## 1.INTRODUCTION

Standard statistical procedures for the estimation of parameters mostly depend on a simple population density or mass function *f*(*x*; *δ*), where *δ* is the related parameter. In situations where the sampling probability is related to any characteristic of the population unit, traditionally observed data mirrors the population characteristics in a distorted proportion resulting in biased and inefficient estimates. This induced sampling bias can be taken care of by using a weighted density in place of a simple or unweighted density.

For a process in which the decision about an unexamined population is constructed on the foundation of an examined sample, extrapolation is used. Such extrapolation seems logical when the selected part is representative of the whole. Same is the case of acceptance sampling. From a homogeneous lot, selection of a representative sample is not an issue, but a sample may not be representative if the lot is heterogeneous in its variable characteristics. Using a weighted distribution to model such a lot can be an appropriate solution if some characteristic is affecting its selection probability. If that characteristic is linear with only one dimension, like length or width, the variable can be termed as one dimensional. If the characteristic is more than 1-dimensional, like the mass or the volume, the variable can be named as 2 or 3-dimensional, respectively.

Fisher introduced weighted distributions to model ascertainment bias in 1934 (Iyengar *et al*., 1999). Cox (1969) discussed the problem of sampling textile fiber and emphasized that each fiber has a chance of selection proportional to its length. Gove and Patil (1998) unified the concept of breast height diameter and basal area of tree (DBH-frequency and basal area-DBH). Mir and Ahmad (2009) introduced some size-biased distributions and discussed their applications.

The weighted density *g*(*x*; *δ*) proportional to an unweighted density *f*(*x*; *δ*) is developed by using a nonnegative weight function *w*(*x*). Mathematically, Dara and Ahmad (2002) defined the weighted density as(1)

Here, *X* is the original random variable with equal chance of inclusion in the sample, provided that the expected value of weight function i.e. *E*[*w*(*X*)] exists. When the weight function *w*(*x*) depends on the size of the unit of interest, the weighted distribution *g*(*x*; *δ*) is called sizebiased. A size-biased m-dimensional weighted distribution is constructed as (Mahfoud and Patil, 1982; Patil and Ord, 1976).

where ${\mu}_{\text{X}}^{\left(\text{m}\right)}$ is the *m*^{th} order moment about the origin calculated from *f*(*x*; *δ*). For *m* = 1, the size-biased distribution given in equation (2) is known as length-biased; and for *m* = 2 it is known as an area-biased distribution.

This study uses two separately developed sets of weighted parametric distributions to model a lifetime random variable in designing acceptance sampling plans. These weighted distributions are derivatives of un-weighted Erlang distribution and un-weighted Lomax distribution. Both Erlang and Lomax distributions are widely used for analysis and modeling in lifetime testing.

Use of statistical tools in establishing a system to monitor and maintain the quality of a process and its output is termed as statistical quality control. Acceptance sampling, as a component of this control system, is applied to sentence a lot as conforming or non-conforming to specifications. An acceptance sampling plan is a scheme that determines a minimum sample size and a criterion or acceptance number to categorize a lot as acceptable or not. In this plan a maximum fraction of defectives is agreed at which the lot, if categorized as non-conforming, is rejected. This fraction is set as being the probability of rejection of a good lot and it is known as the producer’s risk *α*. A minimum fraction of defectives is also agreed upon at which the lot, if categorized as conforming, is accepted by the consumer. This fraction is set as the probability of acceptance of a bad lot and is known as the consumer’s risk *β* .

This study defines the variable of a weighted distribution as having some finite dimensional characteristics of interest, which play a role in determining the sampling probability of that variable. Thus an un-weighted density will be denoted with a suffix of zero dimension i.e. (0*d*) , a length-biased weighted density like the one discussed by Cox (1969) will be denoted with a suffix of (1*d*) , an area-biased weighted density, like the one discussed by Gove (2003) with a suffix of (2*d*). Generalizing this concept, a *j*-dimensional weighted density will be derived and denoted with a suffix of (*jd*).

This paper includes (i) developing families of weighted models of Erlang distribution and Lomax distribution with several properties including the extended relations of bias in mean moment estimator (based on Cox (1969) relation) along with the recursive formulae of all derivatives. These extended recursive relations make it possible to calculate the bias in estimated mean of a (*jd*) weighted population if it is assumed mistakenly to be an (*id*) weighted one where *i* < *j*. (ii) formulating time truncated acceptance sampling plans for application purposes with fixed and known values of shape parameter of both Erlang and Lomax (0*d*), (1*d*), and (2*d*) models. (iii) finding minimum sample size and acceptance number to decide the fate of a production line as conforming or nonconforming to specifications. (iv) use of real data of average dissolved oxygen in water mg/L as an application example of the acceptance sampling plan.

## 2.THE WEIGHTED ERLANG FAMILY

The un-weighted Erlang probability density, denoted by *r*_{0d}(*x*), with shape parameter *k* (a positive integer) and scale parameter *θ* (> 0) is given as;(3)

The *q*^{th} order moment about origin for un-weighted Erlang distribution is obtained by

Particularly the mean and the variance are given by(5)

### 2.1.j-dimensional Weighted Erlang Distribution

Using equation (2) j-dimensional weighted Erlang probability density denoted by *r _{jd}*(

*x*) is developed as below

provided that *E*_{0d} (*X ^{j}*) exists.

Using the result (4) for the expectation, the (*jd*) weighted density with shape parameter *k* and scale parameter * θ* is derived as below.(6)

The *q*^{th} order moment around the origin of the jdimensional Erlang distribution is given by(7)

Particularly, the first moment (or mean) is given by

The second and the third order moments about mean are obtained by(10)

So, the skewness is given by(11)

Hence the distribution is positively skewed and the skewness decreases as the value of shape parameter *k* and /or the dimension of weight increases.

The moment generating function about the origin is derived as(12)

The cumulant generating function is obtained by(13)

The cumulative distribution function (cdf) is given by

Using *θu* = *t*, the cumulative distribution function of j-dimensional weighted Erlang density reduces to

### 2.2.Bias in Estimated Mean of Weighted Erlang Density

Cox (1969) established a relation to express the induced bias in the estimation of mean of a population having a (1*d*) weighted distribution if a (0*d*) un-weighted density is used. He showed that the (1*d*) mean is given by

Clearly the relative bias in estimated mean of (1*d*) population is equal to the squared coefficient of variation of (0*d*) population and the absolute bias is equal to the ratio of variance to mean of (0*d*) population.

To calculate the bias in mean of a population having (1*d*) Erlang density when estimation of mean is based on equal probability sampling using a (0*d*) density, let us first express the (1*d*) mean of weighted Erlang distribution as(16)

Hence it can be concluded that if an un-weighted Erlang density is mistakenly used for estimation of mean of a 1 dimensionally weighted Erlang population, the estimate of mean will be biased. The size of induced absolute bias will be equal to 1/*θ* and the size of induced relative bias will be equal to 1/*k*.

A generalization of relation (15) is suggested for bias in mean of a population having (*jd*) Erlang density *r _{jd}*(

*x*) when the estimation is based on (

*id*) Erlang density

*r*(

_{id}*x*) when

*i*and

*j*both are natural numbers and

*i*<

*j*.

Substituting mean and variance of (*id*) Erlang density using (8) and (9) and then simplifying, we get(17)

Here we have(18)

## 3.WEIGHTED LOMAX DISTRIBUTION FAMILY

The un-weighted (0*d*) Lomax probability density *l*_{d}(*y*) with shape parameter *λ* and scale parameter * σ* is given by(19)

The *q*^{th} order moment about the origin is given as

Particularly, the mean and the variance are given by(21)

### 3.1.j-dimensional Weighted Lomax Distribution:

Using equation (2), the j-dimensional weighted Lomax density *l _{jd}*(

*y*) is obtained as

provided that *E*_{0d} (*Y ^{j}*) exists. Using equation (20) for the expectation, the (

*jd*) weighted Lomax density with shape parameter

*λ*and scale parameter

*σ*is(22)

The *q*^{th} order moment about the origin is given by(23)

The mean of the *j*-dimensional Lomax distribution is given by(24)

The second and the third order moments about mean are given by(25)(26)

(Variance)

Hence, the skewness is obtained by(27)

The cumulative distribution function is given by

### 3.2.Bias in Estimated Mean of Lomax:

Again using Cox relation (15) to calculate the bias in mean of a Lomax (1*d*) population when estimation is based on equal probability sampling using a (0*d*) Lomax density. The (1*d*) mean of weighted Lomax distribution is given by

Simplification gives(29)

Here, we have

relative bias = $=\frac{\lambda}{\lambda -2},\text{\hspace{0.17em}}\lambda \ne 2$ and absolute bias(30)

A generalization of Cox (1969) relation for the bias in the estimated mean of a population having (*jd*) Lomax density when the estimation is based on (*jd*) Lomax density if *i* < *j* is given by

Substitution and simplification for ${\sigma}_{id}^{2}$ and ${\sigma}_{id}^{2}$ gives(31)

Therefore,

## 4.APPLICATION TO DESIGN OF ACCEPTANCE SAMPLING PLANS

To highlight the significance of a weighted model in acceptance sampling plans suppose that a manufacturing unit of toilet rolls packs 12 rolls in a packet and places 200 such packets in a box. Before shipment of all such boxes, a quality inspector selects a sample of 6 packets from each box. By examining all 72 rolls in 6 selected packets, the inspector decides to accept or reject the whole box containing 2,400 rolls. It can be imagined that it is not possible for the inspector to first shuffle all 200 packets in a box and then select 6 out of them. He will be inclined to select the packets from top layer or at the ends of layers near the walls of the box. In this scenario the placement of a packet in a box is a 1-dimensional characteristic of the packet which is affecting its probability of selection. Alternatively, the rolls may be packed in two different arrangements in a packet. Suppose that some are packed in 2 rows with each row having 6 rolls. Some others may be packed in 3 rows with each row having 4 rolls. If a box of 200 packets now has 100 packets of each type, the probability of selection of a packet from the box will be affected by 2-dimensions, namely the placement of the packet in the box and the arrangement of rolls in the packet. Surely, one of the two arrangements will be more convenient to be taken out of the box.

It can thus be stated that probability of acceptance or rejection of a specific box is a function of the dimension/ dimensions used in the process of packing. World War II saw the practical applications of acceptance sampling. Epstein (1954) used an exponential distribution in his acceptance sampling plan based on truncated life tests. Gupta (1962) used normal and log-normal distributions in life tests. A log-logistic model was used for life testing by Kantam *et al*. (2001), Birnbaum Saunders model by Baklizi and El Masri (2004), generalized Rayleigh distribution by Kundu and Raqab (2005). Aslam and Jun (2009a) used a Weibull distribution, Aslam and Jun (2009b) also used inverse Rayleigh and log-logistic distributions. Aslam *et al*. (2013) designed a variable sampling inspection plan for resubmitted lots based on process capability index. Aslam and Jun (2013) designed time truncated acceptance sampling plans by using the two-point approach. Aslam *et al*. (2014) gave a skip-lot sampling plan with the two-stage group acceptance sampling plan as reference. Malik (2012) provided some extension in sampling plans. Leiva *et al*. (2014) discussed Birnbaum- Saunders process application to electronic and food industries.

Usage of weighted models in acceptance sampling is quite new. Leiva *et al*. (2009) presented a length-biased version of the Birnbaum-Saunders distribution with application in water quality. Rao *et al*. (2014) suggested an acceptance sampling plan based on a size biased Lomax model and compared their experimental results with the results obtained by Gupta and Groll (1961) to show that size biased Lomax model gave good results when compared to a gamma model. Rao *et al*. (2014) also designed a hybrid group acceptance sampling plan based on a sizebiased Lomax model.

### 4.1.Time-Truncated Acceptance Sampling Plan

To test the quality of a lot produced, a null hypothesis *H*_{0} :*μ* ≥ *μ*_{0} is formulated against the alternative hypothesis *H*_{1} :*μ* < *μ*_{0} where *μ* is true mean life of the population lot under study and *μ*_{0} is a specified mean. For testing procedure, a sample of *n* units is selected randomly from the population and units in the sample are then subjected to a life test which is terminated at the preassigned time *t*_{0}. The test duration denoted by *t*_{0} is determined as a multiple of *μ*_{0} (that is, *t*_{0} = *aμ*_{0}) where *a* is termed as test termination time multiplier. If more than *c* sample elements fail during *t*_{0}, the lot is labeled as nonconforming to the specifications and *H*_{0} is rejected. If at most *c* units fail up to *t*_{0}, the lot is accepted. Here, c is the acceptance number, which will be determined. This acceptance number represents the maximum possible number of defective units, if found in any selected sample, will decide the conformity of population lot according to standards.

Considering the lot to be large enough, the lot acceptance probability *L*(*p*) is calculated by using binomial sampling. Here *p* denotes the probability of a unit failing in the test before termination time *t*_{0} = *aμ*_{0}.

To safe guard the rights of producer and consumer, producer’s risk *α* and consumer’s risk *β* are agreed upon usually at 5% and 10% respectively. Clearly *α* denotes the probability of rejection of a good lot if unfortunately it fails the test and *β* denotes the probability of unfortunate acceptance of a bad lot. A maximum proportion of defectives in the population, considered to be satisfactory as an overall process average denoted by AQL or *p*_{1}, and a minimum proportion of defective denoted by in any individual lot is denoted by LQL or *p*_{2}.

Thus the sampling plan parameters are decided to meet the following inequalities.

### 4.2.Plan Parameters under Erlang and Lomax Distributions

Acceptance sampling plans are designed for (0*d*), (1*d*), and (2*d*) distributions of both Erlang and Lomax families in this section. We calculate the probability of a unit’s failure in each of all six cases mentioned above. These probabilities are calculated by replacing the unknown scale parameters, *θ* in Erlang and *σ* in Lomax, in their respective cumulative distribution functions with a function of respective means *μ*. The expressions of cumulative distribution functions are obtained for Erlang and Lomax models by assigning respective order of dimension in equations (14) and (28).(37)(38)(39)(40)(41)

For (0*d*) Erlang; replacing ‘θ’ by $\frac{k}{\mu}$

For (1*d*) Erlang; replacing ‘θ’ by $\frac{k+1}{\mu}$

For (2*d*) Erlang; replacing ‘θ’ by $\frac{k+2}{\mu}$

For (0*d*) Lomax; replacing ‘σ’ by μ (λ−1)

For (1*d*) Erlang; replacing ‘θ’ by $\frac{\mu \left(\lambda -2\right)}{2}$

For (2*d*) Erlang; replacing ‘θ’ by $\frac{\mu \left(\lambda -3\right)}{3}$

For the purpose of comparison, the shape parameters of both families have been assigned equal value i.e. *k* = *λ* = 4. Using equations (36) to (41) as failure probabilities and the relation (33) for lot acceptance probability, the plan parameters of our time-truncated acceptance sampling plan are determined to satisfy the relations (34) and (35). The producer’s risk *α* is set at 0.05 with the following variations of remaining parameters.

Consumer’s risk *β* = 0.25, 0.10, 0.05, and 0.01

Test termination time multiplier *a* = 0.50, 0.75, 1.00, and 1.50

Ratio *μ* / *μ*_{0} = 2.0, 2.5, 3.0, 3.5, 4.0, and 5.0.

Table 1 Gives sampling plans for (0*d*), (1*d*), and (2*d*) Erlang distributions

Results show that the sample size *n* is directly related to the dimension of distribution *j* if test termination time multiplier *a* equals 0.5. As the value of *a* approaches 1 this relation starts changing its direction from direct to inverse and stays to be inverse for largest values of a under study. This change is more significant for smaller values of ratio.

Table 2 gives comparison between sampling plans based on (0*d*), (1*d*), and (2*d*) Lomax distributions

Results show that the sample size *n* is inversely related to the dimension of distribution irrespective of the value of test termination time multiplier *a* except for very few and nominal exceptions.

When any two out of three parameters (*β*, *a* and *μ* / *μ*_{0}) are kept constants and the third is varied, minimum sample size *n* and acceptance number *c* exhibit inverse relations to the parameter varied. This fact holds for all models of both families and it is more evident for small values of *a*.

### 4.3.Application Using Real Data

The data in Table 3 shows monthly average dis- solved oxygen in water (mg/L.) recorded at Suffolk County, New York for hydraulic unit # 02030201 by USGS (United States Geological Survey). USGS records daily water data throughout USA and publishes daily mean data and monthly mean data. The data used in this study spans over 1,613 days for the period from May 01, 2008 till September 30, 2010. Amount of dissolved oxygen (DO) in water is one of the major factors that determine the safe quality of water for marine life. According to a scale shared by USGS for cold water fisheries a DO level of less than 7 mg/L, and a DO level of less than 5 mg/L for hot water fisheries and plants is threatening. Dissolved oxygen is expressed in milligrams per liter or parts per million (ppm) of water. At sea level, concentration of DO ranges from 7.56 mg/L at 30 degrees Celsius to 14.62 mg/L at 0 degree Celsius.

In order to test the quality of water in this hydraulic unit all six models used in simulations are fitted to the data at first stage. Kolmogorov-Smirnov (K-S) goodness of fit test is applied to select the model of best fit. Results favored Erlang (1*d*) as the best fit showing maximum value of observed K-S statistic as 0.1804 in comparison to 0.375, the critical value of K-S statistic for *n* = 12 and *α* = 0.05. It is thus concluded with 95% confidence that monthly average amount of dissolved oxygen in water follows the Erlang (1*d*) distribution with shape parameter *k* = 4.

At second stage, from Table 1, the highlighted acceptance sampling plan based on Erlang (1*d*) distribution with *n* = 12 is selected. Other parameters of selected sampling plan are *β* = 0.01, *a* = 0.75, and ratio = 4.0 with acceptance number *c* = 0. Setting the value of observed sample mean 8.96 as *μ*_{0}, the null hypothesis is formulated as *H*_{0} :*μ* ≥ 8.96 against the alternative hypothesis *H*_{1} :*μ* < 8.96. Test termination time *t*_{0} is set at *aμ*_{0} = 0.75(8.96) = 6.72. It is observed that there are three values in water data table lesser than test termination time for the months of July, August, and September i.e. 6.10, 6.40, and 6.60 so these are regarded as failures before test is terminated. As the sampling plan allowed *c* to be zero whereas our plan has *c* = 3, therefore the null hypothesis is rejected and it is concluded that average dissolved oxygen is not more than 8.96 mg/ L in this county. The 95% confidence limits are found to be (7.59-10.32).

As stated earlier, more than 7 mg/L DO level is not threatening according to USGS even for cold water fisheries. It can thus be stated by observing the results that the monthly average amount of DO in this hydraulic unit seems to be safe for aquatic life.

## 5.CONCLUSIONS

Weighted families of Erlang and Lomax distributions are developed in this study. These distributions are named families because of their recursive nature. These recursive relations are very convenient and simple to use as complete probability models with all mentioned properties of a finite dimensional variable following Erlang or Lomax law as the models can be developed by simply specifying the dimension of variable in derived relations. Separate extensions to Cox (1969) relation, for both families, giving bias in moment mean estimator are also suggested. These extensions are valid to the extent of any difference of dimensions for which the bias is desired to be calculated and on which the estimation is based. These extensions can also be very helpful for future reference while studying bias in estimated mean of populations having other probability distributions too.

Results and analyses suggest that application of weighted Erlang with more data sets in acceptance sampling needs more to be probed. It is also recommended that in future studies, application of weighted Lomax should also be sought in parallel to weighted Erlang.