SEQUENTIAL TESTING PROCEDURE FOR THE PARAMETERS OF INVERSE DISTRIBUTION FAMILY Текст научной статьи по специальности «Математика»

SEQUENTIAL TESTING PROCEDURE FOR THE PARAMETERS OF INVERSE DISTRIBUTION FAMILY

K. S. Chauhan1 *, A. Sharma2 •

Ram Lal Anand College, University of Delhi, New Delhi-110021 1*kuldeepsinghchauhan.stat@rla.du.ac.in, 2 anurag.stats@rla.du.ac.in

Abstract

The sequential probability ratio test is a powerful statistical tool that is frequently employed for hypothesis testing, parameter estimation, and statistical inference. The aspect of robustness is of utmost importance when employing SPRTS in practical applications. Past studies have investigated the robustness of SPRTS for specific distributions. We have developed SPRTS for a family of inverse distributions that includes eleven distinct distributions. The primary objective of this study is to investigate and evaluate the robustness of SPRTS under various conditions and distributions, focusing on the parameters of the inverse distribution family. SPRTS efficacy is measured using OC and ASN functions. This study comprehensively covers the construction and rigorous evaluation of SPRTS, particularly in testing simple null hypotheses against simple alternative hypotheses. Additionally, we investigate the robustness of SPRTS under various factors, including the presence of other parameters and specified coefficients of variation. Conclusive results, graphic representations, tables, and acceptance and rejection regions add clarity to the findings.

Keywords: Inverse Distributions Family, Sequential Probability Ratio Tests (SPRT), Operating Characteristics (OC), Average Sample Number (ASN).

1. Introduction

Sequential Probability Ratio Tests (SPRT) are innovative methodologies that prove highly effective for both hypothesis testing and parameter estimation in statistical inference. The foundational work by [21] introduced the concept of SPRT for analyzing simple null hypotheses against simple alternatives. To assess the effectiveness of SPRT, operating characteristic (OC) and average sample number (ASN) functions were developed as performance measures. Sequential probability ratio tests (SPRTS) have long been recognized as valuable tools for making efficient and prompt decisions in various statistical applications. These tests play a crucial role in scenarios where data are collected sequentially over time, and the goal is to make a conclusive determination about a specific hypothesis. Robustness, which ensures the validity and reliability of these tests under varying conditions, is an essential aspect to consider when employing SPRTS in real-world situations. Multiple studies have scrutinized the robustness of SPRTS in disparate scenarios, enhancing our understanding of their performance and versatility. For instance, [1] examined Wald's SPRT for Levy processes, while [3] explored the robustness of sequential testing procedures for generalized life distributions. Other research, such as that by [6] studied the robustness of sequential testing procedures for parameters of zero-truncated negative binomial, binomial and Poisson distributions. Previous works have also assessed the robustness of SPRTS in specific settings, such as [8], considered sequential life tests in the exponential case. [9] examined the robustness of sequential probability ratio tests in the presence of nuisance parameters.[11] evaluated exponential and Weibull test plans, whereas [12] concentrated on investigating the robustness of the SPRT for a negative binomial distribution in cases where the shape parameter

is not specified. Additionally, [13] investigated the robustness of the exponential SPRT when failures from a Weibull distribution were transformed using a known shape parameter. Other relevant research includes [17]discusses the performance analysis of the Sequential Probability Ratio Test (SPRT) under various conditions and [14] explored robustifying the SPRT for a discrete model under "contamination." In contrast, [15] analyzed the performance and robustness of an SPRT for non-identically distributed observations. The robustness of SPRTS has also been examined in the context of exponential life-testing procedures [18] and the scale parameter of gamma and exponential distributions [19].[16] discusses the use of sequential probability ratio tests (SPRTS) for the statistical analysis of simulation outputs generated by computers. The type I and type II errors exponents of sequential probability ratio tests, when the actual distributions differ from the test distributions analyzed by [2]. In light of these studies, this research aims to investigate further and evaluate the robustness of sequential probability ratio tests under various conditions and distributions. In this study, we aim to extend the existing research and contribute to the robustness analysis of SPRTS for parameters of inverse distribution family suggested by [7]. Our focus will be on thoroughly examining the robustness of these tests using OC and ASN functions. We will develop and rigorously evaluate the SPRTS, with specific attention given to their robustness about the OC and ASN functions. Sections 3 and 5 will cover the essential elements of constructing and evaluating the SPRTS, including testing simple null hypotheses against simple alternatives, sequential analyses of composite hypotheses, and comprehensively examining their robustness. Section 4 shall analyze simple null hypotheses established on the parameter 7, taking into account the presence of the illustrious 5. Furthermore, in Section 6, we shall investigate comparable hypotheses founded on the parameter 5, factoring in the existence of 7. In Section 7, we will further investigate the robustness of the SPRTS in the presence of a specified coefficient of variation. Section 8 presents the regions of acceptance and rejection deduced for the null hypothesis H0 compared to the alternative hypothesis H1. Finally, Section 9 will effectively explain the synthesized data and provide conclusive findings using a combination of tables and graphics.

Through this comprehensive analysis, we aim to gain valuable insights into the robustness,

performance, and limitations of SPRTS in the inverse family of distributions.

2. Inverse Distributions Family

Suppose a random variable (rv) x having p.d.f.

c( -1 : a\ tV-1 (x-1;0V(x-1;0) , , \\

f (x; a ' Y, 60) = * v x2p(i) -Z" exp (-7g(x 1; 0)); (1)

0 < x < a-1, y > 0,6 > 0.

Where, g (x-1; 9), is a function of 0 and x. Moreover, g (x-1; 0) real-valued, Strict decreasing the function of x with g(ro; 0) = to and g' (x-1; 0) stances for the derivative of g(x; 0) by x-1. the equation (1) shows that the above distribution can be converted in the following distributions as special cases: If g(x; 0) = x2,6 = k + 1(k > 0), (k = ^r) provide the inverse Half-normal

distribution and (k = 0) the inverse Rayleigh distribution. If g(x; 0) = log + ^r) , b > 0, v >

0,6 = 1, provide the inverse log-logistic model. If g(x; 0) = log + , b > 0, v = 1,6 > 1,

provide the inverse Burr distribution. If g(x; 0) = log + ^ , b = 1, v > 1,6 > 1, provide the

inverse Lomax (distribution. If g(x; 0) = xr, 6 = 5 (h > 0), it becomes inverse Chi-distribution. If

g(x; 0) = log (x) and 6 = 1, obtain inverse Pareto distribution. If g(x; 0) = xr exp(ax), r > 0, a >

2

0,6 = 1, obtain inverse modified Weibull distribution. If g(x; 0) = ^x + , Y = 6 = 1, obtain inverse linear exponential distribution. If g(x; 0) = log x, obtain the inverse of the log-gamma distribution. If g(x; 0) = xp, p > 0,6 > 0, obtained the inverse generalized gamma distribution.

3. SPRT FOR EVALUATING THE HYPOTHESES OF y

Let a series X\, X2,... from (1), assume one needs to assess the simple hypotheses Ho : y = 70 as opposed to H1 : 7 = Yi (> y0). The analysis of SPRT on behalf of H0, expressed in this manner

z=>n{fm^rn}=lJ< - g (*-•2 ) y - y»)

(2)

Admit Ho if 0=1 Zi < InB, refuse H0 if 0=1 Zi > In A, or else, carry on sampling using the value of (n + 1))th. If a and ft belong to the interval (0,1) and represent type I and type II errors sequentially, the work by [21] provides definitions for A and B that are specified as

A

(1 -,

and

Where 0 < B < 1 < A

The OC function is almost specified as

(1 - a)

LY

(Ato - 1)

(A*0 - B*0)

Where t0 is the non-zero result for equation

E (et"zi) = 1

Note 1: Use the statement that g (x-1; 0) follows gamma distribution Using (1) with (3), we find

71Y" 110 (71 - 70)+ Y \-S = 1 Y0) I Y

or,

Y

to (7i - 7o) m)to -1

7o I

To find the values of OC and ASN functions, evaluate (4) as

71 - 7o

toln( 71 ) = ln

1 + to

7

(3)

(4)

(5)

By utilizing the natural logarithm function of (1 + x), which is defined for 1 < x < 1, in (5).we can achieve the desired outcome from (6).

1 71 - 7o

3Y Using (2), provides that

t2 I 1 ( 71 - 7o t2 1

to +

7

71 - 7»\ -ln( 71

7o

o

E (Zi | 7) = S

7o

ln[ 71)- 171 - 70

7

Using (7), we get, the ASN function

E(N | y)

L(7)lnB + {1 - L(7)}lnA

S

ln( 71 - 71-70

(6)

(7)

(8)

f>

B

3

2

Using (8) the ASN function for H0 along with H specified as

(1 - a)1n B + a ln A

and

Eo (N)

Ei (N)

S

l«'71 -

ß lnB + (1 - ß)lnA

S

l«< ^ - P-70

4. SPRT FOR EVALUATING THE HYPOTHESES OF 7 ALTHOUGH 5 IS

CHANGING

Using section (3), The maximum value of ASN gets on behalf of 7 = 7 where 7 is getting from E (Zi | 7) = 0 and the maximum value is specified as

E=y(N) = -

(ln A * ln B) E (z? | 7)

Also

Also, using (7) we get

E (Z2 | 7) =

Utilizing (9) and (11), we find that

E7(N) =

7 =

7i - 7o

ln ( Y1

. 7o

70

lni /7i - 70

7

2 + (71 - 7o)? S

7'

2

(9)

(10)

(11)

-(lnA * ln B)

Sln ( 7^ - ÎH-YYo^}? + (y1-y?0)2S

7o

7

7

Assuming that there has been a modification to the parameter 6 and that (1) has transformed into f (x; a, 7, d, 0), this can be attained by replacing 6 with d. To analyze the robustness of SPRT, suggest t0 as the result of the equation

1 i f (x; a, 71, S, 0) lf°r, , ,

\ tH-^^ r f (xi; a, 7, d, 0) dx;

if (xi ; a, 7o, S, 0 )J ^ v '

(12)

We achieve from (12) and put fa

/^xSto ra-1

V7J r(d^o

exp

-{(71 - 7o) to + 7}g fx, 1; 0

gd-1 (x-1; ^ g' (xr1; 0)

dxi = 1,

or,

or,

(Y1 - Y0) ^ + 1 =(£ T

=

( 71 - 7o) ¿o

71 ^^

70)

To find the values of OC functions, evaluate (13) as

$1 70)=in

1 + to

71 - 7o 7

(13)

(14)

a

1

x

Equation (14), Solve as (5) and find the roots of t0 from (15)

13 (^)3} '0 -12 (^)2} '0+{(- ^ (=0 (15)

where fa = ^). The ASN function coincides with (8)

E (Zi | y) = fa

ln, 7i)_ /71 - 70

70/ V Y

5. SPRT FOR EVALUATING THE HYPOTHESES OF S

(16)

Suppose taking a sequence X1r X2,... from (1) are independently and identically distributed. To analyze the simple null hypotheses in contradiction of the simple alternative hypotheses when y is identified. H0 : S = S0 as opposed to H1 : S = S1 (> S0). We suggest the resulting SPRT

Zi = (Si - S0) ln y + (Si - S0) ln {g (x-1; e)} + ln((17)

Admit H0 on the nth step, if

Yjn {g (x-1; e)} < i^lnB - n (S1 - S0 ) ln Y - nln (f|S°y)} / (S1 - So ) (18)

Reject H0 if

Yjn {g (x-1; e)} > { ln A - n (S1 - S0 ) ln Y - nln (f|Sy)} / (S1 - So ) (19)

Then using the (n + 1)th value carry on sampling if

lnB - n (S1 - S0) ln y - nln( w^)! n , , ..

=--^ < E {g (x-1; e)} <

'lnA - n (S1 - S0 ) ln y - nln ([J!))}

(20) (¿1 - ¿0)

The OC function, A and B same as previously.

(A'0 - 1)

L(^) = (21)

Here to is the positive as well as negative but not zero

E^etoZ^ = 1. (22)

Using Note 1 with (22), we get

r (t0 (S1 - S0)+ S)\ = (f0 r(S) / I r (S0)) .

Taking the logarithm of both sides of (23), with ln(1 + x); -1 < x < 1

ln r(x) = ln Vlñ - x + ^x - ^ ln

(23)

x + ( x - 1 ) ln x (24)

(25)

By using the equation (24) of approximation, we get

T ()3 (6 + D - T ()2 ( 26 + 1) - (60 - 1) ln60 + (61 - 1) ln61

- +ln 6 - 26) (61 - 60)=0

Simplifying terms up to the third degree in t0, we get the roots of t0 from (25).

E {ln(g (X--1; 0))} = Jo" (Znx)x6-1 e-7xdx (26)

We achieved, using [10], that

E {ln(g (xri;0))} = {0(6) - 1n7}, (27)

And 0(6) is specified as

0(6) = W 'nr(6)

Using (7) and (26), we find

E (Zi | 6) = [ln {r (60)} - ln {r (61)}] + (61 - 60) 0(6) (28)

The ASN function for Ho and H using (22) and (27) are specified as

E (N) =_(1 - a)ZnB + a ZnA_ (29)

0( V) {ln (r (60)) - ln (r (61))} + (61 - 60) 0 (6) (9)

and

F (N) = P 'nB + (1 - ft)/nA_

1( V) {ln (r (60)) - ln (r (61))} + (61 - 60) 0 (6) (30)

6. SPRT FOR EVALUATING THE HYPOTHESES OF 6 ALTHOUGH 7 IS

CHANGING

Using Section (5), The greatest value of ASN attained for 6 = 67, where 67 is the result of

E (Zi | 6) = 0

7) = {Znf (61) - InT (60)} (61 - 60)

This gives the highest worth as

E (N) ~ ('nA * Zn B) ts (N) =--

E (z2 | 6)

Using (17) and [10], we get

E (Z2 | 6) = {ln (r (60) /r (61 ))}2 + (61 - 60)2 {(0(6))2 + £(2,7- 1 Where £(z, q) is specified as

( 1 £ (z,q)=£A (F^

Where t0 is the solution of the equation

C { f(S0B }'" f (x,;a,',6,» dx- = 1 ™

We achieve this using (17) and (31),

7 or,

(S1 -S0 )t0

r (S0) 1t» ra-1 g(S1 -So)h+S-1 (x-1; e) g (x-1 : e) exp (-7g (x-1 : e)) dx,

r (S1H r(s) J0

x2

to (S1 -So )tj IM It0 r ((S1 - S» ) t» + S) _ 1 <p2 X r (S1 )/ r(S) 1

(32)

Where 02 _ 7 •

By applying the logarithm function to both sides of the equation (32), and employing the approximation (24), the solutions for the variable t0 are obtained from the following equation,

t»2 ( S1 - So 6

(S + 1) - t» (2S + 1) - (So - 1)lnS» + (S1 - i)lnS1

- (S1 - So) ln 02 - (1 + ln S - ¿) (S1 - So) _ 0

(33)

The ASN function coincides with (8),

E (Zi | S) _ ln{ r|]} + (S1 - S») r(S) + (S1 - S») ln 02

(34)

7. SPRT ROBUSTNESS FOR 7 WITH INDICATED COEFFICIENT OF

VARIATION

If g(x; e)_ ^T, S _ h (h > 0) in (1), the values of y _ for h > 2 and a2 _ (h-22h2h-4), for h >

4. Then, the coefficient of variation (CV)

C

(h - 4)

Assume that the value of the coefficient of variation alters from to c to c*, then 5 becomes

S* _

1

C*2

+2

The OC function is

01 to ln ( 71 ) _ ln

1 + to

71 - 7o 7

Solve (37) as (5) up to the third degree in t0 and find the roots of t0 from (39)

{3 (^ )3} '2 - {2 (^ )2} '0+{(^) -** (Y0

where 01 _ (j*) •

The ASN function coincides with (8)

E (Zi | 7) _ 01 [ln ^ -< 71

(35)

(36)

(37)

o (38)

7o

(39)

1

2

8. ACCEPTANCE AND REJECTION REGION

we need to assess the simple hypotheses H0 : Y = 70 as opposed to Hi : 7 = yi (> y0) having preassigned 0 < a and fi < 1 then Zj is

Zi = s.ln( yO) - g (x-1;£) (yi - yo)

(40)

Define, Z(N) = £f=i Xj and N = initial integer n(> 1), so that the inequality is defined as Z(N) < ci + dn or Z(N) > C2 + dn valid among the parameters.

ln B ln A ^ 3^f Yi

ci = 7-r, C2 = 7-r and d

(yi - y0) (yi - y0) (yi - y0)

9. RESULT AND DISCUSSION

Table 1: H0 : Y0 = 22, : y0 = 26 H0 : S0 = 22, : S0 = 26

Y l(y) E[N ] s L(s) E[ N]

22.0 0.997848 396.3 22.0 0.997500 i6.82

22.2 0.995846 442.9 22.2 0.995382 i8.70

22.4 0.992i0i 499.5 22.4 0.99i5i7 20.98

22.6 0.985i9i 568.6 22.6 0.984524 23.77

22.8 0.972657 653.2 22.8 0.9720i9 27.i6

23.0 0.950427 755.6 23.0 0.950054 3i.27

23.2 0.9i2296 875.9 23.2 0.9i2590 36.08

23.4 0.850i78 i008.3 23.4 0.85i663 4i.38

23.6 0.756664 ii36.4 23.6 0.759744 46.52

23.8 0.63i008 i232.9 23.8 0.635534 50.40

24.0 0.485370 i268.9 24.0 0.490420 5i.83

24.2 0.342685 i233.2 24.2 0.347054 50.32

24.4 0.224024 ii40.9 24.4 0.227029 46.47

24.6 0.i38008 i02i.i 24.6 0.i39693 4i.46

24.8 0.08i636 898.8 24.8 0.082409 36.35

25.0 0.047077 788.0 25.0 0.047344 3i.74

25.2 0.026743 693.5 25.2 0.026777 27.8i

25.4 0.0i5062 6i5.i 25.4 0.0i50ii 24.55

25.6 0.008442 550.6 25.6 0.008374 2i.88

25.8 0.0047i9 497.5 25.8 0.004660 i9.69

26.0 0.002634 453.5 26.0 0.002590 i7.87

Figure 1: OC and ASN Curve for section 3.

1.200000 1 Operating Characteristics

1.000000

J °'800000

1

^ 0.400000

21.5 22.0 22.5 23.0 23.5 24.0 24.5 25.0 25.5 26.0 26.5 Values of 6

Figure 2: OC and ASN Curve for section 5.

I. The values denoted by the OC and ASN functions for sections 3 and 5 under a = ß = 0.05, corresponding to the parameters 7 and 5 can be found in Table 1, while the visuals representing these values are illustrated in Figures 1 and 2. The table mentioned above and curves yield outcomes that are deemed acceptable.

Table 2: OC and ASN Functions for section 4, under a = ß = 0.05, where Ho : 70 = 22, Hi : 71 = 26

01 = 0.95 01 = 0.98 01 = 1 01 = 1.02 01 = 1.05

Y L(Y) E[ N] L(Y) E[N ] L(Y) E[N] L(y) E[N ] l(y) E[ N]

22.0 0.999977 256.109 0.999593 325.533 0.997848 396.275 0.990174 501.269 0.925563 760.351 22.2 0.999949 275.377 0.999182 357.072 0.995846 442.881 0.981647 572.664 0.871228 878.478 22.4 0.999891 297.332 0.998388 394.382 0.992101 499.451 0.966278 659.846 0.787253 998.243 22.6 0.999773 322.565 0.996875 439.007 0.985191 568.593 0.939285 764.602 0.670219 1097.251 22.8 0.999539 351.837 0.994031 492.942 0.972657 653.184 0.893689 885.460 0.528271 1148.101 23.0 0.999084 386.143 0.988758 558.689 0.950427 755.616 0.821178 1013.745 0.382212 1134.421 23.2 0.998208 426.784 0.979124 639.166 0.912296 875.930 0.715853 1129.741 0.255118 1063.405 23.4 0.996548 475.461 0.961857 737.241 0.850178 1008.278 0.581040 1204.816 0.159600 959.524 23.6 0.993440 534.365 0.931752 854.390 0.756664 1136.436 0.433599 1215.202 0.095382 847.741 23.8 0.987698 606.223 0.881433 987.837 0.631008 1232.898 0.297492 1159.170 0.055358 743.688 24.0 0.977243 694.173 0.802653 1126.039 0.485370 1268.878 0.190055 1058.028 0.031567 653.709 24.2 0.958580 801.169 0.690763 1245.161 0.342685 1233.182 0.115203 939.908 0.017819 578.588 24.4 0.926211 928.388 0.551685 1313.657 0.224024 1140.910 0.067448 825.361 0.010001 516.709 24.6 0.872518 1071.916 0.404611 1309.361 0.138008 1021.076 0.038658 724.251 0.005595 465.802 24.8 0.789395 1217.829 0.273264 1236.136 0.081636 898.796 0.021884 639.054 0.003124 423.701 25.0 0.673200 1339.075 0.172445 1120.253 0.047077 788.025 0.012302 568.673 0.001742 388.586 25.2 0.531691 1402.143 0.103641 991.440 0.026743 693.530 0.006888 510.809 0.000971 359.005 25.4 0.385428 1386.471 0.060349 869.573 0.015062 615.103 0.003848 463.075 0.000540 333.831 25.6 0.257639 1300.211 0.034475 763.352 0.008442 550.600 0.002147 423.395 0.000301 312.195 25.8 0.161310 1173.099 0.019477 674.378 0.004719 497.491 0.001197 390.091 0.000167 293.426 26.0 0.096429 1035.896 0.010936 601.031 0.002634 453.477 0.000666 361.853 0.000093 277.004

II. Figure 3 illustrates the numerical values of the OC and ASN curves extracted from Table 2, corresponding to different fa values. When fa < l(fa > 1), the OC curve shifts either towards the right or left direction, while the ASN curve shifts towards the upper right or lower left direction. Both curves demonstrate that the SPRT exhibits a high degree of sensitivity towards alterations in 5.

Figure 3: OC and ASN Curve for section 4.

Table 3: OC and ASN Functions for section 6, under a = ß = 0.05 where H0 : H0 : ¿0 = 22, H1 : ¿1 = 26

<£2 = 0.95

<£2 = 0.99

<£2 = 1

<£2 = 1.02

<£2 = 1.05

s L(s) E[N ] L(s) E[ N ] L(s) E[ N] L(s) E[ N] L(s) E[ N]

22.0 22.2 22.4 22.6 22.8 23.0 23.2 23.4 23.6 23.8 24.0 24.2 24.4 24.6 24.8 25.0 25.2 25.4 25.6 25.8 26.0

0.999949 0.999903 0.999816 0.999653 0.999350 0.998787 0.997751 0.995848 0.992379 0.986097 0.974842 0.955006 0.920963 0.865024 0.779245 0.660606 0.517881 0.372321 0.246826 0.153354 0.091038

12.805 13.769 14.867 16.128 17.592 19.307 21.339 23.773 26.718 30.311 34.709 40.058 46.419 53.596 60.891 66.954 70.107 69.324 65.011 58.655 51.795

0.998800 0.997769 0.995875 0.992418 0.986152 0.974920 0.955114 0.921109 0.865216 0.779487 0.660892 0.518186 0.372612 0.247071 0.153542 0.091170 0.052585 0.029808 0.016731 0.009340 0.005199

16.277 17.854 19.719 21.950 24.647 27.934 31.958 36.862 42.719 49.392 56.302 62.258 65.683 65.468 61.807 56.013 49.572 43.479 38.168 33.719 30.052

0.997500 0.995382 0.991517 0.984524 0.972019 0.950054 0.912590 0.851663 0.759744 0.635534 0.490420 0.347054 0.227029 0.139693 0.082409 0.047344 0.026777 0.015011 0.008374 0.004660 0.002590

16.821 18.704 20.984 23.765 27.161 31.265 36.081 41.381 46.521 50.398 51.829 50.320 46.466 41.460 36.355 31.737 27.807 24.553 21.882 19.688 17.872

0.989824 0.981471 0.966597 0.940663 0.896956 0.827248 0.725144 0.592699 0.445466 0.307359 0.196953 0.119421 0.069790 0.039871 0.022479 0.012580 0.007012 0.003901 0.002167 0.001204 0.000669

25.063 28.633 32.992 38.230 44.273 50.687 56.487 60.241 60.760 57.959 52.901 46.995 41.268 36.213 31.953 28.434 25.540 23.154 21.170 19.505 18.093

0.930141 0.879707 0.800964 0.689215 0.550252 0.403178 0.271799 0.171045 0.102434 0.059402 0.033783 0.018999 0.010619 0.005915 0.003290 0.001828 0.001015 0.000564 0.000313 0.000174 0.000097

38.018 43.924 49.912 54.863 57.405 56.721 53.170 47.976 42.387

37.184 32.685 28.929 25.835 23.290

21.185 19.429 17.950 16.692 15.610 14.671 13.850

Operating Characteristics —•— <|>2 = C .95

1.000000 1 —•— 4>2 = 1 .02

\ \ —«t>2 = .05

— JS I.

1

Vr

22.0 22.5 23.0 23.5 24.0 24.5 25.0 25.5 26.0 26.5 Values of 6

Figure 4: OC and ASN Curve for section 6. III. Figure 4 portrays the values of the operational characteristic (OC) and average sample num-

ber (ASN) curves derived from Table 3 across various magnitudes of <2. When < < 1(< > 1), the OC curve experiences a rightward (leftward) shift, while the ASN curve undergoes an upward rightward (downward leftward) shift. Both curves demonstrate the considerable sensitivity of the sequential probability ratio test (SPRT) to parameter 7 alterations.

IV. Figure 5 illustrates the plotted values of the OC and ASN curves obtained from Table 4 while considering different values of 'ty'. When ty < 1(ty > 1) is taken into account, the OC curve experiences a shift towards the right (left), while the ASN curve shifts upwards (downwards) towards the right. It is evident from both curves that the SPRT demonstrates a considerable level of sensitivity towards variations in 'ty'.

Table 4: OC and ASN Functions for section 7, under a = ß = 0.05 where Hq : Hq : 70 = 22, Hi : 71 = 26

^ = 0.96 $ = 1 $ = 1.04

Y L(Y) E[N] L(Y) E[N ] l(y) e[ n]

22.0 0.999936 275.746 0.997848 396.275 0.960659 658.849

22.2 0.999864 298.188 0.995846 442.881 0.929568 764.064

22.4 0.999719 324.055 0.992101 499.451 0.877735 883.456

22.6 0.999432 354.157 0.985191 568.593 0.796964 1006.162

22.8 0.998873 389.555 0.972657 653.184 0.683060 1110.363

23.0 0.997803 431.636 0.950427 755.616 0.542805 1168.026

23.2 0.995782 482.209 0.912296 875.930 0.396046 1160.562

23.4 0.992010 543.586 0.850178 1008.278 0.266293 1092.897

23.6 0.985068 618.594 0.756664 1136.436 0.167506 988.999

23.8 0.972493 710.346 0.631008 1232.898 0.100472 874.913

24.0 0.950216 821.423 0.485370 1268.878 0.058441 767.625

24.2 0.912031 951.852 0.342685 1233.182 0.033369 674.376

24.4 0.849857 1095.264 0.224024 1140.910 0.018849 596.352

24.6 0.756289 1234.025 0.138008 1021.076 0.010583 532.042

24.8 0.630591 1338.272 0.081636 898.796 0.005922 479.150

25.0 0.484941 1376.797 0.047077 788.025 0.003307 435.440

25.2 0.342284 1337.532 0.026743 693.530 0.001844 399.016

25.4 0.223688 1236.956 0.015062 615.103 0.001028 368.364

25.6 0.137752 1106.603 0.008442 550.600 0.000572 342.305

25.8 0.081457 973.726 0.004719 497.491 0.000318 319.929

26.0 0.046959 853.432 0.002634 453.477 0.000177 300.536

—•—1|)= 0.96

—•— 4) = 1.04

_

»

\

1_

24.0 25.0 26.0 27.0

Values of y

1600.000 1400.000 1200.000 S 1000.000 S 800.000

cd

> 600.000 400.000 200.000

Average Sample Number ~~

--iM

0.000

—•— ij) = l.j

21.0 22.0

23.0 24.0

Values of y

25.0 26.0 27.0

Figure 5: OC and ASN Curve for section 7. V. The acceptance and rejection zones for the null hypothesis Hq, with Hq : 70 = 22 and the

alternative hypothesis H0 : 7o = 26. Both the a and ft significance levels are set to 0.05, and the degrees of freedom 5 are set to 2. The values of the constants ci, c2, and d are -287.0828, 287.0828, and -27.90466, respectively. As a result, if the observed value Z(n) is less than or equal to —27.90466N + 287.0828, we accept the null hypothesis H0, and we accept the alternative hypothesis H1 if Z(n) is higher than or equal to —27.90466N — 287.0828. In the intermediate stages, the sampling procedure continues.

1000

0 5 10 15 20 25 30

n.......>

Figure 6: The Acceptance and Rejection zones for H

References

[1] Buonaguidi, B. and Muliere, P.(2013). On the Wald's Sequential Probability Ratio Test for Levy Processes. Sequential Analysis, 32(3), 267 - 287.

[2] Boroumand, P. and Guillen i Fabregas, A. (2021). Error exponent sensitivity of sequential probability ratio testing. IEEE International Symposium on Information Theory (ISIT), 184"189.

[3] Chaturvedi, A., Tiwari, N. and Tomer, S.K. (2002). Robustness of the sequential testing procedures for the generalized life distributions. Brazilian Journal of Probability and Statistics, 16, 7"24.

[4] Chaturvedi, A., Chauhan, K., and Alam, M. W. (2009). Estimation of the reliability function for a family of lifetime distributions under type I and type II censorings. Journal of reliability and statistical studies, 11-30.

[5] Chaudhary, A. and Chauhan, K. (2009). Estimation and testing procedures for the reliability function of Weibull distribution under type I and type II censoring. Journal of statistics sciences, 1(2), 121-136.

[6] Chaturvedi, A, Chauhan, K., Alam, M. W. (2013). Robustness of the sequential testing procedures for the parameters of zero-truncated negative binomial, binomial, and Poisson distributions. Journal of the Indian Statistical Association, 51(2), 313-328.

[7] Chauhan, K. S. (2022). Estimation and testing procedures of P (Y< X) for the inverse distributions family under type-II censoring. Reliability. Reliability: Theory Applications, 17(3 (69)), 328-339.

[8] Epstein, B. and Sobel, M. (1955). Sequential life test in exponential case. The Annals of Mathematical Statistics, 26, 82-93.

[9] Eger, K.and Tsoy, E.B. (2008). Robustness of sequential probability ratio tests in case of nuisance parameters. Third International Forum on Strategic Technologies, 266-270.

[10] Gradshteyn, I. S. and Ryzhik, I. M. (2007). Tables of Integrals Series and Products, Academic Press, New York.

[11] Harter, H.L. and Moore, A.H. (1976). An Evaluation of Exponential and Weibull Test Plans.IEEE Transactions on Reliability, 25(2), 100-104.

[12] Hubbard, D. J. and Allen, O.B. (1991). Robustness of the SPRT for a negative binomial to misspecification of the dispersion parameter. Biometrics, 47(2), 419-427.

[13] Hauck, D. J. and Keats, J.B. (1997). Robustness of the exponential sequential probability ratio test (SPRT), when Weibull distributed failures are transformed using a known' shape parameter.Microelectronics Reliability, 37(12), 1835-1840.

[14] Kharin, A. Y. (2016). On Robustifying of the Sequential Probability Ratio Test for a Discrete Model under "Contaminations". Austrian Journal of Statistics, 31(4), 267-277.

[15] Kharin, A. Y.and That Tu, T. (2018). Performance analysis and robustness evaluation of a sequential probability ratio test for non-identically distributed observations. Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series 54(2), 179-192.

[16] Kleijnen, J. P. C. and Shi, W.(2021). Sequential probability ratio tests: conservative and robust. SIMULATION, 97(1), 33-43.

[17] Liu, Y.and Li, X. R. (2013). Performance Analysis of Sequential Probability Ratio Test. Sequential Analysis, 32(4), 469-497.

[18] Montage, E. R., and Singaurwalla, N. D. (1985). Robustness of the Sequential Exponential Life-Testing Procedures. Jour. Amer. Statist. Assoc., 80, 715-719.

[19] Pandit, P. V. and Gudaganavar, N. V. (2010). On Robustness of a Sequential Test for Scale Parameter of Gamma and Exponential Distributions. Applied Mathematics-Journal of Chinese Universities Series B, 01, 274-278.

[20] Raghavachari, M. (1965). Operating Characteristic and Expected Sample Size of a Sequential Probability Ratio Test for the Simple Exponential Distribution. Calcutta Statistical Association Bulletin, 14, 65 - 73.

[21] Wald, A. (1947). Sequential Analysis. John Wiley and Sons, New York.