1

A Shorter-Length Confidence-Interval Estimator (CIE) for Sharpe-Ratio Using a Multiplier k* to the

Usual Bootstrap-Resample CIE and Computational Intelligence

Chandra Shekhar Bhatnagar1, Chandrashekhar.Bhatnagar@sta.uwi.edu

Ashok Sahai2, sahai.ashok@gmail.com

Viswanadham Bulusu 3, viswanadhambulusu1970@gmail.com

ABSTRACT

The population value of the Sharpe (1966) performance measure for portfolio i is defined as i f

i

r

for i = 1, 2,..., n. It is simply the

mean excess return over the standard deviation of the excess returns for the portfolio. The sample estimate of any of these portfolio

measures is challenging not only because of the dynamic nature of this measure but also because of the statistical estimation issue

involved therein. This paper has been motivated by the desire to meet the challenge of statistical estimation. A new estimator for

Sharpe‟s ratio is formulated using the optimal value defined as k*, which is used as a multiplier for the usual sample-counterpart

estimator. This formulation has been pivotal to an efficient Confidence-Interval estimator (CIE). „Computational Intelligence‟ has

been deployed for the optimal mixing [using the optimal value of a design parameter, θ] of the proposed estimator and usual

sample-counterpart estimator with the aim of achieving the shortest length of the proposed bootstrap-resample Confidence-Interval

estimator (PBCIE) for Sharpe‟s ratio. We have been successful in achieving a shorter length through the proposed Bootstrap-

resample Confidence Interval Estimator [PBCIE] for Sharpe‟s ratio vis-à-vis the Usual Bootstrap-resample Confidence Interval

Estimator [UBCIE], without paying the usual cost in terms of a greater „Coverage Error‟. An empirical simulation study has been

used to bring forth the potential gain through a more efficient estimation in the context, with the limitation of the normal distribution

being followed by the population for the portfolios. The empirical simulation study is modeled around the values of parameters in

the study by Vinod and Morey (1999)

Key Words & Phrases: Bootstrap resampling; Computational intelligence; Coverage error & the length of the confidence interval

estimation; Empirical simulation study.

1 Department of Management Studies, The University of the West Indies,St. Augustine, Trinidad 2 Department of Mathematics and Statistics, The University of the West Indies, St. Augustine, Trinidad 3 Principal, Aurora‟s Degree & PG College, Chikkadpally; Hydearbad, India.

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I. Introduction

The Sharpe‟s Ratio (1966, 1994) is one of the standard tools in evaluating the performance of portfolios. It is

intuitive and lends itself easily to computation and interpretation. It represents the excess return per unit of

standard deviation of return and can be easily looked up as the slope of the capital allocation line (CAL).

Since the Sharpe‟s ratio gives us the price of risk, it follows that the ratio is widely used in comparing

portfolio performance and identifying superior portfolios. Using Roy‟s (1952) argument to use the reward –

to- risk ratio in appraising portfolio (or any strategy‟s) performance, the Sharpe‟s ratio modifies Markowitz‟s

(1952) efficient frontier of risky assets by incorporating a proxy risk-free rate (or the rate offered by a

benchamrk portfolio resulting in what we understand as the Information Ratio). From there, it facilitates the

task of identifying the optimum portfolio that is tangent to the the efficient frontier, having a property of

maximizing the slope of CAL.

Returns are often not normal (Goetzmann, Ingersoll, Spiegel and Welch (2002)), and comparable over time

(Sharpe (1994), Lo (2002)) and since they serve as the first data admission in computing Sharpe‟s ratio, there

is bound to be an estimation error in the ratio itself. The recognition of non-normality as a contributing factor

in estimation errors has always nagged the researchers, more so since Michaud (1989) examined and

questioned the “optimality” of the “optimized” portfolio in the Markowitz mean-variance paradigm. Further,

due to the presence of a random denominator in the definition of the ratio, the sampling distribution of the

Sharpe ratio is somewhat difficult to determine, even when using data from large samples (Christie (2007)).

Lo (2002) set the ball rolling with his work on the statistical distribution of the Sharpe ratio under various

scenarios for the return distribution. Earlier, Jobson and Korkie (1981) did consider a test of the difference

between Sharpe ratios under the assumption of multivariate normality, but found that their test lacked power.

We tackle the challenge of statistical estimation by attempting to propose a new estimator for Sharpe‟s ratio to

serve as an efficient Confidence-Interval estimator (CIE). We draw from the work of Vinod and Morey (2001)

who proposed a modified version of the Sharpe ratio, called the Double Sharpe ratio, to take into account

estimation risk. We use bootstrap sampling and computational intelligence with the aim of achieving the

shortest length proposed bootstrap-resample Confidence-Interval estimator of the Sharpe‟s Ratio. The

procedure followed is similar but not limited to Sahai and Strepnek (2011)

In the next section, we specify the Sharpe‟s ratio and discuss our proposed bootstrap confidence interval (CI)

estimator for the ratio. Section III illustrates the relative gain from our proposed estimator. Section IV

concludes the paper.

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II. The Proposed Bootstrap Confidence Interval Estimator for Sharpe’s Ratio

Let us say, there is an investment position, P, invested partially in a risky portfolio and a risk-free asset. If w

represents the weight on the risky portfolio i, µ represents the expected return, and rf is the risk-free rate of

return, the expecte return of position P can be described as:

(1 )P i fw w r (1)

We can re-write (1) as follows:

P i f fw wr r , or

[ ]P f i fr w r (2)

Since the risk of combined position P, as measured by the standard deviation of expected return, would

emanate from only the risky portfolio, we can write the standard deviation of position P as:

P iw , which further implies that;

P

i

w

(3)

Substituting (3) in (2) yields the capital allocation line, CAL;

PP f i f

i

r r

(4)

From (4), it can be seen that i f

i

r

is the slope of the CAL, also known as the Sharpe‟s ratio, Sri. The

population value of the Sharpe‟s performance measure for portfolio i is simply the mean excess return over

the standard deviation of the excess returns for the portfolio. Therefore, for any portfolio in general, we have

the population value of the Sharpe‟s ratio as i f

i

r

. In the aforesaid μ is the population mean, rf is the risk-

free rate of return with zero variance and σi is the portfolio‟s standard deviation. It is commonly seen in the

literature, that the sample counterparts that estimate these population parameters, namely the sample mean,

and the sample standard deviation are calculated from a random sample (x1, x2, x3,…, xn) as follows:

4

1

n

i

i

x

xn

,

2

2 1

1

n

i

i

x x

sn

, and

2s s

This leads to an applied point estimate for Sharpe‟s ratio of;

ˆx

Srs

As indicated earlier, one problem with the Sharpe‟s ratio is that its denominator is random, as it is computed

using a data sample of returns on a given history and not the whole population of returns. So, it is difficult to

evaluate its risk estimation. Vinod and Morey (2001) proposed a modified version of the Sharpe‟s ratio, called

the Double Sharpe ratio, to take into account the estimation risk. This ratio is defined as ˆ

ˆˆ

Sr

SrDSr

, where

ˆSr is the standard deviation of the Sharpe ratio estimate, or the estimation risk. They do not use the sample

(x1, x2, x3,…, xn) directly in the calculation of their „Double Sharpe Ratio‟. Instead, to calculate this standard

deviation they use bootstrap methodology to generate 999 resamples from the original returns. We note here

that the sampling error is seminal to the estimation risk or the estimation error. Vinod and Morey (2001)‟s

„Double Sharpe ratio‟ takes care of this „estimation error‟ implicitly, just like the Sharpe‟s ratio.

Our proposed improvement of Vinod and Morey (2001)‟s „Double Sharpe Ratio‟ also uses 999 bootstrap

resamples. These resamples have been used to calculate the bootstrap mean of these estimates and hence

„Confidence Interval Estimate (CIE)‟ of Sharpe‟s ratio. The improved „Confidence Interval Estimate (CIE)‟ of

Sharpe‟s ratio constructed and proposed in this paper consists in estimation of both the numerators & the

denominators of the ratio.

For doing so, our first target is to find an efficient estimator of the inverse of the normal standard deviation 1

. For that we first prove the following result:

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Lemma: For a random sample (x1, x2, x3, …, xn) from a normal population N (μ, σ), * 1.k

is the “Minimum

Mean-Square-Error Estimator (MMSE) of 1

; wherein;

2. 1

( 1) 2*

( 3)

2

n

nk

n

(5)

Proof: It can easily be checked that;

*

2

1 1.

1

Es

k

Es

Rest follows from the well-known fact that (n-1). s2 ~

2n-1.

Q.E.D.

We can handle the efficient estimation of numerator ˆSr of the Double Sharpe ratio, by possibly a more

efficient estimator using bootstrap resamples. This new estimator will take care of the „estimator error‟, rather

less explicitly.

To illustrate our proceedings comprehensibly we, for the ready comparative reference, borrow the study by

Vinod and Morey (1999) and to keep things in perspective as we proceed forward, we reproduce a selection

from their work:

“……the resampling for the Sharpe measure is done “with replacement” of the original excess returns

themselves for j =1,2, …, J or 999 times. Thus, we calculate 999 Sharpe measures from the original excess

return series. The choice of the odd number 999 is convenient, since the rank-ordered 25-th and 975-th values

of estimated Sharpe ratios arranged from the smallest to the largest, yield a useful 95% confidence interval. It

is from these 999 Sharpe measures that we calculate ( rS ˆ ). As an illustration, we have calculated the

Sharpe and Double Sharpe ratios for the 30 largest growth mutual funds. (as of January 1998 in terms of

overall assets managed)”.

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Their work reports the excess mean monthly returns, the standard deviation of the excess monthly returns, the

Sharpe ratio, the mean and standard deviation of the bootstrapped Sharpe ratios, the lower [0.025] and upper

[0.975] confidence values of the bootstrapped Sharpe value, the 95% confidence interval width, and the

Double Sharpe ratio. They mention that in their case, the sampling distribution that represents the estimation

risk is non-normal with positive skewness. This is why the means of the bootstrapped Sharpe ratios are always

slightly higher than the point estimates of Sharpe ratios, which ignore the estimation risk altogether.

In line with their work, we compute the „Usual Bootstrap-resample Confidence-Interval Estimator [UBCIE]

of Sharpe‟s ratio using the “the lower [0.025] and upper [0.975] confidence values of the bootstrapped

Sharpe value”, say LCV & UCV respectively.

Our “Proposed Bootstrap-resample Confidence-Interval Estimator [PBCIE] of Sharpe‟s ratio” has two phases.

In the first phase, we propose the “Modified Bootstrap-resample Confidence-Interval Estimator [MBCIE] of

Sharpe‟s ratio by modifying the LCV & UCV with k*, as below:

MLCV = k*. LCV (6)

and,

MUCV = k*. UCV (7)

In the second phase, we consider the “Proposed Bootstrap-resample Confidence-Interval Estimator [PBCIE]

of Sharpe‟s ratio as the one with the “Lower & Upper Confidence Values” , as below:

PLCV = LCV + θ.(LCV - k*.LCV) (8)

and

PUCV = UCV + θ.(UCV - k*.UCV) (9)

The parameter θ is the “Design Parameter” in the “Proposed Bootstrap-resample Confidence-Interval

Estimator [PBCIE] of Sharpe-Ratio” proposed to be used “Optimally” to design the proposed PBCIE is such a

manner as to lower the its “length” without paying the usual price in terms of the “Coverage Error” with the

CIE. This has been achieved by using the “Computational Intelligence” to determine that “Optimal Value” of

the design-parameter θ. The extensive simulational computational results lead to this sought-after Optimal-

Value of the design-parameter θ to be 15.6.

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III. Empirical Study Illustrating the Relative Gain through our Proposed CI Estimator

The empirical simulation study is carried out using a Matlab 2010b code for various illustrative values of the

sample sizes ~ 11, 21, 31, 41, 51, 71, 85 & 101. The parent population is taken to be normal with the

population Sharpe ratio „Sr‟ as 0.20 [around the values of the case-study in Vinod and Morey (1999)]. The

various values of population standard deviation (likewise in that study) are taken as σ = 3.25, 3.75, 4.25, 4.75,

5.25, 5.75, 6.25, 6.75 & 7.25 for illustration. The number of replications in the simulation is taken to be quite

large ~ „4444‟.

In each of these (4444) replications, for a typical value-combination of “n & σ”, we let the computer generate

the psuedo-random sample from the relevant “Normal Distribution” with the value of σ , and that of the

„Sharpe Ratio (Sr = 0.2)”. That sample is then used for the „999‟ bootstrap-resamples leading to the „Lower &

Upper‟ values of the various (95%) CIs; LCV & UCV for „Usual Bootstrap-resample Confidence-Interval

Estimator [UBCIE] for Sharpe‟s ratio & PLCV & PUCV for „Proposed Bootstrap-resample Confidence-

Interval Estimator [PBCIE] for Sharpe‟s ratio . Thence, from all these 4444 replications, we calculate for

these CIs {[UBCIE] & [PBCIE]}, the following performance-icons‟ values:

Coverage Probability ~ The relative frequency of the actual value being inside the relevant CI.

Coverage Error ~ Abs. (0.95 - Coverage Probability) for the relevant CI.

Length ~ average of the values Abs.(Lower Value of CI – Upper value of the CI) for the relevant CI.

Left Bias ~ average of the values Abs.(Lower Value of CI – Actual Value of SR (0.2)) {iff the actual value is

less than the lower value of the CI} for the relevant CI.

Right Bias ~ average of the values Abs.(Upper Value of CI – Actual Value of SR (0.2)) {iff the actual value is

more than the upper value of the CI} for the relevant CI.

Relative Bias = Abs. Left Bias - Right Bias

Left Bias + Right Bias

The resultant values of these „Features of CIs‟ are tabulated in Tables A.1 through A.9, in the appendix.

As apparent from the tabulated values in the appendix, our „Proposed Bootstrap-resample Confidence-Interval

Estimator [PBCIE] of Sharpe‟s ratio is consistently shorter in its “length” than the „Usual Bootstrap-resample

8

Confidence-Interval Estimator [UBCIE] of Sharpe‟s ratio‟. Nevertheless, we have evidently manipulated

[through our optimal choice of the value of the design-parameter θ in our PBCIE of Sharpe‟s ratio using

“Computational Intelligence”] for the said “Optimal mixing of UBCIE of Sharpe‟s ratio & the “Modified

Bootstrap-resample Confidence-Interval Estimator [MBCIE] of Sharpe‟s ratio.

IV. Conclusion

The Sharpe's ratio is one of the widely prevalent metrics for evaluating and comparing portfolio performance.

It relates the reward of investing with its inherent risk. However, in a world where financial data does not

strictly conform to desired statistical attributes, the ratio‟s reliability may be questioned due to statistical

estimation error. This study seeks to and achieves an improvement in so far as a more robust statistical

estimation is evident by using bootstrap resampling and computational intelligence. The simulation results

demonstrate a marked improvement in terms of a shorter length of the proposed bootstrap-resample

Confidence-Interval estimator (PBCIE) for Sharpe‟s ratio, while keeping the coverage error in check. This has

been achieved by using k* as the multiplier for the usual sample counterpart estimator for the Sharpe‟s ratio

of the population. Using computational intelligence, a unique design parameter, θ, has been arrived at for an

optimal mixing of the proposed estimator and usual sample-counterpart estimator.

References:

Goetzmann, William N, Jonathan E Ingersoll, Matthew I Spiegel, and Ivo Welch, 2002, “Sharpening Sharpe

Ratios”, Yale Working Paper

H.D. Vinod and M.R. Morey, 1999, A Double Sharpe Ratio, Fordham University, New York

H.M. Markowitz, 1952, “Portfolio Selection”, Journal of Finance, 7(1), 77-91

Jobson, J.D. and B.M. Korkie, 1981, "Performance Hypothesis Testing with the Sharpe and Treynor

Measures," Journal of Finance, 36, No. 4, 889-908

Lo, Andrew W, 2002, “The Statistics of Sharpe Ratios”, Financial Analysts Journal, 36-52

Michaud, Richard O, 1989, “The Markowitz Optimization Enigma: Is 'Optimized' Optimal?”, Financial

Analysts Journal, 31-42

9

Roy, Arthur D., 1952, “Safety First and Holding of Assets”, Econometricia, 431-450

Sahai, Ashok and Strepnek, Grant. H, 2011, “An Estimation Error Corrected Sharpe Ratio using Bootstrap

Resampling”, Journal of Applied Finance and Banking, Vol.1, No.2, 189-206

S. Christie, 2007, Beware the Sharpe Ratio, Macquarie University Applied Finance Centre, Sydney

Vinod, H. D. and M. R. Morey, 2001 “A Double Sharpe Ratio” in Advances in Investment Analysis and

Portfolio Management, Vol. 8.

W.F. Sharpe, 1966, “Mutual Fund Performance”, Journal of Business, 39, 119-138

W.F. Sharpe, 1994, “The Sharpe Ratio”, Journal of Portfolio Management, 21(1), 49-58

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APPENDIX

SIMULATION RESULTS [4444REPLICATIONS] FOR 95% CI :

THE PROPOSED VIS-A-VIS THE USUAL BOOTSTRAP CI FOR Sr USING k* AND COMPUTATIONAL INTELLIGENCE

Table A.1

n = 11, k*=0.922746

σ = 3.250000

95% CI Estimator for SR Coverage Probability Coverage Error Length Left Bias Right Bias Relative Bias

UBCIEstimator 0.904365 0.045635 0.775564 0.029478 0.066157 0.383529

PBCIEstimator 0.904365 0.045635 0.304701 0.066157 0.029478 0.383529

σ = 3.750000

UBCIEstimator 0.908191 0.041809 0.770863 0.029703 0.062106 0.352941

PBCIEstimator 0.908191 0.041809 0.304089 0.062106 0.029703 0.352941

σ = 4.25

UBCIEstimator 0.90099 0.04901 0.770903 0.066157 0.032853 0.336364

PBCIEstimator 0.90099 0.04901 0.30294 0.066157 0.032853 0.336364

σ = 4.75

UBCIEstimator 0.90099 0.04901 0.774925 0.032853 0.066157 0.336364

PBCIEstimator 0.90099 0.04901 0.30294 0.066157 0.032853 0.336364

σ = 5.25

UBCIEstimator 0.903915 0.046085 0.775883 0.031953 0.064131 0.334895

PBCIEstimator 0.903915 0.046085 0.305519 0.064131 0.031953 0.334895

σ = 5.75

UBCIEstimator 0.897165 0.052835 0.797595 0.035104 0.067732 0.317287

PBCIEstimator 0.897165 0.052835 0.306319 0.067732 0.035104 0.317287

σ = 6.25

UBCIEstimator 0.89964 0.05036 0.778255 0.033303 0.067057 0.336323

PBCIEstimator 0.89964 0.05036 0.303756 0.067057 0.033303 0.336323

σ = 6.75

UBCIEstimator 0.901665 0.048335 0.778221 0.033078 0.065257 0.327231

PBCIEstimator 0.901665 0.048335 0.304833 0.065257 0.033078 0.327231

σ = 7.25

UBCIEstimator 0.90009 0.04991 0.79186 0.029253 0.070657 0.414414

PBCIEstimator 0.90009 0.04991 0.306329 0.070657 0.029253 0.414414

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Table A.2

n = 21, k*=0.961945

σ = 3.250000

95% CI Estimator

for SR

Coverage

Probability Coverage Error Length Left Bias Right Bias Relative Bias

UBCIEstimator 0.922142 0.027858 0.699992 0.025653 0.052205 0.34104

PBCIEstimator 0.922142 0.027858 0.384994 0.025653 0.052205 0.34104

σ = 3.750000

UBCIEstimator 0.925518 0.024482 0.701911 0.027228 0.047255 0.268882

PBCIEstimator 0.925518 0.024482 0.385653 0.027228 0.047255 0.268882

σ = 4.25

UBCIEstimator 0.928443 0.021557 0.69918 0.024752 0.046805 0.308176

PBCIEstimator 0.928443 0.021557 0.384956 0.024752 0.046805 0.308176

σ = 4.75

UBCIEstimator 0.926193 0.023807 0.699455 0.027228 0.04658 0.262195

PBCIEstimator 0.926193 0.023807 0.385489 0.027228 0.04658 0.262195

σ = 5.25

UBCIEstimator 0.923267 0.026733 0.698154 0.026553 0.05018 0.307918

PBCIEstimator 0.923267 0.026733 0.384651 0.026553 0.05018 0.307918

σ = 5.75

UBCIEstimator 0.930693 0.019307 0.698936 0.025653 0.043654 0.25974

PBCIEstimator 0.930693 0.019307 0.385386 0.025653 0.043654 0.25974

σ = 6.25

UBCIEstimator 0.924842 0.025158 0.695847 0.031503 0.043654 0.161677

PBCIEstimator 0.924842 0.025158 0.384979 0.031503 0.043654 0.161677

σ = 6.75

UBCIEstimator 0.926868 0.023132 0.698519 0.025878 0.047255 0.292308

PBCIEstimator 0.926868 0.023132 0.384726 0.025878 0.047255 0.292308

σ = 7.25

UBCIEstimator 0.930243 0.019757 0.701766 0.023177 0.04658 0.335484

PBCIEstimator 0.930243 0.019757 0.385501 0.023177 0.04658 0.335484

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Table A.3

n = 31, k*=0.974754

σ = 3.250000

95% CI Estimator for SR Coverage Probability Coverage Error Length Left Bias Right Bias Relative Bias

UBCIEstimator 0.932493 0.017507 0.622104 0.027453 0.040054 0.186667

PBCIEstimator 0.932493 0.017507 0.457921 0.027453 0.040054 0.186667

σ = 3.750000

UBCIEstimator 0.936994 0.013006 0.618924 0.026103 0.036904 0.171429

PBCIEstimator 0.936994 0.013006 0.456452 0.026103 0.036904 0.171429

σ = 4.25

UBCIEstimator 0.935869 0.014131 0.621041 0.024077 0.040054 0.249123

PBCIEstimator 0.935869 0.014131 0.457496 0.024077 0.040054 0.249123

σ = 4.75

UBCIEstimator 0.935419 0.014581 0.619503 0.025203 0.039379 0.219512

PBCIEstimator 0.935419 0.014581 0.456784 0.025203 0.039379 0.219512

σ = 5.25

UBCIEstimator 0.939694 0.010306 0.620684 0.026103 0.034203 0.134328

PBCIEstimator 0.939694 0.010306 0.457053 0.026103 0.034203 0.134328

σ = 5.75

UBCIEstimator 0.931818 0.018182 0.618582 0.028578 0.039604 0.161716

PBCIEstimator 0.931818 0.018182 0.456729 0.028578 0.039604 0.161716

σ = 6.25

UBCIEstimator 0.939019 0.010981 0.620812 0.024752 0.036229 0.188192

PBCIEstimator 0.939019 0.010981 0.456905 0.024752 0.036229 0.188192

σ = 6.75

UBCIEstimator 0.932943 0.017057 0.620508 0.024302 0.042754 0.275168

PBCIEstimator 0.932943 0.017057 0.457767 0.024302 0.042754 0.275168

σ = 7.25

UBCIEstimator 0.940819 0.009181 0.620678 0.021377 0.037804 0.277567

PBCIEstimator 0.940819 0.009181 0.457566 0.021377 0.037804 0.277567

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Table A.4

n = 41, k*=0.981112

σ = 3.250000

95% CI Estimator for SR Coverage Probability Coverage Error Length Left Bias Right Bias Relative Bias

UBCIEstimator 0.932043 0.017957 0.562544 0.022952 0.045005 0.324503

PBCIEstimator 0.932043 0.017957 0.456505 0.022952 0.045005 0.324503

σ = 3.750000

UBCIEstimator 0.939919 0.010081 0.560547 0.023627 0.036454 0.213483

PBCIEstimator 0.939919 0.010081 0.456179 0.023627 0.036454 0.213483

σ = 4.25

UBCIEstimator 0.943294 0.006706 0.558505 0.018902 0.037804 0.333333

PBCIEstimator 0.943294 0.006706 0.455138 0.018902 0.037804 0.333333

σ = 4.75

UBCIEstimator 0.940594 0.009406 0.558583 0.024977 0.034428 0.159091

PBCIEstimator 0.940594 0.009406 0.455309 0.024977 0.034428 0.159091

σ = 5.25

UBCIEstimator 0.943744 0.006256 0.558568 0.022502 0.033753 0.2

PBCIEstimator 0.943744 0.006256 0.455104 0.022502 0.033753 0.2

σ = 5.75

UBCIEstimator 0.941044 0.008956 0.560738 0.023177 0.035779 0.21374

PBCIEstimator 0.941044 0.008956 0.455998 0.023177 0.035779 0.21374

σ = 6.25

UBCIEstimator 0.943969 0.006031 0.560158 0.022727 0.033303 0.188755

PBCIEstimator 0.943969 0.006031 0.455534 0.022727 0.033303 0.188755

σ = 6.75

UBCIEstimator 0.939019 0.010981 0.558946 0.024527 0.036454 0.195572

PBCIEstimator 0.939019 0.010981 0.455623 0.024527 0.036454 0.195572

σ = 7.25

UBCIEstimator 0.937219 0.012781 0.56014 0.029253 0.033528 0.0681

PBCIEstimator 0.937219 0.012781 0.456249 0.029253 0.033528 0.0681

14

Table A.5

n = 51, k*=0.984912

σ = 3.250000

95% CI Estimator for SR Coverage Probability Coverage Error Length Left Bias Right Bias Relative Bias

UBCIEstimator 0.937669 0.012331 0.512111 0.024077 0.038254 0.227437

PBCIEstimator 0.937669 0.012331 0.439221 0.024077 0.038254 0.227437

σ = 3.750000

UBCIEstimator 0.949595 0.000405 0.511922 0.022277 0.028128 0.116071

PBCIEstimator 0.949595 0.000405 0.438955 0.022277 0.028128 0.116071

σ = 4.25

UBCIEstimator 0.942844 0.007156 0.512718 0.020027 0.037129 0.299213

PBCIEstimator 0.942844 0.007156 0.438928 0.020027 0.037129 0.299213

σ = 4.75

UBCIEstimator 0.938569 0.011431 0.511413 0.028803 0.032628 0.062271

PBCIEstimator 0.938569 0.011431 0.438753 0.028803 0.032628 0.062271

σ = 5.25

UBCIEstimator 0.934293 0.015707 0.513681 0.024077 0.041629 0.267123

PBCIEstimator 0.934293 0.015707 0.439384 0.024077 0.041629 0.267123

σ = 5.75

UBCIEstimator 0.943969 0.006031 0.512978 0.022952 0.033078 0.180723

PBCIEstimator 0.943969 0.006031 0.439233 0.022952 0.033078 0.180723

σ = 6.25

UBCIEstimator 0.943969 0.006031 0.512852 0.021377 0.034653 0.236948

PBCIEstimator 0.943969 0.006031 0.439163 0.021377 0.034653 0.236948

σ = 6.75

UBCIEstimator 0.94577 0.00423 0.511085 0.024302 0.029928 0.103734

PBCIEstimator 0.94577 0.00423 0.43874 0.024302 0.029928 0.103734

σ = 7.25

UBCIEstimator 0.941719 0.008281 0.511504 0.025878 0.032403 0.111969

PBCIEstimator 0.941719 0.008281 0.438982 0.025878 0.032403 0.111969

15

Table A.6

n = 61, k*=0.987439

σ = 3.250000

95% CI Estimator for SR Coverage Probability Coverage Error Length Left Bias Right Bias Relative Bias

UBCIEstimator 0.942844 0.007156 0.475927 0.024527 0.032628 0.141732

PBCIEstimator 0.942844 0.007156 0.419925 0.024527 0.032628 0.141732

σ = 3.750000

UBCIEstimator 0.940369 0.009631 0.476062 0.024077 0.035554 0.192453

PBCIEstimator 0.940369 0.009631 0.420332 0.024077 0.035554 0.192453

σ = 4.25

UBCIEstimator 0.94532 0.00468 0.475442 0.025878 0.028803 0.053498

PBCIEstimator 0.94532 0.00468 0.419655 0.025878 0.028803 0.053498

σ = 4.75

UBCIEstimator 0.944644 0.005356 0.475901 0.021152 0.034203 0.235772

PBCIEstimator 0.944644 0.005356 0.419931 0.021152 0.034203 0.235772

σ = 5.25

UBCIEstimator 0.943744 0.006256 0.475624 0.023402 0.032853 0.168

PBCIEstimator 0.943744 0.006256 0.420421 0.023402 0.032853 0.168

σ = 5.75

UBCIEstimator 0.938119 0.011881 0.474645 0.028128 0.033753 0.090909

PBCIEstimator 0.938119 0.011881 0.419939 0.028128 0.033753 0.090909

σ = 6.25

UBCIEstimator 0.942394 0.007606 0.475762 0.023402 0.034203 0.1875

PBCIEstimator 0.942394 0.007606 0.419811 0.023402 0.034203 0.1875

σ = 6.75

UBCIEstimator 0.937669 0.012331 0.475538 0.026553 0.035779 0.148014

PBCIEstimator 0.937669 0.012331 0.420101 0.026553 0.035779 0.148014

σ = 7.25

UBCIEstimator 0.937669 0.012331 0.475136 0.025878 0.036454 0.169675

PBCIEstimator 0.937669 0.012331 0.420178 0.025878 0.036454 0.169675

16

Table A.7

n = 71, k*=0.989241

σ = 3.250000

95% CI Estimator for SR Coverage Probability Coverage Error Length Left Bias Right Bias Relative Bias

UBCIEstimator 0.939469 0.010531 0.445152 0.022502 0.038029 0.256506

PBCIEstimator 0.939469 0.010531 0.401157 0.022502 0.038029 0.256506

σ = 3.750000

UBCIEstimator 0.94712 0.00288 0.444846 0.022502 0.030378 0.148936

PBCIEstimator 0.94712 0.00288 0.401085 0.022502 0.030378 0.148936

σ = 4.25

UBCIEstimator 0.942169 0.007831 0.445246 0.024527 0.033303 0.151751

PBCIEstimator 0.942169 0.007831 0.401377 0.024527 0.033303 0.151751

σ = 4.75

UBCIEstimator 0.943069 0.006931 0.444511 0.027903 0.029028 0.019763

PBCIEstimator 0.943069 0.006931 0.401178 0.027903 0.029028 0.019763

σ = 5.25

UBCIEstimator 0.945095 0.004905 0.444467 0.024302 0.030603 0.114754

PBCIEstimator 0.945095 0.004905 0.40127 0.024302 0.030603 0.114754

σ = 5.75

UBCIEstimator 0.943969 0.006031 0.444334 0.021827 0.034203 0.220884

PBCIEstimator 0.943969 0.006031 0.400812 0.021827 0.034203 0.220884

σ = 6.25

UBCIEstimator 0.942844 0.007156 0.444299 0.025653 0.031503 0.102362

PBCIEstimator 0.942844 0.007156 0.401083 0.025653 0.031503 0.102362

σ = 6.75

UBCIEstimator 0.944644 0.005356 0.444934 0.024077 0.031278 0.130081

PBCIEstimator 0.944644 0.005356 0.401544 0.024077 0.031278 0.130081

σ = 7.25

UBCIEstimator 0.939019 0.010981 0.443682 0.026328 0.034653 0.136531

PBCIEstimator 0.939019 0.010981 0.400823 0.026328 0.034653 0.136531

17

Table A.8

n = 85, k*=0.991040

σ = 3.250000

95% CI Estimator for SR Coverage Probability Coverage Error Length Left Bias Right Bias Relative Bias

UBCIEstimator 0.948245 0.001755 0.410266 0.023177 0.028578 0.104348

PBCIEstimator 0.948245 0.001755 0.377535 0.023177 0.028578 0.104348

σ = 3.750000

UBCIEstimator 0.95027 0.00027 0.40935 0.020477 0.029253 0.176471

PBCIEstimator 0.95027 0.00027 0.376814 0.020477 0.029253 0.176471

σ = 4.25

UBCIEstimator 0.952295 0.002295 0.410413 0.020477 0.027228 0.141509

PBCIEstimator 0.952295 0.002295 0.377576 0.020477 0.027228 0.141509

σ = 4.75

UBCIEstimator 0.943519 0.006481 0.410511 0.021827 0.034653 0.227092

PBCIEstimator 0.943519 0.006481 0.377481 0.021827 0.034653 0.227092

σ = 5.25

UBCIEstimator 0.946445 0.003555 0.409964 0.026778 0.026778 0

PBCIEstimator 0.946445 0.003555 0.377319 0.026778 0.026778 0

σ = 5.75

UBCIEstimator 0.95162 0.00162 0.410785 0.021377 0.027003 0.116279

PBCIEstimator 0.95162 0.00162 0.377858 0.021377 0.027003 0.116279

σ = 6.25

UBCIEstimator 0.943744 0.006256 0.410274 0.026328 0.029928 0.064

PBCIEstimator 0.943744 0.006256 0.37727 0.026328 0.029928 0.064

σ = 6.75

UBCIEstimator 0.944869 0.005131 0.410498 0.023627 0.031503 0.142857

PBCIEstimator 0.944869 0.005131 0.377696 0.023627 0.031503 0.142857

σ = 7.25

UBCIEstimator 0.946445 0.003555 0.410325 0.025203 0.028353 0.058824

PBCIEstimator 0.946445 0.003555 0.377411 0.025203 0.028353 0.058824

18

Table A.9

n = 101, k*=0.992478

σ = 3.250000

95% CI Estimator for SR Coverage Probability Coverage Error Length Left Bias Right Bias Relative Bias

UBCIEstimator 0.944194 0.005806 0.379228 0.026103 0.029703 0.064516

PBCIEstimator 0.944194 0.005806 0.35415 0.026103 0.029703 0.064516

σ = 3.750000

UBCIEstimator 0.936994 0.013006 0.379392 0.027678 0.035329 0.121429

PBCIEstimator 0.936994 0.013006 0.354069 0.027678 0.035329 0.121429

σ = 4.25

UBCIEstimator 0.939019 0.010981 0.379455 0.024977 0.036004 0.180812

PBCIEstimator 0.939019 0.010981 0.354343 0.024977 0.036004 0.180812

σ = 4.75

UBCIEstimator 0.946895 0.003105 0.379154 0.021827 0.031278 0.177966

PBCIEstimator 0.946895 0.003105 0.35401 0.021827 0.031278 0.177966

σ = 5.25

UBCIEstimator 0.946445 0.003555 0.379266 0.022727 0.030828 0.151261

PBCIEstimator 0.946445 0.003555 0.354091 0.022727 0.030828 0.151261

σ = 5.75

UBCIEstimator 0.94847 0.00153 0.379468 0.024077 0.027453 0.065502

PUBCIEstimator 0.94847 0.00153 0.354477 0.024077 0.027453 0.065502

σ = 6.25

UBCIEstimator 0.943519 0.006481 0.379636 0.024302 0.032178 0.139442

PBCIEstimator 0.943519 0.006481 0.354252 0.024302 0.032178 0.139442

σ = 6.75

UBCIEstimator 0.945095 0.004905 0.379695 0.022277 0.032628 0.188525

PBCIEstimator 0.945095 0.004905 0.354264 0.022277 0.032628 0.188525

σ = 7.25

UBCIEstimator 0.937894 0.012106 0.37939 0.030378 0.031728 0.021739

PBCIEstimator 0.937894 0.012106 0.354378 0.030378 0.031728 0.021739