1. INTRODUCTION

Overlapping coefficient (OVL) is defined as the common area under two probability density functions, and it is a measure of similarity between two populations. The value of OVL ranges from 0 to 1, where a value 0 indicates that there is no overlap and a value 1 shows that the two populations are identical. It has wide applications as a similarity measure.

Let f1(x) and f2x be the probability density functions of two continuous populations. The OVL can be defined as

following Weitzman [1].

Inference for the OVL has been investigated by several researchers under both normal and exponential distributions. Inman and Bradley [2] considered both interval estimation and hypothesis testing for the OVL under two normal distributions with equal variances. “For normal distributions with equal means, but different variances, Mulekar and Mishra [3] investigated the computation of confidence intervals.” “For two one-parameter exponential distributions, Al-Saleh and Samawi [4] investigated the interval estimation of the OVL, and compared several methods, including the bootstrap and Taylor series approximation.” It should be noted that “Rom and Hwang [5] proposed the OVL as a measure of bioequivalence.”

In this article, we consider both one-parameter and two-parameter exponential distributions, and investigate both interval estimation and hypothesis testing for the OVL using the “concept of a generalized pivotal quantity (GPQ) due to Weerahandi [6–8].” “This concept was earlier applied by Roy and Mathew [9] for the interval estimation of the reliability function of a two-parameter exponential distribution.” In the context of inference concerning the OVL, “Reiser and Faraggi [10] did use the GPQ idea for two normal distributions with equal variances.” In the subsequent sections, we shall take up the interval estimation and hypothesis testing concerning the OVL for one-parameter exponential distributions, for two-parameter exponential distributions having a common scale parameter, and finally for general two-parameter exponential distributions that may not have any common parameters. The performance of the proposed GPQ approach will be assessed using simulations, and the methodology will be illustrated using examples.

2. ONE-PARAMETER EXPONENTIAL DISTRIBUTIONS

Consider two one-parameter exponential distributions with pdf

for i = 1, 2, and let Δ be the OVL between them, as defined in (1). If r=θ1/θ2, then Δ simplifies to where we assume that r≠1. If r=1, the OVL is of course equal to one. The interval estimation of Δ is nontrivial since it is not a monotone function of r; “see the plot of Δ given in Figure 1 in [4].”

Let X1,X2,…,Xn1 and Y1,Y2,…,Yn2 be two independent random samples of sizes n1 and n2 taken from f1⋅ and f2⋅, respectively. Let X¯ and Y¯ be the respective sample means. Now,

are independent chi-square random variables with 2n1 and 2n2 degrees of freedom, respectively, so that F=2n1V2n2U∼F2n2,2n1, an F-distribution with df = 2n2,2n1. In order to derive a GPQ for Δ, we recall that a GPQ is a function of the random variables, the observed data, and the unknown parameters, and it satisfies two conditions: (i) given the observed data, the GPQ has a distribution that is completely free of unknown parameters, and (ii) if the random variables are replaced by the corresponding observed values, the GPQ will simplify to a quantity that is free of any nuisance parameters; very often, the GPQ will simplify to the parameter of interest. Appropriate percentiles of the GPQ can be used to obtain confidence limits for the parameter of interest. We shall obtain a GPQ for Δ after deriving a GPQ for r=θ1/θ2. For this, let x¯ and y¯, respectively, denote the observed values of X¯ and Y¯. Define where w=x¯y¯ and F=2n1V2n2U∼F2n2,2n1, as already noted. It is now easily verified that Tr satisfies the two properties required of a GPQ, and it simplifies to r when the random variables X¯ and Y¯ are replaced by the corresponding observed values x¯ and y¯. A GPQ for Δ, say TΔ, is now given by

The percentiles of TΔ provide confidence limits for Δ. For example, if TΔ0.95 is the 95th percentile of TΔ, then TΔ0.95 is a 95% upper confidence limit for Δ, referred to as a generalized upper confidence limit. A lower confidence limit, or a two-sided confidence interval, can be similarly obtained. Furthermore, for testing H0:Δ=Δ0 against H1:Δ<Δ0, a test procedure consists of rejecting H0 if TΔ0.95<Δ0, when we use a 5% significance level. We note that the required percentiles have to be numerically obtained. For example, in order to compute the 95th percentile of TΔ, we keep the observed data fixed (i.e., in Eq. (3) we keep w fixed), and generate several values (say, 10,000 values) from the F2n2,2n1 distribution. This gives 10,000 values of TΔ for a fixed value of w. The 95th percentile of these 10,000 values is an estimate of the 95th percentile of TΔ.

3. TWO-PARAMETER EXPONENTIAL DISTRIBUTIONS WITH A COMMON SCALE PARAMETER

The pdfs are now given by

for i=1,2. The OVL in this case, denoted by ρ, is given by

It is obvious that when α1=α2, ρ=1. Let X1,X2,…,Xn1 and Y1,Y2,…,Yn2 be two independent random samples of sizes n1 and n2 taken from f1⋅ and f2⋅, respectively. Let X¯ and Y¯ be the respective sample means, and let X1=min X1,X2,…,Xn1 and Y1=min Y1,Y2,…,Yn2. Define

Then it is well known that U1, U2, and V are independent, distributed as χ22, χ22, and χ2n1+2n2−42, respectively. Let x1,y1, and w, be the observed values of X1,Y1, and W, respectively. GPQs for β, α1, and α2, say Tβ, Tα1, and Tα2, respectively, are given by

For ρ given in (4), a GPQ, say Tρ, is then obtained as

The percentiles of Tρ can be used to obtain confidence intervals for ρ, and also to test hypotheses concerning ρ.

4. TWO-PARAMETER EXPONENTIAL DISTRIBUTIONS WITH DIFFERENT LOCATION AND SCALE PARAMETERS

Consider two exponential populations with pdfs fix;αi,βi

for i = 1, 2. The point of intersection of f1⋅ and f2 ⋅ is easily seen to be

The overlaps between the curves f1⋅ and f2⋅ can be calculated by considering the following four cases:

(i) α1<α2, β1<β2, (ii) α1<α2, β1>β2, (iii) α1>α2, β1<β2, and (iv) α1>α2, β1>β2. The corresponding OVLs will be denoted by ηi,i=1,2,3,4 respectively, for the four different cases given above. The ηi′s are as follows:

• Case 1: α1<α2 and β1<β2

  η1=1−eα1−α2β1−β2β2β1β1β1−β2−β2β1β2β1−β2; α2−α1<β1lnβ2β1e−α2−α1β1; α2−α1>β1lnβ2β1.

• Case 2: α1<α2 and β1>β2

  η2=e−α2−α1β1−eα1−α2β1−β2β2β1β2β1−β2−β2β1β1β1−β2.

• Case 3: α1>α2 and β1<β2

  η3=e−α1−α2β2−eα1−α2β1−β2β2β1β1β1−β2−β2β1β2β1−β2.

• Case 4: α1>α2 and β1>β2

  η4=1−eα1−α2β1−β2β2β1β2β1−β2−β2β1β1β1−β2; α2−α1>β2lnβ2β1e−α1−α2β2; α2−α1<β2lnβ2β1.

The expression for OVL, say η, can be written as a single expression, using indicator functions:

where I⋅ denotes the indicator function. We note that the indicator functions are also parameter dependent.

Consider two independent random samples of sizes n1 and n2 from f1⋅ and f2⋅, respectively. Let X¯, Y¯ be the respective sample means and X1, Y1 be the respective sample minima of the two samples. Then

where U1, U2, V1, and V2 are also independent. Also let x¯, y¯, x1, and y1 denote the observed values of X¯, Y¯, X1, and Y1, respectively. Then GPQs dor β1, β2, α1, and α2 are given below:

Since the OVL η is a function of β1, β2, α1, and α2, a GPQ for η, say Tη can be obtained by replacing each parameter by the corresponding GPQ in the expression for the OVL. As was noted in the previous sections, the percentiles of Tη can be estimated by Monte Carlo simulation, and provide confidence limits for η.

5. DISCUSSION

In this brief note, we have presented a unified approach for computing confidence limits for the OVL of two exponential distributions, one-parameter, as well as two-parameter, using the concept of a GPQ. Through numerical results, we have demonstrated the accuracy of the proposed method in terms of meeting the coverage probability requirement. Furthermore, illustrative examples are provided. Though the proposed solutions have to be numerically obtained, the required computations are both simple and straightforward.