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The techniques to find appropriate new models for data sets are very popular nowadays among the researchers of this area where existed models in the literature are not suitable. In this paper, a new distribution, generalized inverted Kumaraswamy (GIKum) distribution is introduced. The main aims of this research are to develop a general form of inverted Kumaraswamy (IKum) distribution which is flexible than the IKum distribution and all of its related and sub models. Some properties of GIKum distribution such as measures of central tendency and dispersion, models of stress-strength, limiting distributions, characterization of GIKum distribution and related probability distributions through some specific transformations are derived. The mathematical expressions of reliability function (r.f) and the hazard rate function (hrf) of the GIKum distribution are found and presented through their graphs. The parameters estimation through the maximum likelihood (ML) estimation method is used and the results are applied to the data set of prices of wooden toys of 31 children.

In the past years, several ways of generating inverted distributions from classic ones were developed and discussed. Calabria and Pulcini [

A number of researchers studied the inverted distributions and its applications; for example, Prakash [

Kumaraswamy [

Abd Al-Fattah et al. [

F ( x ; α , β ) = ( 1 − ( 1 + x ) − α ) β , x > 0 ; α , β > 0 (1)

Iqbal et al. generalized the some continuous distribution by using power transformation. Here we use the same technique to find the cdf of generalized inverted Kumaraswamy distribution (GIKum) and is derived by using transformation t = x γ which has closed form and is as under

F ( x ) = ( 1 − ( 1 + x γ ) − β ) α , α , β , γ > 0 (2)

Assuming X is a random variable with shape parameters, α > 0 , β > 0 and γ > 0 the pdf of GIKum is as

#Math_13# (3)

This model is flexible enough to accommodate both monotonic as well as non-monotonic failure rates.

From

This paper is arranged as follows. In Section 2, some statistical properties of GIKum distribution such as measures of central tendency and dispersion, reliability function (rf) hazard rate functions (hrf) and reverse hazard function, models of stress-strength, mode, moment generating function, the asymptotic mean and variance, incomplete moments, quantile functions, mean deviation and

Renyi entropy are analyzed [

This section is devoted to illustrate some statistical properties of GIKum distribution, through rf, some models of the stress-strength, hrf and reversed hazard (rhrf), measures of central tendency and dispersion, graphical and order statistics (

R 1 ( x ; α , β , γ ) = P ( T > x ) = 1 − ( 1 − ( 1 + x γ ) − α ) β , t > 0 , α , β , γ > 0 (4)

Suppose if G ( x ) the distribution function is defined as

G ( x ) = 1 − ( 1 + x γ ) − α , α , γ > 0

then [

F ( x ) = ( G ( x ) ) β , β > 0

The hazard rate function (hrf) denoted by λ ( x ) and reverse hrf denoted by λ * ( x ) are given, respectively, by

λ F ( x ) = α β γ x γ − 1 ( 1 − ( 1 + x γ ) − α ) β − 1 ( 1 + x γ ) ( α + 1 ) ( 1 − ( 1 − ( 1 + x γ ) − α ) β ) , x > 0 , α , β , γ > 0

and

λ F * ( x ) = α β γ x γ − 1 ( 1 − ( 1 + x γ ) − α ) − 1 ( 1 + x γ ) ( α + 1 ) , x > 0 , α , β , γ > 0

and their relations are shown as under

λ F ( x ) = β ( 1 − δ β ( x ) ) λ G ( x ) , β > 0

where

δ β ( x ) = 1 − ( G ( x ) ) α − 1 1 − ( G ( x ) ) α

For 0 < α < 1 λ F ( x ) ≥ λ G ( x )

For α ≥ 1 0 ≤ λ F ( x ) ≤ λ G ( x ) .

and relation in reverse hrf of F and G as λ F * ( x ) = β λ G * ( x ) and cdf can be expressed as

F ( x ) = λ F ( x ) λ F ( x ) + λ F * ( x ) .

1) Let T be the stress component subject strength Y, the random variables T and Y are independent distributions from GIKum (α, β, γ) respectively, then the reliability function is given by;

R 2 ( t ) = P ( T < Y ) = ∫ 0 ∞ ∫ 0 y α β 2 γ t γ − 1 ( 1 − ( 1 + t γ ) − α ) β 1 + β 2 − 1 ( 1 + t γ ) ( α + 1 ) dtdy = β 2 β 1 + β 2

Let T and Z be two independent random stress variables with known cdfs (𝑡), 𝐺(𝑧), and both follow G I K u m ( α 1 , β 1 , γ ) and G I K u m ( α 2 , β 2 , γ ) respectively, and let Y be, independent of T and Z, a random strength variable follow to G I K u m ( α 2 , β 2 , γ )

R 3 ( t ) = P ( T < Y < Z ) = ∫ 0 ∞ H T ( y ) d F Y ( y ) − ∫ 0 ∞ H T ( y ) G z ( y ) d F Y ( y ) = ∫ 0 ∞ ∫ 0 y α β 2 γ t γ − 1 ( 1 + t γ ) − ( α + 1 ) ⋅ ( 1 − ( 1 + t γ ) − α ) β 1 + β 2 − 1 − ∫ 0 ∞ ∫ 0 y α β 2 γ t γ − 1 ⋅ ( 1 + t γ ) − ( α + 1 ) ⋅ ( 1 − ( 1 + t γ ) − α ) β 1 + β 2 + β 3 − 1 = β 2 β 3 ( β 1 + β 2 ) ( β 1 + β 2 + β 3 )

The mode of the GIKum distribution is given by

∂ ln f ∂ x = ( γ − 1 ) x − ( α + 1 ) ⋅ γ x γ − 1 ( 1 + x γ ) + α ⋅ γ ( β − 1 ) ⋅ x γ − 1 ( 1 + x γ ) ⋅ ( ( 1 + x γ ) α − 1 ) = 0

When γ = 1 , M o d e = ( α + 1 α β + 1 ) − 1 α − 1

The mode of the GIKum distribution is given by

When γ = 1 , M o d e = ( α + 1 α β + 1 ) − 1 α − 1

The quantile function of the GIKum is given by

t q = ( ( 1 − q 1 β ) − 1 α − 1 ) 1 γ , 0 < q < 1

for γ = 1 , t q = ( 1 − q 1 β ) − 1 α − 1 , 0 < q < 1

Special cases can be obtained using [

The r^{th} non central moment of the GIKum (α, β) distribution is given by

μ ′ s = ∑ j = 0 s γ ( s γ j ) ( − 1 ) s γ − j β B ( i α + 1 , β ) , α , β , γ > 0

where B ( . , . ) is the beta function.

The central moments can be obtained by applying the general relation in central and the non-central moments which as follows

μ r = ∑ i = 0 r ( r i ) ( − 1 ) r − i μ i μ ^ r − i , r = 1 , 2 , 3 , ⋯ ,

Thus the mean and variance of GIKum when γ = 1 are given by

μ = β B ( 1 − 1 α , β ) − 1 , α ≥ 1 ,

and

μ 2 = β B ( 1 − 2 α , β ) − ( β B ( 1 − 2 α , β ) ) 2 , α ≥ 1

The Asymptotic Mean and VarianceIf W ∼ exp ( β ) with μ = 1 β and σ 2 = 1 β 2 then the variable

T = g ( W ) = ( ( 1 − e − w ) − 1 α − 1 ) 1 γ ∼ GIKum ( α , β , γ ) . This relation can be used to approximate the mean and variance

E ( T ) ≈ g ( μ ) + 1 2 σ 2 g ′ ( μ )

V ( T ) ≈ σ 2 ( g ′ ( μ ) ) 2

where

g ( μ ) = ( ( 1 − e − μ ) − 1 α − 1 ) 1 γ

g ′ ( μ ) = 1 γ ( ( 1 − e − μ ) − 1 α − 1 ) 1 γ − 1 ⋅ ( − 1 α ( 1 − e − μ ) − ( 1 α + 1 ) e − μ )

E ( T ) ≈ ( ( 1 − e − 1 β ) − 1 α − 1 ) 1 γ + 1 2 γ β 2 ⋅ ( ( 1 − e − 1 β ) − 1 α − 1 ) 1 γ − 1 ⋅ ( − 1 α ( 1 − e − 1 β ) − ( 1 α + 1 ) e − 1 β )

V ( T ) ≈ 1 β 2 ( 1 α 2 γ 2 ( ( 1 − e − 1 β ) − 1 α − 1 ) 2 γ − 2 ( ( 1 − e − 1 β ) − ( 2 α + 2 ) e − 2 β ) )

E ( Y r ) = α B ( 1 + r / γ , α − r ) , r / γ < α

E ( X r ) = ∑ i = 0 ∞ q i ⋅ α ⋅ ( i + 1 ) B ( 1 + r / γ , α ( i + 1 ) − r / γ ) , r / γ < α

M ( t ) = ∑ s = 0 ∞ ∑ i = 0 ∞ q i α ( i + 1 ) B ( 1 + s / γ , α ( i + 1 ) − s / γ ) t s s ! ,for s / γ < α

where q i is defined as

q i = ∑ j = 0 ∞ β ⋅ ( − 1 ) j + i Γ ( a + b ) ⋅ Γ ( β ⋅ ( a + j ) ) i ! Γ ( a ) j ! ( i + 1 ) Γ ( b − j ) Γ ( β ( a + j ) − i )

x = Q ( u ) = [ [ 1 − ( Q a , b ( u ) ) 1 / β ] − 1 / α − 1 ] 1 / γ , u ∈ ( 0 , 1 ) X = [ [ 1 − ( V ) 1 / β ] − 1 / α − 1 ] 1 / γ S K = Q ( 3.0 / 4.0 ) + Q ( 1.0 / 4.0 ) − 2.0 Q ( 1.0 / 2.0 ) Q ( 3.0 / 4.0 ) − Q ( 1.0 / 4.0 ) , K R = [ Q ( 7.0 / 8.0 ) − Q ( 5.0 / 8.0 ) ] + [ Q ( 3.0 / 8.0 ) − Q ( 1.0 / 8.0 ) ] [ Q ( 6.0 / 8.0 ) − Q ( 2.0 / 8.0 ) ]

The r^{th} incomplete moments m r ( z ) of GIKum distribution is

m r ( z ) = ∫ 0 z y r g ( γ , α ) d y = k B z γ ( 1 + r / γ , α − r / γ ) , r / γ < α m r ( z ) = ∑ i = 0 ∞ q i α ( i + 1 ) B z γ ( 1 + r / γ , α ( i + 1 ) − r / γ ) , r / γ < α

where

q i = ∑ j = 0 ∞ β ( − 1 ) j + i Γ ( a + b ) Γ ( β ( a + j ) ) i ! Γ ( a ) j ! ( i + 1 ) Γ ( b − j ) Γ ( β ( a + j ) − i )

The mean deviation D ( μ ) of GIKum

D ( μ ) = E | X − μ | = 2 μ F ( μ ) − 2 m 1 ( μ ) = 2 β B ( 1 − 1 α , β ) ( 1 − ( 1 + ( β B ( 1 − 1 α , β ) − 1 ) γ ) − β ) α − 2 ∑ i = 0 ∞ q i α ( i + 1 ) B ( β B ( 1 − 1 α , β ) − 1 ) γ ( 1 + 1 / γ , α ( i + 1 ) − 1 / γ )

The Rényi entropy of an r.v X is defined as

I R ( δ ) = 1 1.0 − δ log I ( δ ) , δ > 0 , δ ≠ 1

where I ( δ ) = ∫ f δ ( x ) d x , δ > 0 and δ ≠ 1 using the pdf of GIKum

I ( δ ) = α δ β δ γ δ B δ ( a , b ) ∫ 0 ∞ x δ ( γ − 1 ) ( 1 + x γ ) − δ ( α + 1 ) ( 1 − ( 1 + x γ ) − α ) δ ( a β − 1 ) × ( ( 1 − ( 1 + x γ ) − α ) β ) δ ( b − 1 ) d x

expanding the last term of integrand through binomial expansion which simplifies as

I ( δ ) = α δ β δ γ δ B δ ( a , b ) ∑ i = 0 ∞ ( δ ( b − 1 ) i ) ( − 1 ) i ∫ 0 ∞ x δ ( γ − 1 ) ( 1 + x γ ) − δ ( α + 1 ) × ( 1 − ( 1 + x γ ) − α ) δ ( a β − 1 ) + β i d x

Again, applying the same expansion we have

I ( δ ) = α δ β δ γ δ B δ ( a , b ) ∑ i = 0 ∞ ∑ j = 0 ∞ ( δ ( b − 1 ) i ) ( β ( i + a δ ) − δ j ) ( − 1 ) i + j ∫ 0 ∞ x δ ( γ − 1 ) × ( 1 + x γ ) − ( α ( j + δ ) + δ ) d x

Using the transformation y = x γ in above expression and simplifying,

I ( δ ) = α δ β δ γ δ − 1 B δ ( a , b ) ∑ i = 0 ∞ ∑ j = 0 ∞ ( δ ( b − 1 ) i ) ( β ( i + a δ ) − δ j ) ( − 1 ) i + j × B [ ( δ ( γ − 1 ) + 1 γ , α ( δ + j ) γ + ( δ − 1 ) γ ) ]

the Rényi entropy, hence, finally reduces to

I R ( δ ) = ( 1 1 − δ ) [ δ log ( α ⋅ β B ( a , b ) ) + log ∑ i = 0 ∞ ∑ j = 0 ∞ ( δ ( b − 1 ) i ) ( β ( i + a δ ) − δ j ) ( − 1 ) i + j × B [ ( δ ( γ − 1 ) + 1 γ , α ( δ + j ) γ + ( δ − 1 ) γ ) ] ]

This section discussed some sub-models of GIKum distribution, some relations of the GIKum distribution to other distributions, limiting and several IKum G families of distributions.

The Lomax (Pareto type II) distribution is a special case from GIKum distribution, when β = 1 , γ = 1 in (2) with the following pdf

f ( x ) = α ( 1 + x ) α + 1 , α , x > 0

The inverted beta type II (β, 1) is a special case from GIKum distribution, when α = 1 , γ = 1 in (2)

f ( x ; β ) = 1 B ( β , 1 ) x β − 1 ( 1 + x ) ( β + 1 ) , x , β > 0

Also when Y = ( k ( 1 − ( 1 + x γ ) − α ) ) − 1 then Y ∼ Pareto typeI ( β , k )

The log-logistic (Fisk) distribution is a special case from IKum distribution, when α = β = γ = 1 in (2), with the following form

f ( x ) = 1 ( 1 + x ) 2 , x > 0

The inverted distribution can be transformed to several distributions using appropriate transformations such as exponentiated Weibull (exponentiated exponential, Weibull, Burr type X, exponential, Rayleigh), generalized uniform (beta type I, inverted generalized Pareto type I, uniform (0, 1)), left truncated exponentiated exponential (left truncated exponential, exponential), exponentiated Burr type XII (Burr type XII, generalized Lomax, beta type II, F- distribution), Kumaraswamy-Dagum (Dagum, Kumaraswamy-Burr type III, Burr type III, log logistic) and Kumaraswamy-inverse Weibull (Kumaraswamy-inverse exponential, inverse exponential).

1) If X ∼ GIkum ( α , β , γ ) and Y = β − 1 α ( 1 + X γ ) on ( β − 1 α , ∞ ) then the PDF of y is

α\β | 2 | 3 | 4 | 5 |
---|---|---|---|---|

1 | 0.5 | 1 | 1.5 | 2 |

2 | 0.2930 | 0.5276 | 0.7321 | 0.9149 |

3 | 0.2051 | 0.3573 | 0.4813 | 0.5874 |

4 | 0.1583 | 0.2698 | 0.3579 | 0.4316 |

5 | 0.1289 | 0.2168 | 0.2847 | 0.3408 |

f ( y ) = α y − ( α + 1 ) ( 1 − y − α β ) β − 1 , y > 0

If β → ∞ then it is the pdf of the inverted Weibull distribution.

2) If X ∼ GIkum ( α , β , γ ) and Y = α ( 1 − ( 1 + X γ ) − 1 ) then the pdf of y is f ( y ; α , β ) = β ( 1 − y α ) α − 1 ( 1 − ( 1 − y α ) α ) β − 1 , y > 0 and as α → ∞ the pdf of y tends to f ( y ; β ) = β e − y ( 1 − e − y ) β − 1 , y > 0 , which is the pdf of the generalized exponential distribution.

3) If T ∼ GIKUM ( α , β , γ ) and Y = α ( 1 − β 1 α ( 1 + T γ ) − 1 ) on ( α ( 1 − β 1 α ) , α ) , then the pdf of y is f ( y ; α , β ) = ( 1 − y α ) α − 1 ( 1 − ( 1 − ( 1 − y α ) β ) α ) β − 1 , y > 0. As

both β → ∞ and α → ∞ the pdf of y tends to f ( y ; β ) = e − y exp ( − e − y ) β − 1 , y > 0 , which is the pdf of the standard extreme value distribution of the first type (

Characterization of a probability distribution for continuous r.v is important in several research areas and has recently involved many researchers attention. Here we characterize generalized inverted Kumaraswamy distribution based on 1) relationship of two moments based on truncation; 2) truncated moments of the statistic of n^{th} order with certain functions.

Following Hamedani [

Theorem 4.1

Let probability space ( Ω , ℘ , P ) and for a < b with a = − ∞ and b = ∞ ,

Transformation | The resulting distribution | |
---|---|---|

Exponential Weibull distribution | ||

Special cases | ||

i) | Exponentiated exponential (α, β) | |

ii) | Weibull (α, θ) | |

iii) | Burr type X (α, β) | |

iv) | Exp (α) | |

v) | Rayleigh (α) | |

Generalized Uniform (α, β, w, c) | ||

Special cases | ||

i) | BetaType I (1, β) | |

ii) | Inverted generalized Pareto Type I (α, β, w) | |

iii) | Uniform (0, 1) | |

Left truncated exponentiated exp (α, β, θ, b) | ||

Special cases | ||

i) | Left truncated exp (α, θ, b) | |

ii) | Exp (α, θ) | |

Exponentiated Burr type XII (α, β, s, c) | ||

Special cases | ||

i) | Burr type XII (α, s, c) | |

ii) | Generalized Lomax (Pareto type II) (α, s) | |

iii) | Beta type II (1, α) | |

iv) | F-distribution (2, 2α) | |

Kumaraswamy-Dagum (α, β, λ, s) | ||

i) | Dagum (α, λ, s) | |

ii) | Kumaraswamy-Burr type III (α, β, s) | |

iii) | Burr type III (α, s) | |

iv) | Kumaraswamy-Fisk (log logistic) (λ, β, s) | |

Kumaraswamy-inverse Weibull (α, β, 𝜃, 𝑏) | ||

i) | Kumaraswamy-inverse exponential (α, β, 𝜃) | |

ii) | nverse exponential (α, 𝜃) |

where as the interval H is defined as H = [a, b]. Suppose continuous r.v X with cdf F where X is such that X ( Ω ) = H and let both g, h be functions of real type on H satisfying the following expression

E [ g ( X ) | X ≥ x ] = E [ h ( X ) | X ≥ x ] η ( x ) , with x ∈ H ,

where η is a real function. Furthermore, assuming that g, h are continuous η and F are differentiable twicely and F monotone function on H. Finally, assuming the equation h η = g and it has never any solution in real belongs to H. and then F is determined uniquely from the following relation

F ( x ) = c ∫ 0 x | η ′ ( u ) / ( η ( u ) h ( u ) − g ( u ) ) | exp ( − s ( u ) ) d u ,

and the function s ( u ) has a solution of s ′ ( u ) = η ′ ( u ) h ( u ) / ( η ( u ) h ( u ) − g ( u ) ) and C is chosen so that ∫ H d F = 1 .

The condition in which the both functions g n ( X ) and h n ( X ) are integrable uniformly and the cdf { F n } relatively compact form, and { X n } → X in distribution if η n → η

η ( x ) = E [ g ( X ) | X ≥ x ] E [ h ( X ) | X ≥ x ] .

Proposition 4.1

Let X : Ω → ( 0 , ∞ ) be an r.v and let h ( x ) = { 1 − [ 1 + x γ ] − α } 1 − β [ 1 + x r ] − α and g ( x ) = { 1 − [ 1 + x γ ] − α } 1 − β , x ∈ ( 0 , ∞ ) . The pdf of X is (2) if η defined in 2.1 theorem and has the following form η ( x ) = 2 [ 1 + x γ ] α for x > 0 .

Proof. Let X has density (1.3), then

E [ g ( X ) | X ≥ x ] = 1 2 B ( α , β ) F ¯ ( x ) [ 1 + x γ ] − 2 α , x > 0

and E [ h ( X ) | X ≥ x ] = 1 B ( α , β ) F ¯ ( x ) [ 1 + x γ ] − 2 α , x > 0 and finally η ( x ) ⋅ h ( x ) − g ( x ) = ( 1 − [ 1 + x γ ] − α ) 1 − β > 0 for x > 0

Conversely, when η is defined earlier, then

s ′ ( x ) = η ′ ( x ) ⋅ h ( x ) / ( η ( x ) ⋅ h ( x ) − g ( x ) ) = 2 α γ x γ − 1 [ 1 + x γ ] − 1 , x > 0

and finally hence s ( x ) = ln [ 1 + x γ ] 2 α , x > 0

Now, from 2.1 theorem, X has pdf (3)

Corollary 4.1

Let X : Ω → ( 0 , ∞ ) be an r.v with h ( x ) proposition in 4.1. The pdf of X is (2) if there exist g and η functions given in 4.1 theorem with the following differential equation (DE)

s ′ ( x ) = η ′ ( x ) h ( x ) η ( x ) h ( x ) − g ( x ) = 2 α γ x γ − 1 [ 1 + x γ ] − 1 , x > 0

Remark 4.1. (a) The DE has the general solution of 4.1 corollary of the form

η ( x ) = [ 1 + x r ] 2 α [ − ∫ g ( x ) 2 α γ x γ − 1 [ 1 + x r ] − ( α + 1 ) × { 1 − [ 1 + x r ] − α } β − 1 d x + D ]

for x > 0 , where a new constant D is introduced and it may be in particular value D = 0 according to proposition 4.1.

This section contains the characterizations of GIKum distribution based on function of the last order statistics. This characterization is derived through the consequence of the proposition 4.2, which is similar to the Hamedani [

Proposition 4.2

Let X : Ω → ( 0 , ∞ ) be a r.v with cdf F. let ψ ( x ) and q ( x ) be two differentiable functions on ( 0 , ∞ ) such that lim x → 0 ψ ( x ) [ F ( x ) ] n = 0 and ∫ 0 ∞ q ′ ( t ) [ ψ ( t ) − q ( t ) ] d t

Then

E [ ψ ( X n : n ) | X n : n < t ] = q ( t ) , t > 0

implies

F ( x ) = exp { − ∫ x ∞ q ′ ( t ) n [ ψ ( t ) − q ( t ) ] d t } , x ≥ 0

Taking, e.g. ψ ( x ) = 2 { ∫ 0 1 − [ 1 + x r ] − α ω β − 1 d ω } and q ( x ) = 1 2 ψ ( x ) , F ( x ) will be a result in (2).

We present here a ML estimator of the parameters of GIKum distribution

l ( α , β , γ ; x o b s ) = α n β n γ n ∏ j = 1 n x γ − 1 ( 1 + x γ ) − ( α + 1 ) ( 1 − ( 1 + x γ ) − α ) β − 1 , (5)

and

ln ( l ( α , β , γ , x o b s ) ) = n ln α + n ln β + n ln γ + ( γ − 1 ) ∑ ln x − ( α + 1 ) ∑ ln ( 1 + x γ ) + ( β − 1 ) ∑ ln ( 1 − ( 1 + x γ ) − α ) (6)

Taking partial derivatives with respect to α , β and γ respectively from (6), we have

∂ ln ( l ( α , β , γ , x o b s ) ) ∂ α = n α − ∑ ln ( 1 + x γ ) + ( β − 1 ) ∑ ln ( 1 + x γ ) ( ( 1 + x γ ) α − 1 ) (7)

∂ ln ( l ( α , β , γ , x o b s ) ) ∂ β = n β + ∑ ln ( 1 − ( 1 + x γ ) − α ) (8)

∂ ln ( l ( α , β , γ , x o b s ) ) ∂ γ = n γ + ∑ ln x − ( α + 1 ) ∑ x γ ( ln x ) [ ( 1 + x γ ) ] + α ( β − 1 ) ∑ x γ ( 1 + x γ ) − 1 ln x ( ( 1 + x γ ) α − 1 ) (9)

The ML estimators say Θ ^ of Θ , are found through solution of the nonlinear system. This system of nonlinear equations does not provide explicit functions of the estimators of the parameters of GIKum distribution. Therefore, for the solution of this system of equations using software can be estimated numerically with R.

For the inference about model parameters i.e. the point estimation and the testing of hypothesis we require the information matrix of order 3 × 3 which have partial derivatives of second order and they derived from Equations (7)-(9) with again differentiating. Assuming that the regularity conditions holds, the vector n ( θ ^ − θ ) follows the multivariate normal distribution asymptotically i.e. N 3 ( 0 , A − 1 ( Θ ) ) , where A − 1 ( Θ ) = lim n → ∞ I n ( Θ ) and I n ( Θ ) is an information matrix.

We conclude this section by expressing β ^ in terms of a random variable T ′ whose distribution will be derived in the next section.

β ^ = − n ∑ ln [ 1 − ( 1 + x γ ) − α ] = − n T ′

where

T ′ = − ∑ ln [ 1 − ( 1 + x γ ) − α ] = ∑ ln [ 1 − ( 1 + x γ ) − α ] − 1

The following remarks and a theorem illustrate the distributions of T i and T ′ .

The following conclusions can be obtained easily which we present them as remarks.

i) If X ~ GIKumdistribution ( α , β ) with γ known, then T i = − ln [ 1 − ( 1 + x γ ) − α ] follows Exp ( β ) .

ii) T ′ ~ Gamma ( β , n ) T ′ ~ GIKumdistribution ( α , β )

iii) T ¯ = ∑ T i n ~ Gamma ( β n , n ) .

iv) In view of (2), 1 T ′ ~ InvertedGamma .

v) If X 1 , X 2 , ⋯ , X n are i.i.d. Gamma (β, n), then the i^{th} transformed ordered failures are i.i.d. Exp(β).

vi) Y = β ⌢ n β ~ Gamma

The r^{th} moment of the statistic Y = β ⌢ n β is

μ ′ r = E ( Y r ) = Γ ( n − r ) Γ n

and that of β ^ is

E ( β ^ r ) = ( n β ) r Γ ( n − r ) Γ n

Theorem 5.1

Let X 1 , X 2 , ⋯ , X n be i.i.d. random variable with cdf F and let X ( n ) be the n^{th} order statistic. Consider the sequence of random variables Y n = [ 1 − F ( X n ) ] The limiting function of Y n ( Y n > 0 ) is e − Y n for α > 0 and n → ∞ .

Proof:

The pdf of u = X ( n ) is

g ( u ) = n [ F ( u ) ] n − 1 f ( u )

g ( u ) = n α γ β [ 1 − ( 1 + u γ ) − α ] ( n − 1 ) β + β − 1 u γ − 1 ( 1 + u γ ) − α − 1

Let Y n n = 1 − [ 1 − ( 1 + u γ ) − α ] β

Differentiating it w.r.t u , we have

d Y n n = α β γ [ 1 − ( 1 + u γ ) − α ] β − 1 u γ − 1 ( 1 + u γ ) − α − 1 d u .

The pdf of Y n is g ( y ( n ) ) = ( 1 − y ( n ) n ) β ( n − 1 ) and its cdf is

G ( y ( n ) ) = ∫ 0 y ( n ) ( 1 − t n ) β ( n − 1 ) d t

G ( y ( n ) ) = 1 β ( 1 − 1 n ) + 1 n − ( 1 − y ( n ) n ) β ( n − 1 ) β ( 1 − 1 n ) + 1 n

Letting n → ∞ , we arrive at G ( y ( n ) ) = 1 − e − β y ( n ) β and g ( y ( n ) ) = e − β y ( n )

In this section, the proposed distribution is fitted to the data set of prices of wooden toys of 31 children in April 1991 at Suffolk craft shop (

distribution fits better for given data set. The same thing can be confirmed after seeing

In this paper, a new distribution called GIKum distribution is introduced. Some properties of GIKum distribution such as measures of central tendency and dispersion, models of stress-strength, limiting distributions, characterization of GIKum distribution and related probability distributions through some specific transformations are derived. The mathematical expressions of reliability function (r.f) and the hazard rate function (hrf) of the GIKum distribution are found and presented through their graphs. The parameters estimation through the technique of maximum likelihood estimation is used and the results are applied to the data set of prices of wooden toys of 31 children.

Iqbal, Z., Tahir, M.M., Riaz, N., Ali, S.A. and Ahmad,^{ }M. (2017) Generalized Inverted Kumaraswamy Distribution: Properties and Application. Open Journal of Statistics, 7, 645-662. https://doi.org/10.4236/ojs.2017.74045