GUIDE FOR THE USE OF THE DECISION SUPPORT SYSTEM (DSS)*

Size: px
Start display at page:

Download "GUIDE FOR THE USE OF THE DECISION SUPPORT SYSTEM (DSS)*"

Transcription

1 GUIDE FOR THE USE OF THE DECISION SUPPORT SYSTEM (DSS)* *Note: I Frech SAD (Système d Aide à la Décisio) 1. Itroductio to the DSS Eightee statistical distributios are available i HYFRAN-PLUS software to fit data sets that are idepedet, homogeous ad statioary. A Decisio Support System (DSS) is developed to support selectio of the most appropriate class of distributios, with respect to extreme values. Distributios that are usually used i flood frequecy aalysis ca be grouped i three mai classes: - Class C (regularly varyig distributios): Fréchet (EV2), Halphe IB (HIB), Log-Pearso (LP3), Iverse Gamma (IG). - Class D (sub-expoetial distributios): Halphe type A (HA), Halphe type B (B), Gumbel (EV1), Pearso type 3 (P3), Gamma (G). - Class E (Expoetial distributio). Figure 1 presets expoetial (E), sub-expoetial (D) ad regularly varyig (C) distributios. Distributios are ordered from light tailed (from the left) to heavy tailed (to the right). The limitig cases (bottom squares) represeted by distributios i the limits of classes. The tail of the class C distributios is heavier tha that of the class D distributios, which is heavier tha that of the class E. Thus, estimated quatiles ca be ordered equivaletly. Ideed, for a give sample, the T-evet correspods to the quatile of the probability of o-exceedace p 1 1 T estimated by distributios of the classes C, D ad E, are QT (C), QT (D) ad QT (E) respectively, which verify the followig relatio: QT (E) < QT (D) < QT (C). ~ I Busiess Sice 1971 ~ Water Resources Publicatios, LLC. P.O. Box Highlads Rach, CO , USA ifo@wrpllc.com 1

2 Gumbel Halphe A Gamma Pearso type 3 Fréchet Halphe IB Iverse Gamma Log-Pearso type3 Stable Distributios Light tail Class D Class C Normal Logormal Heavy tail Expoetial Pareto Figure 1: distributios ordered with respect to their right tails (El Adloui et al., 2008). The methods developed i the DSS allow the idetificatio of the most adequate class of distributio to fit a give sample, especially for extremes. These methods are (cf. Diagram): - The Log-Log plot : used to discrimiate betwee o the oe had the class C ad o the other had the classes E ad D; - The mea excess fuctio (MEF) to discrimiate betwee the classes D ad E; ad - Two statistics: Hill's ratio ad modified Jackso statistic, for cofirmatory aalysis of the coclusios suggested by the previous two methods. 2

3 Use the log-log plot If the curve is liear No Use the graph of the Mea Excess Fuctio (MEF) This curve is liear for the classes D ad E Yes The distributio with regular variatios (class C) i.e. HIB, EV2, LP3, IG If the slope of the curve is ull, the we suggest Expoetial type distributio (class E) i.e. Exp If the slope of the curve is positive, the we suggest Subexpoetial type distributio (class D) i.e. HA, G, P3, EV1, LN, HB Cofirmatory Aalysis - Hill's report - Statistics of Jackso Cofirmatory Aalysis - Hill's report - Statistics of Jackso Figure 2: Diagram for class selectio used i the DSS More theoretical details of this classificatio ad the criteria are available i El Adloui et al. (2008). This article is available as attachmet i the HYFRAN-PLUS setup. 2. Log-Log plot 3

4 The log-log plot is based o the fact that the survival fuctio F u PX u, is give by u / F u P X u e for expoetial tail with mea, ad for regularly varyig distributio with tail idex, F is equivalet to (for large quatile) : 1 1 x 1 u 1 x 1 (with 1 u F u P( X u ) C dx C C u, which is equivalet to fiite mea). Therefore, takig the logarithm we have regularly varyig distributios log P X u log C 1 log u. This suggests that, for the log-log plot, the tail probability is represeted by a straight lie for power-law (or regularly varyig distributios, class C) but ot for the other sub-expoetial or expoetial distributios (class D or E). As illustrated i figure 3, the curve represeted i the Log-Log plot correspods to a straight lie for the distributios of the class C i.e. Fréchet (EV2), Halphe type IB (HIB), Log-Pearso type 3 (LP3) ad Iverse Gamma (IG), but ot for sub-expoetial or expoetial type tails (class D or E). Whe the diagram is ot liear we suggest the use of the Mea Excess Fuctio (MEF) to discrimiate betwee the classes D ad E. Figure 3: Illustratio of the Log-Log plot to characterize the regularly varyig distributios To check the liearity of the curve i the log-log diagram, a test o the associated correlatio coefficiet is cosidered. Simulatio studies allow the determiatio of critical values correspodig to sigificace levels of 5 % ad 1 %, to test the HYPOTHESIS H0: THE DATA FOLLOW A DISTRIBUTION OF 4

5 THE CLASS C (i.e. THE CURVE IS LINEAR). These critical values are calculated accordig to the size N of the sample (30 N 200). Note that the decisios give by the DSS are based, by default, o the sigificace level 5 %. If the hypothesis H0 is rejected, at the sigificace level 5 %, we suggest the use of the mea excess fuctio plot (MEF). However the critical values at the sigificace level 1 % are give for more flexibility ad to allow the user to make aother decisio tha that based o the sigificace level 5 %. Ideed, if the observed correlatio coefficiet (ro) is greater tha critical value (rc) at the sigificace level 5 %, the we coclude that it is ot sigificatly differet from 1 at the sigificace level 5 % ad the hypothesis H0 of liearity is accepted at this level (Figure 4). I this case, the most adequate choice correspods to the class C of regularly varyig distributios (power-law type): Halphe type IB (HIB), Fréchet (EV2), Log-Pearso type 3 (LP3), Iverse Gamma (IG). Régio de rejet (1%) Régio de rejet (5%) Régio d acceptatio (5%) Régio d acceptatio (1%) r0 (cas1) r0 (cas2) r0 (cas3) Valeur critique (rc5%) au iveau de sigificatio 5% Valeur critique (rc1%) au iveau de sigificatio 1% Figure 4 : Illustratio de la décisio d u test uilatéral de l hypothèse H0. Figure 4 shows, i geeral, the decisio rule for a uilateral test related to two sigificace levels 1% ad 5%. The critical values correspodig to each sigificace level are, respectively, rc1% ad rc5%. These two critical values are obtaied by Mote Carlo simulatios geerated from regularly varyig distributios. For a give dataset, we calculate the correlatio coefficiet r0. To illustrate the use of this test, three cases are cosidered such as the correlatio coefficiets verify: r0(cas1) < rc5% < r0(cas2) < rc1% < r0(cas3). The hypothesis H0 (case1) is rejected for the sigificace levels 1% ad 5%. Ideed, r0(cas1) < rc5% ad r0(cas1) < rc1%. I this case the distributio is ot regularly varyig (the curve is ot liear). For case2, the hypothesis H0 is rejected at the sigificace level 1%, but it is accepted at the sigificace level of 5%. Ideed, r0(cas2) > rc5% ad r0(cas2) < rc1%. For this case, the hypothesis H0 is 5

6 accepted by the SAD ad the use of regularly varyig distributio is suggested (based o the sigificace level 5%). However, the critical value at the sigificace level of 1% is preseted to give more flexibility to the user. The case 3, correspods to the case where r0 is higher tha the two critical values (r0(cas3) > rc5% ad r0(cas3) > rc1%). I this case, ad for the two sigificace levels, the hypothesis H0 is accepted ad the suggested distributio belog to the class C of regularly varyig distributios. 3. The Mea Excess Fuctio Diagram (MEF) The mea excess fuctio method is based o the fuctio eu EX u X u costat for expoetial tail distributios ( eu distributio with tail idex 2: eu. This fuctio is ). However, i the case of regularly varyig u 2. The Mea Excess Fuctio (MEF) allows discrimiatig betwee the class D (sub-expoetial distributios) ad the class E (Expoetial distributio). Ideed, the curve preseted i the MEF diagram is liear for high observed values for distributios of both classes D ad E. If i additio the slope of this curve is (Figure 5): - Equal to zero, the most adequate distributio belogs to the class E (Expoetial law); - Strictly positive, the most adequate distributio belogs to the class D of sub-expoetial distributios: Halphe type A (HA), Gumbel (EV1), Halphe type B (HB), Pearso type 3 (P3), Gamma (G). 1.4 Expoetial distributio 0.5 Sub-expoetial distributio E(X-u X>u) E(X-u X>u) k k Figure 5: Mea excess fuctio for expoetial ad sub-expoetial distributios. 6

7 The use of this diagram i the DSS is based o the slope of the MEF curve for the observatios that exceed the media (50 % of the highest observed value of the sample). Simulatio studies allow the determiatio of critical values correspodig to sigificace levels of 5 % ad 1 %, to test the HYPOTHESIS H0: THE DATA FOLLOW A DISTRIBUTION OF THE CLASS E (i.e. THE SLOPE OF THE MEF IS EQUAL TO ZERO). These critical values are calculated accordig to the size N of the sample (30 N 200). Note that the decisios give by the DSS are based, by default, o the sigificace level 5 %. Whe the hypothesis H0 is accepted we suggest the use of the Expoetial distributio (class E). However, whe it is rejected at the sigificace level 5 %, we suggest the use of a distributio of the class D (HA, EV1, HB, P3, G). Note that the critical values at the sigificace level 1 % are give for more flexibility ad to allow the user to make possibly aother decisio tha that suggested for the sigificace level of 5% (Figure 4). Remark: - The Logormal distributio (LN) does t belog to ay of these classes. It has a asymptotic behaviour which is i the frotier of the classes C ad D. Ideed, the LN tail is lighter (respectively, heavier) tha that of a distributio of the class C (respectively, class D). Thus, the quatiles (QT) estimated by a distributio belogig to the classes C, D ad the LN, verify the followig relatio: QT ( D ) < QT (LN) < QT ( C ). Cosequetly: - If the paret distributio is regularly varyig (class C), ad the LN distributio is cosidered for the fit, thus the estimated quatile, for a fixed retur period, will be lower tha the real value ad there is a risk to uderestimate this quatile; - If the true distributio is sub-expoetial (class D), ad the LN distributio is cosidered for the fit, thus the estimated quatile, for a fixed retur period, will be higher tha the real value ad there is a risk to overestimate this quatile. I the DSS, ad to have a safe choice, LN is cosidered by default as a distributio of the class D. However, the user could make a differet decisio ad associate it to the class C. 7

8 4. Hill's ratio plot [for the theoretical details cf. El Adloui et al. 2008] The Hill ratio is defied by 1 if where X i x 0 if a i x X i x. X x i X 1 i x i X x X x 1 log / i i This method is based o the fact that a is a cosistet estimator of if the tail is regularly varyig (Class C) with tail idex (Hill, 1975). I the expressio of the Hill ratio, x is chose to be large such that PX x 0 ad PX x, ad is the idicator fuctio. The stadard Hill estimator, of the tail idex, correspods to the particular case where the observatios are ordered X X ad x X 1 k 1, where k is a iteger which teds to ifiity as teds to ifiity. I practice, oe plots a x as a fuctio of x ad looks for some stable regio from which a x ca be cosidered as a estimator of. Figure 4, presets the Hill ratio plot for a sample geerated from the regularly varyig (a) ad Expoetial (b) distributios. Figure 4: Geeralized Hill ratio plot for (a) regularly-varyig ad (b) sub-expoetial distributios. 8

9 This statistics is used i the DSS to cofirm the suggested choice give by the first two diagrams (the distributio belogs to the class C, D or E). - If the curve coverges to a o-ull costat value, the most adequate distributio belogs to the class C (regularly varyig distributio). We suggest the the use of a distributio of the class C: Fréchet (EV2), Halphe type B Iverse (HIB), Log-Pearso type 3 ( LP3), Iverse Gamma (IG). - If the curve decreases to zero, the distributio belog to the Sub-expoetial class (class D: Halphe type A, Gamma, Pearso type 3, Halphe type B, Gumbel); ad the Expoetial class (class E: Expoetial distributio). Note that (cf. sectio 3) to discrimiate betwee the classes D ad E, we suggest the use of the MEF method. 5. Jackso Statistic [for the theoretical details cf. El Adloui et al. 2008] This method is preseted by Beirlat et al. (2006) ad is based o the Jackso statistic. It allows to test whether the sample is cosistet with Pareto type distributios (Class B). Note that the distributios of the class C (regularly varyig distributio) have asymptotically the same behaviour as that of the Pareto distributio. Origially the Jackso statistic (Jackso, 1967) was proposed as a goodess-of-fit statistic for testig expoetial behaviour, ad give the lik betwee the Expoetial ad the Pareto distributio (if X has a Pareto distributio the logarithmic trasformatio Y log X is expoetially distributed) this statistic is used to assess Pareto-type behaviour. The Jackso statistic is further modified by takig ito accout the secod-order tail behaviour of a Pareto-type model. Beirlat et al. (2006) give the limitig distributio of this statistic with corrected bias versio for fiite size samples. The modified Jackso statistic coverges to 2 for regularly varyig distributio (Power-law) ad has a irregular behaviour for sub-expoetial or expoetial distributios (Figure 5). 9

10 Figure 5: Modified Jackso statistic for (a) regularly varyig ad (b) sub-expoetial distributios. I the DSS this method is cosidered as a cofirmatory method for suggested decisio based o the Log- Log ad the MEF. So: - If the curve coverges clearly ad regularly to 2, the studied distributio belogs to the class C (regularly varyig distributio). We suggest the, the use of: Fréchet (EV2), Halphe type IB (HIB), Log- Pearso type 3 (LP3), Iverse Gamma (IG); - If the curve presets some irregularities for the distributio tail, tha we suggest the sub-expoetial class (class D: Halphe type A, Gamma, Pearso type 3, Halphe type B, Gumbel); or expoetial (class E: Expoetial distributio). Note that (cf. sectio 3) to discrimiate betwee the classes D ad E, we suggest the use of the MEF method. Remarque: Eve if the modified Jackso statistic was developed to test Pareto type behaviour, it is used i the DSS to check if the of the studied distributio has similar tail as regularly varyig distributio (class C). I deed, distributios of the class C have asymptotically Pareto type tail. I practice, the Geeralized Pareto distributio (GPD) is used i the Peaks-over-threshold model (POT). However, the GPD is available i HYFRAN ad ca be used to fit ay data sets that are idepedet, homogeous ad statioary. 10

11 Referece: Beirlat, J., de Wet, T., Goegebeur, Y., (2006). A goodess-of-fit statistic for Pareto-type behaviour. Joural of Computatioal ad Applied Mathematics, 186, El Adloui, S., Bobée, B. et Ouarda, T. B.M.J (2008). O the tails of extreme evet distributios i Hydrology. Accepted i Joural of Hydrology. Jackso, O.A.Y., (1967). A aalysis of departures from the expoetial distributio. Joural of the Royal Statistical Society B, 29,

Introduction to Extreme Value Theory Laurens de Haan, ISM Japan, Erasmus University Rotterdam, NL University of Lisbon, PT

Introduction to Extreme Value Theory Laurens de Haan, ISM Japan, Erasmus University Rotterdam, NL University of Lisbon, PT Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 Itroductio to Extreme Value Theory Laures de Haa Erasmus Uiversity Rotterdam, NL Uiversity of Lisbo, PT Itroductio to Extreme Value Theory

More information

Properties and Hypothesis Testing

Properties 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 information

Goodness-Of-Fit For The Generalized Exponential Distribution. Abstract

Goodness-Of-Fit For The Generalized Exponential Distribution. Abstract Goodess-Of-Fit For The Geeralized Expoetial Distributio By Amal S. Hassa stitute of Statistical Studies & Research Cairo Uiversity Abstract Recetly a ew distributio called geeralized expoetial or expoetiated

More information

Table 12.1: Contingency table. Feature b. 1 N 11 N 12 N 1b 2 N 21 N 22 N 2b. ... a N a1 N a2 N ab

Table 12.1: Contingency table. Feature b. 1 N 11 N 12 N 1b 2 N 21 N 22 N 2b. ... a N a1 N a2 N ab Sectio 12 Tests of idepedece ad homogeeity I this lecture we will cosider a situatio whe our observatios are classified by two differet features ad we would like to test if these features are idepedet

More information

Math 2784 (or 2794W) University of Connecticut

Math 2784 (or 2794W) University of Connecticut ORDERS OF GROWTH PAT SMITH Math 2784 (or 2794W) Uiversity of Coecticut Date: Mar. 2, 22. ORDERS OF GROWTH. Itroductio Gaiig a ituitive feel for the relative growth of fuctios is importat if you really

More information

Groupe de Recherche en Économie et Développement International. Cahier de Recherche / Working Paper 10-18

Groupe de Recherche en Économie et Développement International. Cahier de Recherche / Working Paper 10-18 Groupe de Recherche e Écoomie et Développemet Iteratioal Cahier de Recherche / Workig Paper 0-8 Quadratic Pe's Parade ad the Computatio of the Gii idex Stéphae Mussard, Jules Sadefo Kamdem Fraçoise Seyte

More information

Efficient GMM LECTURE 12 GMM II

Efficient GMM LECTURE 12 GMM II DECEMBER 1 010 LECTURE 1 II Efficiet The estimator depeds o the choice of the weight matrix A. The efficiet estimator is the oe that has the smallest asymptotic variace amog all estimators defied by differet

More information

A statistical method to determine sample size to estimate characteristic value of soil parameters

A 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 information

Lecture 6 Simple alternatives and the Neyman-Pearson lemma

Lecture 6 Simple alternatives and the Neyman-Pearson lemma STATS 00: Itroductio to Statistical Iferece Autum 06 Lecture 6 Simple alteratives ad the Neyma-Pearso lemma Last lecture, we discussed a umber of ways to costruct test statistics for testig a simple ull

More information

3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense,

3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense, 3. Z Trasform Referece: Etire Chapter 3 of text. Recall that the Fourier trasform (FT) of a DT sigal x [ ] is ω ( ) [ ] X e = j jω k = xe I order for the FT to exist i the fiite magitude sese, S = x [

More information

4. Partial Sums and the Central Limit Theorem

4. Partial Sums and the Central Limit Theorem 1 of 10 7/16/2009 6:05 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 4. Partial Sums ad the Cetral Limit Theorem The cetral limit theorem ad the law of large umbers are the two fudametal theorems

More information

Resampling 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.

Resampling 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 information

S Y Y = ΣY 2 n. Using the above expressions, the correlation coefficient is. r = SXX S Y Y

S Y Y = ΣY 2 n. Using the above expressions, the correlation coefficient is. r = SXX S Y Y 1 Sociology 405/805 Revised February 4, 004 Summary of Formulae for Bivariate Regressio ad Correlatio Let X be a idepedet variable ad Y a depedet variable, with observatios for each of the values of these

More information

The Sampling Distribution of the Maximum. Likelihood Estimators for the Parameters of. Beta-Binomial Distribution

The Sampling Distribution of the Maximum. Likelihood Estimators for the Parameters of. Beta-Binomial Distribution Iteratioal Mathematical Forum, Vol. 8, 2013, o. 26, 1263-1277 HIKARI Ltd, www.m-hikari.com http://d.doi.org/10.12988/imf.2013.3475 The Samplig Distributio of the Maimum Likelihood Estimators for the Parameters

More information

Lecture 2: Monte Carlo Simulation

Lecture 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 information

GUIDELINES ON REPRESENTATIVE SAMPLING

GUIDELINES ON REPRESENTATIVE SAMPLING DRUGS WORKING GROUP VALIDATION OF THE GUIDELINES ON REPRESENTATIVE SAMPLING DOCUMENT TYPE : REF. CODE: ISSUE NO: ISSUE DATE: VALIDATION REPORT DWG-SGL-001 002 08 DECEMBER 2012 Ref code: DWG-SGL-001 Issue

More information

Lecture 19: Convergence

Lecture 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 information

MOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND.

MOST 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

UNIVERSITY OF TORONTO Faculty of Arts and Science APRIL/MAY 2009 EXAMINATIONS ECO220Y1Y PART 1 OF 2 SOLUTIONS

UNIVERSITY OF TORONTO Faculty of Arts and Science APRIL/MAY 2009 EXAMINATIONS ECO220Y1Y PART 1 OF 2 SOLUTIONS PART of UNIVERSITY OF TORONTO Faculty of Arts ad Sciece APRIL/MAY 009 EAMINATIONS ECO0YY PART OF () The sample media is greater tha the sample mea whe there is. (B) () A radom variable is ormally distributed

More information

Problem Set 4 Due Oct, 12

Problem Set 4 Due Oct, 12 EE226: Radom Processes i Systems Lecturer: Jea C. Walrad Problem Set 4 Due Oct, 12 Fall 06 GSI: Assae Gueye This problem set essetially reviews detectio theory ad hypothesis testig ad some basic otios

More information

[ ] ( ) ( ) [ ] ( ) 1 [ ] [ ] Sums of Random Variables Y = a 1 X 1 + a 2 X 2 + +a n X n The expected value of Y is:

[ ] ( ) ( ) [ ] ( ) 1 [ ] [ ] Sums of Random Variables Y = a 1 X 1 + a 2 X 2 + +a n X n The expected value of Y is: PROBABILITY FUNCTIONS A radom variable X has a probabilit associated with each of its possible values. The probabilit is termed a discrete probabilit if X ca assume ol discrete values, or X = x, x, x 3,,

More information

POWER COMPARISON OF EMPIRICAL LIKELIHOOD RATIO TESTS: SMALL SAMPLE PROPERTIES THROUGH MONTE CARLO STUDIES*

POWER COMPARISON OF EMPIRICAL LIKELIHOOD RATIO TESTS: SMALL SAMPLE PROPERTIES THROUGH MONTE CARLO STUDIES* Kobe Uiversity Ecoomic Review 50(2004) 3 POWER COMPARISON OF EMPIRICAL LIKELIHOOD RATIO TESTS: SMALL SAMPLE PROPERTIES THROUGH MONTE CARLO STUDIES* By HISASHI TANIZAKI There are various kids of oparametric

More information

Confidence interval for the two-parameter exponentiated Gumbel distribution based on record values

Confidence interval for the two-parameter exponentiated Gumbel distribution based on record values Iteratioal Joural of Applied Operatioal Research Vol. 4 No. 1 pp. 61-68 Witer 2014 Joural homepage: www.ijorlu.ir Cofidece iterval for the two-parameter expoetiated Gumbel distributio based o record values

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 19 11/17/2008 LAWS OF LARGE NUMBERS II THE STRONG LAW OF LARGE NUMBERS

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 19 11/17/2008 LAWS OF LARGE NUMBERS II THE STRONG LAW OF LARGE NUMBERS MASSACHUSTTS INSTITUT OF TCHNOLOGY 6.436J/5.085J Fall 2008 Lecture 9 /7/2008 LAWS OF LARG NUMBRS II Cotets. The strog law of large umbers 2. The Cheroff boud TH STRONG LAW OF LARG NUMBRS While the weak

More information

Comparison of Minimum Initial Capital with Investment and Non-investment Discrete Time Surplus Processes

Comparison of Minimum Initial Capital with Investment and Non-investment Discrete Time Surplus Processes The 22 d Aual Meetig i Mathematics (AMM 207) Departmet of Mathematics, Faculty of Sciece Chiag Mai Uiversity, Chiag Mai, Thailad Compariso of Miimum Iitial Capital with Ivestmet ad -ivestmet Discrete Time

More information

Monte Carlo Integration

Monte Carlo Integration Mote Carlo Itegratio I these otes we first review basic umerical itegratio methods (usig Riema approximatio ad the trapezoidal rule) ad their limitatios for evaluatig multidimesioal itegrals. Next we itroduce

More information

Investigating the Significance of a Correlation Coefficient using Jackknife Estimates

Investigating 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 information

Recurrence Relations

Recurrence Relations Recurrece Relatios Aalysis of recursive algorithms, such as: it factorial (it ) { if (==0) retur ; else retur ( * factorial(-)); } Let t be the umber of multiplicatios eeded to calculate factorial(). The

More information

A Note on Box-Cox Quantile Regression Estimation of the Parameters of the Generalized Pareto Distribution

A Note on Box-Cox Quantile Regression Estimation of the Parameters of the Generalized Pareto Distribution A Note o Box-Cox Quatile Regressio Estimatio of the Parameters of the Geeralized Pareto Distributio JM va Zyl Abstract: Makig use of the quatile equatio, Box-Cox regressio ad Laplace distributed disturbaces,

More information

Expectation and Variance of a random variable

Expectation 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 information

Simulation. Two Rule For Inverting A Distribution Function

Simulation. 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 information

Frequency Response of FIR Filters

Frequency Response of FIR Filters EEL335: Discrete-Time Sigals ad Systems. Itroductio I this set of otes, we itroduce the idea of the frequecy respose of LTI systems, ad focus specifically o the frequecy respose of FIR filters.. Steady-state

More information

Discrete-Time Systems, LTI Systems, and Discrete-Time Convolution

Discrete-Time Systems, LTI Systems, and Discrete-Time Convolution EEL5: Discrete-Time Sigals ad Systems. Itroductio I this set of otes, we begi our mathematical treatmet of discrete-time s. As show i Figure, a discrete-time operates or trasforms some iput sequece x [

More information

Goodness-Of-Fit For The Generalized Exponential Distribution. Abstract

Goodness-Of-Fit For The Generalized Exponential Distribution. Abstract Goodess-Of-Fit For The Geeralized Expoetial Distributio By Amal S. Hassa stitute of Statistical Studies & Research Cairo Uiversity Abstract Recetly a ew distributio called geeralized expoetial or expoetiated

More information

EECS564 Estimation, Filtering, and Detection Hwk 2 Solns. Winter p θ (z) = (2θz + 1 θ), 0 z 1

EECS564 Estimation, Filtering, and Detection Hwk 2 Solns. Winter p θ (z) = (2θz + 1 θ), 0 z 1 EECS564 Estimatio, Filterig, ad Detectio Hwk 2 Sols. Witer 25 4. Let Z be a sigle observatio havig desity fuctio where. p (z) = (2z + ), z (a) Assumig that is a oradom parameter, fid ad plot the maximum

More information

Exam II Review. CEE 3710 November 15, /16/2017. EXAM II Friday, November 17, in class. Open book and open notes.

Exam II Review. CEE 3710 November 15, /16/2017. EXAM II Friday, November 17, in class. Open book and open notes. Exam II Review CEE 3710 November 15, 017 EXAM II Friday, November 17, i class. Ope book ad ope otes. Focus o material covered i Homeworks #5 #8, Note Packets #10 19 1 Exam II Topics **Will emphasize material

More information

1 Introduction to reducing variance in Monte Carlo simulations

1 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 information

Describing the Relation between Two Variables

Describing the Relation between Two Variables Copyright 010 Pearso Educatio, Ic. Tables ad Formulas for Sulliva, Statistics: Iformed Decisios Usig Data 010 Pearso Educatio, Ic Chapter Orgaizig ad Summarizig Data Relative frequecy = frequecy sum of

More information

Regression, Inference, and Model Building

Regression, Inference, and Model Building Regressio, Iferece, ad Model Buildig Scatter Plots ad Correlatio Correlatio coefficiet, r -1 r 1 If r is positive, the the scatter plot has a positive slope ad variables are said to have a positive relatioship

More information

Module 18 Discrete Time Signals and Z-Transforms Objective: Introduction : Description: Discrete Time Signal representation

Module 18 Discrete Time Signals and Z-Transforms Objective: Introduction : Description: Discrete Time Signal representation Module 8 Discrete Time Sigals ad Z-Trasforms Objective:To uderstad represetig discrete time sigals, apply z trasform for aalyzigdiscrete time sigals ad to uderstad the relatio to Fourier trasform Itroductio

More information

Math 155 (Lecture 3)

Math 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 information

Polynomial identity testing and global minimum cut

Polynomial identity testing and global minimum cut CHAPTER 6 Polyomial idetity testig ad global miimum cut I this lecture we will cosider two further problems that ca be solved usig probabilistic algorithms. I the first half, we will cosider the problem

More information

Probability and statistics: basic terms

Probability 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 information

Problem Set 2 Solutions

Problem Set 2 Solutions CS271 Radomess & Computatio, Sprig 2018 Problem Set 2 Solutios Poit totals are i the margi; the maximum total umber of poits was 52. 1. Probabilistic method for domiatig sets 6pts Pick a radom subset S

More information

Lecture 7: October 18, 2017

Lecture 7: October 18, 2017 Iformatio ad Codig Theory Autum 207 Lecturer: Madhur Tulsiai Lecture 7: October 8, 207 Biary hypothesis testig I this lecture, we apply the tools developed i the past few lectures to uderstad the problem

More information

5. Likelihood Ratio Tests

5. Likelihood Ratio Tests 1 of 5 7/29/2009 3:16 PM Virtual Laboratories > 9. Hy pothesis Testig > 1 2 3 4 5 6 7 5. Likelihood Ratio Tests Prelimiaries As usual, our startig poit is a radom experimet with a uderlyig sample space,

More information

32 estimating the cumulative distribution function

32 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 information

Chapter 6 Principles of Data Reduction

Chapter 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 information

Worksheet 23 ( ) Introduction to Simple Linear Regression (continued)

Worksheet 23 ( ) Introduction to Simple Linear Regression (continued) Worksheet 3 ( 11.5-11.8) Itroductio to Simple Liear Regressio (cotiued) This worksheet is a cotiuatio of Discussio Sheet 3; please complete that discussio sheet first if you have ot already doe so. This

More information

Chapter 6 Sampling Distributions

Chapter 6 Sampling Distributions Chapter 6 Samplig Distributios 1 I most experimets, we have more tha oe measuremet for ay give variable, each measuremet beig associated with oe radomly selected a member of a populatio. Hece we eed to

More information

Statistics 3858 : Likelihood Ratio for Multinomial Models

Statistics 3858 : Likelihood Ratio for Multinomial Models Statistics 3858 : Likelihood Ratio for Multiomial Models Suppose X is multiomial o M categories, that is X Multiomial, p), where p p 1, p 2,..., p M ) A, ad the parameter space is A {p : p j 0, p j 1 }

More information

Rank tests and regression rank scores tests in measurement error models

Rank tests and regression rank scores tests in measurement error models Rak tests ad regressio rak scores tests i measuremet error models J. Jurečková ad A.K.Md.E. Saleh Charles Uiversity i Prague ad Carleto Uiversity i Ottawa Abstract The rak ad regressio rak score tests

More information

Random Walks on Discrete and Continuous Circles. by Jeffrey S. Rosenthal School of Mathematics, University of Minnesota, Minneapolis, MN, U.S.A.

Random Walks on Discrete and Continuous Circles. by Jeffrey S. Rosenthal School of Mathematics, University of Minnesota, Minneapolis, MN, U.S.A. Radom Walks o Discrete ad Cotiuous Circles by Jeffrey S. Rosethal School of Mathematics, Uiversity of Miesota, Mieapolis, MN, U.S.A. 55455 (Appeared i Joural of Applied Probability 30 (1993), 780 789.)

More information

Approximate Confidence Interval for the Reciprocal of a Normal Mean with a Known Coefficient of Variation

Approximate Confidence Interval for the Reciprocal of a Normal Mean with a Known Coefficient of Variation Metodološki zvezki, Vol. 13, No., 016, 117-130 Approximate Cofidece Iterval for the Reciprocal of a Normal Mea with a Kow Coefficiet of Variatio Wararit Paichkitkosolkul 1 Abstract A approximate cofidece

More information

7.1 Convergence of sequences of random variables

7.1 Convergence of sequences of random variables Chapter 7 Limit Theorems Throughout this sectio we will assume a probability space (, F, P), i which is defied a ifiite sequece of radom variables (X ) ad a radom variable X. The fact that for every ifiite

More information

Department of Mathematics

Department of Mathematics Departmet of Mathematics Ma 3/103 KC Border Itroductio to Probability ad Statistics Witer 2017 Lecture 19: Estimatio II Relevat textbook passages: Larse Marx [1]: Sectios 5.2 5.7 19.1 The method of momets

More information

Lecture 7: Properties of Random Samples

Lecture 7: Properties of Random Samples Lecture 7: Properties of Radom Samples 1 Cotiued From Last Class Theorem 1.1. Let X 1, X,...X be a radom sample from a populatio with mea µ ad variace σ

More information

x iu i E(x u) 0. In order to obtain a consistent estimator of β, we find the instrumental variable z which satisfies E(z u) = 0. z iu i E(z u) = 0.

x iu i E(x u) 0. In order to obtain a consistent estimator of β, we find the instrumental variable z which satisfies E(z u) = 0. z iu i E(z u) = 0. 27 However, β MM is icosistet whe E(x u) 0, i.e., β MM = (X X) X y = β + (X X) X u = β + ( X X ) ( X u ) \ β. Note as follows: X u = x iu i E(x u) 0. I order to obtai a cosistet estimator of β, we fid

More information

Sample Size Estimation in the Proportional Hazards Model for K-sample or Regression Settings Scott S. Emerson, M.D., Ph.D.

Sample Size Estimation in the Proportional Hazards Model for K-sample or Regression Settings Scott S. Emerson, M.D., Ph.D. ample ie Estimatio i the Proportioal Haards Model for K-sample or Regressio ettigs cott. Emerso, M.D., Ph.D. ample ie Formula for a Normally Distributed tatistic uppose a statistic is kow to be ormally

More information

NANYANG TECHNOLOGICAL UNIVERSITY SYLLABUS FOR ENTRANCE EXAMINATION FOR INTERNATIONAL STUDENTS AO-LEVEL MATHEMATICS

NANYANG TECHNOLOGICAL UNIVERSITY SYLLABUS FOR ENTRANCE EXAMINATION FOR INTERNATIONAL STUDENTS AO-LEVEL MATHEMATICS NANYANG TECHNOLOGICAL UNIVERSITY SYLLABUS FOR ENTRANCE EXAMINATION FOR INTERNATIONAL STUDENTS AO-LEVEL MATHEMATICS STRUCTURE OF EXAMINATION PAPER. There will be oe 2-hour paper cosistig of 4 questios.

More information

1 of 7 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 6. Order Statistics Defiitios Suppose agai that we have a basic radom experimet, ad that X is a real-valued radom variable

More information

1 Inferential Methods for Correlation and Regression Analysis

1 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 information

Power and Type II Error

Power and Type II Error Statistical Methods I (EXST 7005) Page 57 Power ad Type II Error Sice we do't actually kow the value of the true mea (or we would't be hypothesizig somethig else), we caot kow i practice the type II error

More information

Power Comparison of Some Goodness-of-fit Tests

Power Comparison of Some Goodness-of-fit Tests Florida Iteratioal Uiversity FIU Digital Commos FIU Electroic Theses ad Dissertatios Uiversity Graduate School 7-6-2016 Power Compariso of Some Goodess-of-fit Tests Tiayi Liu tliu019@fiu.edu DOI: 10.25148/etd.FIDC000750

More information

ECO 312 Fall 2013 Chris Sims LIKELIHOOD, POSTERIORS, DIAGNOSING NON-NORMALITY

ECO 312 Fall 2013 Chris Sims LIKELIHOOD, POSTERIORS, DIAGNOSING NON-NORMALITY ECO 312 Fall 2013 Chris Sims LIKELIHOOD, POSTERIORS, DIAGNOSING NON-NORMALITY (1) A distributio that allows asymmetry differet probabilities for egative ad positive outliers is the asymmetric double expoetial,

More information

MOMENT-METHOD ESTIMATION BASED ON CENSORED SAMPLE

MOMENT-METHOD ESTIMATION BASED ON CENSORED SAMPLE Vol. 8 o. Joural of Systems Sciece ad Complexity Apr., 5 MOMET-METHOD ESTIMATIO BASED O CESORED SAMPLE I Zhogxi Departmet of Mathematics, East Chia Uiversity of Sciece ad Techology, Shaghai 37, Chia. Email:

More information

Optimally Sparse SVMs

Optimally Sparse SVMs A. Proof of Lemma 3. We here prove a lower boud o the umber of support vectors to achieve geeralizatio bouds of the form which we cosider. Importatly, this result holds ot oly for liear classifiers, but

More information

6 Sample Size Calculations

6 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 information

FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures

FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING Lectures MODULE 5 STATISTICS II. Mea ad stadard error of sample data. Biomial distributio. Normal distributio 4. Samplig 5. Cofidece itervals

More information

Gini Index and Polynomial Pen s Parade

Gini Index and Polynomial Pen s Parade Gii Idex ad Polyomial Pe s Parade Jules Sadefo Kamdem To cite this versio: Jules Sadefo Kamdem. Gii Idex ad Polyomial Pe s Parade. 2011. HAL Id: hal-00582625 https://hal.archives-ouvertes.fr/hal-00582625

More information

1.3 Convergence Theorems of Fourier Series. k k k k. N N k 1. With this in mind, we state (without proof) the convergence of Fourier series.

1.3 Convergence Theorems of Fourier Series. k k k k. N N k 1. With this in mind, we state (without proof) the convergence of Fourier series. .3 Covergece Theorems of Fourier Series I this sectio, we preset the covergece of Fourier series. A ifiite sum is, by defiitio, a limit of partial sums, that is, a cos( kx) b si( kx) lim a cos( kx) b si(

More information

Goodness-of-Fit Tests and Categorical Data Analysis (Devore Chapter Fourteen)

Goodness-of-Fit Tests and Categorical Data Analysis (Devore Chapter Fourteen) Goodess-of-Fit Tests ad Categorical Data Aalysis (Devore Chapter Fourtee) MATH-252-01: Probability ad Statistics II Sprig 2019 Cotets 1 Chi-Squared Tests with Kow Probabilities 1 1.1 Chi-Squared Testig................

More information

Dr. Maddah ENMG 617 EM Statistics 11/26/12. Multiple Regression (2) (Chapter 15, Hines)

Dr. Maddah ENMG 617 EM Statistics 11/26/12. Multiple Regression (2) (Chapter 15, Hines) Dr Maddah NMG 617 M Statistics 11/6/1 Multiple egressio () (Chapter 15, Hies) Test for sigificace of regressio This is a test to determie whether there is a liear relatioship betwee the depedet variable

More information

Tests of Hypotheses Based on a Single Sample (Devore Chapter Eight)

Tests of Hypotheses Based on a Single Sample (Devore Chapter Eight) Tests of Hypotheses Based o a Sigle Sample Devore Chapter Eight MATH-252-01: Probability ad Statistics II Sprig 2018 Cotets 1 Hypothesis Tests illustrated with z-tests 1 1.1 Overview of Hypothesis Testig..........

More information

Convergence of random variables. (telegram style notes) P.J.C. Spreij

Convergence of random variables. (telegram style notes) P.J.C. Spreij Covergece of radom variables (telegram style otes).j.c. Spreij this versio: September 6, 2005 Itroductio As we kow, radom variables are by defiitio measurable fuctios o some uderlyig measurable space

More information

Control Charts for Mean for Non-Normally Correlated Data

Control Charts for Mean for Non-Normally Correlated Data Joural of Moder Applied Statistical Methods Volume 16 Issue 1 Article 5 5-1-017 Cotrol Charts for Mea for No-Normally Correlated Data J. R. Sigh Vikram Uiversity, Ujjai, Idia Ab Latif Dar School of Studies

More information

Chapter 13: Tests of Hypothesis Section 13.1 Introduction

Chapter 13: Tests of Hypothesis Section 13.1 Introduction Chapter 13: Tests of Hypothesis Sectio 13.1 Itroductio RECAP: Chapter 1 discussed the Likelihood Ratio Method as a geeral approach to fid good test procedures. Testig for the Normal Mea Example, discussed

More information

Econ 325/327 Notes on Sample Mean, Sample Proportion, Central Limit Theorem, Chi-square Distribution, Student s t distribution 1.

Econ 325/327 Notes on Sample Mean, Sample Proportion, Central Limit Theorem, Chi-square Distribution, Student s t distribution 1. Eco 325/327 Notes o Sample Mea, Sample Proportio, Cetral Limit Theorem, Chi-square Distributio, Studet s t distributio 1 Sample Mea By Hiro Kasahara We cosider a radom sample from a populatio. Defiitio

More information

[412] A TEST FOR HOMOGENEITY OF THE MARGINAL DISTRIBUTIONS IN A TWO-WAY CLASSIFICATION

[412] A TEST FOR HOMOGENEITY OF THE MARGINAL DISTRIBUTIONS IN A TWO-WAY CLASSIFICATION [412] A TEST FOR HOMOGENEITY OF THE MARGINAL DISTRIBUTIONS IN A TWO-WAY CLASSIFICATION BY ALAN STUART Divisio of Research Techiques, Lodo School of Ecoomics 1. INTRODUCTION There are several circumstaces

More information

Assessment of extreme discharges of the Vltava River in Prague

Assessment of extreme discharges of the Vltava River in Prague Flood Recovery, Iovatio ad Respose I 05 Assessmet of extreme discharges of the Vltava River i Prague M. Holický, K. Jug & M. Sýkora Kloker Istitute, Czech Techical Uiversity i Prague, Czech Republic Abstract

More information

Journal of Multivariate Analysis. Superefficient estimation of the marginals by exploiting knowledge on the copula

Journal 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 information

17. Joint distributions of extreme order statistics Lehmann 5.1; Ferguson 15

17. 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 information

Chapter 13, Part A Analysis of Variance and Experimental Design

Chapter 13, Part A Analysis of Variance and Experimental Design Slides Prepared by JOHN S. LOUCKS St. Edward s Uiversity Slide 1 Chapter 13, Part A Aalysis of Variace ad Eperimetal Desig Itroductio to Aalysis of Variace Aalysis of Variace: Testig for the Equality of

More information

Random Variables, Sampling and Estimation

Random 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 information

Double Stage Shrinkage Estimator of Two Parameters. Generalized Exponential Distribution

Double 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 information

NCSS Statistical Software. Tolerance Intervals

NCSS Statistical Software. Tolerance Intervals Chapter 585 Itroductio This procedure calculates oe-, ad two-, sided tolerace itervals based o either a distributio-free (oparametric) method or a method based o a ormality assumptio (parametric). A two-sided

More information

The standard deviation of the mean

The 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 information

Access to the published version may require journal subscription. Published with permission from: Elsevier.

Access 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 information

CEE 522 Autumn Uncertainty Concepts for Geotechnical Engineering

CEE 522 Autumn Uncertainty Concepts for Geotechnical Engineering CEE 5 Autum 005 Ucertaity Cocepts for Geotechical Egieerig Basic Termiology Set A set is a collectio of (mutually exclusive) objects or evets. The sample space is the (collectively exhaustive) collectio

More information

A goodness-of-fit test based on the empirical characteristic function and a comparison of tests for normality

A goodness-of-fit test based on the empirical characteristic function and a comparison of tests for normality A goodess-of-fit test based o the empirical characteristic fuctio ad a compariso of tests for ormality J. Marti va Zyl Departmet of Mathematical Statistics ad Actuarial Sciece, Uiversity of the Free State,

More information

PAijpam.eu ON TENSOR PRODUCT DECOMPOSITION

PAijpam.eu ON TENSOR PRODUCT DECOMPOSITION Iteratioal Joural of Pure ad Applied Mathematics Volume 103 No 3 2015, 537-545 ISSN: 1311-8080 (prited versio); ISSN: 1314-3395 (o-lie versio) url: http://wwwijpameu doi: http://dxdoiorg/1012732/ijpamv103i314

More information

Application to Random Graphs

Application to Random Graphs A Applicatio to Radom Graphs Brachig processes have a umber of iterestig ad importat applicatios. We shall cosider oe of the most famous of them, the Erdős-Réyi radom graph theory. 1 Defiitio A.1. Let

More information

ECE 8527: Introduction to Machine Learning and Pattern Recognition Midterm # 1. Vaishali Amin Fall, 2015

ECE 8527: Introduction to Machine Learning and Pattern Recognition Midterm # 1. Vaishali Amin Fall, 2015 ECE 8527: Itroductio to Machie Learig ad Patter Recogitio Midterm # 1 Vaishali Ami Fall, 2015 tue39624@temple.edu Problem No. 1: Cosider a two-class discrete distributio problem: ω 1 :{[0,0], [2,0], [2,2],

More information

The Random Walk For Dummies

The Random Walk For Dummies The Radom Walk For Dummies Richard A Mote Abstract We look at the priciples goverig the oe-dimesioal discrete radom walk First we review five basic cocepts of probability theory The we cosider the Beroulli

More information

ON POINTWISE BINOMIAL APPROXIMATION

ON POINTWISE BINOMIAL APPROXIMATION Iteratioal Joural of Pure ad Applied Mathematics Volume 71 No. 1 2011, 57-66 ON POINTWISE BINOMIAL APPROXIMATION BY w-functions K. Teerapabolar 1, P. Wogkasem 2 Departmet of Mathematics Faculty of Sciece

More information

Spurious Fixed E ects Regression

Spurious Fixed E ects Regression Spurious Fixed E ects Regressio I Choi First Draft: April, 00; This versio: Jue, 0 Abstract This paper shows that spurious regressio results ca occur for a xed e ects model with weak time series variatio

More information

Correlation Regression

Correlation Regression Correlatio Regressio While correlatio methods measure the stregth of a liear relatioship betwee two variables, we might wish to go a little further: How much does oe variable chage for a give chage i aother

More information

Signal Processing. Lecture 02: Discrete Time Signals and Systems. Ahmet Taha Koru, Ph. D. Yildiz Technical University.

Signal Processing. Lecture 02: Discrete Time Signals and Systems. Ahmet Taha Koru, Ph. D. Yildiz Technical University. Sigal Processig Lecture 02: Discrete Time Sigals ad Systems Ahmet Taha Koru, Ph. D. Yildiz Techical Uiversity 2017-2018 Fall ATK (YTU) Sigal Processig 2017-2018 Fall 1 / 51 Discrete Time Sigals Discrete

More information

Chapter 5: Hypothesis testing

Chapter 5: Hypothesis testing Slide 5. Chapter 5: Hypothesis testig Hypothesis testig is about makig decisios Is a hypothesis true or false? Are wome paid less, o average, tha me? Barrow, Statistics for Ecoomics, Accoutig ad Busiess

More information

Section 9.2. Tests About a Population Proportion 12/17/2014. Carrying Out a Significance Test H A N T. Parameters & Hypothesis

Section 9.2. Tests About a Population Proportion 12/17/2014. Carrying Out a Significance Test H A N T. Parameters & Hypothesis Sectio 9.2 Tests About a Populatio Proportio P H A N T O M S Parameters Hypothesis Assess Coditios Name the Test Test Statistic (Calculate) Obtai P value Make a decisio State coclusio Sectio 9.2 Tests

More information