Introduction. Why One-Pass Statistics?
|
|
- Jack Johnston
- 6 years ago
- Views:
Transcription
1
2 BERKELE RESEARCH GROUP Ths manuscrpt s program documentaton for three ways to calculate the mean, varance, skewness, kurtoss, covarance, correlaton, regresson parameters and other regresson statstcs. Although nformaton contaned n ths manuscrpt s beleved to be accurate, the documentaton s offered wthout warranty, and users agree to assume all responsbltes and consequences from usng ths documentaton. Introducton Formulas for common statstcs are generally well known, and users have access to natve routnes n Mcrosoft Excel and most programmng languages to calculate many statstcs. Under most crcumstances and wth most data, these routnes provde dentcal results. That s, they produce dentcal results wthn the mathematcal precson avalable n that envronment. However, these algorthms can be constructed n at least three ways, and sometmes the results dffer because the algorthms exceed the precson of the envronment. Stated dfferently, the three methods place unequal demands on the precson avalable for the calculatons. Some data also put more demands on the precson avalable for calculatons. For most data, the choce nvolves convenence; for some data, choosng the rght algorthm s mportant. Why One-Pass Statstcs? The standard defntons of the statstcal formulas descrbed below requre two passes through the data. At tmes t s mpossble or nconvenent to wat untl all data s avalable to make the calculatons. Ths mght occur because t s necessary to calculate a statstc wth all avalable data up to a pont and recalculate after recevng each addtonal data pont. Some data sets are large enough that retanng the data to make two passes s ether mpractcal or mpossble. These condtons argue for usng one of the one-pass methods descrbed below. One example n whch usng one-pass statstcs may be valuable nvolves Monte Carlo smulaton, where the number of samples can quckly become very large. In ths case, t s convenent to calculate dstrbuton parameters such as the mean, standard devaton, sample skewness, or kurtoss usng a one-pass method to avod havng to retan all of the data for ex post analyss. For the same reason, t s convenent to embed the statstcal calculatons n lne n the same code that generates the Monte Carlo test rather than to rely on natve statstcal routnes. A second example n whch one-pass statstcs may be valuable also nvolves Monte Carlo smulaton, where the tests are repeated untl a certan level of statstcal confdence s acheved. For example, the standard error of a Monte Carlo result generally declnes proportonate wth the square root of the number of trals. When the standard devaton of path results s known n advance, t s possble to also determne n advance how many trals are requred. When the standard devaton of sample paths s not known n advance ( for example, f ths uncertanty depends on nputs to the smulaton), t s convenent to run the test untl the standard error of the estmate falls below a targeted level. A one-pass method that can be ncrementally updated makes such a smart stop possble. For the rest of ths manuscrpt, sample skewness wll just be called skewness. In general, ths manuscrpt wll not assume that kurtoss wll actually mean excess kurtoss unless labeled as such explctly. In all cases, kurtoss wll refer to the kurtoss of a sample. IMPLEMETIG A TRIOMIAL COVERTIBLE BOD PRICIG MODEL
3 BERKELE RESEARCH GROUP WHITE PAPER Two-Pass Statstcs The standard defntons of varance, skewness, kurtoss, covarance, and smple lnear regresson begn by assumng that the mean of data to be analyzed s already known. An algorthm frst calculates the mean. In the nterest of completeness and to ntroduce the notaton, that mean s shown n Equaton : µ () To calculate the mean, μ, of a vector, add all the values for and dvde by the number of observatons. The mean s sometmes called the frst sample moment of a statstcal dstrbuton. The unt of measure that apples to μ s the same unt that apples to. For example, f s measured n feet, the mean produced by Equaton wll be n feet. The standard defnton of sample varance appears n Equaton : ( ) - µ () An algorthm that frst calculates the results of Equaton and then Equaton s called a two-pass algorthm for calculatng varance. The varance s sometmes called the second sample moment of a statstcal dstrbuton and the numerator s called the sum of squares. The unt of measure that apples to s the square of the unt that apples to. The defnton of sample standard devaton appears n Equaton : ( µ ) - () Alternatvely, the standard devaton can be descrbed as the square root of varance, n whch case the algorthm bulder doesn t really need a separate formula as n Equaton. Each of the one-pass methods descrbed below follows that pattern: fnd the varance, then transform t nto standard devaton f needed. The unt of measure that apples to s the same unt that apples to. For example, f s measured n feet, the standard devaton produced by Equaton wll be n feet. Equatons and dvde the sum of squares by the number of observatons reduced by. Ths adjustment makes these sample statstcs unbased. A smlar bas adjustment s requred for skewness and kurtoss but often does not appear n publshed formulas. Ths manuscrpt wll follow that conventon and then dscuss how to adjust the results to be unbased. The standard defnton of skewness appears n Equaton : Skew ( µ ) * ()
4 BERKELE RESEARCH GROUP The denomnator n Equaton can be descrbed as ether the standard devaton rased to the thrd power or the varance rased to the.5 power. Of course, that denomnator requres a pass through the data, and the calculaton of the denomnator must be made before the skewness s calculated. However, the summatons requred to calculate the denomnator usng Equaton or Equaton can be talled on the same pass through the data requred to calculate the sum n the numerator of Equaton. For ths reason, t s stll generally descrbed as a two-pass formula. The skewness s sometmes called the thrd moment of a statstcal dstrbuton. The unt of measure that apples to Skew s ndependent of the unt that apples to. For any data, a skewness of 0 s consdered not skewed, whle postve values are descrbed as skewed rght and negatve values as skewed left. The standard defnton of kurtoss appears n Equaton 5: Kurtoss ( µ ) * (5) Ths kurtoss formula could be descrbed as a two-pass formula, because t reles on a pror step to calculate the mean, then a second step that sums values for both the numerator and the denomnator. The kurtoss s the fourth sample moment of a statstcal dstrbuton. The unt of measure that apples to Kurtoss x s ndependent of unt that apples to. For any data, a kurtoss of about s consdered typcal of normally dstrbuted values and descrbed as mesokurtc. Kurtoss larger than about s descrbed as leptokurtc ( fat tals), and kurtoss smaller than about s platykurtc (thnner tals). A measure called excess kurtoss subtracts approxmately so that excess kurtoss s centered around 0. The standard defnton for covarance appears as Equaton 6:, ( µ )( µ ) (6) Equaton 6 closely resembles the defnton of varance n Equaton. In fact, Equaton 6 becomes Equaton (except for the mnor dfference n the denomnators) when Equaton 6 s used to measure the covarance between a varable and tself. The covarance s not consdered a moment. The unts that apply to Equaton 6 lack ntutve clarty. For ths reason, correlaton s calculated as a knd of standardzed or normalzed covarance. See Equaton 7: ρ,, (7) See a longer descrpton on how the adjustment dffers from the bas secton. See Appendx D for a more precse defnton of the excess kurtoss adjustment. IMPLEMETIG A TRIOMIAL COVERTIBLE BOD PRICIG MODEL
5 BERKELE RESEARCH GROUP WHITE PAPER Textbook Formulas for One-Pass Statstcs Each of the four moments descrbed above and the covarance can be restated n a format that s conducve for buldng a onepass algorthm. Each of the formulas s algebracally equvalent to the standard formulas. Ths means that f mathematcal routnes could produce the exact values called for n the equatons above and below, the results would dentcally match the results from the equatons above. The equvalent one-pass formula for the varance 5 appears as equaton 8: (8) Equaton 8 s sometmes called the textbook formula because t s frequently ncluded n statstcal textbooks. 6 It permts constructon of a one-pass algorthm because the mean s only needed at the end of a pass through the data. That algorthm sum both and. The equvalent one-pass textbook formula for the skewness 7 appears as Equaton 9: Skew (9) Although ths author has not seen Equaton 9 publshed, t s convenent to descrbe t as the textbook formula for skewness. It permts constructon of a one-pass algorthm because the mean s only needed at the end of a pass through the data. That algorthm must sum,, and. The equvalent textbook one-pass formula for the kurtoss 8 appears as Equaton 0: 6 Kurt (0) Although ths author has not seen Equaton 0 publshed, t s convenent to descrbe t as the textbook formula for kurtoss. It permts constructon of a one-pass algorthm because the mean s only needed at the end of a pass through the data. That algorthm must sum,,, and. 5 The dervaton of Equaton 8 appears n Appendx A. 6 Chan, Tony F., Gene H. Golub, and Randall J. LeVeque, Algorthms for Computng Sample Varance, Analyss and Recommendatons, The Amercan Statstcan 7: (August 98), 7. 7 The dervaton of Equaton 9 appears n Appendx B. 8 The dervaton of Equaton 0 appears n Appendx C.
6 BERKELE RESEARCH GROUP The textbook methodology lends tself to a one-pass method for calculatng the covarance. Equaton 9 follows the textbook strategy and requres the sums of,, and. Of course, a one-pass textbook algorthm to calculate the correlaton coeffcent follows, usng Equaton and Equaton 7. Calculate the covarance of two vectors usng Equaton and the square root of Equaton 8 (varance) to calculate the standard devaton of each vector wth a sngle pass. It s also possble to develop a smlar one-pass formula for a regresson slope, β, for a sngle 0 ndependent varable. Equaton requres the sums of,,, and and requres knowledge of the mean, but that mean s not requred to complete the other calculatons, so the terms needed to evaluate Equaton can be accumulated on a sngle pass through the data. () β () The ntercept, α, n Equaton can be found usng terms evaluated for Equaton. Equaton reles on the knowledge that the means of the and values represent a pont on the regresson lne. β α α m Therefore, the ntercept can be determned from a sngle pass f the slope s known. By relyng on Equaton, whch permts a one-pass algorthm, the regresson lne can be determned wth a one-pass methodology smlar to the textbook algorthms above. umercal Precson of Textbook One-Pass Algorthms The textbook algorthms are vulnerable to computatonal errors for certan types of data. For example, f the magntude of that data s large, requrng much of the avalable precson of a computer system, and f the varance s small relatve to the underlyng data, t s not dffcult to construct a hypothetcal data set where algorthms based on the textbook formulas for varance, skewness, and kurtoss produce unrelable results. Some authors have advsed aganst usng the textbook algorthm because t s more prone to errors ntroduced by the computatonal lmts of computer mathematcal operatons. Although two-pass methods are much less lkely to exceed the computatonal precson of a computer, t s also possble to fnd data where the two-pass method can produce unrelable results. Some strateges can mprove the accuracy of two-pass methods. For example, from all data, subtract a large number somewhat close to the expected mean. A thrd method ntroduced by Welford s generally least lkely to requre arthmetc operatons that exceed the precson of the computer. () () 9 The dervaton of Equaton appears n Appendx D. 0 The dervaton of Equaton appears n Appendx E. See for example, Cook, John D., Comparng Three Methods of Computng Standard Devaton, John D. Cook blog (September 6, 008), accessed at: IMPLEMETIG A TRIOMIAL COVERTIBLE BOD PRICIG MODEL 5
7 BERKELE RESEARCH GROUP WHITE PAPER Onlne One-Pass Statstcs Welford ntroduced a way to calculate varance wthout requrng the pror calculaton of the mean. Knuth desgned a well-tested algorthm for calculatng varance, relyng on the Welford formulaton. Welford defned the mean conventonally. M (5) A thoughtful algorthm accumulates the sum n the numerator. The mean s found by dvdng ths accumulaton by the prevalng. The Onlne varance accumulates the sum of squares and uses that sum to calculate varance. Equaton 6 defnes the sum of squares as an ncrement based on the prevous sum of squares. S S ( M ) (6) Calculate varance of the data ponts wth the sum of squares usng Equaton 7: S - ote that ths formulaton supports an ncremental algorthm that works as long as the latest sum of the s and the sum of squares s preserved. As wth Equaton 8, the standard devaton s calculated usng Equaton 7 then takng the square root of the varance. Chan et al. extended the Onlne methodology to allow the accumulated analyss from one block of data to be merged wth the accumulated analyss of a second block of data. Tmothy B Terrberry 5 extended the Chan methodology to permt mergng of data sets used to calculate skewness and kurtoss. The Terrberry equatons reduce to the followng methodology when one addtonal data pont s added to a seres: M (8) δ, Equaton 8 calculates the devaton of the next data pont from the mean prevalng before, updatng the mean to reflect that data pont. M of course equals for. δ M, M, (9) ext, the mean s updated usng Equaton 9. otce that Equatons 8 through ntroduce nterm sums. M s equal to the mean of. M, M, and M provde a convenent way to calculate standard devaton, skew, and kurtoss. (7) M, M, M, δ (0A) (0B) Welford, B. P., ote on a Method for Calculatng Corrected Sums of Squares and Products, Technometrcs : (August 96), 9 0. Donald E. Knuth, The Art of Computer Programmng, Volume : Semnumercal Algorthms, thrd ed. (998),. Chan et al. (98). 5 Terrberry, Tmothy, Computng Hgher-Order Moments Onlne, 008, accessed 8//5. 6
8 BERKELE RESEARCH GROUP Use Equaton 0A and 0B to update the standard devaton or to calculate varance. otce that the value of M s not altered and can be contnually used to accumulate more data. M, Skew M, δ M / M,, ( -)( - ) δ*m -,- (A) (B) Update the skewness usng Equatons A and B. M, Kurt M, M M δ ( -)( - ) 6δ M,- δ*m -,- (A) (B) Fnally, update the kurtoss usng Equatons A and B. It s also possble to derve a Welford-lke formula for covarance. 6 The algorthm used heren reles on Equaton 7 : ( ) ( ( )( )/),, Onlne Regresson Parameters It s possble to calculate a regresson beta usng a formula smlar to the Onlne varance formula. The algorthm s summarzed n Equaton 8. Here, values for the numerator rely on prevous values, whch follow a now-famlar pattern because the algorthm adapts the prevous sum to the new mean. The denomnator s the sum of squared devatons accumulated for the ndependent valuaton, shown as a varaton on the sum of squares formulated n Equaton. () Beta Sumx ( ) ( ( ) Sum ), () As descrbed n the Appendx G, the x n Sumx refers to the data ponts mnus the mean of but Equaton nevertheless presents a methodology that allows for ncremental updatng. As before, calculate alpha usng the means of both the ndependent and dependent varables. Ths relatonshp reles on the fact that a ft lne passes through the coordnate equal to the means of and n Equatons 5 and 6. α µ β * µ β * (5) (6) 6 Pebay, Phllp, Formulas for Robust, One-Pass Parallel Computaton of Covarances and Arbtrary-Order Statstcal Moments, Sanda Report, SAD008-6 (September 008). 7 The dervaton of Equaton appears n Appendx F. 8 The dervaton of Equaton appears n Appendx G. IMPLEMETIG A TRIOMIAL COVERTIBLE BOD PRICIG MODEL 7
9 BERKELE RESEARCH GROUP WHITE PAPER These means or summatons can be calculated usng the algorthm n Equaton 5, because no other part of the calculatons depend on the prevalng mean. The β s calculated wth Equaton. Other Statstcs A large number of statstcs nvolved wth lnear regresson could potentally be calculated wth a one-pass algorthm: total sum of squares, error sum of squares, regresson sum of squares, R-square, F statstc, the standard error of estmate, the standard error of the slope, the standard error of the ntercept, and t-tests of regresson parameters. Ths manuscrpt wll not seek to derve one-pass methods to calculate these addtonal statstcal values. Multple regresson s almost always conducted wth the use of matrx operatons: the nverse of a matrx, the transpose of a matrx, and matrx multplcaton. The format does not appear to lend tself to one-pass algorthms. The formula for beta, for example, appears n Equaton 7: In Equaton 7, {} refers to a matrx that contans two or more ndependent varables, multplcaton refers to matrx multplcaton, the symbol { } refers to the transpose of a matrx, the symbol {} refers to the nverse of a matrx, and the vector {y} represents a vector of the devatons from the average of all the values. Ths manuscrpt wll not attempt to ncrementally adapt to, for example, data ponts followng the analyss of data ponts. Exponental Smoothng Exponentally smoothed data s nherently one-pass n nature. A weghted average of prevous values s descrbed n Equaton 8, where an updated average equals a combnaton of the latest sample and the prevous estmated average: Exponental smoothng may offer computatonal effcences over other one-pass methods. By pckng a relatvely low value for α, the statstc should approxmate an average of all data n the sample. Alternatvely, by selectng a relatvely hgh value for α, the statstc can be calbrated to match recent observatons. It follows that another way to create a one-pass method of calculatng the varance s to adopt the method nto the standard defnton of varance. One example of such a hybrd s Equaton 9: The use of α and α make the result an expected value of the sum of squares, whch s also the ntent n Equaton, where the sum of squares s dvded by. To calculate the analogue to the standard devaton, take the square root of the statstc n Equaton 9. Equaton 0 apples the exponental weghtng to the elements n the numerator of the skewness formula n Equaton : SumS Ŝkew (0) ˆ 8 β n ( ' ) ' y ( α) ˆ α Where 0 α ˆ ( ˆ ) ( α) ˆ ˆ α where SumS α ( ˆ ) ( α) SumS (7) (8) (9)
10 BERKELE RESEARCH GROUP Here, the exponentally skewed average n Equaton 8 s substtuted for the sample mean, and a power of the statstc calculated n Equaton 9 substtutes for the standard devaton. Equaton apples the exponental weghtng to elements n the numerator of the kurtoss formula n Equaton 5: SumK Kˆ urt () ˆ Where SumK α ˆ α SumK As n Equaton 0, the exponentally skewed average n Equaton 8 s substtuted for the sample mean, and the statstc calculated n Equaton 9 substtutes for the standard devaton. Bas adjustments ( ) Takng the mean of a dstrbuton removes a degree of freedom from a sample. Ths s why the formula for varance n Equaton and the formula for standard devaton n Equaton use rather than n the denomnator. A smlar adjustment 9 s necessary to make the formulas for skewness (Equaton 0) unbased for samples of data. Equaton matches the value of skewness as calculated by Mntab. Skew Unbased Skew Based () Lkewse, the sample kurtoss adjusted wth Equaton should match the Mntab measure of unbased excess kurtoss. Excess Kurt Unbased Kurt Based () A slghtly dfferent adjustment s requred to match the skewness calculated by SAS, SPSS, and Excel Skew Unbased Skew Based * () Equaton 5 shows the bas adjustment of kurtoss to match SAS, SPSS, and Excel: Kurt Unbased Kurt Based * ( )( ) (5) Fnally, Equaton 6 provdes the adjustment needed to match the kurtoss calculated wthn SAS, SPSS, and Excel 0 : Kurt These adjustments matter prmarly for small sample szes. Ths adjustment factor would equal.00 f the prelmnary estmate (that s, before adjustng for the bas) s essentally unbased. Fgure shows that these adjustments are neglgble for larger sample szes. 9 Joanes, D.., and C.A. Gll, Comparng Measures of Sample Skewness and Kurtoss, Journal of the Royal Statstcal Socety, Seres D (The Statstcan), 7: (998), Joanes, p 8-89 (( ) *Kurt 6) (6) Based Unbased IMPLEMETIG A TRIOMIAL COVERTIBLE BOD PRICIG MODEL 9
11 BERKELE RESEARCH GROUP WHITE PAPER FIGURE. BIAS ADJUSTMET ADJUSTMET Mntab Skew Mntab Kurt SAS Skew SAS Kurt SAMPLE SIZE 00 0 Standard Error of the Mean, Varance, Skewness, Kurtoss Monte Carlo smulatons frequently report the average of the outcomes. Because these are sampled estmates of the true mean, t s mportant to measure the standard error of the mean. The standard error s defned by Equaton 7 : (7) The standard error can be derved from the standard devaton of the outcomes and s therefore avalable as a one-pass statstc. The standard error of varance s defned n Equaton 8 : S s Where s represents the varance of a sample (8) The standard error of varance can be derved from the varance of a sample and s therefore avalable as a one-pass statstc. For large values of, the standard error of the sample standard devaton s approxmated by Equaton 9 : s s ( ) (9) Ahn, Sangtae, and Jeffrey Fessler, Standard Error of Mean, Varance, and Standard Devaton Estmators, EECS Department, Unversty of Mchgan (00). Accessed 8//5. Ahn and Fessler (00). 0 Ahn and Fessler (00).
12 BERKELE RESEARCH GROUP The standard error of the sample standard devaton can be approxmated from the sample standard devaton and s therefore avalable as a one-pass statstc. The standard error of the skewness s defned n Equaton 0 : SES 6( ) ( )( )( ) (0) The standard error of skewness s derved from the sample sze and s therefore avalable as a one-pass statstc. The standard error of the sample kurtoss s defned n Equaton 5 : SEK SES ( )( 5) () The standard error of sample kurtoss s derved from the sample sze and s therefore avalable as a one-pass statstc. Conclusons Statstcal routnes bult nto spreadsheets and statstcal packages generally return numercally ndstngushable results for most data sets. Certan data sets create measurement problems usng one or two methods descrbed heren. For these data sets, the algorthms bult around the methodology ntroduced by Welford may provde more accurate results. Ths documentaton prmarly descrbes an applcaton of one-pass methodologes to Monte Carlo trals. In these applcatons, a two-pass method may be mpractcal. Many such Monte Carlo samples are not problematc for ether the textbook or Onlne method. Where the results are the same, t s dffcult to argue that one method s better than the other. Whle the textbook method can produce accurate results most of the tme, a level of uncertanty remans that perhaps a partcular tral pushes nto an area where the textbook method s naccurate. One way to be more confdent about statstcal measurements s to perform them wth two or three dfferent algorthms and confrm that the results are equvalent for whatever precson s requred. Cramer, Duncan, Fundamental Statstcs for Socal Research, (997) p Cramer, p. 89 IMPLEMETIG A TRIOMIAL COVERTIBLE BOD PRICIG MODEL
13 BERKELE RESEARCH GROUP Appendx A Dervaton of Textbook Varance Formula (A)... (A) (A) * (A)
14 BERKELE RESEARCH GROUP Appendx B Dervaton of Textbook Skewness Formula ( ) Skew ( ) s (B) ( )( )( ) ( )( )( )... (B) Skew ()( )s [( )( ) ( )( )...] (B) Skew ( )s Skew ()( )s... (B) Skew ()( )s (B5) Skew ()( )s (B6) Skew ()( )s (B7) IMPLEMETIG A TRIOMIAL COVERTIBLE BOD PRICIG MODEL
15 BERKELE RESEARCH GROUP Appendx C Dervaton of Textbook Kurtoss Formula Equaton C6 replaces the numerator of Equaton C and Equaton A provdes the denomnator. (C) Kurt (C)... um (C)... um (C)... um 5) ( 6 6 um C 6) ( 6 um C (C7) 6 Kurt
16 BERKELE RESEARCH GROUP Appendx D Dervaton of Textbook Covarance Formula ρ, ( )( ) - (D) ρ, ( ) - (D) ρ, - (D) ρ, - (D) Appendx E Dervaton of Textbook Regresson Beta Formula The slope of a lnear regresson lne ncludes two terms. 6 The numerator equals the sum of the products of the observatons tmes the amount that the values devate from ther mean. The denomnator s the sum of squared devatons. β ( n ) ( n ) (E) β (E) β (E) 6 Wonnacott, Thomas H., and Ronald J. Wonnacott, Introductory Statstcs for Busness and Economcs, second ed. (977),, Equaton -6. IMPLEMETIG A TRIOMIAL COVERTIBLE BOD PRICIG MODEL 5
17 BERKELE RESEARCH GROUP 6 Appendx F Dervaton of Onlne Covarance Formula (F), (F), (F) *, (F),,
18 IMPLEMETIG A TRIOMIAL COVERTIBLE BOD PRICIG MODEL WHITE PAPER BERKELE RESEARCH GROUP 7 Appendx G Dervaton of Welford Regresson Beta Formula (G) Sumx (G) Sumx n ` (G) Sum Sumx Sumx (G) Sum Sumx Sumx (G5) n β (G6) Sumx, β (G7) Sum Sumx, β
19
Tests for Two Correlations
PASS Sample Sze Software Chapter 805 Tests for Two Correlatons Introducton The correlaton coeffcent (or correlaton), ρ, s a popular parameter for descrbng the strength of the assocaton between two varables.
More informationMgtOp 215 Chapter 13 Dr. Ahn
MgtOp 5 Chapter 3 Dr Ahn Consder two random varables X and Y wth,,, In order to study the relatonshp between the two random varables, we need a numercal measure that descrbes the relatonshp The covarance
More informationEvaluating Performance
5 Chapter Evaluatng Performance In Ths Chapter Dollar-Weghted Rate of Return Tme-Weghted Rate of Return Income Rate of Return Prncpal Rate of Return Daly Returns MPT Statstcs 5- Measurng Rates of Return
More informationMode is the value which occurs most frequency. The mode may not exist, and even if it does, it may not be unique.
1.7.4 Mode Mode s the value whch occurs most frequency. The mode may not exst, and even f t does, t may not be unque. For ungrouped data, we smply count the largest frequency of the gven value. If all
More informationAn Application of Alternative Weighting Matrix Collapsing Approaches for Improving Sample Estimates
Secton on Survey Research Methods An Applcaton of Alternatve Weghtng Matrx Collapsng Approaches for Improvng Sample Estmates Lnda Tompkns 1, Jay J. Km 2 1 Centers for Dsease Control and Preventon, atonal
More informationWhich of the following provides the most reasonable approximation to the least squares regression line? (a) y=50+10x (b) Y=50+x (d) Y=1+50x
Whch of the followng provdes the most reasonable approxmaton to the least squares regresson lne? (a) y=50+10x (b) Y=50+x (c) Y=10+50x (d) Y=1+50x (e) Y=10+x In smple lnear regresson the model that s begn
More information4. Greek Letters, Value-at-Risk
4 Greek Letters, Value-at-Rsk 4 Value-at-Rsk (Hull s, Chapter 8) Math443 W08, HM Zhu Outlne (Hull, Chap 8) What s Value at Rsk (VaR)? Hstorcal smulatons Monte Carlo smulatons Model based approach Varance-covarance
More informationCHAPTER 9 FUNCTIONAL FORMS OF REGRESSION MODELS
CHAPTER 9 FUNCTIONAL FORMS OF REGRESSION MODELS QUESTIONS 9.1. (a) In a log-log model the dependent and all explanatory varables are n the logarthmc form. (b) In the log-ln model the dependent varable
More informationLinear Combinations of Random Variables and Sampling (100 points)
Economcs 30330: Statstcs for Economcs Problem Set 6 Unversty of Notre Dame Instructor: Julo Garín Sprng 2012 Lnear Combnatons of Random Varables and Samplng 100 ponts 1. Four-part problem. Go get some
More informationLikelihood Fits. Craig Blocker Brandeis August 23, 2004
Lkelhood Fts Crag Blocker Brandes August 23, 2004 Outlne I. What s the queston? II. Lkelhood Bascs III. Mathematcal Propertes IV. Uncertantes on Parameters V. Mscellaneous VI. Goodness of Ft VII. Comparson
More informationoccurrence of a larger storm than our culvert or bridge is barely capable of handling? (what is The main question is: What is the possibility of
Module 8: Probablty and Statstcal Methods n Water Resources Engneerng Bob Ptt Unversty of Alabama Tuscaloosa, AL Flow data are avalable from numerous USGS operated flow recordng statons. Data s usually
More informationChapter 3 Student Lecture Notes 3-1
Chapter 3 Student Lecture otes 3-1 Busness Statstcs: A Decson-Makng Approach 6 th Edton Chapter 3 Descrbng Data Usng umercal Measures 005 Prentce-Hall, Inc. Chap 3-1 Chapter Goals After completng ths chapter,
More informationChapter 3 Descriptive Statistics: Numerical Measures Part B
Sldes Prepared by JOHN S. LOUCKS St. Edward s Unversty Slde 1 Chapter 3 Descrptve Statstcs: Numercal Measures Part B Measures of Dstrbuton Shape, Relatve Locaton, and Detectng Outlers Eploratory Data Analyss
More informationMeasures of Spread IQR and Deviation. For exam X, calculate the mean, median and mode. For exam Y, calculate the mean, median and mode.
Part 4 Measures of Spread IQR and Devaton In Part we learned how the three measures of center offer dfferent ways of provdng us wth a sngle representatve value for a data set. However, consder the followng
More informationTests for Two Ordered Categorical Variables
Chapter 253 Tests for Two Ordered Categorcal Varables Introducton Ths module computes power and sample sze for tests of ordered categorcal data such as Lkert scale data. Assumng proportonal odds, such
More informationSimple Regression Theory II 2010 Samuel L. Baker
SIMPLE REGRESSIO THEORY II Smple Regresson Theory II 00 Samuel L. Baker Assessng how good the regresson equaton s lkely to be Assgnment A gets nto drawng nferences about how close the regresson lne mght
More informationII. Random Variables. Variable Types. Variables Map Outcomes to Numbers
II. Random Varables Random varables operate n much the same way as the outcomes or events n some arbtrary sample space the dstncton s that random varables are smply outcomes that are represented numercally.
More informationInterval Estimation for a Linear Function of. Variances of Nonnormal Distributions. that Utilize the Kurtosis
Appled Mathematcal Scences, Vol. 7, 013, no. 99, 4909-4918 HIKARI Ltd, www.m-hkar.com http://dx.do.org/10.1988/ams.013.37366 Interval Estmaton for a Lnear Functon of Varances of Nonnormal Dstrbutons that
More information/ Computational Genomics. Normalization
0-80 /02-70 Computatonal Genomcs Normalzaton Gene Expresson Analyss Model Computatonal nformaton fuson Bologcal regulatory networks Pattern Recognton Data Analyss clusterng, classfcaton normalzaton, mss.
More informationCalibration Methods: Regression & Correlation. Calibration Methods: Regression & Correlation
Calbraton Methods: Regresson & Correlaton Calbraton A seres of standards run (n replcate fashon) over a gven concentraton range. Standards Comprsed of analte(s) of nterest n a gven matr composton. Matr
More information3: Central Limit Theorem, Systematic Errors
3: Central Lmt Theorem, Systematc Errors 1 Errors 1.1 Central Lmt Theorem Ths theorem s of prme mportance when measurng physcal quanttes because usually the mperfectons n the measurements are due to several
More information15-451/651: Design & Analysis of Algorithms January 22, 2019 Lecture #3: Amortized Analysis last changed: January 18, 2019
5-45/65: Desgn & Analyss of Algorthms January, 09 Lecture #3: Amortzed Analyss last changed: January 8, 09 Introducton In ths lecture we dscuss a useful form of analyss, called amortzed analyss, for problems
More informationA Bootstrap Confidence Limit for Process Capability Indices
A ootstrap Confdence Lmt for Process Capablty Indces YANG Janfeng School of usness, Zhengzhou Unversty, P.R.Chna, 450001 Abstract The process capablty ndces are wdely used by qualty professonals as an
More informationECONOMETRICS - FINAL EXAM, 3rd YEAR (GECO & GADE)
ECONOMETRICS - FINAL EXAM, 3rd YEAR (GECO & GADE) May 17, 2016 15:30 Frst famly name: Name: DNI/ID: Moble: Second famly Name: GECO/GADE: Instructor: E-mal: Queston 1 A B C Blank Queston 2 A B C Blank Queston
More informationEDC Introduction
.0 Introducton EDC3 In the last set of notes (EDC), we saw how to use penalty factors n solvng the EDC problem wth losses. In ths set of notes, we want to address two closely related ssues. What are, exactly,
More informationCopyright 2017 by Taylor Enterprises, Inc., All Rights Reserved. Dr. Wayne A. Taylor
Taylor Enterprses, Inc. ormalzed Indvduals (I ) Chart Copyrght 07 by Taylor Enterprses, Inc., All Rghts Reserved. ormalzed Indvduals (I) Control Chart Dr. Wayne A. Taylor Abstract: The only commonly used
More informationProblem Set 6 Finance 1,
Carnege Mellon Unversty Graduate School of Industral Admnstraton Chrs Telmer Wnter 2006 Problem Set 6 Fnance, 47-720. (representatve agent constructon) Consder the followng two-perod, two-agent economy.
More informationSpatial Variations in Covariates on Marriage and Marital Fertility: Geographically Weighted Regression Analyses in Japan
Spatal Varatons n Covarates on Marrage and Martal Fertlty: Geographcally Weghted Regresson Analyses n Japan Kenj Kamata (Natonal Insttute of Populaton and Socal Securty Research) Abstract (134) To understand
More informationCOS 511: Theoretical Machine Learning. Lecturer: Rob Schapire Lecture #21 Scribe: Lawrence Diao April 23, 2013
COS 511: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture #21 Scrbe: Lawrence Dao Aprl 23, 2013 1 On-Lne Log Loss To recap the end of the last lecture, we have the followng on-lne problem wth N
More informationFORD MOTOR CREDIT COMPANY SUGGESTED ANSWERS. Richard M. Levich. New York University Stern School of Business. Revised, February 1999
FORD MOTOR CREDIT COMPANY SUGGESTED ANSWERS by Rchard M. Levch New York Unversty Stern School of Busness Revsed, February 1999 1 SETTING UP THE PROBLEM The bond s beng sold to Swss nvestors for a prce
More informationCreating a zero coupon curve by bootstrapping with cubic splines.
MMA 708 Analytcal Fnance II Creatng a zero coupon curve by bootstrappng wth cubc splnes. erg Gryshkevych Professor: Jan R. M. Röman 0.2.200 Dvson of Appled Mathematcs chool of Educaton, Culture and Communcaton
More informationRandom Variables. b 2.
Random Varables Generally the object of an nvestgators nterest s not necessarly the acton n the sample space but rather some functon of t. Techncally a real valued functon or mappng whose doman s the sample
More informationChapter 5 Student Lecture Notes 5-1
Chapter 5 Student Lecture Notes 5-1 Basc Busness Statstcs (9 th Edton) Chapter 5 Some Important Dscrete Probablty Dstrbutons 004 Prentce-Hall, Inc. Chap 5-1 Chapter Topcs The Probablty Dstrbuton of a Dscrete
More informationTeaching Note on Factor Model with a View --- A tutorial. This version: May 15, Prepared by Zhi Da *
Copyrght by Zh Da and Rav Jagannathan Teachng Note on For Model th a Ve --- A tutoral Ths verson: May 5, 2005 Prepared by Zh Da * Ths tutoral demonstrates ho to ncorporate economc ves n optmal asset allocaton
More informationCS 286r: Matching and Market Design Lecture 2 Combinatorial Markets, Walrasian Equilibrium, Tâtonnement
CS 286r: Matchng and Market Desgn Lecture 2 Combnatoral Markets, Walrasan Equlbrum, Tâtonnement Matchng and Money Recall: Last tme we descrbed the Hungaran Method for computng a maxmumweght bpartte matchng.
More informationAppendix - Normally Distributed Admissible Choices are Optimal
Appendx - Normally Dstrbuted Admssble Choces are Optmal James N. Bodurtha, Jr. McDonough School of Busness Georgetown Unversty and Q Shen Stafford Partners Aprl 994 latest revson September 00 Abstract
More informationTCOM501 Networking: Theory & Fundamentals Final Examination Professor Yannis A. Korilis April 26, 2002
TO5 Networng: Theory & undamentals nal xamnaton Professor Yanns. orls prl, Problem [ ponts]: onsder a rng networ wth nodes,,,. In ths networ, a customer that completes servce at node exts the networ wth
More informationCapability Analysis. Chapter 255. Introduction. Capability Analysis
Chapter 55 Introducton Ths procedure summarzes the performance of a process based on user-specfed specfcaton lmts. The observed performance as well as the performance relatve to the Normal dstrbuton are
More informationClearing Notice SIX x-clear Ltd
Clearng Notce SIX x-clear Ltd 1.0 Overvew Changes to margn and default fund model arrangements SIX x-clear ( x-clear ) s closely montorng the CCP envronment n Europe as well as the needs of ts Members.
More informationLecture Note 2 Time Value of Money
Seg250 Management Prncples for Engneerng Managers Lecture ote 2 Tme Value of Money Department of Systems Engneerng and Engneerng Management The Chnese Unversty of Hong Kong Interest: The Cost of Money
More informationSpurious Seasonal Patterns and Excess Smoothness in the BLS Local Area Unemployment Statistics
Spurous Seasonal Patterns and Excess Smoothness n the BLS Local Area Unemployment Statstcs Keth R. Phllps and Janguo Wang Federal Reserve Bank of Dallas Research Department Workng Paper 1305 September
More informationA Comparison of Statistical Methods in Interrupted Time Series Analysis to Estimate an Intervention Effect
Transport and Road Safety (TARS) Research Joanna Wang A Comparson of Statstcal Methods n Interrupted Tme Seres Analyss to Estmate an Interventon Effect Research Fellow at Transport & Road Safety (TARS)
More informationOCR Statistics 1 Working with data. Section 2: Measures of location
OCR Statstcs 1 Workng wth data Secton 2: Measures of locaton Notes and Examples These notes have sub-sectons on: The medan Estmatng the medan from grouped data The mean Estmatng the mean from grouped data
More informationPhysics 4A. Error Analysis or Experimental Uncertainty. Error
Physcs 4A Error Analyss or Expermental Uncertanty Slde Slde 2 Slde 3 Slde 4 Slde 5 Slde 6 Slde 7 Slde 8 Slde 9 Slde 0 Slde Slde 2 Slde 3 Slde 4 Slde 5 Slde 6 Slde 7 Slde 8 Slde 9 Slde 20 Slde 2 Error n
More informationHewlett Packard 10BII Calculator
Hewlett Packard 0BII Calculator Keystrokes for the HP 0BII are shown n the tet. However, takng a mnute to revew the Quk Start secton, below, wll be very helpful n gettng started wth your calculator. Note:
More informationNotes are not permitted in this examination. Do not turn over until you are told to do so by the Invigilator.
UNIVERSITY OF EAST ANGLIA School of Economcs Man Seres PG Examnaton 2016-17 BANKING ECONOMETRICS ECO-7014A Tme allowed: 2 HOURS Answer ALL FOUR questons. Queston 1 carres a weght of 30%; queston 2 carres
More informationThe Mack-Method and Analysis of Variability. Erasmus Gerigk
The Mac-Method and Analyss of Varablty Erasmus Gerg ontents/outlne Introducton Revew of two reservng recpes: Incremental Loss-Rato Method han-ladder Method Mac s model assumptons and estmatng varablty
More information3/3/2014. CDS M Phil Econometrics. Vijayamohanan Pillai N. Truncated standard normal distribution for a = 0.5, 0, and 0.5. CDS Mphil Econometrics
Lmted Dependent Varable Models: Tobt an Plla N 1 CDS Mphl Econometrcs Introducton Lmted Dependent Varable Models: Truncaton and Censorng Maddala, G. 1983. Lmted Dependent and Qualtatve Varables n Econometrcs.
More informationMidterm Exam. Use the end of month price data for the S&P 500 index in the table below to answer the following questions.
Unversty of Washngton Summer 2001 Department of Economcs Erc Zvot Economcs 483 Mdterm Exam Ths s a closed book and closed note exam. However, you are allowed one page of handwrtten notes. Answer all questons
More informationMultifactor Term Structure Models
1 Multfactor Term Structure Models A. Lmtatons of One-Factor Models 1. Returns on bonds of all maturtes are perfectly correlated. 2. Term structure (and prces of every other dervatves) are unquely determned
More informationPrice and Quantity Competition Revisited. Abstract
rce and uantty Competton Revsted X. Henry Wang Unversty of Mssour - Columba Abstract By enlargng the parameter space orgnally consdered by Sngh and Vves (984 to allow for a wder range of cost asymmetry,
More informationThe Integration of the Israel Labour Force Survey with the National Insurance File
The Integraton of the Israel Labour Force Survey wth the Natonal Insurance Fle Natale SHLOMO Central Bureau of Statstcs Kanfey Nesharm St. 66, corner of Bach Street, Jerusalem Natales@cbs.gov.l Abstact:
More informationEXAMINATIONS OF THE HONG KONG STATISTICAL SOCIETY
EXAMINATIONS OF THE HONG KONG STATISTICAL SOCIETY HIGHER CERTIFICATE IN STATISTICS, 2013 MODULE 7 : Tme seres and ndex numbers Tme allowed: One and a half hours Canddates should answer THREE questons.
More informationCracking VAR with kernels
CUTTIG EDGE. PORTFOLIO RISK AALYSIS Crackng VAR wth kernels Value-at-rsk analyss has become a key measure of portfolo rsk n recent years, but how can we calculate the contrbuton of some portfolo component?
More informationModule Contact: Dr P Moffatt, ECO Copyright of the University of East Anglia Version 2
UNIVERSITY OF EAST ANGLIA School of Economcs Man Seres PG Examnaton 2012-13 FINANCIAL ECONOMETRICS ECO-M017 Tme allowed: 2 hours Answer ALL FOUR questons. Queston 1 carres a weght of 25%; Queston 2 carres
More informationMonetary Tightening Cycles and the Predictability of Economic Activity. by Tobias Adrian and Arturo Estrella * October 2006.
Monetary Tghtenng Cycles and the Predctablty of Economc Actvty by Tobas Adran and Arturo Estrella * October 2006 Abstract Ten out of thrteen monetary tghtenng cycles snce 1955 were followed by ncreases
More informationAnalysis of Variance and Design of Experiments-II
Analyss of Varance and Desgn of Experments-II MODULE VI LECTURE - 4 SPLIT-PLOT AND STRIP-PLOT DESIGNS Dr. Shalabh Department of Mathematcs & Statstcs Indan Insttute of Technology Kanpur An example to motvate
More informationEconomic Design of Short-Run CSP-1 Plan Under Linear Inspection Cost
Tamkang Journal of Scence and Engneerng, Vol. 9, No 1, pp. 19 23 (2006) 19 Economc Desgn of Short-Run CSP-1 Plan Under Lnear Inspecton Cost Chung-Ho Chen 1 * and Chao-Yu Chou 2 1 Department of Industral
More informationSkewness and kurtosis unbiased by Gaussian uncertainties
Skewness and kurtoss unbased by Gaussan uncertantes Lorenzo Rmoldn Observatore astronomque de l Unversté de Genève, chemn des Mallettes 5, CH-9 Versox, Swtzerland ISDC Data Centre for Astrophyscs, Unversté
More informationConstruction Rules for Morningstar Canada Dividend Target 30 Index TM
Constructon Rules for Mornngstar Canada Dvdend Target 0 Index TM Mornngstar Methodology Paper January 2012 2011 Mornngstar, Inc. All rghts reserved. The nformaton n ths document s the property of Mornngstar,
More informationSupplementary material for Non-conjugate Variational Message Passing for Multinomial and Binary Regression
Supplementary materal for Non-conjugate Varatonal Message Passng for Multnomal and Bnary Regresson October 9, 011 1 Alternatve dervaton We wll focus on a partcular factor f a and varable x, wth the am
More information- contrast so-called first-best outcome of Lindahl equilibrium with case of private provision through voluntary contributions of households
Prvate Provson - contrast so-called frst-best outcome of Lndahl equlbrum wth case of prvate provson through voluntary contrbutons of households - need to make an assumpton about how each household expects
More informationAlternatives to Shewhart Charts
Alternatves to Shewhart Charts CUSUM & EWMA S Wongsa Overvew Revstng Shewhart Control Charts Cumulatve Sum (CUSUM) Control Chart Eponentally Weghted Movng Average (EWMA) Control Chart 2 Revstng Shewhart
More informationISyE 512 Chapter 9. CUSUM and EWMA Control Charts. Instructor: Prof. Kaibo Liu. Department of Industrial and Systems Engineering UW-Madison
ISyE 512 hapter 9 USUM and EWMA ontrol harts Instructor: Prof. Kabo Lu Department of Industral and Systems Engneerng UW-Madson Emal: klu8@wsc.edu Offce: Room 317 (Mechancal Engneerng Buldng) ISyE 512 Instructor:
More informationSequential equilibria of asymmetric ascending auctions: the case of log-normal distributions 3
Sequental equlbra of asymmetrc ascendng auctons: the case of log-normal dstrbutons 3 Robert Wlson Busness School, Stanford Unversty, Stanford, CA 94305-505, USA Receved: ; revsed verson. Summary: The sequental
More informationISE High Income Index Methodology
ISE Hgh Income Index Methodology Index Descrpton The ISE Hgh Income Index s desgned to track the returns and ncome of the top 30 U.S lsted Closed-End Funds. Index Calculaton The ISE Hgh Income Index s
More informationUNIVERSITY OF VICTORIA Midterm June 6, 2018 Solutions
UIVERSITY OF VICTORIA Mdterm June 6, 08 Solutons Econ 45 Summer A0 08 age AME: STUDET UMBER: V00 Course ame & o. Descrptve Statstcs and robablty Economcs 45 Secton(s) A0 CR: 3067 Instructor: Betty Johnson
More informationRisk and Return: The Security Markets Line
FIN 614 Rsk and Return 3: Markets Professor Robert B.H. Hauswald Kogod School of Busness, AU 1/25/2011 Rsk and Return: Markets Robert B.H. Hauswald 1 Rsk and Return: The Securty Markets Lne From securtes
More informationElements of Economic Analysis II Lecture VI: Industry Supply
Elements of Economc Analyss II Lecture VI: Industry Supply Ka Hao Yang 10/12/2017 In the prevous lecture, we analyzed the frm s supply decson usng a set of smple graphcal analyses. In fact, the dscusson
More informationASSESSING GOODNESS OF FIT OF GENERALIZED LINEAR MODELS TO SPARSE DATA USING HIGHER ORDER MOMENT CORRECTIONS
ASSESSING GOODNESS OF FIT OF GENERALIZED LINEAR MODELS TO SPARSE DATA USING HIGHER ORDER MOMENT CORRECTIONS S. R. PAUL Department of Mathematcs & Statstcs, Unversty of Wndsor, Wndsor, ON N9B 3P4, Canada
More informationFinancial mathematics
Fnancal mathematcs Jean-Luc Bouchot jean-luc.bouchot@drexel.edu February 19, 2013 Warnng Ths s a work n progress. I can not ensure t to be mstake free at the moment. It s also lackng some nformaton. But
More informationAC : THE DIAGRAMMATIC AND MATHEMATICAL APPROACH OF PROJECT TIME-COST TRADEOFFS
AC 2008-1635: THE DIAGRAMMATIC AND MATHEMATICAL APPROACH OF PROJECT TIME-COST TRADEOFFS Kun-jung Hsu, Leader Unversty Amercan Socety for Engneerng Educaton, 2008 Page 13.1217.1 Ttle of the Paper: The Dagrammatc
More informationQuiz on Deterministic part of course October 22, 2002
Engneerng ystems Analyss for Desgn Quz on Determnstc part of course October 22, 2002 Ths s a closed book exercse. You may use calculators Grade Tables There are 90 ponts possble for the regular test, or
More informationFinancial Risk Management in Portfolio Optimization with Lower Partial Moment
Amercan Journal of Busness and Socety Vol., o., 26, pp. 2-2 http://www.ascence.org/journal/ajbs Fnancal Rsk Management n Portfolo Optmzaton wth Lower Partal Moment Lam Weng Sew, 2, *, Lam Weng Hoe, 2 Department
More informationPrinciples of Finance
Prncples of Fnance Grzegorz Trojanowsk Lecture 6: Captal Asset Prcng Model Prncples of Fnance - Lecture 6 1 Lecture 6 materal Requred readng: Elton et al., Chapters 13, 14, and 15 Supplementary readng:
More informationStandardization. Stan Becker, PhD Bloomberg School of Public Health
Ths work s lcensed under a Creatve Commons Attrbuton-NonCommercal-ShareAlke Lcense. Your use of ths materal consttutes acceptance of that lcense and the condtons of use of materals on ths ste. Copyrght
More informationarxiv: v1 [q-fin.pm] 13 Feb 2018
WHAT IS THE SHARPE RATIO, AND HOW CAN EVERYONE GET IT WRONG? arxv:1802.04413v1 [q-fn.pm] 13 Feb 2018 IGOR RIVIN Abstract. The Sharpe rato s the most wdely used rsk metrc n the quanttatve fnance communty
More informationChapter 10 Making Choices: The Method, MARR, and Multiple Attributes
Chapter 0 Makng Choces: The Method, MARR, and Multple Attrbutes INEN 303 Sergy Butenko Industral & Systems Engneerng Texas A&M Unversty Comparng Mutually Exclusve Alternatves by Dfferent Evaluaton Methods
More informationIt is important for a financial institution to monitor the volatilities of the market
CHAPTER 10 Volatlty It s mportant for a fnancal nsttuton to montor the volatltes of the market varables (nterest rates, exchange rates, equty prces, commodty prces, etc.) on whch the value of ts portfolo
More informationInformation Flow and Recovering the. Estimating the Moments of. Normality of Asset Returns
Estmatng the Moments of Informaton Flow and Recoverng the Normalty of Asset Returns Ané and Geman (Journal of Fnance, 2000) Revsted Anthony Murphy, Nuffeld College, Oxford Marwan Izzeldn, Unversty of Lecester
More informationConstruction Rules for Morningstar Canada Dividend Target 30 Index TM
Constructon Rules for Mornngstar Canada Dvdend Target 0 Index TM Mornngstar Methodology Paper January 2012 2011 Mornngstar, Inc. All rghts reserved. The nformaton n ths document s the property of Mornngstar,
More informationCHAPTER 3: BAYESIAN DECISION THEORY
CHATER 3: BAYESIAN DECISION THEORY Decson makng under uncertanty 3 rogrammng computers to make nference from data requres nterdscplnary knowledge from statstcs and computer scence Knowledge of statstcs
More information02_EBA2eSolutionsChapter2.pdf 02_EBA2e Case Soln Chapter2.pdf
0_EBAeSolutonsChapter.pdf 0_EBAe Case Soln Chapter.pdf Chapter Solutons: 1. a. Quanttatve b. Categorcal c. Categorcal d. Quanttatve e. Categorcal. a. The top 10 countres accordng to GDP are lsted below.
More informationIntroduction. Chapter 7 - An Introduction to Portfolio Management
Introducton In the next three chapters, we wll examne dfferent aspects of captal market theory, ncludng: Brngng rsk and return nto the pcture of nvestment management Markowtz optmzaton Modelng rsk and
More informationParallel Prefix addition
Marcelo Kryger Sudent ID 015629850 Parallel Prefx addton The parallel prefx adder presented next, performs the addton of two bnary numbers n tme of complexty O(log n) and lnear cost O(n). Lets notce the
More informationCorrelations and Copulas
Correlatons and Copulas Chapter 9 Rsk Management and Fnancal Insttutons, Chapter 6, Copyrght John C. Hull 2006 6. Coeffcent of Correlaton The coeffcent of correlaton between two varables V and V 2 s defned
More informationFinance 402: Problem Set 1 Solutions
Fnance 402: Problem Set 1 Solutons Note: Where approprate, the fnal answer for each problem s gven n bold talcs for those not nterested n the dscusson of the soluton. 1. The annual coupon rate s 6%. A
More informationFacility Location Problem. Learning objectives. Antti Salonen Farzaneh Ahmadzadeh
Antt Salonen Farzaneh Ahmadzadeh 1 Faclty Locaton Problem The study of faclty locaton problems, also known as locaton analyss, s a branch of operatons research concerned wth the optmal placement of facltes
More informationNotes on experimental uncertainties and their propagation
Ed Eyler 003 otes on epermental uncertantes and ther propagaton These notes are not ntended as a complete set of lecture notes, but nstead as an enumeraton of some of the key statstcal deas needed to obtan
More informationRisk Reduction and Real Estate Portfolio Size
Rsk Reducton and Real Estate Portfolo Sze Stephen L. Lee and Peter J. Byrne Department of Land Management and Development, The Unversty of Readng, Whteknghts, Readng, RG6 6AW, UK. A Paper Presented at
More informationTesting for Omitted Variables
Testng for Omtted Varables Jeroen Weese Department of Socology Unversty of Utrecht The Netherlands emal J.weese@fss.uu.nl tel +31 30 2531922 fax+31 30 2534405 Prepared for North Amercan Stata users meetng
More informationApplications of Myerson s Lemma
Applcatons of Myerson s Lemma Professor Greenwald 28-2-7 We apply Myerson s lemma to solve the sngle-good aucton, and the generalzaton n whch there are k dentcal copes of the good. Our objectve s welfare
More informationPASS Sample Size Software. :log
PASS Sample Sze Software Chapter 70 Probt Analyss Introducton Probt and lot analyss may be used for comparatve LD 50 studes for testn the effcacy of drus desned to prevent lethalty. Ths proram module presents
More informationLecture 7. We now use Brouwer s fixed point theorem to prove Nash s theorem.
Topcs on the Border of Economcs and Computaton December 11, 2005 Lecturer: Noam Nsan Lecture 7 Scrbe: Yoram Bachrach 1 Nash s Theorem We begn by provng Nash s Theorem about the exstance of a mxed strategy
More informationUsing Conditional Heteroskedastic
ITRON S FORECASTING BROWN BAG SEMINAR Usng Condtonal Heteroskedastc Varance Models n Load Research Sample Desgn Dr. J. Stuart McMenamn March 6, 2012 Please Remember» Phones are Muted: In order to help
More informationGlobal sensitivity analysis of credit risk portfolios
Global senstvty analyss of credt rsk portfolos D. Baur, J. Carbon & F. Campolongo European Commsson, Jont Research Centre, Italy Abstract Ths paper proposes the use of global senstvty analyss to evaluate
More informationProblems to be discussed at the 5 th seminar Suggested solutions
ECON4260 Behavoral Economcs Problems to be dscussed at the 5 th semnar Suggested solutons Problem 1 a) Consder an ultmatum game n whch the proposer gets, ntally, 100 NOK. Assume that both the proposer
More informationIntroduction to PGMs: Discrete Variables. Sargur Srihari
Introducton to : Dscrete Varables Sargur srhar@cedar.buffalo.edu Topcs. What are graphcal models (or ) 2. Use of Engneerng and AI 3. Drectonalty n graphs 4. Bayesan Networks 5. Generatve Models and Samplng
More information>1 indicates country i has a comparative advantage in production of j; the greater the index, the stronger the advantage. RCA 1 ij
69 APPENDIX 1 RCA Indces In the followng we present some maor RCA ndces reported n the lterature. For addtonal varants and other RCA ndces, Memedovc (1994) and Vollrath (1991) provde more thorough revews.
More informationISE Cloud Computing Index Methodology
ISE Cloud Computng Index Methodology Index Descrpton The ISE Cloud Computng Index s desgned to track the performance of companes nvolved n the cloud computng ndustry. Index Calculaton The ISE Cloud Computng
More information