_{1}

^{*}

In the series of quality monitoring schemes with exponentially weighted moving average, the exponentially weighted moving average distance square scheme was introduced for joint monitoring of process mean and variance. This scheme claims that it has a special feature that the control limits of the scheme are independent of sample size and therefore it gives more freedom to the users. However, this claim was not studied in detail. In this study, the control limits were found for this scheme through simulations, for different sample sizes with different combination of other scheme parameters. This study concludes that the control limits for designing this scheme are independent of sample size.

The exponentially weighted moving average (EWMA) chart was introduced for monitoring the sample mean of a quality parameter by Robrts in 1959 [_{t} is plotted against the sample number t(

Normally A_{0}, is considered as the target mean µ_{0}, l_{m} is a constant such that 0 < l_{m} < 1 and it is selected based on the shift in the mean to be detected quickly for any process for a given in-control average run length (ARL). ^{th} sample mean of the quality parameter, to be monitored. This chart issues an out-of-control signal if A_{t} is greater than the upper control limit (UCL) or lower than the lower control limit (LCL). For designing this chart, l_{m} values and control limits for detecting different shifts in mean under different sample sizes and in- control ARLs, can be found in Crowder (1989) [

Chang and Gan introduced an EWMA chart for monitoring sample variance of a quality parameter in 1993 [_{t} is plotted against the sample number t (

Normally, the value of B_{t} is taken as

interested. l_{v} is a positive constant which has the possible values of 0 < l_{v} < 1 and it is based on the shift in the variance to be detected quickly for a given in-control ARL. Like in the EWMA chart for monitoring sample mean, this chart also issues an out-of-control signal if B_{t} is greater than UCL or the lower than the LCL.

The above discussed two EWMA charts are used for monitoring the process mean and variance independently. In 1997, it was understood that monitoring the sample mean and variance was a bivariate problem and these two had to be monitored jointly [

Subsequently, a new joint monitoring scheme named EWMA distance square scheme (EWMAD2) was introduced with a claim that its control limits were independent of sample size [_{t} and variance V_{t} such that

and

the chi-square distribution with v degrees of freedom and ɸ(.) is the cumulative distribution function of a standard normal random variable [_{t} is the sample standard deviation, σ_{0} is the population standard deviation and n is the sample size.

A statistics

The EWMAD2 scheme is obtained by plotting the EWMAD2 statistic C_{t} against the sample number t where

mum

For this study, commonly used in-control ARLS100, 250, 300, 370, 500 and 1000 were selected. The sample sizes (n) studied were 5, 10, 50, 100 and 150. Samples were simulated in SAS using proc RANNO with sample sizes n. For each sample, the statistics _{t} and v_{t}. The statics C_{t} was obtained for different values of

Initially arbitrary control limits (CL) were assumed for each combination of _{1} < CL, then the second sample was simulated. This procedure was continued till C_{t} > CL where C_{t} is an out-of-control point. Each time, the number of samples generated till to find an out-of-control point was recorded and it is the run length. In the same way 100,000 runs were performed and the ARLs were found. Subsequently, the assumed CL values were adjusted till to obtain the required in-control ARLS of 100, 250, 300, 370, 500 and 1000 for different

The obtained CLs were plotted against the lambda value for the selected in-control ARLS and sample sizes. From the Figures 1-5, it could be observed that the control limits was increased with

The outputs for numerous programs were recorded and arranged by Ms. Iresha Dilhani.

Athambawa Mohamed Razmy, (2016) Effect of Sample Size on the Control Limits of Exponentially Weighted Moving Average Distance Square Scheme. Open Access Library Journal,03,1-7. doi: 10.4236/oalib.1102663

data raw;

mu0 = 0;

sigma = 1.0;

sigmasq = sigma*sigma;

nnn = 5;

nnnsqrt = sqrt(nnn);

nnn1 = nnn -1;

totalrun = 100,000;

maxsamp = 1,000,000;

seed = 86,924,569;

lam = 0.7;

onelm= 1.0 − lam;

bigh = 7.0856;

dobigd = 0;

dosmalld = 1;

actuals = smalld * sigma;

actualm = mu0 + bigd * sigma/nnnsqrt;

message = 'OK';

dorunnum = 1tototalrun;

qqq = 2;

runlen = 0;

do sample = 1tomaxsamp;

x1 = actualm + actuals * rannor (seed);

x2 = actualm + actuals * rannor (seed);

x3 = actualm + actuals * rannor (seed);

x4 = actualm + actuals * rannor (seed);

x5 = actualm + actuals * rannor (seed);

xsum = x1 + x2 + x3 + x4 + x5;

x2sum = x1*x1 + x2*x2 + x3*x3 + x4*x4 + x5*x5;

xbar = xsum/nnn;

s = x2sum − xsum*xsum/nnn;

s = s/nnn1;

uu = (xbar − mu0)/sigma*nnnsqrt;

uuu = uu*uu;

h = nnn1*s/sigmasq;

hhh= CDF('CHISQUARE', h, nnn1);

ifhhh > 0.0000000000001 and hhh < 0.999999999999999then

vv = probit(hhh); else

vv = 8.21;

vvv = vv*vv;

rrr = uuu + vvv;

runlen = runlen + 1;

qqq = onelm * qqq + lam * rrr;

ifqqq > bighthendo;

output;

goto DONE1;

end;

keep lam runlenbigdsmalldbighxbarqqqrrrvvvuuu message ;

end;

Message = 'Maximum # of sample Exceeded';

output;

DONE1:

end;

end;

end;

procmeans data = raw;

by bigdsmalld;

run;