DISTRIBUTIONAL PROPERTIES OF ORDER STATISTICS AND RECORD STATISTICS FROM ERLANG-TRUNCATED EXPONENTIAL FAMILY OF DISTRIBUTION AND ITS CHARACTERIZATIONS Текст научной статьи по специальности «Биологические науки»

DISTRIBUTIONAL PROPERTIES OF ORDER STATISTICS AND RECORD STATISTICS FROM ERLANG-TRUNCATED EXPONENTIAL FAMILY OF DISTRIBUTION AND ITS CHARACTERIZATIONS

Imtiyaz A. Shah

Department of Community Medicine (Bio Statistics), Sher-I-Kashmir Institute of Medical Sciences, Srinagar (J&K) India [email protected]

Abstract

Erlang Truncated Exponential Distributions are characterized by distributional properties of order statistics. These characterizations include known results for ordinary order statistics based on two non-adjacent order statistics coming from two independent Erlang truncated exponential distributions. Using this method and compared with an efficient recent method given by [20], three examples of real lifetime data-sets are analyzed by that deals with non-random samples. Such type of examples predicts the accumulative new cases per million foe infection of the new COVID-19. Corollaries for Pareto and power function distributions are also derived.

Keywords: Order statistics; characterization of distributions; reliability characteristics; Erlang truncated exponential; random translation

1. Introduction

Various characterizations of Erlang truncated exponential distributions based on distributional properties of order statistics are found in the literature. Let X1/n, X2/n, ■■■ Xn/n denote the order statistics of a identically independent distributed (i.i.d) random variables X1,X2, ■■■ ,Xn, n> 2, each with distribution function Fx(x). Furthermore, a variety of other models of ordered random variables are contained in this concept. For a detailed discussion of several of these models, such as sequential order statistics, kth record values and Pfeifer's record model.

In this paper we present characterizations of Erlang truncated exponential distributions DF exp(0a^), with mean , P > 0,«> 0, X > 0. via distributional properties of generalized order statistics including the known results for ordinary order statistics.

Consider a sequence of real numbers X1,X2, — ,Xn which are independently and identically, distributed with common cumulative distribution

and

FXu(r)(X) = 1 - FXu(r)(X) = (2)

The

fxr Jx) = ^^ [FWH1 - F(X)r~r№

and

Fy.

CO = YJ]=r (;) [F(x)]j[ 1 - F(x)]

n—J

(4)

(5)

2. Model

The cumulative distribution function CDF Fx(x)and probability density function PDF fx(x) of

the Extended Erlang-Truncated Exponential (EETE) distribution are given by

Fx(x) = [1 - e-P(a^>x]a , 0 < x < œ, a, p, À > 0, (6)

and

fx(x) = a p (ax) e~P(a^x[l - e-P(a*)x]a -1 , 0 < x < m, a, p, X > 0 where a and p are the shape parameters and X is the scale parameter.

(7)

a=25,p=25,ï=12

/ s / \ X" --¥=35^=2-5,^1.2

/ /

/ / > /' ..<'■'."■. . r,=S,p'=?SlT=1 ?

/ ■■' / / : ; / / / / 1 1 •■- v.: |i

/// / l-'f ! /-"'• / /

Figure 1. Possible shapes of the probability density function f(x) (left) and cumulative distribution function F(x) (right) of the Extended Erlang-Truncated Exponential (EETE) distribution for fixed parameter values of ft and X.

The Extended Erlang-Truncated Exponential (EETE) distribution reduces to Erlang-Truncated Exponential (ETE) when a = 1.

Erlang-Truncated Exponential (ETE) distribution was originally introduced by [15] as an extension of the standard one parameter exponential distribution. The Erlang-Truncated Exponential (ETE) distribution results from the mixture of Erlang distribution and the left truncated one-parameter exponential distribution. The cumulative distribution function CDF Fx(x), and probability density function PDFfx(x) of the Erlang-Truncated Exponential (ETE) distribution are given by

Fx(x) = [1 - e~ß(a^)x] , 0 < x < œ, ß, X > 0, where ax = (l —

and

fx(x) = ß (aj e-^* ,

(8) (9)

0 < x < œ, ß, X > 0

respectively, where p is the shape parameter and X is the scale parameter. The Erlang-Truncated Exponential (ETE) distribution collapses to the classical one-parameter exponential distribution with parameter p and X ^ ot.

X ~ Par(P(ax))

if X has a Pareto distribution with the DF

F(x) = [1- x"^)] , 1 < x < ot , p> 0,aA > 0 (10)

X ~ pow GS(aA))

if X has a power function distribution with the DF

F(x) = , 0 < x < 1 , p> 0,aA > 0 (11)

It may further be noted that

if log X ~ Erlang-truncated exp (P(ax)) then X ~ Par (P(aJ) (12)

if -log X ~ Erlang-truncated exp (p(a^)) then X ~ pow (p(a^)) (13)

3. RELIABILITY CHARACTERISTICS

The reliability function R(x) is an important tool for characterizing life phenomenon. R(x) is analytically expressed as R(x) = 1 — F(x). Under certain predefined conditions, the reliability function R(x) gives the probability that a system will operate without failure until a specified time x. The reliability function of the Extended Erlang-Truncated Exponential (EETE) distribution is given by

£(x) = !-(!-e-^A)*)" ,0<x<ot, p, X> 0 (14)

Another important reliability characteristics is the failure rate function. The failure rate function gives the probability of failure for a system that has survived up to time x. The failure rate function h(x) is mathematically expressed h(x) = f(x)/R(x) . The failure rate function the Extended Erlang-Truncated Exponential (EETE) distribution is given by:

ft(x) =

0 < x < œ, a,ß, X > 0

Figure 2. Possible shapes of the reliability function £(x) (left) and failure rate function ft(x) (right) of the Extended Erlang-Truncated Exponential (EETE) distribution for fixed parameter values of p and A

4. CHARACTERISTION RESULTS BASED ON UPPER RECORDS

In this section we consider a relation characterizing the Erlang-Truncated Exponential distribution based on order statistics and record statistics. This generalizes some previous characterization results and uses upper as well as lower order statistics. It has been assumed here throughout that the df is differentiable w.r.t. its argument.

THEOREM 4.1 :-

A random variable X0(r) be a sequence of i.i.d. non-negative random variables with an absolutely continuous distribution having the rth upper statistic from a sample of size n drawn from a continuous DF Fx (x) with PDF fx(x). Furthermore, let Y0(j) be the rth upper statistic based on a sample of size n, which is drawn from a continuous DF = P(Z < z), where Y is independent of X. Finally, let the relation

Xu(N0) = %U(R) + Z (15)

be satisfied for all 1 < R < N0 < n, Then, z t Xu(n0-r) and Z ~ Erlang truncated exponential

(PoJ if and if Y ~ Erlang truncated exponential(Pa^), p > 0, a > 0,X > 0.

Proof. We first prove the necessary part. Let the moment generating function (MGF) of

Xu(n0) be MXu(Wo)(t). Then, (15) implies that

= (16)

Let us now derive the MGF of the X^^based on Erlang truncated exp(Pa^). Clearly, in view of (15), we get

M (t) = r e-xMai-t^R-ite = (!7)

XU(R)K ' (r-l)!J0 \x-tj V '

Where r(.) is the usual gamma function. On the other hand, in view of (16)

M*IHN )(t) i K \N0~R

Mz(0 = = ( —) (18)

M*U(R)(t) VK~tJ

On comparing (18) with (17), we deduce that M^(t) is the MGF of Y(N0 - R), i.e., the (N0 - R)th upper record statistics from a sample of size R and is independent of drawn from the DF Erlang truncated exp(^(a^). Hence the proved Necessity part.

W To prove the sufficiency part. In view of (15) be satisfied with Z i yU(N0-R) and Y ~ exp(^(a^)). Furthermore, let X^^and in (15) be upper statistic, which are based on an unknown DF Fx(x) and they are independent of YU(Ng). Therefore, the convolution relation (3.1) implies that

fxu(N0)№ = /0 f*U (R)

= x fr-y^"""^^ (19)

Differentiating both the sides of (19) w. r. t. x, we get

d _ (KaA)r°~R

d^W*) = (/V(-/)-2)! / ^^ x [» - y]Wo"fl-2/^(.)(y)rfy

gS(«a))n°

0

N0-R+l

(Wo_fl_1)! -/„"V^*"^ X [x-yf^-V^Cy^y (20)

and by using the representation (19), we get

o-^--* x [x-yr-«-^^ (21)

(ß(gÄ))NQ-R-1

JXu(N0-1)^ - 0vo-fl-2)! Jo c........л L* /J - ;хЦ(Г)\

and by combing (20) and (21), weget

Tjxu(N0)(x) = ^«лХ/^цМ - fxu(N0)(x)] or equivalently, by integrating from 0 to x

W0)M = ^«яЛ^-цМ - (22)

2) = 1) + Y

Then, f t and Y ~ exp(pa^) if and if X ~ exp(1), p > 0,«> 0,X > 0 .

Now, by using the relation (II) of [5] and [9] on page 75, we get

^Wo« - = igSr №)] (23)

Therefore, by combing (1), (22) and (323), we have

/¿CP nf X =T7T =

Hence, the complete sufficient part, Fx(x) = [1 - e"^)x] ,x > 0, p > 0,«> 0,X > 0.

Remark 4.1. ([7], Remark 1) have shown that for two adjacent upper records

d v

U( 2) = AU( 1)

U( 1)

Remark 4.2. [14] and [6] have shown that

%U(R +1) = %U(R) + ^

Then, Z t and Y ~ exp(pa^) if and if X ~ exp(1), p > 0,«> 0,X > 0. Remark 4.3. have shown

d y U(N0) = AU(R)

U(R)

Corollary 4.1. Assume that the RVs X and Y are independent, as we assumed in Theorem 4.1. By replacing the additive relation (15) by the multiplication relation

Xu(N0) = %U(R) + % (24)

be satisfied for all 1 < P < N0 < n, Then, 2 i and Y ~ exp(pa^) if and if X ~ Par (pax),

p > 0,«> 0,X> 0.

Proof. Here the proof immediately follows, by noting that if X ~ Pareto (ft (a^)), then log exp (ft (a^)) and

log XU(N0) = log ^U(R) + log %

which implies

Xu(N0) = %U(R) + Z

Corollary 4.2. Assume that the RVs X and Y are independent, as we assumed in Theorem 4.1. By replacing the additive relation (15) by the multiplication relation

XL(N0) = %L(R) + % (25)

be satisfied for all 1 < P < N0 < n, Then, 2 t Xi(Wo_s) and Y ~ exp(pa^) if and if X ~ Pow(pa^), p > 0,«> 0,X> 0.

Proof. The Corollary can be proved by considering if X ~ Power (ft (a^)) , then -logX ~ exp (ft (a^)) and

£ - - log r

which implies

V"* d y* v*

AL(W0) = AL(R) ■ r

Xu(N0) = %U(R) + ^

Then, J? t and Y ~ Ga(N0 - P, 1) if and if X ~ exp(1), p > 0,«> 0,X > 0.

5. CHARACTERISTION RESULTS BASED ON ORDER STATISTICS THEOREM 5.1 :-

A random variable XRm be a sequence of i.i.d. non-negative random variables with an absolutely continuous distribution having the Rth order statistics from a sample of size n drawn from a continuous DF Fx (x) with PDF (x). Furthermore, let Vr:n be the rth order statistics based on a sample of size n , which is drawn from a continuous DF p^(y), where Y is independent of X. Finally, let the relation

XN0:n = %R:n + ^ (26)

be satisfied for all 1 < P < N0, Then, 2 f? XNg_R.n_R and Y ~ exp(pa^) if and if X ~ exp(pa^), p > 0,«> 0,X > 0.

Proof. Namely, let in view of (26) be satisfied with XR.n be MXRn(t). Then, (26) implies that

MxNo-.n (t)=Mx„0JO^ Ms(t) (27) Let us now derive the MGF of the Xx based on Erlang truncated exp(pa^). Clearly, in view of (26), we get

(t) = P^) r(n+l) p (^n-z+L _ e.x(pax]R.i e-X(Hax)dx (28)

XR.n V ' (R-l)! r(n-R + l)J0 L J L J V I

Which by using the transformation y = e~x(^ax) takes the form

,, r(n+i) r(n-R--+i)

MXb (t) = --—-^ (29)

XRm W r(n-R + l) r(n-«+l) V '

Where r(.) is the usual gamma function. On the other hand, in view of (28)

MXN .„ (t) r(n-R + i) r(n-Nn--+i)

Mg(t) =—2*2— = --—-(30)

MXR m(t) r(n-W0 + l)r(n-R-i+l) V '

On comparing (30) with (29), we deduce that M2(t) is the MGF of YNo_R:n_R, i.e., the (N0 _ R)th order statistics from a sample of size (n _ R)drawn from the DF Erlang truncated exp(fi(ax)) and is independent of XRm drawn from . This completes the proof of the necessity part. while the proof of the sufficiency part follows closely as the sufficiency part of Theorem 5.1. Namely, let the representation (26) be satisfied with Y = XNo_R.n_R and Y ~ exp( ). Furthermore, let XN .n and XNg_R.n_R in (26) be order statistics, which are based on an unknown DF Fx(x) and they are independent of XR.n. Therefore, the convolution relation (26) implies that

fxNo:n(x) = I fxR]n(y)fxNo-R]n-R(x - y)dy

= Pia^) (n—R)! rx e_p{ax)(x_y) (e_p{ax)(x_y))]n_No + 1 , (y)dy (31)

(N0—R — l)! (n—N0)! 0 J1 JXR.n'S' y v >

By differentiating both the sides of (31) with respect to x, we get

dfxN0.n(x) = iPta^2 jN0-R-l) (n—R)l fX[e-p(ax) (x-y)](n-N0 + 2) x[1 _ e-P(ax)(X-y)]N0-R-2 f (v)d dx = (N0-R-l)! (n-N0)! J0[e ] e ] JxRn(y)d

(,n-N0 +1) (n-r)\ fX[c-B(ai)(x-v)]n-N„ + l x [1 _ ^e-P(ax)(x-y))m+l]N0-R-l^(y)dy

= p(ax) (n_N0 + 1) [fXNo_lm(x) _ fXNo.n(x)] Or equivalently, by integrating from 0 to x,

fx(N0,n)(x) = P(ax)(n _ N0 + 1)[Fx(No_1:n_1)(x) _ FX(No,n)(x)] (32)

Now, by using the relation of [13],

FxW HK U

which implies that

Fx(x) = [1_ e-P^y], p > 0,« > 0,1 > 0,x> 0 This complete the proof of the sufficiency part, as well as the proof of Theorem 4.1.

Corollary 5.1. A random variables (RVs) X and Y are independent, as we assumed in Theorem 5.1. By replacing the additive relation (26) by the multiplicative relation

XN0:n = XR.n (33)

Then, Z t YNo_R:n_rR and Y ~ Pareto(P(ax)) if and only if X ~ Pareto(fi(ax)) Proof. The proof follows exactly as the proof of Corollary 4.1.

Remark 5.1. [7] have proved that

xRm = xR_1.n + u

Where U ~ exp (n — R + 1) if and only if Xt~exp (1). Remark 5.2.

^N0:n = XR.n + U Where U t —log M W ~ Be (n — R + 1,N0 — R if and only if X1~exp (1).

Corollary 5.2. A random variables (RVs) X and Y are independent, as we assumed in Theorem 5.1. By replacing the additive relation (26) by the multiplicative relation

X'R.n £ **Wo:n • Z* , (34)

Then, Z* i r*r:s_! and Y*~ Power(£(ax)) if and if X* ~ Power (J](ax), p > 0,«> 0,X > 0 . Proof. To prove the corollary, we note that —logXNo.n = —log XRm — logXNo_Rm_R implies

Xn-N0 + l:n = Xn_R + lm + Xn_No + 1.n_R Or,

v d v _i_ v

An-N0 + l:n = An-R + l:n + An-N0 + l:n-R

THEOREM 5.2 :-

A random variable XR:n be a sequence of i.i.d. non-negative random variables with an absolutely continuous distribution having the Rth order statistics from a sample of size n drawn from a continuous DF Fx (x) with PDF fx(x). Furthermore, let YR:n be the Rth order statistics based on a sample of size n , which is drawn from a continuous DF Fz(z), where Z is independent of X. Finally, let the relation

^Ng.n = XN0-R:n-R + Z, (35)

be satisfied for all 1< R < N0, Then, 2 ± XR.n and Y ~ exp(pa^) if and if X ~ exp(pa^), p > 0, «> 0,X> 0.

Proof. We first prove the necessary part. Let the moment generating function (MGF) of XR,n be MXRn(t). Then, (38) implies that

MxNo-.n (t)=Mx„oJ0^ Ms(t) (36)

Let us now derive the MGF of the Xx based on Erlang truncated exp(pa^). Clearly, in view of (26), we get

(t) = p(ax) r(n+i) (paA)]n-R+L [1 — e-xiPaX]R-i e-x(Hax)dx (37)

xR:n V ' (R-l)! r(n-R + l)J0 L J L J V I

Which by using the transformation y = takes the form

MXb (t) = --—-^ (38)

XR.n W r(n-R+1) r(n-L+i) x '

Where r(.) is the usual gamma function. On the other hand, in view of (3.14)

, ^ MXN .„ (t) r(n-R + i) r(n-s--+i)

Mz(t) = , = -(-—-(39)

MXR.n (t) r(n-N0 + l) r(n-R-±+1) V '

On comparing (39) with (38), we deduce that M^(t) is the MGF of YNg_R.n_R, i.e., the (N0 — R)th order statistics from a sample of size (n — R)drawn from the DF Erlang truncated exp(fi(ax)) and is independent of XRm drawn from . This completes the proof of the necessity part. while the proof of the sufficiency part follows closely as the sufficiency part of Theorem 4.1. Namely, let the representation (26) be satisfied with 2 i XNg_R.n_R and Y ~ exp( pa^ ). Furthermore, let XN .n and XNg_R.n_R in (26) be order statistics, which are based on an unknown DF Fx(x) and they are independent of XR.n. Therefore, the convolution relation (26) implies that

X

fxNo.n(x) = J fxNo.R]n.R(y)fxr]n(x _ y)dy 0

= x [1 _ ie-eM*-»)]*-^^^ (40)

By differentiating both the sides of (40) with respect to x, we get

^if^ = (x-y)](n-R+2) x [1 _ e-^ia^)ix-У)]R-2fXNo_Rn_R(y)dy

X

-J+1J [e-fi(a,)(X-y)]n-R + l x[1_ (e-Pia^-y))]*-If^^tydy

(p(gx)Y (n_R + 1)

B(n_R + 1, R)

0

= (n) [fXNo_1.n_1M_fXNoJx)] Or equivalently, by integrating from 0 to x,

fxNo.n(x) = P(ai)(n)[Fx(N0-i,n-D(x) _ Fx(N0,n)(x)] (41)

Now, by using the relation of [13], we get

;X(No-ljn-i)(x) _ FxM(x) — (¡¡o _11) [Fx(x)]No-![1 _ Fx(x)]n~No+1 (42)

Therefore, by combing (1), (41) and (42), we get fx(x) af .

Fx(x)

which implies that

Fx(x) — [1_ e-P^y], p > 0,« > 0,1 > 0,x > 0

This complete the proof of the sufficiency part, as well as the proof of Theorem 4.1.

Corollary 5.1. A random variables (RVs) X and Y are independent, as we assumed in Theorem 5.2. By replacing the additive relation (35) by the multiplicative relation

XN0:n = XR:n (43)

Then, Z t YNn

,-R.n-R and Y ~ Pareto(fi(ax)) if and only if X ~ Pareto(fi(ax)). Proof. The proof follows exactly as the proof of Corollary 4.1.

Corollary 5.2. A random variables (RVs) X and Y are independent, as we assumed in Theorem 3.2. By replacing the additive relation (26) by the multiplicative relation

X'R.n £ X\o:n • Z* , (44)

Then, Z* i Y'r^ and Y*~ Power(£(ax)) if and if X* ~ Power(P(ax), p > 0,«> 0,1 >0 . Proof. To prove the corollary, in view of (11) and (20).

6. APPLICATIONS

Many authors have considered prediction problems based on samples of random sizes, The importance of the order statistics in the reliability theory is attributed to the fact that the rth order statistics (n _ r + 1) out-of-n system made up of n identical components with independent life lengths. On the other hand, in dealing with censored samples, where the lifetest is terminated after observing the rth failure (Type II censoring), or the termination of the test occurs after a given time lapse (Type I censoring), the complete survival times can not usually be observed (due to time or cost). In many biological and agriculture problems, we often come across a situation where the sample size is not deterministic because either some observations get lost

for various reasons, or the size of the target population and its representative sample cannot be determined well. For example, assume that the inhabitants of a populous town are exposed to a dose of radiation resulting from an atomic accident, or exposed to an infection of an unknown epidemic. Furthermore, assume that our interest focuses on the time at which r persons would die among a big random sample of size n that is drawn from the residents of this town. Since the number of infected people in this town is unknown and changes randomly with time, the drawn sample contains a random number of infected and non-infected people. Accordingly, the sample size of the sub-sample of the infected people will be a non-negative integer valued RV, e.g.N, and it will be described by a sequence of independent and identically distributed RVs X1,X2, •••,XN. Therefore, the rth smallest order statistic will be denoted by Xr.N, which represents the time at which r persons will die.

7. CONCLUSIONS

In this paper we consider the equality by distribution of the form Y £ XV, where X and V are two independent random variables. It should be noted that the random contraction-dilation schemes have important applications in many areas such as economic modeling and reliability. The characterization results given in Section 4 can be used in developing goodness-of-fit tests for the corresponding probability distributions. This paper deals with the generalized order statistics and dual generalized order statistics within a class of Erlang-Truncated Exponential distribution. Two theorems for characterizing the general form of distribution based on generalized order statistics dual generalized order statistics are given. Special cases are also deduced.

References

[1] Okorie, I. E., Akpanta, A.C and Ohakwe J. (2016). Transmuted Erlang-truncated exponential

distribution, Econ. Qual. Control 31 (2) 71-84.

[2] Okorie, I.E, Akpanta, A.C. and Ohakwe, J. (2017). Marshall-Olkin generalized Erlang-truncated exponential distribution: properties and applications, Cogent Math. 4 (1) 1285093.

[3] Nasiru, S., Luguterah, A. and Iddrisu, M.M. (2016). Generalized Erlang-truncated exponential distribution, Adv. Appl. Stat. 48 (4) 273-301.

[4] Alosey-EL, R. A. (2007). Random sum of new type of mixture of distribution. Int. J. Stat. Syst. 2: 49-57.

[5] Ahsanullah, M. (1995). Record Statistics. New York: Nova Science Publishers, Inc. Commack.

[6] Ahsanullah, M. (2006). The generalized order statistics from exponential distribution. Pak. J.

[7]

[8]

New York: Wiley, USA. [10] [11]

translations, contractions and dilations of order statistics and records. Statist.:

[12]

[13]

Alosey-EL, R. A. (2007). Random sum of new type of mixture of distribution. Int. J. Stat. Syst. 2: 49-57.

[16] Rao, C. R. & Chanabhng, D. N. (1998), Recent Approaches to Characterizations Based on Order Statistics and Record Values (Handbook of Statistics, 16, 231- 256), Elsevier.

[17] Tavangar, M. & Hashemi, M. (2013), 'On Characterizations of the Generalized Pareto Distributions Based on Progressively Censored Order Statistics', Statistical Papers 54, 381390.

[18] Fagiuoli, E., Pellerey, F. and Shaked, M. (1999). A characterization of the dilation order and its applications. Statist. Papers 40: 393-406.

[19] Rao, C. R. & Chanabhng, D. N. (1998), Recent Approaches to Characterizations Based on Order Statistics and Record Values (Handbook of Statistics, 16, 231- 256), Elsevier.

[20] Barakat et. Al. (2021), Predicting future order statistics with random sample size. AIMS Mathematics, 6(5): 5133-5147.