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CHARACTERIZATION OF GENERALIZED DISTRIBUTIONS BASED ON CONDITIONAL EXPECTATION OF ORDER STATISTICS
Abu Bakar1, Haseeb Athar2 and Mohd Azam Khan3
Department of Statistics and Operations Research, Aligarh Muslim University
Aligarh - 202002, India [email protected], [email protected] Corresponding author: [email protected]
Abstract
Characterization of probability distributions plays a significant role in the field of probability and statistics and attracted many researchers these days. Characterization refers to the process of identifying distributions uniquely based on certain statistical properties or functions. The various characterization results have been established by using different methods. The paper aims to characterize two general forms of continuous distributions using the conditional expectation of order statistics. Further, the results obtained are applied to some well-known continuous distributions. Finally, some numerical calculations are performed.
Keywords: Order statistics, conditional expectation; truncated moments, characterization, continuous distributions.
I. Introduction
The characterization of probability distributions is indeed crucial in statistical studies as it allows us to understand and utilize various distributions effectively. The distributions can be characterized using different statistical properties like moments, truncated moments, order statistics, record values, reliability functions, characteristic function etc. Each of these approaches leverages different statistical properties to uniquely identify or characterize specific probability distributions. This allows statisticians and researchers to model data effectively, understand underlying processes, and make informed decisions based on statistical analyses.
The various characterization results using truncated moment and conditional expectation of order statistics are available in the literature. Grudzien and Szynal [15] characterized the uniform distribution in terms of moments of order statistics when the sample size is random whereas Balasubramanian and Beg [11] focused on distribution characterizations using moments and order statistics. Khan and Abu-Salih [18] characterized a general class of distribution through conditional expectations of order statistics. Further, Khan and Abouammoh [17] extended the results of Khan and Abu-Salih [18] by characterized the general form of distribution for higher order gap. Khan and Athar [23] characterized some continuous distributions by examining the linearity of regression when conditioned on a pair of order statistics. Nassar [27] characterized a mixture of two generalized power function distributions based on the conditional expectation of order statistics. Similarly, Lee
et al. [25] provided a characterization of mixtures of Weibull and Pareto distributions through the conditional expectation of order statistics and upper record values. The recent interest on characterizing probability distributions via truncated moments has led to significant contributions from various authors. For example, Ahsanullah et al. [3] explore the characterization of Lindley distribution based on a relation between truncated moments and failure rate function. Kilany [24] established the characterization of the Lindley distribution using the truncated moments of order statistics. Athar and Abdel-Aty [6] characterized a class of continuous distributions based on left and right truncated moments whereas Athar et al. [9] studied the characterization of some generalized continuous distribution by doubly truncated moments. Bashir and Khan [12] employed a range of techniques to characterize the weighted power function distribution, using mean inactivity times, mean residual function, conditional moments, conditional variance, doubly truncated mean, incomplete moments, and the reverse hazard function. For more studies on characterization, one can refer to Khan and Khan [21], Khan and Masoom Ali [22], Franco and Ruiz [14], Ali and Khan [4], Khan and Athar [20], Khan and Alzaid [19], Athar et al. [10], Ahsanullah and Hamdani [1], Huang and Su [16], Ahsanullah and Shakil [2], Athar and Akhtar [7], Athar et al. [8], Ansari et al. [5] and references given therein.
Let Xi,Xj,...,Xn be a random sample of size n(> 2) from a continuous population having probability density function (pdf) f (x) and cumulative distribution function (cdf) F(x) and its corresponding order statistics be the Xy.n < Xj-n <... < Xnn respectively. It is well Known that the
conditional pdf of rth order statistic (Xrn = x) given that sth order statistic (Xs.n = y) for s > r,
is the same as the distribution of the rth order statistics obtained from a sample of size (s — 1) from
th
a population whose distribution is truncated on the right side at y while the conditional pdf of s order statistic (Xs.n = y) given that rth order statistic (Xr.n = x) for r < s, is the same as the
distribution of the (s — r)th order statistics from a sample of size (n — r) from a population whose distribution is simply truncated on the left side at x (David and Nagaraja [13]).
Therefore, the conditional pdf of rth order statistic (Xrn = x) given that sth order statistic
(Xs.n = yX1 < r < s < n ^
f (x | y)
(s — 1)! ( F (x) )r—1 ( F (y) — F (x) )
s—r—1
f (x) x <y
(1.1)
(r — 1)!(s — r — 1)! ( F (y) )s—1
and the conditional pdf of s order statistic given r order statistic is
th
th
(n — r)! ( F (y) — F (x) )s—r—1 (1 — F (y))
n—s
f (y), x <y . (1.2)
(s — r — 1)!(n — s)!
Also note that for any monotonic and differentiable function ^ (x) of X over the support (a, 0) . Here a may be —ro and 0 may be .
E[£(Xnn) | Xn_1.n = x] = E[£(X) | X > x]
(1.3)
and
E[£(XVn)|X2n = x] = E£(X) | X < x].
(1.4)
Abu Bakar, Haseeb Athar, and Mohd. Azam Khan RT&A, No 4(80) CHARACTERIZATION OF GENERALIZED DISTRIBUTIONS..._V°lume 19, Decemter, 2024
In this paper, first we have characterized two general form of distributions F(x) = e~a^(x\ a * 0 and F (x ) = 1 - x\ a * 0 through the conditional expectation of
order statistics E [£(X) | Xr.n = x], where Xr.n is rth order statistics. Further, these results are applied to some well-known continuous distributions.
II. Characterization Theorems
Before presenting the main result, we will discuss the following propositions that were established by Athar et al. [10]:
Proposition 2.1: Let X be a random variable and E [%(X) | Xr.n = x] = /ur (x), then for 1 < r < s < n
f ^ _ n(s-X)^r(x)-n(r-\)v's(x)-(s-r)g{x)
1 - F(x) n(s -1)jur (x) - n(r -1)j (x) - n(s - r)£(x) where £(x) is a monotonic and differentiable of x e (a, P).
Proposition 2.2: Under the conditions as stated in Proposition 2.1,
f (x) = n(n - r)y's (x) - n(n - s)y'r (x) - (s - r'(x) F(x) n(n - r)js (x) - n(n - s)jr (x) - n(s - r)£(x)
(2.2)
Theorem 2.1: Let X be a absolutely continuous (w.r.t Lebesgue measure) random variable with CDF F(x) and PDF f (x) on the support (a, ¡), where a and 3 may be finite or infinite. Then for 1 < r < s < n
E [aX)IXm = x] = vr (x) = To(x) + — Ti(x) + — T2(x), (2.3)
n n
if and only if
F(x) = e~a^(x), x e (a, ¡3), a * 0, (2.4)
where,
F ( x) = 1 - F ( x)
M. /
n
x)
_ | x
T0( x) =
Ti( x) = E [Ç( X )\X < x] = T2( x) -
1 - e~a%(x) ' 1
T2( x)=e [a X )\x > x]=a x)+-,
a
and a(x) is a monotonic and differentiable function of x, such that ¿¡(x) ^ 0 as x ^ a and ¿;(x)F(x) ^ 0as x ^ p.
Abu Bakar, Haseeb Athar, and Mohd. Azam Khan RT&A, No 4(80) CHARACTERIZATION OF GENERALIZED DISTRIBUTIONS..._V°lume 19, December, 2024
Proof. To prove necessary part, the value of T (x) and 22 (x) can be obtained easily by integrating
by parts and noting that
1 - F(x) = —^ f (x). aç (x)
To prove the sufficiency part, from (2.1) we have
f (x) n(s -1K (x) - n(r -ï)p's (x) - (s - r)ç'( x) N
1 — F (x) n(s — 1)^r (x) — n(r — 1)^s (x) — n(s — r)£( x) D
Now,
n(s — 1) /ur (x) — n(r — 1)^s (x) = (s — r )£( x) + (n — 1)(s — r)T2 (x).
Thus,
D=(n — 1)(s — r) a
and N = (n — 1)(s — r)£' (x). Therefore,
t—FX=<(x) •
Integrating both the sides with respect to x, we get
1 — F (x) = e"ai( x), and hence the sufficiency part.
Theorem 2.2: Under the conditions as stated in Theorem 2.1.
E[£(X) | XrM = x] = Ar(x) = To(x) + —Ti(x) + — T2(x), (2.5)
n n
if and only if
F(x) = e"<(x); x e (a, 0), a * 0, (2.6)
where,
Ti(x) = E[4(X) | X < x] = £(x) + T,
a
and
Z( x)
T2( x) = E [Ç( X )| X > x] = Ti( x)--
1 - Q~aÇ(x) '
such that Ç(x)F(x) ^ 0 when x ^a and Ç(x) ^ 0 when x ^ 0.
Proof. Necessary part can be proved on the lines of Theorem 2.1 after noting the relation
F ( x) = e"aç( x > =-fx-aÇ ' (x)
To prove sufficiency part, in view of (2.2), we have
f ( x) _ N
F ( x) D '
Now,
n [(n - r)As (x) - (n - s)Ar (x)] = (s - r)£(x) + (n - 1)(s - r)E [£(x) | X < x]
Differentiating the above equation w.r.t. x, and after rearranging, we get, N = (n -1) (s - r' (x)
D = (n - 1)(s - r) 111
F$>=-D
Now, integrating both the sides w.r.t x, we get F(x) = e~a^(x), a > 0.
Hence the result is proved.
III. Examples and Applications
I. Examples Based on Theorem 2.1
1. Weibull distribution
Let the CDF of Weibull distribution is given as
F(x) = 1 - e~AxP , x > 0,A> 0, p > 0. On comparison of (2.7) with (2.4), we get
(2.7)
a = X and %(x) = xp
Thus,
Ti( x) = E
T2( x) = E Therefore,
Xp | X < x Xp | X > x
1 e
-Xxp
A
1 - e
-Axp
= xp +-. A
E
XP 1 Xrn = x
i \ xp r -1 = Vr (x ) =-+-<
n
-Axp
A
1 - e
-Axp
K ^ i xp + - j n I A\
= xp +
n -1 r -1 xp
nA n 1 - e~Axp if and only if
F (x) = 1 - e~AxP, 0 < x <»; A, p > 0.
2. Pareto Distribution
Suppose random variable X follows Pareto distribution with CDF given as
F(x) = 1 -vpx p, v <x <œ,p > 0
(2.8)
By comparing (2.8) with (2.4), we get
/
a = - p,Ç( x ) = log
V x J
Therefore,
or
or
and
v
log V X F >x
22 ( x) = E T2( x) = E [log X | X > x] = log x + p,
1
=--+ log
p
V x J
T1( x) = E
log
X < x
log
v X j
T(x) = E [log X | X < x] = log
= 22 ( x) -
V x J
V x J
Vr (x) = 1log I -1 + — -jlog ( V n V x J n I v x
1 -vpx-p "2 - vpx~p '
1 -vpx~p
2 - vpx -p 1 -vpx~p
p,
- p- + ( log x + p ) .
n
Hence,
E
log( X 11 Xr:n = x
= r log (v
n V x J
2 - vpx~p^ V 1 -vpx-p J
n-1 n - r. +--p +--log x,
or
n
E [log X|Xr:n = x] = logv- r log (V
J
2 - vp x~p 1 - vp x~p
nn
n -1 n - r,
p--log x,
A
n
n
if and only if
F(x) = 1 -vpx~p, v <x<œ.
3. Gumbel Extreme Value I
Let the CDF of Gumbel extreme value I distribution is given as
F(x) = 1 -e~e , -œ< x <œ. Now on comparison of (2.9) with (2.4), we get a = 1, Ç(x) = ex.
Therefore,
(2.9)
E
E
eX | X > x
eX | X < x
= T2( x) = ex +1
= T1( x) = 1 -
exe~e
1 - e"
x
e
and E
X IF e I X rn = x
= Vr (x) =
n - r + 1 x r - 1
-e +--
i
n
n
x -e
ee
1 -
V 1 - e y
n-r
if and only if
F (x ) = 1 — e~e, x > 0.
Similarly, with proper choice of a and £(x) several other distributions can be characterized using Theorem 2.1. For more distribution belonging to this class, one may refer to Khan and Abu-Salih [18] and Noor and Athar [28].
II. Examples Based on Theorem 2.2 1. Inverse Weibull Distribution
Let random variable X follows inverse Weibull distribution with CDF
F (x) = eXP ,0 < x . On comparison of (2.10) with (2.6), we get
Here, a = X and £(x) = x"p . Therefore,
E
E
X- p | X <
x
X- p I X >.
= 21( x) = x -p +1, A
T. 1 x-pe~Ax -= T2( x) =---
A i --aT p
and E (X"p | Xr:n = x) = Ar (x) :
1 - e"
n-1 +fr-e~A P
nA V n
f - p \ x p
i -Ax- p
V1 - e y
if and only if
F (x) = p ,0 < x <œ .
2. Power Function Distribution
(2.10)
Let the CDF of power function distribution is F(x) = v~pxp ,0 < x <v;v, p > 0.
(2.11)
Now on comparing (2.11) with (2.6), we get
a = — p and £(x) = log Therefore,
V^y
or
or
E
log W <x
1
= T1(x) = log x - log v--
p
E [log X | X < xl = T1(x) = log x - -.
p
E
log
X
v
X > x
rr ( N , 1 log(^ /v) = T2( x) = log x-------
p 1 - (x/v)p
E [log X | X > x ] = T2( x) = log v + log x - - -
log( x /v) p 1 - (x / v)p
and hence X
E
log I — 11 Xr.n = x
=K{ x )=^oÉâlvl+r— 1 1
n
n
1
A
log x--
p J
+ -
n — r
1 vp log(x fv)
log vx----—--
p vp — xp )
or
_ _ 2n — r — 1 n — r v E [log X | Xr:n = x\ = log x +--logv —
n vp — xp )
p n—1
log( x fv)--
np
if and only if
F(x) = v~pxp ,0 < x <v; v, p > 0.
3. Burr type II
Let the CDF of Burr type II distribution is given as
F(x) = (1 + e-x)-k -»< x <» . Now on comparing (2.12) with (2.6), we get
a = k and x) = log(1 + e~x). Therefore,
"log(1 + e~X ) I X < x] = T1(x) = log (1 + e"x ) +1,
(2.12)
and
E E
E
log(1 + e~X ) | X > x] = T2 (x) = log(1 + e~x ) +1 — ^^Ç^, J k 1 — (1 + e~x)—k
, ^ — „ 1 . . . n — 1 , — ^ n — r log(1 + e~x )
log(1 + e )| XKn = x =Xr(x) = -— + log(1 + e x)---^-.
J nk n 1 — (1 + e~x)—k
IV. Numerical computation
In this section, we have carried out some numerical computation. In Table 1 estimated value of X using Theorem 2.1 and based on Weibull, Pareto, and Gumbel distributions for different randomly chosen truncation points are listed while Table 2 is based on Theorem 2.2 for inverse Weibull, power function and Burr type II. A random number is used to choose the various random truncation points. For the purpose of computation work, we have taken the real data set represents the failure times of 50 components (per 1000 hours) [Merovei et al. [26]].
0.036 0.058 0.061 0.074 0.078 0.086 0.102 0.103 0.114 0.116
0.148 0.183 0.192 0.254 0.262 0.379 0.381 0.538 0.570 0.574
0.590 0.618 0.645 0.961 1.228 1.600 2.006 2.054 2.804 3.058
3.076 3.147 3.625 3.704 3.931 4.073 4.393 4.534 4.893 6.274
6.816 7.896 7.904 8.022 9.337 10.940 11.020 13.880 14.730 15.080
For the power function distribution, the values in the given data set are not within an interval of [0, 1]. Thus, the original values are divided by the maximum value (15.080) and changed into the interval [0, 1]. Similarly, for the Pareto distribution, the original values are divided by the minimum value (0.036) of the data set and shifted them in the interval of (1,
Table 1. Estimated values of X for different parameters
Distribution Parameters r x T)( x) t1( x) T2( x) Hr( X) X
3 0.061 0.005 0.048 1.292 1.221 1.492
8 0.103 0.006 0.099 1.330 1.138 1.294
13 0.192 0.009 0.168 1.387 1.076 1.157
p = 0.5,2 = 0.5 18 0.538 0.015 0.317 1.515 1.092 1.193
22 0.618 0.016 0.341 1.536 1.019 1.038
3 0.061 0.005 0.046 1.118 1.058 1.119
8 0.103 0.006 0.094 1.182 1.013 1.026
p = 0.5,2 = 1.0 13 0.192 0.009 0.158 1.282 0.996 0.992
18 0.538 0.015 0.292 1.524 1.089 1.187
Weibull 22 0.618 0.016 0.313 1.566 1.024 1.048
3 0.061 0.001 0.020 2.052 1.931 1.931
8 0.103 0.002 0.045 2.093 1.767 1.767
p = 1.0,2 = 0.5 13 0.192 0.004 0.091 2.182 1.640 1.640
18 0.538 0.011 0.256 2.523 1.714 1.714
22 0.618 0.012 0.292 2.606 1.594 1.594
3 0.061 0.001 0.019 1.061 0.999 0.999
8 0.103 0.002 0.044 1.103 0.935 0.935
p = 1.0,2 = 1.0 13 0.192 0.004 0.089 1.192 0.907 0.907
18 0.538 0.011 0.243 1.538 1.078 1.078
22 0.618 0.012 0.276 1.618 1.034 1.034
5 2.167 0.016 0.314 1.438 1.334 0.137
10 3.222 0.023 0.422 1.832 1.565 0.172
p = 1.5 15 7.278 0.040 0.560 2.636 2.042 0.277
25 34.111 0.071 0.649 4.041 2.402 0.398
45 259.361 0.111 0.665 2.959 0.993 0.097
5 2.167 0.015 0.198 0.996 0.927 0.091
Pareto 10 3.222 0.023 0.216 1.392 1.176 0.117
p = 4.5 15 7.278 0.040 0.222 2.207 1.647 0.187
25 34.111 0.071 0.222 3.752 2.053 0.281
45 259.361 0.111 0.222 5.057 0.812 0.081
5 2.167 0.016 0.131 0.907 0.842 0.084
10 3.222 0.023 0.133 1.303 1.090 0.107
p = 7.5 15 7.278 0.040 0.133 2.126 1.565 0.172
25 34.111 0.071 0.133 3.663 1.966 0.257
45 259.361 0.111 0.133 5.522 0.781 0.079
4 0.074 0.022 0.022 2.077 1.934 0.659
8 0.103 0.022 0.034 2.108 1.798 0.587
13 0.192 0.024 0.091 2.212 1.683 0.520
Gumbel 21 0.590 0.036 0.312 2.804 1.787 0.581
29 2.804 0.330 0.722 17.511 8.089 2.091
33 3.625 0.751 0.722 38.602 14.338 2.663
47 11.020 1221.674 0.722 0.000 1222.338 7.108
Table 2. Estimated values of X for different parameters
Distribution Parameters r x Tq( x) T1( x) T2( x) Ar (x) X
4 0.074 0.270 14.513 0.998 2.059 0.486
15 0.262 0.076 4.817 0.912 2.063 0.485
p = 1.0 26 1.600 0.013 1.625 0.276 0.957 1.045
A = 1.0 37 4.393 0.005 1.228 0.099 0.914 1.094
48 13.880 0.002 1.072 0.005 1.009 0.991
4 0.074 0.001 14.180 0.664 1.463 0.684
15 0.262 0.001 4.484 0.651 1.713 0.584
Inverse
M e p = 1.0 26 1.600 0.013 1.292 0.259 0.783 1.278
Weibull
A = 15 37 4.393 0.005 0.894 0.097 0.674 1.485
48 13.880 0.001 0.739 0.005 0.696 1.437
4 0.074 0.001 50.677 0.999 3.962 0.399
15 0.262 0.149 8.457 0.996 3.214 0.459 p = 1.5 26 1.600 0.010 1.494 0.226 0.867 1.101 A = 1.0 37 4.393 0.002 1.109 0.052 0.814 1.147
_48 13.880 0.001 1.0193 0.002 0.960 1.028
7 0.008 -0.097 -4.025 -0.961 -1.406 3.696
19 0.038 -0.065 -3.891 -0.871 -2.006 2.029
p = 1.0 28 0.136 -0.040 -2.889 -0.686 -1.902 2.251
39 0.325 -0.023 -2.080 -0.459 -1.704 2.744
_46 0.726 -0.006 -1.296 -0.150 -1.185 4.611
7 0.008 -0.097 -4.635 -0.663 -1.223 4.438 19 0.038 -0.065 -3.854 -0.642 -1.851 2.369 ower p = 1.5 28 0.136 -0.040 -2.649 -0.561 -1.718 2.707 UnCti0n 39 0.325 -0.023 -1.787 -0.411 -1.471 3.463 _46 0.726 -0.006 -0.986 -0.147 -0.905 6.098
7 0.008 -0.097 -5.022 -0.400 -1.043 5.313 19 0.038 -0.065 -3.666 -0.399 -1.633 2.947
p = 3.5 28 0.136 -0.040 -2.395 -0.386 -1.503 3.354 39 0.325 -0.023 -1.524 -0.328 -1.253 4.308 _46 0.726 -0.006 -0.720 -0.139 -0.666 7.750
8 0.103 0.013 0.011 0.304 0.270 1.172
16 0.379 0.010 0.044 0.249 0.193 1.545 =a5 27 2.006 0.003 0.093 0.062 0.080 2.489
38 4.534 0.000 0.092 0.005 0.069 2.633
8 0.103 0.013 0.041 0.254 0.232 1.341
Burr 16 0.379 0.010 0.157 0.216 0.205 1.482
Type II =20 27 2.006 0.003 0.234 0.061 0.152 1.806
_38 4.534 0.000 0.201 0.005 0.149 1.827
8 0.103 0.013 0.071 0.210 0.199 1.513
16 0.379 0.010 0.246 0.186 0.211 1.451
= ' 27 2.006 0.003 0.271 0.058 0.171 1.683
38 4.534 0.000 0.203 0.005 0.151 1.811
Table 1 presents the estimated values of X for various parameters under three different distributions: Weibull, Pareto, and Gumbel. These distributions exhibit different behaviors as their parameters change. For the Weibull distribution, increasing the shape parameters (p = 0.5,1.0) leads to an increase in the estimated values of X . Conversely, increasing the scale parameters ( A = 0.5,1.0) and the r - th order statistics results in a decrease in the estimated values of X . In the
Pareto distribution, increasing the parameters causes a decrease in the estimated values of X , while an increase in the r - th order statistics results in higher estimated values of X . For the Gumbel distribution, no specific pattern is observed in the behavior of the estimated values of X . In Table 2, the estimated values X for the inverse Weibull, power function, and Burr type II distributions increase as the parameter values increase.
V. Summary
The paper contributes to the field of probability and statistics by proposing a method to characterize continuous distributions based on the conditional expectation of order statistics. This approach not only enhances theoretical understanding but also provides practical insights into modelling and analyzing data using well-known distributional forms. Further, the results are applied to some well-known continuous distributions, like Weibull, Pareto, extreme value I, inverse Weibull, power function, Burr type II and Lindley distribution. One may utilize our results to characterize more distributions belonging to these classes. The numerical computations serve to support the theoretical findings and demonstrate the applicability of the method.
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