A Note on β-Content γ-Level Tolerance Interval for IFR Class Distribution

A Note on -Content -Level Tolerance Interval for IFR Class Distribution

R. R. Kumbhar D.T. Shirke

Department of Statistics Department of Statistics

P.D.V.P. Mahavidyalaya, Tasgaon. Shivaji University, Kolhapur.

Dist.: Sangali. India.416312 India. 416004.

ABSTRACT

Barlow and Proschan (1966) have obtained -content -level Tolerance Intervals for IFR as well as IFRA class of distributions. Their results are based on inequalities for linear combinations of ordered statistics from IFR and IFRA families. The model considered in this chapter belongs to IFR class of distribution. We obtain parametric

-content -level Tolerance Interval for the same following Kumbhar and Shirke (2004). We further compare performance of parametric Tolerance interval with the one given by Barlow and Proschan (1966).

Key Words: -content -level Tolerance Interval, IFR class of distribution.

1. Introduction

   In general, term Tolerance Interval (TI) is an interval determined from observed values of a random sample for the purpose of drawing inferences about the proportion of a distribution contained in that interval. Usually TI is designed to capture at least a given proportion of some distribution. Two types of TI have received considerable attention in the literature; -expectation TI and -content – level TI. In order to be more specific about the meaning of TI, let X be a measurable characteristic having a distribution function F(x;), . Let L(X) and U(X) be two functions of observations such that L( X) < U(X). Then (L(X), U(X)) is called a content level TI, if for given , (0, 1),

   U(X)

   P{ f(t; )dt } , for every , (1.1)

   L(X)

   where

   f(x; )

   is probability density function (pdf) of X. The quantity

   U( X)

   f(x; ) dx is

   L(X)

   called the sample coverage and L(X) and U(X) are called lower and upper tolerance limits, respectively. If we set L(X) = and obtain U(X) satisfying (1.1) then we get upper -content -level TI. Similarlly if we set U(X) = and obtain L satisfying (1.1) then we get lower -content -level TI. It is easy to observe that lower -content – level tolerance limit is also an upper (1-)-content (1-)-level tolerance limit. In the present study, we obtain only upper tolerance limits.

   Wilks (1941) treated the problem of determining TIs in pioneer article. Since then a large number of papers dealing with this and other aspects of tolerance limits have appeared in the literature. Jilek (1981) classifies papers according to general results, distribution free results, normal and multivariate normal distributions, gamma, exponential, Weibull and other continuous and discrete distributions. Patel (1986) provided a review, which contains a large collection of known results on content

   level TIs for some continuous and discrete univariate distributions.

   Barlow and Proschan (1966) have obtained TIs for IFR as well as IFRA class of distributions. Their results are based on inequalities for linear combinations of ordered statistics from IFR and IFRA families. The model considered in this article belongs to IFR class of distribution (Please see Lemma below). Therefore, it is interesting to compare content level TI based on MLE with respect to the one given by Barlow and Proschan (1966).

   The Model

   The cumulative distribution function (cdf) and probability density function (pdf) of lifetime distribution of largest of k independent and identically distributed (i.i.d.) exponential with mean are

   X

   X

   F x; 1exp x/ k x0, 0,

   (1.2)

   0 otherwise

   and k exp x/1exp x/ k-1

   x0, 0,

   (1.3)

   fX x;

   0

   otherwise

   respectively. The corresponding survival function is

   S(x;)1(1exp( x / )) k , (1.4)

   and the hazard function is

   h(x; )

   k exp( x / )(1exp( x / )) k

   (1(1exp( x / )) k

   . (1.5)

   The graph of the harazad function is given in Fig.2.1. The following lemma proves that distribution defined in (1.2) is IFR.

   Lemma 1.1: F(t; )=(1exp(t/))k is a member of IFR class of distribution, for k>1.

   Proof: Consider Harzard function defined as h(t)f (t;) / F(t;) ,

   where f(t; ) is pdf as given in (2.4.2) and

   F(t;)1F(t;) .

   Therefore,

   log h(t) logk (k 1) log1exp( t / ) t

   log11exp( t / )k .

   (1.6)

   If y1exp( t / ) , equation (1.6) reduces to

   logh(t) = logh(y) = c + (k – 1)log y log (1- y) – log (1- yk).

   Differentiating with respect to y we get

   d log h( y) (k1) 1 ky k1 . (1.7)

   dy y

   (1y)

   (1yk )

   When k=1, it corresponds to exponential distribution. For k 2, (k-1) 1 and since

   0 < y <1 ;

   d log h( y) 0

   dt

   k2.

   Hence the Lemma.

   We further note that as long as k 1 (even not integer) IFR property is retained for this distribution.

2. -content -level TI for lifetime distribution of k-unit parallel system

   Let I(X) = (0 , ) be an upper content level TI for the distribution having distribution function (1.2). The constant ( > 0) for (0,1), (0,1) is to be determined such that

   P{F ( ; ) } = . (2.1)

   Using asymptotic normality of , (2.1) can be equivalently written as P {Z ( / ()) [1 + log(11/ k ) / ] } = 1 ,

   where Z N (0, 1). This gives

   = log (1 1/ k ) / [1 + Z1 ()/],

   where Z1 is the 100(1)th lower percentile of the standard normal distribution.

   Define

   Ak

   k

   j j

   (j1) 2

   k(k 1)

   j j

   (j 2) 2

   k(k 1)

   j j

   ,

   (j 2)3

   k1 k1

   0

   (1) j

   k2 k2

   0

   (1) j

   k3 k3

   0

   (1) j

   where aCb = a!/(b!(ab)!).

   Then an upper tolerance limit of – content – level TI, I2 (X) is given by,

   U(X) = { log(11/k) / [1 + Z1 ()/] }, (2.2)

   where

   2

   ()

   2 / n 0.55312 / n

   k 1

   k 2 ; (2.3)

   (2A k 1) 1 2 / n

   k3

   Figure 2.1 : Hazard Function of F(.,.)

   Figure 2.1 : Hazard Function of F(.,.)

   Vol. 2 Issue 9, September – 2013

   2.5

   2

   k=.5

   1.5

   h(t)

   h(t)

   1 k=1

   k=2

   0.5

   k=3

   k=5

   0

   0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

   t

   It is clear that ()/ in (2.5) is free from for all k 1. Furthermore, I(X) is not an exact content level upper TI, since the constant is determined using asymptotic normality. This necessitates study of confidence level of I(X) for various values of n, and . [Details of performance of (2.2) is reported in Kumbhar and Shirke (2004).]

   IFR Class Tolerance Interval

   Upper TI for F IFR class of distributions following Barlow and Proschan (1966) is as follows:

   IBP (X)(0,

   C, ,n n ) ; (2.3)

   where

   C, ,n 2n log(1)

   2

   2

   2n,

   2n,

   2n,

   with 2 is the upper -percent percentile of the chi-square distribution with 2n

   degrees of freedom and

   n

   n

   1 n ni1X ni1

   (i)

   X(i1)

   ; (2.4)

   with X(i) is the ith ordred statistic in the sample X1, X2 ,, Xn . Thus for the model (2.1) we have two TIs. Therefore it is interesting to see relative performance of these TIs.

3. Comparison of MLE based and IFR class Tolerance intervals

   In order to compare the two TIs, we conduct simulation experiment for various values of n, , and =1 and 2, when k=2. We generate 25000 samples of size n each from (1.2). Upper TLs of IBP(X) and I(X) are computed for each sample and the average upper TLs of the same are taken. Please refer to Table 3.1 and Table

   3.2 for the simulated average upper TLs. We have also conducted simulation experiment to observe the effect of number of units in the parallel system on width of both the above TIs. The results are tabulated in Table 3.3.

   Table 3.1:Comparison of MLE based and IFR class Upper TLs for =1 and k=2.

   Sample Size (n)	=0.90
   	=0.90			=0.95
   	UMLE(X)	UBP(X)	Difference	UMLE(X)	UBP(X)	Difference
   10	4.278070	5.189807	0.911737	5.280729	6.734745	1.454016
   20	3.779746	4.594466	0.814720	4.681151	5.980370	1.299219
   25	3.671747	4.455863	0.784116	4.548275	5.803499	1.255224
   50	3.434548	4.138115	0.703537	4.252196	5.384721	1.132525
   =0.95
   10	4.859227	5.933962	1.074735	6.015581	7.724343	1.708762
   20	4.085189	5.027972	0.942783	5.065436	6.552964	1.487528
   25	3.928513	4.828962	0.900449	4.867146	6.281099	1.413953
   50	3.59487	4.376463	0.781593	4.442412	5.685691	1.243279

   Table 3.2:Comparison of MLE based and IFR class Upper TLs for =2 and k =2.

   Sample Size (n)	=0.90
   	=0.90			=0.95
   	UMLE(X)	UBP(X)	Difference	UMLE(X)	UBP(X)	Difference
   10	8.539951	10.353682	1.813731	10.559595	13.452091	2.892496
   20	7.568627	9.201962	1.633335	9.363304	11.967651	2.604347
   25	7.331238	8.899210	1.567972	9.104967	11.616709	2.511742
   50	6.858973	8.263653	1.40468	8.510165	10.776218	2.266053
   =0.95
   10	9.694414	11.843263	2.148849	12.009997	15.429117	3.41912
   20	8.183687	10.072673	1.888986	10.098490	13.067099	2.968609
   25	7.863148	9.663821	1.800673	9.735890	12.561660	2.82577
   50	7.172979	8.732775	1.559796	8.893104	11.381074	2.48797

   Table 3.3: Comparison of MLE based and IFR class Upper TLs when =1, = 0.90 and = 0.90 .

   Sample Size (n)	Number of units (k)	UMLE(X)	UBP(X)	Difference
   10	2	4.263407	5.547987	1.284581
   	3	4.546516	6.774717	2.228201
   	5	4.927414	8.428308	3.500894
   20	2	3.778254	4.750770	0.972516
   	3	4.121627	5.813017	1.691390
   	5	4.559469	7.228765	2.669296
   25	2	3.670547	4.577007	0.906461
   	3	4.023212	5.597181	1.573968
   	5	4.469737	6.957966	2.488229
   50	2	3.434350	4.192794	0.758436
   	3	3.810167	5.131015	3.320848
   	5	4.279542	6.382337	2.102795

4. Conclusions

   From the Table 3.1 to Table 3.3 we observe that

   1. An average upper TL based on MLE is below the average upper TL based on Barlow and Proschan (1966) procedure for all combinations of n, , and . The difference in length increases as increases.

   2. As n increases the difference between these average TLs does not reduce rapidly.

   3. Increase in number of units in the parallel system; k has significant effect on the length of TI due to Barlow and Proschan (1966). The length of IBP(X) increases rapidly as compared to the length of I(X). as k increases the hazard function of F(., ) moves away from hazard function of exponential distribution.[Please see Fig.2.1).]

Therefore, while using IBP(X) it should be kept in mind that, length of TI has significant effect as distribution moves away from exponentiality. It should be interesting to see performance of IBP(X) with respect to hazard rate ingeneral. One of the reasons for this observation about IBP(X) could be use of stochastic order between exponential distribution and distribution belonging to IFR class. Therefore as distribution moves away from exponential distribution with respect to hazard rate, TI will havs significant effect on the length.

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2. Cramer (1946): Mathematical methods of statistics, Princeton University press , NJ.

3. Jilek, M. (1981): A bibliography of statistical tolerance regions. Math. operat. fcrsch. Statist. Ser. Statist., 12, 441-456.

4. Kumbhar R. R. and Shirke D. T. (2004): Tolerance limits for lifetime distribution of k- unit parallel system, JSCS Vol.74(3), 201-213.

5. Patel J. K.(1986): Tolerance Interval- A Review, Communications in Statistics, Theory and method-!5(19), 2719-2762.

6. Wilks, S. S. (1941): Determination of sample sizes for setting tolerance liits. Ann. Math. Statist., 12, 91-96.