On robustness of a sequential probability ratio test for shape parameters of Lomax and Burr distributions under parameter variations

Lomax distribution and Burr distribution as given above. The paper is organized as below: Section 2 deals with the problem of constructing Sequential Probability Ratio Test (SPRT) for evaluating the hypothesis concerning the shape parameter by deriving equations for the Operating Characteristic (OC) and Average Sample Number (ASN) functions. Section 3 examines the robustness of the SPRT concerning OC functions when the distribution experiences a change in the scale parameter. In Section 4, we examined the robustness of the SPRT concerning the shape parameter of the Burr distribution in relation to the change in the other shape parameter.

Methodology 2.1 SPRT for Shape Parameter of Lomax Distribution

Let 𝑿 = (𝑋 1 , 𝑋 2 , … ) be an infinite sequence of independent and identically random variables from a Lomax distribution with shape parameter 𝜃 (> 0) and known scale parameter 𝜆 (> 0) with probability density function is

In the following, SPRT is developed for the problem of testing simple null against simple alternative hypothesis of shape parameter in Lomax distribution.

Let (𝛼, 𝛽) be the desired strength of SPRT, then according Wald [4] , 𝐴 and 𝐵 are approximately given by 𝐴 ≈ , where 𝛼 and 𝛽 are both within the interval (0,1). The Operating Characteristic (OC) function 𝐿(𝜃) is given by

Where ℎ(𝜃) is the non-zero solution of

From ( 1) and ( 2), we have

The Average Sample Number (ASN) function is given by

The maximum value of ASN occurs in the neighbourhood of 𝜃 = 𝜃 ̃, say where 𝜃 ̃ is the solution of 𝐸 𝜃 ̃(𝑍) = 0 and this value is given by

Table 1 contains the values of 𝐸 0 (𝑁), 𝐸 1 (𝑁) and 𝐸 𝜃 ̃(𝑁) for different values of 𝛼, 𝛽 and ratios of 𝜃 0 and 𝜃 1 .

Robustness of the SPRT for shape parameter of Lomax distribution

In this section, the possibility of a misspecification for of the scale parameter λ and its effect on testing for the shape parameter using the SPRT in the Lomax distribution is are analysed. Then the probability density function of (1) becomes 𝑓(𝑥; 𝜃, 𝜆 * ). To study the robustness of the SPRT developed in Section 2 with respect to OC function, consider ℎ(𝜃) as the solution of the equation:

We can solve above equation for different values of 𝜃, ℎ(𝜃) [Using Mathematica and R Software] and the corresponding OC function is obtained. The robustness of SPRT with respect to ASN function can be studied by replacing the denominator of ( 6) by

We consider the cases 𝜆 < 𝜆 * and 𝜆 > 𝜆 * to study the robustness of the SPRT. In Table 2, we present the ASN function for the cases 𝜆 < 𝜆 * for the SPRT of testing the hypothesis 𝐻 0: 𝜃 = 5 against 𝐻 1: 𝜃 = 10. The values of OC function for the cases 𝜆 < 𝜆 * and 𝜆 > 𝜆 * respectively are plotted in Figure 1 and Figure 2.

SPRT for shape parameter of Burr distribution

In this sub section, SPRT for the testing for the shape parameter of Burr distribution. The random variable 𝑋 is said to follow Burr distribution with shape parameters 𝜃 and 𝑐, if its probability density function is given by

Where the parameter 𝑐 is assumed to be known. Given a sequence of observations 𝑋 1 , 𝑋 2 , …, from (13), we wish to test the null hypothesis 𝐻 0: 𝜃 = 𝜃 0 against the alternative 𝐻 1: 𝜃 = 𝜃 1 (> 𝜃 0 ). We propose the following SPRT.

Let 𝑍 = ln ( 𝑓(𝑥;𝜃 1 ) 𝑓(𝑥;𝜃 0 ) ) = 𝑙𝑛 ((

The OC function 𝐿(𝜃) is given by

Where ℎ(𝜃) is the solution of 𝐸(𝑒 ℎ(𝜃)𝑧 ) = 1.

Robustness of the SPRT for shape parameter of Burr distribution

The present subsection is concentrated on robustness of the SPRT for testing the hypothesis about shape parameter. Let us suppose that the other shape parameter undergone change and the probability density function (13) becomes 𝑓(𝑥; 𝜃, 𝑐 * ). To study the robustness of the SPRT developed here with respect to OC function, we consider ℎ(𝜃) as the solution of the equation

Conclusion

• The values of the OC function for the scenarios where 𝜆 < 𝜆 * and 𝜆 > 𝜆 * are illustrated in Figure 1 and Figure 2, respectively. The figures demonstrate that when 𝜆 < 𝜆 * , the OC curves shifts to the right, whereas for 𝜆 > 𝜆 * the OC curves shift to the left of the curve corresponding to 𝜆 = 𝜆 * . It is evident from the figures that the SPRT is robust for 𝜆 * = 𝜆 ± 0.1, as the variation in the OC function is minimal. • We have analysed the SPRT for evaluating the null hypothesis 𝐻 0: 𝜃 = 5 against the alternative 𝐻 1: 𝜃 = 10 for 𝛼 = 𝛽 = 0.05. The operating characteristic (OC) function values for scenarios where 𝑐 exceeds 𝑐 * and where 𝑐 is less than 𝑐 * are illustrated in Figures 3 and4, respectively. The figures demonstrate that when 𝑐 > 𝑐 * , the OC curves shift to the right, whereas for the case 𝑐 < 𝑐 * OC curves shift to the left of the curve corresponding to 𝑐 = 𝑐 * . It is evident from the figures that the SPRT maintains its robustness for 𝑐 * = 𝑐 ± 0.005, as the variations in the OC function are minimal.

Acknowledgement

AcknowledgementThe authors acknowledge the support of Bangalore University though a project.

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