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17
CHARACTERIZATION OF SOME CONTINUOUS DISTRIBUTIONS BY CONDITIONAL VARIANCE OF
RECORD VALUES
Zaki Anwar1, Mohd Faizan2, Zakir Ali3
•
department of Statistics and Operations Research (Women's College), Aligarh Muslim University, Aligarh, India. 2,3Department of Statistics and Operations Research, Aligarh Muslim Universty, Aligarh, India. [email protected] [email protected] 3Corrosponding Author: [email protected]
Abstract
Characterization of a probability distribution gives a unique property enjoyed by that distribution. Various approaches are available in the literature to characterize distributions through record values. Many researchers have characterized Exponential, Pareto, and Power function distributions using moments, conditional expectation, and some other characteristics of record values. In this paper, we have characterized these three distributions through conditional variance of adjacent record values. The results have been verified using numerical computation.
Keywords: Characterization of continuous distributions, conditional variance, record values.
1. Introduction
Let X1, X2,... be a sequence of independent, identically distributed random variables with distribution function (df )F(x) and probability density function (pdf)f (x). Let XU(r) be the r th upper record value, then the conditional pdf of XU(r+1) given XU(r) = x, 1 < r < s is given by (Ahsanullah, 2004)[1]
f (XU(r+i) = y I XU(r) = x) = ^ (L1)
where F(x) = P(X > x) = 1 - F(x).
One can transform the upper record into lower record values by replacing the original sequence of (Xj) by (-Xi, j > 1) (Ahsanullah, 2004) [1]. Let XL(r) be the r-th lower record value, then the conditional pdf of XL(r+1) given XL(r) = x, 1 < r < s is given by
f (XL(r+i) 1 XL(r) = x) = F^j. (L2)
The record values have been extensively studied in literature. For an excellent review, one may refer to Ahsamullah (2004) [1]. Arnold at al. (1998) [2] and Nevzorov (2001) [3] amongst others. Characterization of distributions through conditional expectations of record values have been considered, among others, by Nagaraja, H.N. and Nevzorov, V.B. (1997) [4], Franco and Ruiz(1997)
[5], Athar et al. (2003) [6], Khan et al. (2010) [7] and Faizan and Khan (2011) [8].
Beg, M.I. and Kirmani. S.N.U.A. (1978) [9] characterized exponential distribution by a weak homoscedasticity. Khan and Beg (1987) [10] extended the result of Beg and Kirmani (1978) for Weibull distribution. Khan et al. (2008) [11] characterized a general class of distribution by conditional variance of order statistics and Shah et al. (2018) [12] characterized Pareto and power function distributions by conditional variance of order statistics, In this paper we have characterized exponential, Pareto and power function distributions by conditional variance of record values.
2. Characterization Results
Theorem 2.1: Let x be a random variable with df F(x) and E (X2) < to. Then for r < s
V
Xx(r+1) 1 Xv(r) — x
02
for some 0 > 0 if and only if
F(x) — e-re; x > 0.
Proof: First we will prove (2.2) implies (2.1). It is easy to see that from (1.1) and (2.2)
and
E
Xu(r+1) 1 Xu(r)
X2(r + 1) 1 Xu(r)) — x
— x + 0
x2 + 2x0 + 202
Now, using (2.3) and (2.4), we have
V
For sufficiency part, we have from (2.2)
Xu(r+1) 1 Xu(r) — x
if(y),
f (y),
(2.1) (2.2)
(2.3)
(2.4)
Jx VW) dy "I Jx VW) dV) — 0
f- to / f- TO \ 2
F(x) jx y2f (y)dy jx yf (y)d^ — 02(x) Differentiating (2.5) twice w.r.t. x and simplifying, we get
p TO
/ yf (y)dy — xFF (x) + 02f (x)
x
Now differentiate (2.6) again w.r.t. x, we get
F (x) — -02f' (x)
and hence the result.
Theorem 2.2: Let X be a random variable with df F(x) and E (X2) < to. Then, for some r < s and 0 < p < 1, we have
(2.5)
(2.6)
V
Xu(r+1) 1 Xu(r) — x
p
(p - 2)(p - 1)2'
if and only if
p
F(x) — VxJ ; a - x < to.
(2.7)
(2.8)
E
x
2
0
2
to
to
2
Proof: First we will prove (2.8) implies (2.7). By using (1.1) and (2.8), it is easy to show that
E
and
E
which gives
Xu(r+1) I Xu(r) — x
Xl(r+1) 1 Xu(r) — x
p
r x
p -1
p 2 rX2
p - 2
V
Xu(r+1) 1 Xu(r)
x
p2 r X2.
(p - 2)(p - 1)2
Now, to prove (2.7) implies (2.8), we have using (1.1) and (2.7)
/•TO / pTO \ 2
F(x)Jx y2f (y)dy yf (y)dyj — cx2F2(x) (2.9)
where
p
(p - 2)(p - 1)2'
Differentiating (2.9) twice w.r.t. x and simplifying, we get
C to C TO _ F2( x)
Jx y2f (yy)dy - 2xJx yf (y)dy — (2c - 1)x2F(x) - 2cxj(-). (2.10) Now, after differentiating (2.10) w.r.t. x, we get
£ yf (y)dy — cx2f (x) - (4c - 1)xF(x) + c^ - cx^fff1 ■ (2.11) Again differentiating (2.11), we get
2xHfxl_ 2 F (x)f' (x) F(x)f'' (x) + f' (x) f3 (x) 2 f2 (x) x f'2(x) + 2xf (x)
2 f (x) , f (x) , 1
+ + 6x- 6 ^ — 0.
F(x) F(x) c
Let Fx) — y — y(x) bearing in mind that f(x) — F'(x), f'(x) — F''(x), f'(x) — F'''(x),
F~m— y' + y2, iFw— y" + 3yy' + y3, we get
y'' + 3yy' + y3 „ (y' + y2)2 ( 2 2 2x\ ( , , 2 „ 2 xl-w-- 2x——--x2--^--y' + y1) - 6xy + p2 - 4p---1 — 0.
y2 - y3 - - y2 - y - - - p -
(2.12)
There exists a unique solution of the differential equation (2.12) that satisfies the prescribed initial conditions
that y'(a) — - £
and
y(a) — p J a
where a is any finite point in the support of F. Thus by the existence and uniqueness theorem (Boyce and Diprima, 2012) [13], we get
F'(x) _ _ p l{x)— y — - x
which implies that
F(x) — (Ax)-p; a < x < to.
where A is a constant to be determined and hence the Theorem.
Theorem 2.3: Let X be a random variable with dfF(x) and E (X2) < to. Then for r < s
V
XL(r+1) 1 XL(r) — x
p
(p+2)(p+1)2
if and only if
F(x)— (0Y; 0 - x < 0 <
to.
(2.13)
Proof: This can be proved on lines of Theorem 2.2
Table 1: Verification of the characterization results in case of Exponential distribution.
2
0 X L.H.S. R.H.S. |L.H.S. - R.H.S.| 1 L.H.S.-R.H.S. | 1 R.H.S 1
1.5 0.4 2.2499 2.25 0.0001 0.00005
2.5 0.8 6.2497 6.25 0.0003 0.00005
4.5 1.6 20.2504 20.25 0.0004 0.00002
5.5 2.0 30.2454 30.25 0.0046 0.00015
6.5 2.4 42.2430 42.25 0.0070 0.00017
7.5 2.8 56.2436 56.25 0.0064 0.00011
8.5 3.2 72.2603 72.25 0.0103 0.00014
9.5 3.6 90.2302 90.25 0.0198 0.00022
10.5 4.0 110.2485 110.25 0.0015 0.00001
11.5 4.4 132.2380 132.25 0.0120 0.00009
12.5 4.8 156.1910 156.25 0.0590 0.00038
13.5 5.2 182.2197 182.25 0.0303 0.00017
14.5 5.6 210.2026 210.25 0.0474 0.00023
15.5 6.0 240.2801 240.25 0.0301 0.00013
Table 2: Verification of the characterization results in case of Pareto distribution.
a p X L.H.S. R.H.S. |L.H.S. - R.H.S.| 1 L.H.S.-R.H.S. | | R.H.S |
0.3 3 0.2156 0.0347 0.0348 0.0001 0.0029
0.7 4 0.5797 0.0745 0.0747 0.0002 0.0027
1.1 5 0.8523 0.0756 0.0757 0.0001 0.0013
1.5 6 1.1692 0.0838 0.0820 0.0018 0.0220
1.9 7 1.4536 0.0699 0.0822 0.0123 0.1496
2.3 8 1.7510 0.0836 0.0834 0.0151 0.1530
2.7 9 2.0642 0.0943 0.0856 0.0087 0.1016
3.1 10 2.3171 0.0829 0.0812 0.0017 0.0209
3.5 11 2.6390 0.0850 0.0851 0.0001 0.0011
3.9 12 2.9604 0.0878 0.0869 0.0009 0.0103
4.3 13 3.2612 0.0847 0.0873 0.0026 0.0002
4.7 14 3.5998 0.0899 0.0895 0.0004 0.0045
5.1 15 3.8431 0.0862 0.0869 0.0007 0.0080
5.5 16 4.2212 0.0871 0.0905 0.0034 0.0376
5.9 17 4.5505 0.0917 0.1147 0.0230 0.2005
Conclusions:
This paper introduces a study of the Exponential, Pareto, and Power function distributions, showcasing their characterizations based on the conditional variance of adjacent record values. The validity of our findings has been confirmed through some numerical computation.
References
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