ite
More informationMachine Learning Brett Bernstein
Machie Learig Brett Berstei Week Lecture: Cocept Check Exercises Starred problems are optioal. Statistical Learig Theory. Suppose A = Y = R ad X is some other set. Furthermore, assume P X Y is a discrete
More informationKernel density estimator
Jauary, 07 NONPARAMETRIC ERNEL DENSITY ESTIMATION I this lecture, we discuss kerel estimatio of probability desity fuctios PDF Noparametric desity estimatio is oe of the cetral problems i statistics I
More informationEntropy and Ergodic Theory Lecture 5: Joint typicality and conditional AEP
Etropy ad Ergodic Theory Lecture 5: Joit typicality ad coditioal AEP 1 Notatio: from RVs back to distributios Let (Ω, F, P) be a probability space, ad let X ad Y be A- ad B-valued discrete RVs, respectively.
More informationDirection: This test is worth 250 points. You are required to complete this test within 50 minutes.
Term Test October 3, 003 Name Math 56 Studet Number Directio: This test is worth 50 poits. You are required to complete this test withi 50 miutes. I order to receive full credit, aswer each problem completely
More informationBecause it tests for differences between multiple pairs of means in one test, it is called an omnibus test.
Math 308 Sprig 018 Classes 19 ad 0: Aalysis of Variace (ANOVA) Page 1 of 6 Itroductio ANOVA is a statistical procedure for determiig whether three or more sample meas were draw from populatios with equal
More informationMachine Learning Brett Bernstein
Machie Learig Brett Berstei Week 2 Lecture: Cocept Check Exercises Starred problems are optioal. Excess Risk Decompositio 1. Let X = Y = {1, 2,..., 10}, A = {1,..., 10, 11} ad suppose the data distributio
More informationChapter 6 Principles of Data Reduction
Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Chapter 6 Priciples of Data Reductio Sectio 6. Itroductio Goal: To summarize or reduce the data X, X,, X to get iformatio about a
More informationMOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND.
XI-1 (1074) MOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND. R. E. D. WOOLSEY AND H. S. SWANSON XI-2 (1075) STATISTICAL DECISION MAKING Advaced
More information( θ. sup θ Θ f X (x θ) = L. sup Pr (Λ (X) < c) = α. x : Λ (x) = sup θ H 0. sup θ Θ f X (x θ) = ) < c. NH : θ 1 = θ 2 against AH : θ 1 θ 2
82 CHAPTER 4. MAXIMUM IKEIHOOD ESTIMATION Defiitio: et X be a radom sample with joit p.m/d.f. f X x θ. The geeralised likelihood ratio test g.l.r.t. of the NH : θ H 0 agaist the alterative AH : θ H 1,
More informationResampling modifications for the Bagai test
Joural of the Korea Data & Iformatio Sciece Society 2018, 29(2), 485 499 http://dx.doi.org/10.7465/jkdi.2018.29.2.485 한국데이터정보과학회지 Resamplig modificatios for the Bagai test Youg Mi Kim 1 Hyug-Tae Ha 2 1
More informationThere is no straightforward approach for choosing the warmup period l.
B. Maddah INDE 504 Discrete-Evet Simulatio Output Aalysis () Statistical Aalysis for Steady-State Parameters I a otermiatig simulatio, the iterest is i estimatig the log ru steady state measures of performace.
More informationSTA6938-Logistic Regression Model
Dr. Yig Zhag STA6938-Logistic Regressio Model Topic -Simple (Uivariate) Logistic Regressio Model Outlies:. Itroductio. A Example-Does the liear regressio model always work? 3. Maximum Likelihood Curve
More informationDiscrete Mathematics for CS Spring 2008 David Wagner Note 22
CS 70 Discrete Mathematics for CS Sprig 2008 David Wager Note 22 I.I.D. Radom Variables Estimatig the bias of a coi Questio: We wat to estimate the proportio p of Democrats i the US populatio, by takig
More informationA RANK STATISTIC FOR NON-PARAMETRIC K-SAMPLE AND CHANGE POINT PROBLEMS
J. Japa Statist. Soc. Vol. 41 No. 1 2011 67 73 A RANK STATISTIC FOR NON-PARAMETRIC K-SAMPLE AND CHANGE POINT PROBLEMS Yoichi Nishiyama* We cosider k-sample ad chage poit problems for idepedet data i a
More informationOutput Analysis (2, Chapters 10 &11 Law)
B. Maddah ENMG 6 Simulatio Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should be doe
More informationStochastic Simulation
Stochastic Simulatio 1 Itroductio Readig Assigmet: Read Chapter 1 of text. We shall itroduce may of the key issues to be discussed i this course via a couple of model problems. Model Problem 1 (Jackso
More informationModule 1 Fundamentals in statistics
Normal Distributio Repeated observatios that differ because of experimetal error ofte vary about some cetral value i a roughly symmetrical distributio i which small deviatios occur much more frequetly
More information4 Multidimensional quantitative data
Chapter 4 Multidimesioal quatitative data 4 Multidimesioal statistics Basic statistics are ow part of the curriculum of most ecologists However, statistical techiques based o such simple distributios as
More informationAlgebra of Least Squares
October 19, 2018 Algebra of Least Squares Geometry of Least Squares Recall that out data is like a table [Y X] where Y collects observatios o the depedet variable Y ad X collects observatios o the k-dimesioal
More informationStat 200 -Testing Summary Page 1
Stat 00 -Testig Summary Page 1 Mathematicias are like Frechme; whatever you say to them, they traslate it ito their ow laguage ad forthwith it is somethig etirely differet Goethe 1 Large Sample Cofidece
More informationChapter 2 The Monte Carlo Method
Chapter 2 The Mote Carlo Method The Mote Carlo Method stads for a broad class of computatioal algorithms that rely o radom sampligs. It is ofte used i physical ad mathematical problems ad is most useful
More informationAccess to the published version may require journal subscription. Published with permission from: Elsevier.
This is a author produced versio of a paper published i Statistics ad Probability Letters. This paper has bee peer-reviewed, it does ot iclude the joural pagiatio. Citatio for the published paper: Forkma,
More informationChapter 3. Strong convergence. 3.1 Definition of almost sure convergence
Chapter 3 Strog covergece As poited out i the Chapter 2, there are multiple ways to defie the otio of covergece of a sequece of radom variables. That chapter defied covergece i probability, covergece i
More informationThe Bootstrap, Jackknife, Randomization, and other non-traditional approaches to estimation and hypothesis testing
The Bootstrap, Jackkife, Radomizatio, ad other o-traditioal approaches to estimatio ad hypothesis testig Ratioale Much of moder statistics is achored i the use of statistics ad hypothesis tests that oly
More information6 Sample Size Calculations
6 Sample Size Calculatios Oe of the major resposibilities of a cliical trial statisticia is to aid the ivestigators i determiig the sample size required to coduct a study The most commo procedure for determiig
More informationAbstract. Ranked set sampling, auxiliary variable, variance.
Hacettepe Joural of Mathematics ad Statistics Volume (), 1 A class of Hartley-Ross type Ubiased estimators for Populatio Mea usig Raked Set Samplig Lakhkar Kha ad Javid Shabbir Abstract I this paper, we
More informationThe standard deviation of the mean
Physics 6C Fall 20 The stadard deviatio of the mea These otes provide some clarificatio o the distictio betwee the stadard deviatio ad the stadard deviatio of the mea.. The sample mea ad variace Cosider
More informationChapter 8: Estimating with Confidence
Chapter 8: Estimatig with Cofidece Sectio 8.2 The Practice of Statistics, 4 th editio For AP* STARNES, YATES, MOORE Chapter 8 Estimatig with Cofidece 8.1 Cofidece Itervals: The Basics 8.2 8.3 Estimatig
More informationEstimating the Population Mean using Stratified Double Ranked Set Sample
Estimatig te Populatio Mea usig Stratified Double Raked Set Sample Mamoud Syam * Kamarulzama Ibraim Amer Ibraim Al-Omari Qatar Uiversity Foudatio Program Departmet of Mat ad Computer P.O.Box (7) Doa State
More informationModel-based Variance Estimation for Systematic Sampling
ASA Sectio o Survey Researc Metods Model-based Variace Estimatio for Systematic Samplig Xiaoxi Li ad J D Opsomer Ceter for Survey Statistics ad Metodology, ad Departmet of Statistics, Iowa State Uiversity,
More information