Power size biased exponential distribution with applications in bio-science

In this article, a modified version of size biased exponential distribution has been introduced and named as Power Size Biased Exponential Distribution (PSBED). Various statistical properties of the said distribution have been derived. The pdf, CDF and HRF of the distribution has been represented through the graphs. The method of maximum likelihood estimation MLE has been used to estimate the parameters of the distribution. Finally the performance of the distribution is checked through the data set where it shows the better results than the parallel distributions.

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PSBED Conceptual FrameworkClose ×

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AIC and CAIC and more: PSBE vs WAD vs ABAD vs LBAD vs ADClose ×

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AIC and CAIC and more: PSBE vs WAD vs ABAD vs LBAD vs ADClose ×

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Introduction

In recent developments, researchers focused on generating more flexible, tractable and meaningful distributions and modeled various types of lifetime data with monotone failure rates. In spite of their simplicity in solving many problems of lifetime data and reliability studies, such existing distributions are not useful to model bathtub and multimodal shaped failure rates and also fail to provide sound parametric fit to some practical application. In recent past, new families of probability distributions have been defined that are extension of well-known families of distributions. These newly developed families/classes of distributions provide greater flexibility in modeling complex data. Weighted in general and size biased in particular distributions arise in practice when observations from a sample are recorded with unequal probability and provide unifying approach for the problems when the observations fall in the non-experimental, non replicated and non-random categories, first introduced by to model ascertainment bias these are later formulized in uniform theory by. Discussed that weighted distributions have various statistical applications, especially in analysis of data relating to ecology and human populations. For the first, time applied weighted distribution to connect with sampling wood cells. Gove (2003) [9] studied some of the latest results on sizebiased distributions especially conceder the Weibull family relating to application and parameter estimation with method of moments and maximum likelihood in forestry. A weighted version of exponential distribution is discussed by Mir et al. (2013) [5] . They derive some mathematical properties and estimate the parameter with method of moments, maximum likelihood and Bayesian method. The probability density function of size biased exponential distribution is given by;

And the corresponding cdf is given as;

The Power Size Biased Exponential Distribution

If 𝑌 be a random variable that follows Size Biased Exponential distribution, then the random variable 𝑋 = 𝑌

b) Moments

Suppose x be a random variable follows PSBED. Then the r moment denoted by 𝜇 𝑟 ′ is given as;

Substituting 𝜃𝑥 𝛼 = 𝑧,then solving the integral we get;

By substituting𝑟 = 1, 2, 3 and 4 we obtain the first four moments of the distribution about origin

Variance of the distribution is given by;

Substituting the values of 𝜇 2 ′ and 𝜇 1 ′ from the above equations, we get;

c) Moment generating function

Let 𝑥 be a random variable follows power biased exponential distribution then the moment generating function denoted by 𝑀 𝑥 (𝑡) is obtained as;

Using Taylor's theorem, we get

d) Characteristics function of power size biased exponential distribution

Let 𝑥 be a random variable follows power size biased exponential distribution then the moment generating function denoted by 𝜙 𝑥 (𝑡) is obtained as.

Using Taylor's theorem, we get

e) Harmonic mean of power size biased exponential distribution The harmonic mean (H) is given as.

Substituting 𝜃𝑥 𝛼 = 𝑚,then solving the integral we get;

Order Statistics of Power size biased exponential Distribution

Let us suppose 𝑥 1 , 𝑥 2 , 𝑥 3 … , 𝑥 𝑛 be random samples of size n from power size biased exponential distribution with p.d.f 𝑓(𝑥) and c.d.f F(𝑥). Then the probability density function of 𝑘 𝑡ℎ order statistics is given as.

Now using the values of 𝑓(𝑥) and F(𝑥) in the above equation.Then the probability of 𝑘 𝑡ℎ order statistics of Power size biased exponential distribution is given as;

Then, the p.d.f of first order 𝑥 1 Power size biased exponential distribution is given as;

Then, the p.d.f of first order 𝑥 𝑛 Power size biased exponential distribution is given as

Method of Maximum Likelihood Estimation

The estimation of parameters of Power size biased exponential distribution is doing by using the method of maximum likelihood estimation. Suppose

Data Analysis

The usefulness of the formulated distribution has been explained through two bioscience related data sets. The data sets are as follows Data set 1: The data set represents the remission times (in months) of a random sample of 128 bladder cancer patients. In order to compare the efficiency of the formulated distribution, we consider various criterions for goodness of fit such as-log 𝑙, AIC, CAIC and BIC. The established model is compared with Weighted Ailamujia distribution (WAD), Area Biased Ailamujia Distribution (ABAD), Length Biased Ailamujia Distribution (LBAD), Ailamujia Distribution (AD). The distribution having lower value of AIC, CAIC and BIC is considered to be best fit. The descriptive statistics of data set 1 and data set 2 are presented in table 1 and 3. The estimates of the parameters, log-likelihood, Akaike information criteria (AIC) for the data sets are generated and presented in Table 2 and4. Hence power size biased exponential distribution leads better fit.

Model	Parameter Estimation	Standard Error	−𝐥𝐨𝐠𝒍	AIC	CAIC	BIC
PSBE	𝛼̂ = 0.72031 𝛽̂ = 0.43880	𝛼̂ = 0.0287 𝛽̂ = 0.0470	397.79	799.59	799.69	805.23
WAD	𝛼̂ = 0.1074 𝛽̂ = 0.0010	𝛼̂ = 0.0143 𝛽̂ = 0.2359	413.49	830.99	831.095	836.63
ABAD	𝛼̂ = 0.2148	𝛼̂ = 0.0096	489.35	980.713	980.74	983.53
LBAD	𝛼̂ = 0.1074	𝛼̂ = 0.0068	413.470	828.94	828.97	831.76
AD	𝛼̂ = 0.1074	𝛼̂ = 0.0068	413.47	828.94	828.97	831.762

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Performance of distributions for first Data set

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Data summary for the first data set

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Conclusion

In this article, a new version of size biased exponential distribution has been introduced and named as Power Size Biased Exponential Distribution (PSBED). Various statistical properties of the said distribution have been derived. The PDF, CDF and HRF of the distribution have been represented through the graphs. The method of maximum likelihood estimation MLE has been used to estimate the parameters of the distribution. Finally, the performance of the distribution is checked through the data set where it shows better results than the parallel distributions.

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Table 3Close ×

Data summary for the first data set

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Model	Parameter Estimation	Standard Error	−𝐥𝐨𝐠𝒍	AIC	CAIC	BIC
PSBE	𝛼̂ = 0.5122 𝛽̂ = 0.3583	𝛼̂ = 0.0510 𝛽̂ = 0.067	153.50	311.01	311.41	314.006
WAD	𝛼̂ = 0.0244 𝛽̂ = 0.0010	𝛼̂ = 0.0063 𝛽̂ = 0.4557	171.79	347.58	347.98	347.58
ABAD	𝛼̂ = 0.0489	𝛼̂ = 0.0042	217.79	437.58	437.71	439.08
LBAD	𝛼̂ = 0.0244	𝛼̂ = 0.0030	171.77	345.54	345.67	347.03
AD	𝛼̂ = 0.0244	𝛼̂ = 0.0030	171.77	345.54	345.67	347.03

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Performance of distributions for second data set

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1: PSBE Fitting

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2: PSBE Fitting

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